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Final Answers
© 2000-2023   Gérard P. Michon, Ph.D.

Western Art Music
Mathematical Aspects of Music

Leibniz arms
Music is the pleasure the human mind experiences from counting, without
being aware that it is counting.

G. G. Leibniz  (1646-1716)

 Michon
 
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Related Links (Outside this Site)

Art and Science of Music Acoustics  by  Joe Wolfe  (UNSW,  Physics).
 
The Scale Omnibus: 399 scales in 12 keys.   Francesco Balena  (2014-06-08).
AllTheScales.org  &  Musica Arcana  by  William Zeitler (1954-).
A Study of Scales  by  Ian Ring.   |   Scale Finder
Fretboard Topologies (1117 Scales and Modes)  by  Nicholas Puryear  (2010).
All the heptatonic scales  by  Richard Repp.
 
Sibelius  vs.  Finale  by  Ron Hess  (Film Music Magazine, 2008-09-17).
MuseScore 2, Notion 6, Finale v25, Sibelius 8  by  Freixas  (2016-20-22).
 
The Ultimate Guide to the Circle of Fifths  (Musical U,  2017-07-18).

short history  of tunes:

Hurrian Hymn to Nikkal (h.6).  Hurrian songs  (Ras Shamrah, c. 1400 BC).
 
Psalm 95  (c.1000 BCHebrew chant attributed to David.
 
Seikilos Epitaph  (1st Century AD)  by  Seikilos  of Tralles.
Hymn to Nemesis  (2nd Century AD)  by  Mesomedes of Crete.
 
Te Deum  (Ambrosian hymn, c.387)  under  Ambrose of Milan (c.340-397).
 
Medieval Instrumental MusicDerek and Brandon Fiechter (2014-08-20).
 
Farai un vers pos mi sonelh  by  Guilhem de Peitieu (1071-1127).
 
Dies Irae, Dies Illa  (bef. 1250).  Gregorian chant in D-Dorian:  F-E-F-D...
 
Nuper rosarum flores (1436-03-25)  by  Guillaume Du Fayt  (1397-1474).
 
Scaramella  by  Josquin des Prez  (c.1450-1521).
 
Tant que vivray (1527)  by  Claudin de Sermisy  (1490-1562).
La guerre (1537)  by  Clément Janequin  (1485-1558).
 
Aux marches du palais  (cf.  La Flamande.  Jehan Chardavoine, 1576).
OriginalDans la cour du palais, il y a une Flamande.
Elle a tant d'amoureux qu'elle ne sait lequel prendre...

 
Greensleeves  by  Richard Jones  (1580-09-03).  History.
Flow my tears (1596)  by  John Dowland (1563+1629).
L'Orfeo (1607Claudio Monteverdi (1567-1643) & Striggio the Younger.
Daphne (1644)  by  Jacob van Eyck (1590-1657).
Au clair de la lune  by  Jean-Baptiste Lully (1632-1687).
Canon in D major (c.1685Johann Pachelbel (1653-1706).  Revived in 1980.
Rondo from Abdelazer (1695)  by  Henry Purcell (1659-1695).
Toccata and Fugue in D minor  (c.1704Johann Sebastian Bach (1685-1750).
The Violin Concertos (1:19:50)  by  Johann Sebastian Bach (1685-1750).
The Four Seasons  (1717)  by  Antonio Vivaldi (1678-1741).
Hallelujah Chorus  (Messiah1741George Frideric Handel (1685-1759).
Le bon roi Dagobert (c.1755).  Poking fun at Royalty,  for the first time?
Yankee Doodle Dandy (c.1755)  by  Richard Shuckburgh.
Frère Jacques  (c.1760)  by  Jean-Philippe Rameau (1683-1764).
Minuet  (1771)  by  Luigi Boccherini (1743-1805).
Amazing Grace  (1779)  by  John Newton (1725-1807).
Turkish March  (c.1783)  by  Wolfgang Amadeus Mozart (1756-1791).  Spoofed.
Eine kleine Nachtmusik  (c.1787)  by  Wolfgang Amadeus Mozart (1756-1791). 
Auld Lang Syne  (1788)  by  Rabbie Burns  (1759-1796).
The Magic Flute  (1791)  by  Wolfgang Amadeus Mozart (1756-1791). 
 
Twinkle, Twinkle, Little_Star:  Lyrics by  Jane Taylor (1806)  to the tune of  Ah vous dirai-je, Maman (c.1740).  First published in 1761  (melody)  and 1774  (with lyrics).  Rearranged by  many  composers,  including  Mozart (1781).
 
Ça Ira  (May 1790)  evolved from tame lyrics by the street-singer  Ladré,
on a popular  tune,  known as  Carillon national,  due to  Bécourt (1785).
 
La Marseillaise  (1792-04-25)  by  Rouget de Lisle  (1760-1836).
La Carmagnole  (Anonymous, 1792-08-10)
Chant du départ  (1794)  by  Chénier (1764-1811)  and  Méhul (1763-1817).
 
Emperor Quartet  (1797)  by  Joseph Haydn (1732-1809).
 
Moonlight Sonata  (1801)  by  Ludwig van Beethoven (1770-1827).
5th Symphony, in C Minor (c.1806Ludwig van Beethoven (1770-1827).
Für Elise  (1810)  by  Ludwig van Beethoven (1770-1827).
Unfinished Symphony  (1822)  by  Franz Schubert (1797-1828).
Military March  (1822)  by  Franz Schubert (1797-1828).
Ode to Joy  (1824)  by  Beethoven.  Lyrics  (1785)  by  Schiller (1759-1805).
 
William Tell Overture  (1828)  by  Gioachino Rossini (1792-1868).
La Campanella  (1838, 1851)  by  Franz Liszt (1811-1886).
Ride of the Valkyries  (1854)  by  Richard Wagner (1813-1883).
 
Sweet Betsy from Pike  (PA).  Lyrics by  John A. Stone (c. 1858)
to the tune of the British  broadside ballad  entitled  William and Dinah.
 
Battle Hymn of the Republic (1861) by Julia Ward Howe (1819-1910)
to the tune of  John Brown's Body, from the folk hymn  Oh!  Brothers.
 
When Johnny Comes Marching Home (1863)  by  Patrick Gilmore.
 
Le temps des cerises (1866)  by
Jean-Baptiste Clément (1836-1903)  and  Antoine Renard (1825-1872).
 
Night on Bald Mountain  (1867Modest Mussorgsky (1839-1881).  Fantasia.
Brahms' Lullaby (1868Johannes Brahms (1833-1897).  Celine Dion (1998).
Fifth Hungarian dance  (1869Johannes Brahms (1833-1897) & Béla Kéler.
The Moldau = Vltava  (1874)  by  Bedrich Smetana (1824-1884).
1812 Overture  (1880)  by  Pyotr Ilyich Tchaikovsky (1840-1893).
Dance of the Sugar Plum Fairy  (The Nutcracker, 1892)  by  Tchaikovsky.
Gymnopedia  (1888)  by  Erik Satie (1866-1925).
New World Symphony  (1893)  by  Antonín Dvorák (1841-1904).
Prélude à l'après-midi d'un faune  (1894)  by  Claude Debussy (1862-1918).
Also sprach Zarathustra  (1896)  by  Richard Strauss (1864-1949).
Entry of the Gladiators  (1897)  by  Julius Fucík (1872-1916).
Maple Leaf Rag  (1899)  by  Scott Joplin (c.1868-1917).
Flight of the Bumblebee  (1900).  Nikolai Rimsky-Korsakov (1844-1908).
Second piano concerto  (1900)  by  Sergei Rachmaninoff (1873-1943).
La Mer  (1905)  by  Claude Debussy (1862-1918).
 
Sous les ponts de Paris  (1913)  by
Jean Rodor (1881-1967)  and  Vincent Scotto  (1874-1952).
 
El Cóndor Pasa  (1913)  by  Daniel Alomía Robles (1871-1942).  [sheet music]
Colonel Bogey March  (1914)  by  Kenneth J. Alford (1881-1945).
Tico-Tico no fubá  (1917)  by  Zequinha de Abreu (1880-1935).
 
Rhapsody in Blue (1924)  by  George Gershwin  (1898-1937).
Tea for Two (1924): Vincent Youmans (1898-1946) Irving Caesar (1895-1996).
Sweet Georgia Brown (1925)  by  Ben Bernie  (1891-1943).
Nessun Dorma.  Aria from the opera  Turandot  by  Giacomo Puccini  (1924)  and  Franco Alfano  (1926).
I Wanna Be Loved by You (1928Helen Kane  in Good BoyStothart & Ruby.
Malagueña  (1933)  by  Ernesto Lecuona (1896-1963).
 
Summertime (1934) in Gershwin's Porgy and Bess; most recorded song ever!
Billie Holiday, Ella Fitzgerald + Louis Armstrong, Nina Simone, Norah Jones.
 
Chime Bells  (1939)  by  Elton Britt (1913-1972).  Cover by  Jewel (1974-).
Dance of the Knights  (1935)  by  Sergei Prokofiev (1891-1953).
Sing, Sing, Sing  (1936)  by  Louis Prima (1910-1978).  Benny Goodman.
(O Fortuna)  Carmina Burana  (1936)  by  Carl Orff (1895-1982).
When the Saints Go Marching In  (1923).  Louis Armstrong (1938-05-13).
Orange Blossom Special  (1939)  by  Ervin T. Rouse (1917-1981).
 
In the Mood,  popularized in  1939  by  Glenn Miller (1904-1944)
was a cover of Tar Paper Stomp (1929) by Wingy Manone (1900-1982).
 
Concierto de Aranjuez (1939)  by  Joaquín Rodrigo (1901-1999).
Sabre Dance (1942)  by  Aram Ilyich Khachaturian  (1903-1978).
 
Chant des Partisans (1943)  by  Anna Marly  (1917-2008).
Lyrics due to  Maurice Druon (1918-2009) & Joseph Kessel (1898-1979).
 
La vie en rose (May 1945)  by  Edith Piaf  (1915-1963),  music by  Louiguy.
Padam, padam...  (1951Edith Piaf (1915-1963).  Glanzberg & Contet.
Rock around the Clock  (1952)  by  Max_C._Freedman & James E. Myers.
Second Waltz  (1955)  by  Dmitri Shostakovich (1906-1975).
 
Tutti Frutti  (1955)  by  Little Richard  (1932-2020)
and  Dorothy LaBostrie  (1928-2007).
Fever  (1956)  by  Eddie Cooley (1933-2020) & Otis Blackwell (1931-2002).
Recorded by  Little Willie John in 1956 and Peggy_Lee (1920-2002) in 1958.
 
Que sera sera  (1956Doris Day (1922-2019).  By  Livingston & Evans.
Elvis PresleyHound Dog (1956) & Jailhouse Rock (1957).  Leiber & Stoller.
Johnny B. Goode  (1958)  by  Chuck Berry (1926-2017).
La Bamba  (1958)  by  Ritchie Valens (1941-1959).  Mexican folk song.
So What?  (1959)  by  Miles Davis (1926-1991).
Take Five  (1959)  by  Paul Desmond (1924-1977).  Dave Brubeck quartet.
Volare  (1959Domenico Modugno (1928-1994)  in  Nel blu dipinto di blu.
 
Ta Pediá tou Pirée (1959)  Manos Hadjidakis  (1925-1994).  Melina Mercouri.
Les enfants du Pirée (1960):  Cover by  Dalida (1933-1987).
(Never on Sunday)  Ann-Margret, The Chordettes, Connie Francis, Don Costa, Margarita, Konstantina, Nana Mouskouri (1968), Bouzouki, etc.
 
What a Diff'rence a Day Makes  (1934, in Spanish)  María Grever (1885-1951).
Made famous in English by the  1959 cover of  Dinah Washington (1924-1963).
 
Michael Row the Boat Ashore  (1960)  by  The Highwaymen.
The Flintstones  (1961)  by  Hoyt Curtin (1922-2000)  of  Hanna-Barbera.
James Bond Theme  (1962)  by  Monty Norman (1928-).
The Girl from Ipanema  (1962Antônio Carlos Jobim  (1927-1994).  Analysis.
Ring of Fire (1963June Carter-Cash  (1929-2003).  Johnny Cash (1932-2003).
You Really Got Me  (1964)  by  Ray Davies  (1944-)  The KinksMixolydian.
My Girl  (1964)  by  Smokey Robinson  (1940-)  for  The Temptations.
Satisfaction (1965Mick Jagger  (1943-) & Keith Richards (1943-).
Mustang Sally (1965Mack Rice  (1933-2016).  Wilson Pickett's  1966 cover.
Yesterday (August 1965)  by  Paul McCartney & John LennonThe Beatles.
Mission Impossible  (1966) by  Lalo_Schifrin  (1932-).
The Good, the Bad and the Ugly  (1966) by  Ennio Morricone  (1946-2020).
Purple Haze  (March 1967) by  Jimi Hendrix  (1942-1970).  Analysis.
Brown-Eyed Girl  (March 1967) by  Van Morrison  (1945-).
A Whiter Shade of Pale  (May 1967) debut album of  Procol Harum  [Organ].
All you need is love (July 1967John Lennon.  Worldwide TV: 1967-06-25.
Les moulins de mon cœur (1968)  by  Michel Legrand (1932-2019).
Hey Jude (August 1968Paul McCartney.  First song recorded for  Apple.
Sugar, Sugar (1969)  by  Jeff Barry (1938-)  &  Andy Kim (1952-).
Aquarius (1967The Fifth Dimension.  By  Rado and Ragni for Hair (1967).
Bridge over Troubled Water (1970) Paul Simon (1941-).  Simon & Garfunkel.
In the Summertime (1970)  by  Mungo JerryRay Dorset (1946-).
Rivers of Babylon (The Melodians1970).  Cover by  Boney M (1978).
Mamy Blue (1970)  by  Hubert Giraud (1920-2016).  In English by Pop-Tops.
Ain't no sunshine (1971)  by  Bill Withers (1938-2020).
(Where do I begin?)  Love story (1971)  by  Andy Williams (1927-2012).
Stairway to Heaven (1971)  by  Led Zeppelin  (together 1968-1980).
A Horse with no Name (1971)  by  Dewey Bunnell (1952-).  America.
Godfather Theme (1972)  by  Nino Rota (1911-1979)  from Fortunella (1958).
Killing Me Softly with His Song (1972Roberta Flack (1937-).  Fox & Gimbel.
I Shot the Sheriff (1973Bob Marley (1945-1981).  Eric Clapton's cover (1974).
Angie (August 1973)  by  Keith Richards (1943-).  The Rolling Stones.
La Jument de Michao (1973)  revived by  Kouerien.
All by Myself (1975; 1996 cover by Céline Dion)  by  Eric Carmen (1949-).
Hotel California (1977)  by  the  The Eagles  (1971-).
We will rock you (1977)  by  Brian May (1947-) and Queen.
Star Wars (1977)  by  John Williams (1932-).
Stayin' Alive (1977)  by  the  Bee Gees  for  Saturday Night Fever.
You're The One That I Want (1978)  by  John_Farrar (1947-) for  Grease.
Superman March (1978John Williams (1932-)  for  Superman: The Movie.
I will survive (1978Gloria Gaynor (1943-).  By  Perren & Fekaris.
Heart of Glass (Blondie1978Debbie Harry (1945-) & Chris Stein (1950-).
Girls Just Want To Have Fun (1979 by Robert Hazard).  Cyndi Lauper  in 1983.
Highway to Hell (1979, AC/DC)  by Bon Scott, Angus & Malcolm Young.
The wall (1979)  by  Pink Floyd (active 1965-1995, 2005).
Whip It (1980)  by  Devo (est. 1973).
On the Road Again (July 1980)  by  Willie Nelson (1933-).
Chariots of Fire (1981)  by  Vangelis (1943-).
Sweet Dreams (1982)  by  Annie Lennox (1954-)  &  Dave Stewart (1952-).
Beat it (1982)  by  Michael Jackson (1958-2009).  Quincy Jones (1933-).
Every Breath You Take (1983)  by  Sting (1951-)  and  The Police.
What a Feeling (1983)  by  Irene Cara (1959-)  for the film  Flashdance.
Time After Time (1983)  by  Cyndi Lauper (1953-)  and  Rob Hyman (1950-).
Like a Virgin (1984Madonna (1958-).  By  Kelly and Steinberg.
Hallelujah (1984).  Leonard Cohen (1934-2016).  2016 cover by Pentatonix.
What's love got to do with it? (1984Tina Turner (1939-).
Axel F (film Beverly Hills Cop, 1984)  by  Harold Faltermeyer (1952-).
Sweetest Taboo (1985)  by  Sade (1959-).
We are the World (1985Michael Jackson, Lionel Richie, USA for Africa.
Take my breath away  (1986)  by  Berlin  for the film  Top Gun.
Time of my Life (1987)  by  Bill Medley (1940-)  &  Jennifer Warnes (1947-).
 Homer Simpsons The Simpsons (1989Danny Elfman (1953-).  Lydian Dominant.
Smells Like Teen Spirit (1991)  by  Nirvana (est. 1987).
Macarena (1993)  by  Los del Río (est. 1962).
Livin' La Vida Loca (1999)  by  Ricky Martin (1971-).
All Star (1999)  by  Greg Camp (1967-) for  Smash Mouth.
I Kissed a Girl (2008)  by  Katy Perry (1984-)  &  Max Martin (1971-).
Bad Romance (2009)  by  Lady Gaga (1986-)  &  Nadir Khayat (1972-).
We No Speak Americano (2010).  Yolanda Be Cool (2009-) & DCUP (1985-).
Rolling in the Deep (2010)  by  Adele (1988-)  &  Paul Epworth (1974-).
Rescue (2019)  by  Lauren Daigle (1991-).  Label:  Centricity Music.

Wikipedia :   Music theory   |   Music in Western culture   |   Euler-Fokker genera   |   Young temperament
List of scales and modes   |   Acoustics   |   Musical acoustics   |   Johann Sebastian Bach (1685-1750)

Harmonic Dualism  by  Hugo Riemann (1849-1919).
Twentieth-century Harmony  (1961)  by  Vincent Persichetti (1915-1987).
Gödel, Escher, Bach (1979)  by  Douglas R. Hofstadter (1945-).
A Theory of Harmony  (1985)  by  Ernst Levy (1895-1981).
The  New  Harvard Dictionary of Music,  edited by  Ron Handel  (1986).
The Mathematical Theory of Sound Systems  by  Jan Haluska.
Arpeggio & Scale Resources:  A Guitar Encyclopedia  by  Rich Cochrane.
Tonal Harmony (2004 / 7th edition 2013)  by  Kostka, Payne & Almén.
The Topos of Music  (2002, 2018, 4 volumes)  by  Guerino Mazzola (1947-).
 
The Sounds of Music (1:48:36)  by  Walter Lewin  (MIT, 1996-06-25).
Resonance, Musical Instruments (1:20:24)  Walter Lewin  (8.03 #9, 2004-10-07).
Clarinet-Flute hybrids (2:39)  by  Joe Wolfe  (UNSW, 2011-05-05).
On the Music of Shapes (1:11:25)  by  Alain Connes  (Lille, 2012-09-26).
Physics of Music (10:34)  by  Shini Somara  (CrashCourse, 2016-08-11).
How Computers Ruined Rock Music (19:53)  by  Rick Beato  (2019-04-13).
The Fascinating World Inside of a Piano (30:52)  Damon Groves  (2019-06-10).
What do the pedals on a piano do? (6:54)  Hugh Sung  (Cunningham Piano).
Synthesizers, Digested by a Classical Musician (16:54)  Nahre Sol  (2019-08-06).
György Ligeti:  The M.C.Escher of Music (20:42)  David Bruce  (2020-07-14).
Music & Symmetry (8:08)  Santa Fe Institute  (2014-02-28).
Mathematical Music (49:01)  by  Alan Stewart  (Numberphile, 2013-06-04).
Math & Music by Jamie York (2012):  13:11c, 13:34c, 13:59c, 14:19c, 14:40c.
The Rule of the Octave (24:03)  Early Music Sources  (2021-04-16).
The Greatest Solo of All Time (17:25)  Oscar Peterson  (Rick Beato, 2021-07-20).
Evolution of Music: AD 1680-2017 (6:17)  by  Lord Vinheteiro  (2017-10-03).

 Rowland RD-64
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Mathematics of Music

Pythagoras
Pythagoras

(c.569 - c.495 BC)
Philolaus
Philolaus

(c.475 - c.385 BC)
Mersenne
Mersenne

(1588-1648)
Helmoltz
Helmholtz

(1821-1894)

All children raised in the modern Western tradition are repeatedly exposed at an early age to the progression of the seven notes in the key of  C-major  (used in all nursery-school songs).
 
Very few  (about 0.01% or one in ten thousands)  will ever associate each note with its exact pitch in absolute terms  (this rare ability is called  perfect pitch).  Most, however, will eventually learn the relative position of the notes in the scale.
 
Those  7  notes form an uneven progression within the regularly-spaced  12  notes of the  chromatic scale,  which is the basis of all current music in the Western world.
 
Let's brush up on the basics of that system and explore how it was born from two opposing forces:  The desire for universal  keyboards  and the need for  harmony  (two tones are harmonious only when their frequencies are nearly in the same ratio as two  small  integers).

Music Theory Distilled:  (10:3311:324:08)  by  Casey Connor  (2018).


(2020-07-27)   The many Greek  tonoi.
A codification inherited from the late  Bronze Age.

LydiaPhrygiaLocrisIoniaAeolisDoris

 Come back later, we're
 still working on this one...

Musical system of ancient Greece   |   Mixolydian tonos   |   Sappho of Lesbos (c.630-670 BC).
 
Why C?  The Convoluted History of Note Names (6:40)  by  David Kulma  (Music Corner, 2015-08-10).


(2018-07-27)   Military Music.  Battle Songs and Marching Drums.
An overlooked ingredient in the evolution of music.

 Come back later, we're
 still working on this one...

Roman march   |   Did the Roman Army Use Marching Songs? (7:30)  Invicta  (2010-08-10).
March of Cambreadth (4:47)  by  Heather Alexander, 1962-  (1997, 2006 best battle song).
 
Loin de chez nous (3:25)   |   Veronika (1:27)   |   Le boudin (2:00)   |   Veronika (1:27)
Les Dragons de Noailles (2:36)   |   Au 31 du mois d'Août (3:28)   |   Tri Martolod (3:08)
 
Battle Dawn (2:49)  by  Mercedes Ritchie Lackey (1950-)  & ; Debra F. Sanders  (2012-07-17).


(2018-02-23)   Occidental origins of diatonicity,  in the Middle Ages.
Musical notation helped crystallize the evolution of  Western art music.

From AD 600 to 1500 or so  (when neumes  came into wide use)  Gregorian chants  were mostly transmitted orally.  The Church recognized eight official modes  (the first and the last one being identical)  each nominally spanning  eight  notes  (hence the name  octave  for the doubling  interval  so spanned).

The common  note  starting and ending an octave was called the  final  (now dubbed  tonic or tonal center)  because of the rule that a chant must end on that note to provide a sense of  resolution.  That expectation is created by giving prominence to the tonic,  not necessarily by starting with it  (e.g.,  Dies Irae  is in D-Dorian but starts with the ominous four notes F-E-F-D).  The eight Gregorian modes span a combined range of two vocal octaves.

In this ancient scheme,  the lowest note involved is an "A",  which may ultimately explain how the notes first got their alphabetical names,  already found in  De Musica (c. AD 510)  by  Boëthius (c.477-524).

G4 and A4 are shared by all modes.  The latter became the  standard  for tuning.  The former is what the  treble clef  points to  (G-clef, clef de sol ).

Traditional Symmetrical 8-way Definition of the Church Modes :
I Dorian  D  EF G4   A4  BC D  
II Hypodorian A3 BC D EFG4 A4 
III Phrygian  EFG4 A4  BC D  E 
IV Hypophrygian  BC D  EFG4 A4B 
V Lydian  FG4 A4BC D  EF 
VI Hypolydian  C D  EFG4 A4BC 
VII Mixolydian  G4 A4BC D  EF G5
VIII  Hypomixolydian   D  EF G4 A4BCD  

Each even-numbered  plagal mode  is named by adding the prefix  hypo-  to the  authentic  odd-numbered mode which precedes it.  In  De Harmonica  (c. AD 880)  Hucbald  specified the plagal mode as running from one fourth below the authentic mode's  final  to one fifth above it,  as tabulated above.  The last mode  (Hypomixolydian)  is identical to the first one  (Dorian).

The arbitrary distinction between authentic and plagal modes is now all but lost and three new words have been coined for the former  hypo-  modes,  namely the two most popular  modern modes  (ionian  and  aeolian; better known as Major and Minor)  and the least popular one  (locrian).  As all of the above Gregorian modes are  diatonic,  they can be matched with the modern  modes of the major scale.  Actually, the Medieval modes denoted not only a mode  (in the modern sense)  but a specific  key  (at a fixed pitch)  according to the following equivalences:

Modern equivalents of the  7  unique Medieval Gregorian modes
Medieval mode  Mode of C-Major  Circle of fifths
VLydian 4F-LydianF
VIHypolydian 1  C-Ionian   (C major)  C
VIIMixolydian 5G-MixolydianG
  VIII    Hypomixolydian     2  D-DorianD
IDorian
IIHypodorian 6  A-Aeolian  (A minor)  A
IIIPhrygian 3E-PhrygianE
IVHypophrygian 7B-LocrianB

The rotation placing  Dorian = Hypomixolydian  in the middle,  rather than at both ends,  reveals a deeper symmetry (mirror inversion of the  interval structure)  among the seven  distinct  modes.  This ordering also puts them in strict order of  brightness,  with  B-Locrian last and least  (all but unused).

The Beginning of Western Musical Notation :

The  musical staff  was devised by  Guido of Arezzo  early in the 11-th century.  He used a  4-line  staff and recommended the use of yellow and red ink.

 Come back later, we're
 still working on this one...

1000 Years of Classical Music History (11:21)  by  David Bruce  (2018-03-30).
 
Gregory I (c.540-604)   |   Gregorian chant   |   8 Gregorian modes   |   Mensural notation
De harmonica institutione (AD 880)   |   Hucbald (AD 840-930)
Letter notation:   Dialogue on Music  (Enchiridion musices)   |   Musicae artis disciplina
Odo de Saint-Maur-des-Fossés (-1030)
Gamut   |   Solmization   |   Guido of Arezzo (AD 992-1050)
 
Why did C become the middle note?  by  DukeOfBoom  (2010-12-01).
 
The Convoluted History of Note Names (6:40)  by  David Kulma  (2015-08-15).


(2018-02-16)   Duration of notes and rests
Binary progression of standard durations  (dotting prolongs by  50%).

note  is an elementary music element with nearly constant pitch and prescribed duration.

A quarter-note is represented by a black oval with either an upward stem to the right or a downward stem to the left  (the French just call it "a black"; une noire).  A half-note has twice the duration and is represented by a void oval with a stem  (French:  une blanche).  A whole note is a void oval without stem;  it's worth two half-notes or four quarter-notes.

Moving in the other direction,  a flag on the stem of a quarter-note reduces its duration by a factor of two and makes it an  eighth-note  (a  quaver  to the British,  une croche  to the French).  Two flags indicate a  sixteenth  (French:  double-croche).  Occasionally,  three flags are used to denote a  thirty-second  (French:  triple-croche).  Four or five flags are rarely used.

The shortest value ever used in the classical repertoire is denoted with six flags.  It's the  two hundred fifty sixth note,  which the French call a  sextuple croche.  The British name is  demisemihemidemisemiquaver.

Note Durations   (and the corresponding silence periods)
DurationAmericanBritishFrench
  8LargaLarge Maxime
(silence de maxime)
  4Quadruple note
(quadruple rest)
Longa
(longa rest)
Longue
(silence de longue)
  2Double note
(double rest)
Breve
(breve rest)
Carrée
(bâton de pause)
  1Whole note
(whole rest)
Semibreve
(semibreve rest)
Ronde
(pause)
  1/2Half note
(half rest)
Minim
(minim rest)
Blanche
(demi-pause)
 1/4Quarter note
(quarter rest)
Crotchet
(crotchet rest)
Noire
(soupir)
  1/8Eighth note
(eighth rest)
Quaver
(quaver rest)
Croche
(demi-soupir)
  1/16Sixteenth note
(sixteenth rest)
Semiquaver
(semiquaver rest)
Double croche
(quart de soupir)
  1/3232nd note
(32nd rest)
DemisemiquaverTriple croche
  1/6464th note
(64th rest)
HemidemisemiquaverQuadruple croche

A dot after a symbol extends its duration by  50%.  A double-dot by  75%.

Two or more consecutive flag-bearing notes are  beamed  together with as many  beams  touching each note's stem as the number of flags it ought to have.  Rest symbols are allowed between beamed notes.

 Come back later, we're
 still working on this one...

Wikipedia :   Note value   |   Rest   |   Caesura   |   128th note   |   256th note
 
Counting Tutorial for Beginning Musicians (10:09)  by  Karen Cuneo Ramirez  (2008-10-11).
 
How to Read Music (14:39)  by  Sarah Jeffery  (Team Recorder, 2019-02-14).


(2018-02-16)   Tempo
How fast to play.

In Western culture,  a musical piece is a sequence of monophonic or polyphonic tones,  timed by regular  beats.  The tempo is either indicated by a traditional Italian locution or given precisely in  beats per minute  (bpm). 

Metronomes  are traditionally marked at the following values,  in bpm:  40,  42,  44,  46,  48,  50,  52,  54,  56,  58,  60,  63,  66,  69,  72,  76,  80,  84,  88,  92,  96,  100,  104,  108,  112,  116,  120,  126,  132,  138,  144,  152,  160,  168,  176,  184,  192,  200,  208.

  Traditional    Beats per Minute  
Larghissimo24 bpm and below
Grave24 bpm - 40 bpm
Largo40 bpm - 60 bpm
Larghetto60 bpm - 66 bpm
Adagio66 bpm - 76 bpm
Andante76 bpm - 100 bpm
Moderato100 bpm - 120 bpm
Allegro120 bpm - 168 bpm
Presto168 bpm - 200 bpm
Prestissimo200 bpm and over

There's little need to perform below  30  bpm  (one beat every other second)  which is roughly the slowest tempo at which the human brain still links the elements of a sequence as parts of a whole.  Slower changes in tonality are perceived as separate discrete events and the melody is just  lost in time.

At the other extreme,  too fast a tempo will not give the brain enough time to grasp subdivisions in individual beats.  Ultimately,  when something changes more than  20  times per second  (20 Hz or 1200 rpm)  it's simply heard as a buzz.  That's when rapid clicks morph into a continuous pitch.

Historically,  the  earliest scientific unit of time  chosen by  Galileo  was roughly the shortest time interval at which he couldn't perceive two percussions as distinct  (about  11 ms  in modern terms).  At face value,  that would imply that we perceive as  separate  two cycles of a 9 Hz sound, if the conditions are right  (although  540 bpm  is musically meaningless).

What is the slowest music humanly possible? (15:53)  by  Adam Neely  (2018-01-22).
What is the fastest music humanly possible? (11:17)  by  Adam Neely  (2017-03-27).
New Horizons in Music: Polyrhythms (45:30)  by  Adam Neely  (2018-03-05).
 
Wikipedia :   Tempo


(2018-02-16)   Beats  (counts)  bars  (measures)  and  phrases.
4/4  common time :  4  beats to a bar  and  4  or  8  bars to most phrases.

Beats are regularly-spaced time intervals.  In dance-music,  they follow the  kick drum.  Otherwise,  a  metronome  can be used,  which delivers regular clicks and visual cues.

A whole number of beats make up a  measure  (also called  bar  because the limits of all measures are indicated by vertical bars on  sheet music).  That number depends on the  time signature,  discussed in the next section.

A rhythm where some notes are stressed on the  upbeat  (between main  downbeats)  is called  syncopated.  In the rare cases where a note straddles two measures,  it is said to be  offbeat.

Time signature  (French: chiffrage)

On  sheet music,  this is indicated at the beginning of the first  staff  by two superposed numbers which summarize the rhythm.  In  simple time  (as opposed to compound time, discussed next)  the top number gives the  number of beats per bar.  The bottom number says which type of  note  counts for  one beat  (2 for a half-note per beat,  4 for a quarter-note per beat,  8 for an eighth-note per beat).  For example,  3/4  is often read  "3 beats per bar and every quarter-note gets a beat".

March time  is simply  1/2.  The  4/4  signature is called  common time.  The  3/4  time-signature is  waltz time.

Less common time signatures include  5/4,  of which a prime example  (with offbeat notes)  is  Take Five  by the late  Dave Brubeck (1920-2012).

Pink Floyd's  Money  (1973)  alternates  3/4  and  4/4,  with offbeat notes.

Compound Time :

When the  time signature's  top number  is a  proper  multiple of three  (6, 9, 12, 15, 18, 24, etc.)  each beat is understood to be divided into three equal divisions.  The number itself indicates how many such divisions there are in a bar  (not  the number of beats per bar,  as is the case with  simple time).

Thus,  a bar in  6/8  has two beats divided into three equal parts,  worth an eighth-note each.  A famous example of  6/8  is the  folk song  Rising Sun Blues,  popularized as  House of the Rising Sun,  by  The Animals  (1964).

How to Count in Beats, Bars & Phrases (12:58)  by  djTLMtv  (2013-10-21).
Improve your Rhythm with Metronome Exercises (11:10)  by  A.J. Block  (Didge Project, 2017-01-20).
Fun and easy way to improve your timing (6:02)  by  Paul Davids  (2016-02-17).
Singing is all about the timing (6:36)  by  Tiffany  (Sing Better in 3 Days, 2014-09-01).
Time Signatures Illustrated on Piano Keyboard (10:34)  by  Brett Gossage  (2013-11-26).
How time signatures work (13:25)  by  Victoria Williams  (2014-03-24).
5/4, the Coolest Odd Time Signature (13:03) Tommaso Zillio  (Music Theory for Guitar, 2019-12-09).
Advanced Time Signatures (14:54)  by  Michael New  (2013-11-13).
Understanding 4/6 and 5/6 Time Signatures (2:09)  by  Jason Adler  (2014-06-04).
How to Play Music in 9/8 (12:37)  by  Adam Neely  (2019-01-15).
Thomas Adès: "Irrational" Time Signatures & Rhythms (7:59)  by  David Bruce  (2017-11-14).
Irrational Time Signatures (4:50)  by  Brian Krock  (Adam Neely, 2018-03-19).
4/20 Time Signatures (10:31)  by  Adam Neely  (2019-11-25).
Songs That Skip a Beat (9:25)  by  David Bennett  (2019-04-06).
How and why classical musicians feel rhythm differently (6:46)  by  Adam Neely  (2016-01-25).
Sightreading Devillish Time Signatures (11:15)  by  Brett Yang & Eddy Chen  (TwoSet Violin, 2020-02-16).
Negative Time Signatures? (6:15)  by  Adam Neely  (2018-03-12).
On Negative Time Signatures (6:51)  by  Cory Arnold  (12tone, 2018-04-20).
Triple Time Signatures? (10:54)  by  Adam Neely  (2019-01-21).
 
Wikipedia :   Time signatures   |   Compound time   |   Unusual time signatures   |   Entrainment


(2018-02-16)   Triplets  and  offbeat  Triplets.
Trained musicians have trouble playing offbeat Triplets at a slow tempo.

Triplet  is usually just a group of three notes of equal durations meant to be played with the same total duration as  two  notes of the indicated kind.  A Triplet is indicated by a bracket with the numeral 3.

Thus,  the duration of a  Whole-note  is divided equally into three  Triplet-halves.  Likewise,  a  Half  is split into three equal  Triplet-quarters.

More generally,  a triplet bracket  (i.e.,  a bracket with the numeral 3)  reduces all note durations within it by a factor of  2/3.  The bracket itself is optional if the notes are already  beamed  together.

Likewise,  a bracket  (or a beam)  bearing the numeral  n  reduces the duration of the notes it spans by a fixed fraction of denominator  n.  That construct is known as a  tuplet  (or an  n-tuplet,  when  n  is specified).  Numerical Greek prefixes  (and/or some Latin alteration thereof)  can also be used:

  • Triplets.  Factor of  2/3  (always).
  • Pentuplets,  Quintuplets  or  Quintolets.  Factor of  4/5  (3/5  in compound time).
  • Heptuplets,  Septuplets,  Septolets  or 

 Come back later, we're
 still working on this one...

Offbeat Triplets:  The "un-performable" rhythm? (6:52)  by  Adam Neely  (2017-05-22).
 
Wikipedia :   Tuplets


(2018-02-13)   Dynamics:  Prescribed loudness of a performance.
Eight possible  performance marksppp   pp   p   mp   mf   f   ff   fff

 Come back later, we're
 still working on this one...

Dynamics   |   Performance marks
 
How to Read Music (16:27)  by  Julian Bradley   (Jazz Tutorial, 2015-09-21).


(2018-02-13)   The Frequency Domain
Describing sound in terms of a superposition of frequencies.

 Come back later, we're
 still working on this one...

The frequencies of music (53:47)  by  Rick Beato.
Why Adults Can't Develop Perfect Pitch (16:25)  by  Rick Beato   (2017-01-01).
The Convoluted History of Note Names (6:40)  by  David Kulma   (Music Corner, 2015-08-10).
 
Numericana :   The Fourier transform


(2018-02-12)   Perfect Pitch   (Absolute Pitch)
The codified language of Western music has its native speakers.

About one in 10000 people have developed native familiarity with the language of music by being exposed to its complexity at a very young age.  The most striking ability they develop is called  perfect pitch  (or  absolute pitch)  which is the ability to instantly name a note or a combination of notes with perfect accuracy without the benefit of prior tuning.  This ability  cannot  be acquired later in life.

 Come back later, we're
 still working on this one...
 Note names on the Grand Staff (Great Stave)

 One octave on the piano keyboard

A take-home message of this diagram is that D, G and A have no other names.

The syllabic names below  (first column)  are used in Romance and Slavic languages.  In English,  Sol  is pronounced  So  and  Si  is called  Ti  (thus avoiding a possible confusion with the letter "C",  for  Ut  or  Do).

Equally-Tempered Frequencies of Western Notes,  in  Hz   (A440  pitch)
 Octave name Contra Low Bass Middle Treble High Supra  
ISO # 012345678
DoC 16.3532.7065.41130.81261.63 523.251046.502093.004186.01
 C# 17.3234.6569.30138.59277.18 554.371108.732217.464434.92
D 18.3536.7173.42146.83293.66 587.331174.662349.324698.64
 D# 19.4538.8977.78155.56311.13 622.251244.512489.024978.03
MiE 20.6041.2082.41164.81329.63 659.261318.512637.025274.04
FaF 21.8343.6587.31174.61349.23 698.461396.912793.835587.65
 F# 23.1246.2592.50185.00369.99 739.991479.982959.965919.91
SolG 24.5049.0098.00196.00392.00 783.991567.983135.966271.93
 G# 25.9651.91103.83207.65415.30 830.611661.723322.446644.88
LaA 27.5 55110220440 880176035207040
 A# 29.1458.27116.54233.08466.16 932.331864.663729.317458.62
Si/TiB 30.8761.74123.47246.94493.88 987.771975.533951.077902.13

C4  is called  middle C  and standard   concert A  (A,  440 Hz)  is dubbed  A above middle C.  Each octave starts at a  C  and ends with the  B  above it.  Treble C  (C5)  is better known to Opera afficionados as  Tenor High C  since it's the highest note in the classical male repertoire.

 Full 88-key piano keyboard

On an  88-key piano,  the lowest note is  A0  ( 27.5 Hz ).  The highest is  C8  ( 4186.009 Hz )  which is the lowest note of  Octave 8,  in ISO numbering.

In the  above  scientific pitch notation  (SPN, used everywhere with growing popularity)  the  ISO  number of each octave is used after the name of a pitch  (following the # or b accidentals, if any)  to denote a particular tone unambiguously.  Subscripting is optional:  A4  and  A4  both mean  440 Hz.

Other competing systems are still in use,  which are mutually incompatible to some degree.  In all cases,  tones in the same octave  (always from C to B, mercifully)  are denoted alike and we give only the notation corresponding to "C"  (Do, Ut)  in the following table.  Musicians routinely speak of a particular tone by identifying the "C" just below it  (e.g., 440 Hz  is "A above middle-C" or concert A,  less often "middle A").

Competing ways of naming an octave  (C to B)  and/or the  C  tone it starts with :
Subcontra  Contra Low Bass Middle Treble High Supra  
Octave # 012345678
Scientific C0C1C2C3C4 C5C6C7C8
Yamaha C-1C0C1C2 C3C4C5C6C7
Helmholtz
(1863)
Helmholtz
C,,
,,C
CCC
C2
2C
C,
,C
CC
C1
1C
Ccc'
c1
c''
c2
c'''
c3
c''''
c4
c'''''
c5
LilyPond c,,,c,,c,cc' c''c'''c''''c'''''
Octocontra
(nonmusical)
Sub-
contra
Contra Great Small 2nd
Small
3rd
Small
4th
Small
5th
Small
6th
Small
  1-line 2-line 3-line 4-line 5-line
Alternate Triple
Pedal
Double
Pedal
Pedal Bass Middle Treble Top Double
Top
Triple
Top

The  "lowoctave,  between  contra  and  bass,  is also called  "great bass".

The usual female classical opera repertoire extends to  Soprano Bb  (Bb5; one whole tone below C6; Soprano C, High C or Top C).  Legendary prima donnas have routinely gone beyond this, up to and including  G#6.

The highest note ever sung in a regular performance at  New-York's Met Opera  was  A above high-C  (A,  1760 Hz)  by soprano  Audrey Luna  as the very first note of  Leticia  in  The Exterminating Angel  by  Thomas Adès  (2017).  She can  sustain  C above high-C  (C,  2093 Hz)  at full voice.

In 2003,  Maria Carey  hit a  G7  (3136 Hz)  during a public rendition of  The Star Spangled Banner For the  Guinness book of world records,  singer  Adam Lopez  smashed his own previous record for a male vocalist  (D,  2349 Hz)  by almost a full octave. when he hit a  C#8  (4435 Hz)  in front of a live studio audience  (2008).  That's one semitone  beyond  the piano range.

An urban legend says that the record for a female vocalist is a  G10  by Brazilian singer  Georgia Brown.  This is silly;  a  G10  would be squarely  ultrasonic  and inaudible  (25 kHz).  In the video,  she does hit a very respectable  A#7  (3729.31 Hz).  Three semitones above Maria Carey's  Gnot  three octaves above it.  (Did someone confuse semitones and octaves?)

Scientific pitch notation (SPN)   |   Helmholtz pitch notation (1863)   |   Hermann von Helmholtz (1821-1894)
Pitch class   |   C = Do = Ut   |   Sharp (dièse)   |   Flat (bémol)   |   Natural (bécarre)   |   Blue note (note bleue)
 
Octave nomenclature   |   Pitch Notation & Octave Naming  by  Brandy Kraemer  (2018-12-03).
Modal voice   |   Falsetto   |   Whistle register (typically above D6 )   |   Yma Sumac (1922-2008)
 
Developing Perfect Pitch :  16:25 (2016-06-21)  10:08 (2016-06-29)  15:54 (2016-07-05)  by  Rick Beato.
Why Adults Can't Develop Perfect Pitch (16:25)  by  Rick Beato   (2017-05-25).
 
Developing Perfect Pitch (12:55)  by  Aimee Nolte   (2017-01-20)
 
One hour of Perfect pitch programming (1:00:00)  Music Training Club  (2015-05-30).
 
The Same Song Played in Every Key (2:24)  Fabricio André Bernard Di Paolo  (Lord Vinheteiro, 2020-03-24).


(2020-06-18)   Legacy Pitch.
Alternate choices of frequencies for  "A above middle C"  (A4 ).

The  tuning fork  was invented in 1711 by  John Shore (c.1662-1752).  Before that time,  the only records of bygone tunings are extant  organ pipes.

There's a slight complication with organ pipes:  Their frequencies are  proportional to the square root of the absolute temperature  of ambiant air.  For Church organs,  the "standard" temperature was then assumed to be  59°F = 15°C = 288.15 K  (much cooler than what scientists use as "room temperature").  Any actual measurement should be adjusted to that temperature,  according to the square-root law.
 
The frequency of steel tuning-forks varies with temperature in the opposite direction  (they sound flatter when hot)  but about 20 times less so than wind instruments  (about 86 ppm/°C,  which translates musically to  0.15 cent/°C).  That's negligible under ordinary circumstances.  It might be possible to devise an  alloy  to suppress the temperature-dependence of tuning-forks,  which would be intermediate between  invar (64% Fe, 36% Ni)  and  elinvar (52% Fe, 36% Ni, 12% Cr).  To the best of my knwledge,  this has never been perfected  (as there are cheaper ways to generate a very accurate sound frequency, in this electronic age).

415HzBaroque low-pitch  (modern convention).
422HzClassical pitch  (Mozart, Beethoven, ...).
423.2HzDresden Opera House  (1815).
430.54HzPhilosophical pitch.  Sauveur  pitch (1713).
432HzPromoted as "Verdi tuning" by  LaRouche followers.
435HzFrench standard  (1859-02-16)  until 1936.
440HzNormal pitch (since 1936).  Stuttgart pitch.  A440.  ISO 16.
442HzSymphonic pitch.
452.5HzPhilarmonic pitch.  British Army.  Brass-bands.
457HzUS military high-pitch.
466HzRenaissance high-pitch  (modern convention).
506HzGreat organ at  Halberstadt  (Nicholas Faber, 1361).

The only  (musically irrelevant)  virtue of the  philosophical pitch  is to make any  "C"  a whole number of Hz, down to  C-4 = 1 Hz,  using:

A4   =   2 8.75 Hz   =   430.5389646099018460319362438314... Hz

Likewise,  432 Hz  gives a whole value in Hz to any  A  above A0 = 27 Hz  (compared to  A0 = 27.5 Hz  under the standard  A440  tuning system).

The 415 and 466 Hz tuning are recent conventions corresponding to just one semitone lower or higher than the A440 modern standard  (1936).  This appeals to musicians who want a better historical authenticity for periods when standard tuning was notoriously higher or lower than today.  That solution is also compatible with electronic instruments which routinely offer the ability to transpose up or down by a whole number of semitones.

Part of this is a fad.  Most of us,  mere mortals not blessed or cursed with perfect pitch,  can't tell the difference when everything is transposed by a semitone.  However,  it's true that a violin tuned one semitone higher will have a slightly richer harmonic content which can be perceptible  (at least by the same kind of people, with or without perfect pitch, who can tell a Stradivarius from a lesser instrument).  This once drove  pitch inflation.

A440   |   Philosophical pitch
 
History of pitch for brass instruments  by  Philip Holcomb.
 
The History of Musical Pitch  by  Alexander J. Ellis  (Nature, pp 550-554, 1880-04-08).
 
Baroque and modern pitch (10:49) by  Sarah Jeffery  (Team Recorder, 2019-08-22).
 
432hz is a sham, please don't buy into it, thanks (21:04) by  Adam Neely  (2017-07-31).
 
How a London Orchestra Broke International Law (9:59)  by  Cory Arnold  (12tone, 2019-01-11).


(2018-02-23)   Musical staves and clefs.  Grand staff.
A staff consisting of 5 lines (4 spaces) can be extended with  ledger lines.

Between the bass and treble staff,  there would normally be room for just a single  ledger line  corresponding to  middle C.  However, the two staves are normally interpreted on the piano by the two hands  (bass staff for the left hand and treble staff for the right hand)  and they are printed with enough room between them to allow for  several  ledger lines.  Middle C  and the adjoining notes are printed either with the bass staff or with the treble staff,  depending on which hand is meant to play them.

Wikipedia :   Staff (Stave)   |   Rastrum   |   Tablature
 
How to Read Piano Notes:  Treble & Bass Clefs (7:22)  by  Mantius Cazaubon   (2014-10-27). <
Using Landmarks to Read Notes Faster (9:48)  by  Rob   (Musicians Inspired, 2017-11-04). <
How to Read Music (31:52)  by  whmozart   (Measure by Measure, 2016-05-20).
How to Read Music (14:39)  by  Sarah Jeffery  (Team Recorder, 2019-02-14).
 


(2018-02-18)   25,  32,  37,  44,  49,  54,  61,  64,  73,  76  or  88  keys.
A grand-piano keyboard has 88 keys  (some rare pianos have 92 or 97).  Two-octave 97-key keyboard

 9 extra bass keys  

The  Imperial Bösendorfer Model 290  (290 cm)  was introduced in 1909 spanning eight octaves (97 keys from C0 to C8)  as suggested by composer  Ferruccio Busoni (1866-1924)  to accomodate some organ works of  Johann Sebastian Bach (1685-1750):  Some 32-foot registers of large organs do go down to  C at 16.35 Hz.
 
As shown at left,  the extension of nine extra bass keys is signaled visually by five dark-brown tops on keys which could be expected to be white.

Organ  manuals  almost always go from  C  to  C.  Full-sized ones span  five  octaves  (61 keys)  more rarely  six  (73 keys)  or  seven  (85 keys)  as found only in a few very large organs meant to play  C0  (16 Hz or so)  the lowest note in the classical repertoire,  which is  felt  more than it is  heard :

 Full 88-key piano keyboard

Historically,  most organ keyboards spanned only  four  octaves  (49 keys).  Small  37-key  manuals are also found.  The  pedalboards  of traditional organs have between  12  and  32  keys.

Twelve sizes of electronic keyboards are widely available in the sixteen different layouts illustrated below.  As these keyboards can often be shifted at will by a whole number of octaves,  the highlighted positions may not always play as  middle C  or  concert A .

8 white
keys
  13-key keyboard C-C
25 keys   Two-octave 25-key keyboard
 
Small accordion
C-C
32 keys   32-key keyboard
Casio Rapman
(1991)
F-C
32 keys   32-key keyboard C-G
37 keys   37-key keyboard F-F
37 keys   37-key keyboard C-C
44 keys   44-key keyboard F-C
49 keys   49-key keyboard C-C
54 keys   54-key keyboard C-F
61 keys   61-key keyboard C-C
64 keys   64-key keyboard A-C
  73-key keyboard 73
  73-key keyboard 73
  73-key keyboard 73
  76-key keyboard 76
 Full 88-key piano keyboard

Both ends of the  64-key  keyboard match the layout of a grand piano.  This pattern helped make  all  Wurlitzer electric pianos  popular,  from 1954 to 1984  (besides a rare 44-key simplified classroom model).  The design was revived in the recently-discontinued  Roland RD-64  (introduced in 2013)  which was  unique in its class,  with  64  weighted hammer-action keys.  With controls to the left, the RD-64 is about as long as a 73-key keyboard.

Roland RD-64, 64-Key Digital Stage Piano & MIDI Keyboard (4:45)  by  Matt Watson   (2014-09-23).
Roland RD-64 Digital Piano Sound Demo (12:31)  RADIO EMA Educational  (2014-09-11).
Roland RD-64 (6:13)  muzykujkropkacom  (2013-08-06).
 
Piano Keyboard Diagram  (layout of a piano keyboard)  by  Mantius Cazaubon
The Layout of Piano Keys on the Keyboard (10:50)  by  Mantius Cazaubon   (2014-10-28).
What Keyboard Should I Buy? (9:28)  by  Mantius Cazaubon   (PlayPianoTODAY.com, 2013-01-11).
 
Wikipedia :   Musical keyboards (from early 8+4 to modern 7+5)   |   Manuals   |   Pedalboards


(2017-04-09)   Relative Pitch  &  Tone Intervals
Ratios of sound frequencies.

Harmony is perceived when two tones are heard whose frequencies are in a ratio close to the ratio of two small integers.  The smaller the integers,  the greater the impression of harmony.  Thus,  the octave is the most harmonious interval  (2:1 ratio)  besides unison  (1:1 ratio).  The fifth  (3/2 ratio)  is not far behind.

Pure Natural Consonant Intervals :

Two  consonant  tones are characterized by frequencies in a simple ratio  (i.e., the ratio of two  small  integers).  A musical interval is a frequency ratio.  Natural consonant intervals are thus ratio of small integers.  The most important ones have traditional  musical  names:

Pure Intervals and Equal-Tempered Approximations Thereof
RatioEnglishFrenchApproximationSemitones
1 : 1UnisonUnisson10
16 : 15Half-toneDemi-ton1.059463094361  (+0.11731)
10 : 9Lesser tone1.122462048312  (-0.17596)
9 : 8Whole tone2  (+0.03910)
8 : 7Septimal tone2  (+0.31174)
7 : 6Septimal Minor Third 1.189207115003  (-0.33129)
6 : 5Minor thirdTierce mineure3  (-0.15641)
5 : 4Major thirdTierce majeure1.259921049894  (+0.13686)
4 : 3FourthQuarte1.334839854175  (-0.01955)
7 : 5Lesser Septimal Tritone 1.414213562376  (-0.17488)
10 : 7Greater Septimal Tritone6  (+0.17488)
3 : 2FifthQuinte1.498307076887  (+0.01955)
8 : 5Minor SixthSixte mineure1.587401051978  (-0.13686)
5 : 3Major SixthSixte majeure1.681792830519  (+0.15641)
12 : 7Septimal Major Sixth 9  (+0.33129)
7 : 4(Septimal) Harmonic Seventh 1.7817974362810  (-0.31174)
16 : 9Minor Seventh 10  (-0.03910)
9 : 5Greater Minor Seventh 10  (+0.17596)
15 : 8Major Seventh 1.8877486253611  (-0.11731)
2 : 1OctaveOctave212

Note the perfect vertical symmetry of the above table with respect to the horizontal line which separates the two kind of tritones.  The product of the ratios (from the first column) in symmetrical pairs is equal to 2.  Equivalently, their values in semitones  (last columns)  add up to 12.  Musicians call that symmetry an  inversion  (French:  renversement).  For example,  the  septimal major sixth  is the inversion of the  septimal minor third  (both of those are quite rare).

One percent of a semitone is called a  cent.  The  fourth  and the  fifth.  have excellent chromatic approximations, which are less than  2 cents  off  (that's impossible to detect by ear).  Likewise for the whole tone (9:8) and minor seventh (16:9)  which are less than 4 cents off.

The other important musical intervals appearing in unshaded entries have good enough chromatic counterparts,  which are respectively off by only  11.7,  15.6  and  13.7 cents,  for the  half-toneminor thirdmajor third  and their respective inverses.  This is the happy coincidence  (a minor mathematical miracle)  which made equal-tempered Western music possible.

A chromatic whole step is also close enough to the  10:9  natural harmony  (lesser tone).

The septimal intervals  (involving the seventh harmonic of fundamental sounds)  are rarely found outside of  Blues  nowadays,  starting with the  lesser septimal tritone  (7:5)  also known as the septimal of Huygens  and its inverse,  the  greater septimal tritone  (10:7)  also called  Euler's tritone.

For intervals greater than an octave  (but less than two octaves)  musicians often talk about  compound  intervals and will use a locution like  compound major third  (compound M3)  instead of  major tenth  (M10).  Lesser musical names have been given to some intervals greater than an octave,  including:

  • 11:5   Neutral Ninth.  (A  neutral second  is 12:11.  Very close to 1.5 semitones.)
  • 9:4   Major Ninth.
  • 7:3   Minimal Tenth.
  • 3:1   (Bohlen-Pierce) Tritave.

Besides unison and octaves,  there are  11  possible intervals in equal-tempered Western music.  Some are good approximations to the consonant   pure  intervals.  Others are  dissonant.  Here's the complete list:

  • m2:   Minor second.  Half step  =  1 semitone.  Twelfth of an octave.
  • M2:   Major second.  Whole step  =  2 semitones.  (10/9 or 9/8 ratio).
  • m3:   Minor third.  Augmented second.  Quarter of an octave.  (» 6/5)
  • M3:   Major thirdThird of an octave.  (Approx. 5/4 ratio.)
  • P4:   Fourth.  Five semitones.  (Approx. 4/3 ratio.)
  • d5, A4:   Diminished fifth or augmented fourth.  Tritone.  6 semitones.
  • P5:   Fifth.  Seven semitones.  (Approx. 3/2 ratio.)
  • m6:   Minor sixth.  Eight semitones.  (Approx. 8/5 ratio.)
  • M6:   Major sixth.  Nine semitones.  (Approx. 5/3 or 12/7 ratio.)
  • m7:   Minor seventh.  Ten semitones.  (Approx. 9/5 or 16/9 ratio.)
  • M7:   Major seventh.  Eleven semitones.  (Approx. 15/8 ratio.)
  • P8:   Octave.  Twelve semitones.  (Exactly 2/1 ratio.)

Six Spooky Secrets Composers use to SCARE you (5:36)  by  Jake Lizzio   (Signals Music Studio, 2017-10-27).

 Come back later, we're
 still working on this one...

Major, Minor, Perfect, Augmented, Diminished... (14:39) Tommaso Zillio  (Music Theory for Guitar, 2020-07-20).
 
Using stepping stones to check intervals (12:48)  by  Julian Bradley   (Jazz Tutorial, 2017-11-18).
Perfect Pitch  vs.  Relative Pitch (12:00)  by  Rick Beato   (2017-01-16).
Just intonation in the Renaissance (14:10)  by  Elam Rotem   (Early Music Sources, 2016-10-06).
Why Music Only Uses 12 Different Notes (14:40)  by  David Bennett   (2020-07-27).
 
Wikipedia :   Music   |   Pitch   |   Unison   |   Octave   |   Decade   |   Major third   |   Minor third
Equal temperament   |   Just intonation   |   List of pitch intervals
Perfect fifth   |   Circle of fifths   |   Wolf fifth (quinte du loup)   |   Superparticular ratios (epimoric ratios)
Bohlen-Pierce scale (1972, 1984)   |   Heinz Bohlen (1935-2016)   |   John R. Pierce (1910-2002)


 Leonhard Euler (2020-08-20)   Euler's Gradus Function   (Euler, 1730)
Suavitatis Gradus   =   Degree of harmony  (suavity).

Arguably,  music builds on the ancient notion of  commensurability:  Two numbers are  commensurable  when they are proportional to two integers.  They are  consonant  when those two integers are  small.  Consonance is thus not a quality that commensurable quantities possess or not;  it's something they have to a greater or lesser  degree  (Latin:  gradus).

Euler's musical treatise  Tentamen Theorae Novae Musicae  was completed in 1730  (when he was 23)  but only published in 1939.  In it,  Euler tried to quantify how  pleasant  two integers are when they measure musical frequencies heard together.  This work didn't attract much interest at the time.  It was said to be too mathematical for musicians and too musical for mathematicians.

For the elusive thing he was trying to quantify,  Euler used the Latin term  suavitas,  which generally indicates how something appeals to the senses or the mind.  Possible translations include:  Charm, appeal, beauty, attractiveness, pleasantness, agreeableness or sweetness.  In a musical context, that could also be called tunefulness, melodiousness, harmoniousness, consonance, smoothness or softness  (as opposed to harshness or spiciness).  Take your pick.  I'll use the direct descendant "suavity" in the rest of this article.

It's clear at the outset that any two-variable function  f  grading the consonance of two numbers must be symmetrical and invariant by scaling  (as the unit of frequency is entirely arbitrary):

f (p,q)   =   f (q,p)   =   f (kp,kq)   =   f (p/q, 1)

using the single-variable grading function  g(x) = f (x, 1)  symmetry implies:

g (x)   =   g(1/x)

Presented as a consequence of interchanging two tones whose consonance is being compared,  this symmetry is obvious.  However,  this implies the nontrivial fact that inverting all tones in a musical piece gives a totally different piece which is equally harmonious.  A fact which was well-known to  Johann Sebastian Bach (1685-1750), who made extensive use of that symmetry in  some of his compositions.  contemporary authors have dubbed related topics  negative harmony.

 Come back later, we're
 still working on this one...

n 123456 789101112 131415161718 19202122232425
gE (n) 123354 7456115 13875176 1979122369
g0 (n) 012243 6345104 12764165 1868112258

As  gE(n)  is always equal to g0(n)+1,  either function is equally suited for comparisons.  However  g0  possesses the rare property of being a  totally additive function  which is to say that it ressembles a  logarithm:

g0 ( x y )   =   g0 (x)  +  g0 (y)

By contrast gE  seems flawed:   gE ( x y )   =   gE (x)  +  gE (y)  +  1

g0  is the difference of two standard (totally) additives functions,  best defined by their effects on the power of a prime  (pn ):

  • A001414:  "Sum of prime factors"   sopfr (pn )  =  n p
  • A001222:  "Number of prime factors"   W (pn )  =  n

p-adic metric   |   A275314:  Euler's gradus function.   |   A001414MacMahon's potency
Tentamen Novae Theoriae Musicae (304 pages, 1739)  by  Leonhard Euler (1707-1783).
Gradus Suavitatis  by  Oliver Knill
Euler's  Degree of Agreeableness  for Musical Chords  by  Oliver Knill (2018-08-09).
Euler and music:  A forgotten arithmetic function  by  József Sándor  (Octogon, April 2009).
 
Number theory underlying Euler's theory of music  by  Brian Rushton  (MathOverflow, 2013-07-13).
 
Leonhard Euler's Magical Consonance Formula (9:07)  by  Cory Arnold  (12tone, 2019-01-18).
 
On Musical Scale Rationalization  by  Albert Gräf  (2006).
 
On the quantification of harmony and metre (2001)  by  Clarence Barlow  (1945-).


(2018-02-15)   The  Circle of Fifths  was born an  Archimedean  spiral...
Two conflicting musical intervals:  Perfect fifth  (3:2)  &  octave  (2:1).

perfect  fifth is a musical interval corresponding to a frequency ratio of  exactly 3:2.  Twelve of those intervals is  slightly over  seven octaves:

1.5 12   =   129.746337890625             vs.             2 7   =   128

Therein lies the secret of Western music;  this simple statement is the reason why we have  12  notes but only use  7  of them in a given  key.

 Come back later, we're
 still working on this one...

Using the Circle of Fifths (15:16)  by  Michael New   (2015-03-13).
Circle of Fifths Explained (18:02)  by  Julian Bradley   (Jazz Tutorial, 2013-02-05).
Beato Basics of Music Theory (10:41)  by  Rick Beato   (2018-04-09).
Circle of Fifths and Key Signatures (21:45)  by  Tim  (Piano Lessons On The Web, 2016-08-05).
Circle of Fifths and Key Signatures (25:32)  by  Gareth Green  (Music Matters, 2014-05-14).
8 Facts about the Circle of Fifths (21:45)  by  Brian Kelly  (Zombie Guitar, 2017-04-11).
Why It's Impossible to Tune a Piano (4:19)  by  Henry Reich   (2015-09-17).
 
Keys, Scales & Intervals (51:12)  by  Declan Plummer  (Decky Pipes #1, 2015).
 
Circle of fifths   |   Heptatonic scales   |   Pentatonic scales


(2018-02-17)   Tritone:  Augmented fourth (A4) or diminished fifth (d5).
Diabolus in Musica.  The sound of a police siren.  Half an octave.

As an octave is a factor of  2,  half  an octave is the  square root of two.

1.4142135623730950488016887242...

The  convergents of the square root of  2  are easily obtained from its  continued fraction expansion  [1;2,2,2,2,2,2,2...].  They are:

1,  3/2,  7/5,  17/12,  41/29,  99/70,  239/169,  577/408,  1393/985,  ...

The  perfect fifth  (3/2)  isn't close enough to a tritone and  the integers in the ratio  17/12  (1.416666...)  are not small enough to qualify as  harmony.  So,  the closest harmonious approximation to a tritone is  7/5 = 1.4.  That unusual interval is technically called the  lesser septimal tritone.  It's used in  Blues.

Its musical inverse  (10/7)  is the  greater septimal tritone.  The ratio of those two tritones is 49/50 = 0.98.  The lesser one is thus exactly  2%  below the greater one.  That makes them about 0.35 semitones apart.

The Devil in Music:  Untold History of the Tritone (9:58)  by  Adam Neely  (2017-06-05).


(2018-02-19)   Pentatonic,  diatonic (heptatonic) and chromatic  scales.
Current keyboards were made for the key of  C major  (or  A minor).

In the modern equal-temperament formally introduced by J.S. Bach,  the  chromatic scale  consists of the twelve pitches whose frequencies form a geometric progression of constant ratio  1.059463...  (the twelfth root of 2)  modulo factors of any power of two  (that's the learned way to say that frequencies separated by any whole number of octaves represent the same pitch and have the same name in Western art music).

The qualifier  chromatic  also apply to any subset thereof which is not strictly contained in a  diatonic scale,  as described next.

Diatonic Scale :

The Western diatonic scale is the  heptatonic  scale formed by the white keys of the piano,  or any transposition thereof.  It includes only seven notes per octave.  The first one is called  tonic,  the fifth one  dominant.

There's more to a scale than the notes it consists of.  The tonic plays a central role  (the melody normally starts and ends with the tonic).  The dominant is a close second in importance.  The specific seven notes which appear in the octave of a diatonic scale are called its  key signature.  Most famously,  C-major  and  A-minor  share the same key signature  (you play them using only the white keys on a piano).

There are  12  major scales and  12  (natural) minor scales.  Six of those  (3 major, 3 minor)  have two enharmonic names  (played alike but denoted differently).  So,  the 24 common  diatonic  scales have 30 different names:

A  7-note  diatonic scale is named after its first note  (the tonic).  E.g.,  C-major.
Modes which share the same tonic are said to be  parallel.
Major ScaleMinor Scale
1 2  34 5 6 7 1 23 4  56 7 
F G  ABb C  D E F GAb  Bb  CDb Eb 
CDEFGAB CDEbFGAbBb
GABCDEF# GABbCDEbF
DEF#GABC# DEFGABbC
ABC#DEF#G# ABCDEFG
EF#G#ABC#D# EF#GABCD
B
Cb
C#
Db
D#
Eb
E
Fb
F#
Gb
G#
Ab
A#
Bb
BC#DEF#GA
F#
Gb
G#
Ab
A#
Bb
B
Cb
C#
Db
D#
Eb
E#
F
F#G#ABC#DE
C#
Db
D#
Eb
E#
F
F#
Gb
G#
Ab
A#
Bb
B#
C
C#D#EF#G#AB
AbBbCDbEbFG G#
Ab
A#
Bb
B
Cb
C#
Db
D#
Eb
E
Fb
F#
Gb
EbFGAbBbCD D#
Eb
E#
F
F#
Gb
G#
Ab
A#
Bb
B
Cb
C#
Db
BbCDEbFGA A#
Bb
B#
C
C#
Db
D#
Eb
E#
F
F#
Gb
G#
Ab

The above is best called the  natural minor scale  (or  descending minor scale).  It's a true diatonic scale  (in the key of A, it uses only the white keys of the piano)  but it suffers from a flaw in ascending order in the sense that the last note,  called the  subtonic,  is no longer just a  semitone  away from the tonic  (as is the  leading tone  in the major scale)  but a whole tone away...

To remedy that,  the  harmonic minor scale  is introduced which raises the subtonic one semitone, back to leading tone status.  However,  this fix introduces an unusual large gap of  three  semitones between the previous note  (the  submediant)  and the subtonic/leading tone.

Another flavor of minor scale,  called the  melodic minor scale  is introduced which closes the gap by raising that previous note too.  All told,  the melodic minor scale is obtained from the  major  scale simply by lowering just the third note  (so that the last four notes are separated by a whole tone).  This is sometimes called the  ascending melodic minor scale  and it's so often used in Jazz  that it's most commonly known as the  Jazz minor scale.

The semi-obsolete locution  descending melodic minor scale  simply refers to the classical  natural minor scale  we've already discussed (which is merely the sixth  mode  of the major scale,  also known as  Aeolian).

Neither the harmonic nor the melodic minor scales obey the diatonic pattern :
Harmonic Minor ScaleMelodic Minor Scale
1 23 4  56  7 1 23 4  5 6 7
F GAb  Bb  CDb    E F GAb  Bb  C  D  E
CDEbFGAbB CDEbFGAB
GABbCDEbF# GABbCDEF#
DEFGABbC# DEFGABC#
ABCDEFG# ABCDEF#G#
EF#GABCD# EF#GABC#D#
BC#DEF#GA# BC#DEF#G#A#
F#G#ABC#DE# F#G#ABC#D#E#
C#D#EF#G#AB# C#
Db
D#
Eb
E
Fb
F#
Gb
G#
Ab
A#
Bb
B#
C
AbBbCbDbEbFbG AbBbCbDbEbFG
EbFGbAbBbCbD EbFGbAbBbCD
BbCDbEbFGbA BbCDbEbFGA

It's usually considered undesirable to have different accidentals on the key signature of modern sheet music, as would happen in three of the above cases  (G harmonic or melodic, and D harmonic).  The problem can be solved in those cases by using only flats in the key and raising the last note with an individual sharp whenever it occurs.

Minor scales

Pentatonic Scales :

One particular example of a pentatonic scale is  C-D-E-G-A.  Another one is formed by the black keys of the piano,  starting with  F#.

Many examples of similar pentatonic scales exist outside of Western music.

 Come back later, we're
 still working on this one...

How to Find the Key of a Song, by Ear (14:28)  by  Rhythmic Canada   (2012-04-23).
Three Scales Every Musician Should Know (36:17)  by  Corey Taylor   (Skilled Musician, 2018-01-10).
Guitar solo beyond the minor pentatonic scale


(2018-02-26)   The Seven Diatonic Modes  (modes of the major scale)
key  is given by a  tonic  and a  mode  (e.g.,  C-major  or  F-Lydian).
The modern diatonic modes described below are named after ancient Greek  harmoniai  (or tonoi)  and/or the medieval church modes.  However, those concepts are only loosely related.

By definition, a diatonic scale consists of seven steps; two half-tones (H) separated by alternating groups of two or three whole-tones (W).  There are seven ways to choose a starting point in such a progression.  Each such way is called a  mode.  The most common ones are the aforementioned major and minor modes,  also called Ionian and Aeolian.  Here's the complete list:

The Seven Diatonic Modes,  Ordered from Brightest to Darkest :
 Mode Name Interval structure:  7  Steps8  White Notes
M
A
J
O
R
Lydian 2221221WWWHWWHFGABCDEF
Ionian  (major) 2212221WWHWWWHCDEFGABC
Mixolydian 2212212WWHWWHWGABCDEFG
M
I
N
O
R
Dorian 2122212WHWWWHWDEFGABCD
Aeolian  (natural minor) 2122122WHWWHWWABCDEFGA
Phrygian 1222122HWWWHWWEFGABCDE
Locrian 1221222HWWHWWWBCDEFGAB

105 names for the  84  diatonic scales:  7  modes in 12 keys  (15 allowed names).
(For each mode,  the  tonic  can be one of the  12  choices in column "1".)
1IV:   Lydian7 18 
1I:   Ionian  (= Major)7 17 
1V:   Mixolydian7 16 
1II:   Dorian7 15 
1VI:   Aeolian  (= Minor)7 14 
1III:   Phrygian7 13 
1VII:   Locrian7 12 
F  G  ABb  C  D E F  G  ABb  C  D E
CDEFGAB CDEFGAB
GABCDEF# GABCDEF#
DEF#GABC# DEF#GABC#
ABC#DEF#G# ABC#DEF#G#
EF#G#ABC#D# EF#G#ABC#D#
B
Cb
C#
Db
D#
Eb
E
Fb
F#
Gb
G#
Ab
A#
Bb
B
Cb
C#
Db
D#
Eb
E
Fb
F#
Gb
G#
Ab
A#
Bb
F#
Gb
G#
Ab
A#
Bb
B
Cb
C#
Db
D#
Eb
E#
F
F#
Gb
G#
Ab
A#
Bb
B
Cb
C#
Db
D#
Eb
E#
F
C#
Db
D#
Eb
E#
F
F#
Gb
G#
Ab
A#
Bb
B#
C
C#
Db
D#
Eb
E#
F
F#
Gb
G#
Ab
A#
Bb
B#
C
AbBbCDbEbFG AbBbCDbEbFG
EbFGAbBbCD EbFGAbBbCD
BbCDEbFGA BbCDEbFGA
MmmMMmmo MmmMMmmo
1st2nd 3rd4th5th 6th7th  <<< mode of the major scale.

In  Numericana  tables like this one,  the next-to-last line gives the  quality (major, minor, diminished, augmented)  of the in-scale triad rooted in the corresponding column.  The last line repeats the rank of the mode named & numbered as a 1-7 span at the top of the table.

Two scales which share the same diatonic mode are said to be modes of each other.  For example,  F-Lydian is the 4-th mode of C-major  (i.e., C-Ionian)  because its tonic (F) is the 4-th note in the C-major scale.  Likewise,  G-Mixolydian is the fifth mode of C-major or the second mode of F-Lydian.  In modern practice,  this type of reference is most commonly used with respect to the relevant major scale only, as indicated in the last line of the above table.

Dorian  is the only  palindromic mode  (2122212).  In all other cases,  a mirror mode is obtained by  inversion:

  • Mixolydian (2212212) and Aeolian (2122122).
  • Ionian (2212221) and Phrygian (1222122).
  • Lydian (2221221) and Locrian (1221222).  Brightest and darkest.

For each of those mirror modal pairs,  the ascending version of one is the same as the descending version of the other.

Mixed Modes and Polymodal chromaticism :

Béla Bartók (1881-1945) remarked that all 12 chromatic tones are obtained by mixing some pairs of diatonic modes on the same tonic  (e.g., C-Lydian with C-Phrygian).  More generally, the above table cab be used to count how many tones are obtained when mixing two diatonic modes on the same tonic.

Number of tones obtained by mixing two modes
 IVIVIIVIIIIVII
FLydianIV   7  8910111212
CIonianI 8  7  89101112
  G    Mixolydian  V 98  7  891011
DDorianII 1098  7  8910
AAeolianVI 111098  7  89
EPhrygianIII 12111098  7  8
BLocrianVII 1212111098  7  

Among many other things,  this provides a name for a half-dozen special octatonic scales,  which are actually the first 6 modes of the  Bebop (dominant) scale:

  • Lydian-Ionian   adds Bb to F-Lydian or F# to C-Ionian.
    Mode IV of the Bebop scale
  • Ionian-Mixolydian   adds Bb to C-Ionian or F# to G-Mixolydian.
    Mode I of the Bebop scale.
  • Mixolydian-Dorian   adds Bb to G-Mixolydian or F# to D-dorian.
    Mode V of the Bebop scale
  • Dorian-Aeolian   adds Bb to D-dorian or F# to A-Aeolian.
    Mode II of the Bebop scale.
  • Aeolian-Phrygian   adds Bb to A-Aeolian or F# to E-Phrygian.
    Mode VI of the Bebop scale.
  • Phrygian-Locrian   adds Bb to E-Phrygian or F# to B-Locrian.
    Mode III of the Bebop scale.

What is a mode? (53:26)  by  Leonard Bernstein  (Young Peoples Concerts, 1966-11-21).
The Lydian Mode (10:50)  by  Rick Beato  (2020-01-05).
The Dorian Mode (11:50)  by  Rick Beato  (2020-03-05).
The Mixolydian Mode: The sound of Rock (12:09)  by  Rick Beato  (2020-02-17).
The Sound of Modes (11:00)  by  Rick Beato  (2019-10-05).
Why C major and A minor are Not the Same  (14:59)  by  Michael New   (2015-11-10).
Why is major  happy ?  (7:35)  by  Adam Neely   (2016-05-30).
The 7 modes and modal interchange  (10:40)  by  Rob  (Musicians Inspired, 2016-08-19).
Interesting chord progressions using modal interchange (4:25)  by  Myles Yang   (2017-05-30).
Demonstrating all 7 modes and their home chords (16:04)  Signals Music Studio   (2018-03-16).
Songs That Actually Use Locrian (8:33)  by  David Bennett Piano   (2019-02-21).
Scales vs. Modes:  What's the Difference? (33:08)  by  Rick Beato   (2020-01-06).
 
Wikipedia :   The diatonic scale   |   Modes   |   Degrees   |   Heinrich Glaeran (1488-1563)


(2020-07-20)   Naming Non-Diatonic Scales and Modes
A systematic nomenclature to define precisely lesser-known modes.

The diatonic scale  (especially its Major and Minor modes)  is the backdrop for almost all classical music and a good chunk of modern tunes, although 6 of the common diatonic modes are now in common use  (locrian  is left out).

Only four heptatonic scales in the Western chromatic system can be expressed  canonically  in all 12 keys,  by naming all seven notes  (letters)  once and only once,  with at most one sharp or one flat each.  Namely:

This property,  which is taken for granted by most casual students of Western music,  is indeed a rare one.  It fails whenever the scale includes two consecutive half-steps  (HINT:  For every mode, there's a key where A and G are included with the tone between them, which can be called neither A# nor Gb without repeating a letter).  There are only two other heptatonic scales for which this doesn't happen  (Hungarian major and Romanian major)  but they both fail in two keys,  for less obvious reasons.

This fact is lost on most composers and almost all practicing musicians, with little or no consequences:  Transposing a piece in writing for all possible keys is rarely required, if ever.

So, there's little or no obstacle to experimentation with a  huge number  of exotic scales.  The most popular ones eventually get a colorful name.  For others, the standard practice is to use a known name  (preferably one of the  diatonic modes  and indicate what modification(s), sharp of flat, is to be applied to what degree(s).

Some are queasy about using degree 1 in this scheme.  I beg to differ but accomodate those concerns by putting such names inside square brackets.

  • [ Mixolydian #1 ]   =   Ultralocrian.
  • [ Mixolydian b1 ]   =   Mode V of  Major b5.
  • [ Locrian b4 ]   =   Super-Locrian.
  • [ Aeolian b1 ]   =   Lydian Augmented #2.

William Zeitler (1954-) has taken the radical creative step of assigning separate  greek-sounding names to all possible modes. The words he coined remain unused,  except as welcome identifiers for some obscure entries of the  comprehensive electronic catalog  put together by  Ian Ring.

The systematic exploration of non-diatonic scales and their modes started in 1907 with the investigations of  synthetic scales  by  Ferruccio Busoni (1866-1924)  who first considered all scales which could be derived from a diatonic mode by lowering or raising a single degree by one semitone.

Each of the  7  diatonic modes features  5  skipped notes which can be included in one of two ways.  Therefore,  70  names can be attributed to new scales this way  (for a total of 77 names)  However,  some of those scales may have more than one name  (as is the case for any mode of the melodic minor scale).  For example,  the  Assyrian scale  (mode II of the  melodic minor scale)  possesses two distinct systematic names:  either  Dorian b2  or  Phrygian #6.  In the key of G,  that's:

G   Ab   Bb   C   D   E   F

As  G-Dorian is   G A Bb C D E F   and  G-Phrygian is   G Ab Bb C D Eb F.  Conversely,  some of the 66 different heptatonic scales can't be named at all with just one modification from a diatonic mode.

The scope was later expanded to include  all possible scales  and their respective modes.

Dominant  means the 3 is major and the 7 is flat.

 Come back later, we're
 still working on this one...

The whole problem is of greater theoretical interest than of practical worth.
J. Murray Barbour (1929) 

Synthetic scales (1907)   |   Synthetic modes   |   Ferruccio Busoni (1866-1924).
 
Thirteen scale families and all their relationships  by  Matt Matheson  (Metagloria/Reddit, 2016).


 Gerard Michon (2020-06-26)   Defining Brightness and Darkness
A nice quantification of an elusive concept.

The white numbers which appear in the rightmost column of our modal charts for heptatonic chromatic scales (including the seven diatonic modes from the previous section)  are a numerical evaluation of  brightness  computed as follows:

For each of the seven notes allowed in a given mode,  we count the number of  disallowed  chromatic tones between the root and itself.  Adding all those counts give a sum of seven integers  (always starting with 0 for the root itself)  totaling between 0 and 30  (those extremes are reached for two modes of the pathological scale where consecutives notes are separated by either one semitone or seven).

For example, the Lydian mode of the major scale has a brightness of 18 (namely 0+1+2+3+3+4+5) whereas the brigthness of the major scale itself (or rather  Ionian,  its first mode) is only 17 = 0+1+2+2+3+4+5.

This evaluation can be proved to be correct for diatonic scales through the miracle of the  circle of fifths  which shows that the above count actually measures precisely  set inclusion...  In other words, changing diatonic modes with a fixed root will never raise some notes and lower others.  It does one or the other, for all individual notes in all pieces of music ever written without individual accidentals.

That wonderful fact isn't true for the modes of non-diatonic scales analysed below.  Yet, we may keep the above measure as a standard indication of the otherwise elusive concept of brightness,  under the debatable simplification that brightness only depends on the difference between the number of notes raised and the number of notes lowered.

We may boldly use that number to compare the brightness of different modes of the same scale.  Any scale.

The brightness of an n-tonic scale goes from  0 to (n-1)(12-n).  That is:

  • 0 for the monotonic scale.
  • 0 to 10 for a ditonic scale,
  • 0 to 18 for a tritonic scale.
  • 0 to 24 for a tetratonic scale.
  • 0 to 28 for a pentatonic scale.
  • 0 to 30 for an hexatonic scale.
  • 0 to 30 for an heptatonic scale.
  • 0 to 28 for an octatonic scale.
  • 0 to 24 for an enneatonic scale.
  • 0 to 18 for a decatonic scale.
  • 0 to 10 for an hendecatonic scale.
  • 0 for the dodecatonic scale.

The brightnesses of two inverse n-tonic modes add up to  (n-1)(12-n).

Linear Brightness is Tonal.  Quadratic Clarity is Atonal.

The perceived brightness of a rooted scale is probably a clever balance between the two.  For the classification of modes within a scale,  I am only retaining the linear score described above which is easy to compute and meets expectations in the only case everybody agrees on  (diatonic modes).

Barely more complicated would be the introduction of the variance (or its square root, the deviation).  Any mode could be represented as a point in the plane with linear brightness as the x-coordinate and quadratic brightness as the y-coordinate.  Experts should then be asked to draw an arrow from one point the another whenever the former is clearly brighter than the latter.

To my knowledge,  this has never been done.  A sensible formula might emerge this way.  A priori,  I'd guess that subjective brightness is an increasing function of  both  the (linear) mean and (quadratic) deviation.

We may use a linear appoximation to the ideal function so described:

f (x,y)   =   a x  +  b y           where     a ≥ 0     and     b ≥ 0

The Garklein Theorem :

Two modes of an heptatonic scale have  different  linear brightness scores.

Diatonic Modes:  From Bright to Dark   by  Frank Jargstorff  (2011-08-26).
 
Musical Palindromes & Negative Harmony (11:26)  by  Rick Beato   (2017-11-16).
 
The 7-Mode Song, Bright to dark (5:11)  by  Music Muncher   (2020-01-21).
The 7-Mode Song, Dark to bright (4:38)  by  Music Muncher   (2020-01-26).
 
The Modes Ranked by Brightness  by  David Bennett   (2019-09-13).
 
Harmonic Brightness & Darkness  by  Anton Schwarz   (2012-11-25).


(2020-06-22)   The two most common non-diatonic minor scales.
Modes of the melodic minor scale.  Modes of the harmonic minor scale.

The  natural minor scale  is diatonic.  As such,  it's just another mode of the major scale  (namely, the 6th mode of the major scale, also called  Aeolian).  Therefore,  all modes of the natural minor scale would also be modes of the major scale and they are not considered separately.

However,  neither the  harmonic minor scale,  nor the  melodic minor scale  are diatonic and they are not modes of each other either.  Therefore,  both give rise to a full set of 7 distinct modes in 12 possible keys.  We may tabulate them as we did the more common diatonic modes  (i.e, the modes of the major scale).

2 1 2 2 2 2 1 2 1 2 2 2 2 1 :   Melodic Minor Scale
All  7  modes ranked by brightness, in 12 keys  (tonic in mode's column "1")
1III:   Lydian augmented7 19 
1IV:   Lydian dominant7 17 
1I:   Jazz minor scale7 16 
1V:   Hindu scale7 15 
1II:   Phrygidorian, Assyrian 14 
1VI:   Half diminished7 13 
1VII:   Altered dominant7 11 
F GAb  Bb C  D E F GAb  Bb C  D E
CDEbFGAB CDEbFGAB
GABbCDEF# GABbCDEF#
DEFGABC# DEFGABC#
ABCDEF#G# ABCDEF#G#
EF#GABC#D# EF#GABC#D#
BC#DEF#G#A# BC#DEF#G#A#
F#G#ABC#D#E# F#G#ABC#D#E#
C#
Db
D#
Eb
E
Fb
F#
Gb
G#
Ab
A#
Bb
B#
C
C#
Db
D#
Eb
E
Fb
F#
Gb
G#
Ab
A#
Bb
B#
C
AbBbCbDbEbFG AbBbCbDbEbFG
EbFGbAbBbCD EbFGbAbBbCD
BbCDbEbFGA BbCDbEbFGA
mmM+MMmomo mmM+MMmomo
1st2nd 3rd4th5th 6th7th  <<< mode of melodic minor.

The 7 modes of the melodic minor scale are known under various names:

  1. Dorian #7.  Ionian b3.  (Ascending) melodic minor scale.  Jazz minor
  2. Phrygian #6.  Dorian b2.  Dorian b9.  Phrygidorian.  Assyrian.
  3. Lydian #5.  [ Phrygian b1.]  Lydian augmented.
  4. Mixolydian #4.  Lydian b7.  Lydomixian.  Lydian dominant.  Super Lydian.  Acoustic scale.  Overtone scale.  Bartók scale.  Simpsons ScalePrometheus heptatonicVachaspati  (64th mela).
  5. Aeolian #3.  Mixolydian b6.  Aeolian dominant.  Hindu scale.  Mixaeolian.   Melodic Major.  Major-Minor.
  6. Locrian #2.  Aeolian b5.  Minor b5.  Semilocrian.  Minor Locrian.  Half diminished.  Aeolocrian.  Altered diminished.  Overtone Inverse.
  7. [ Ionian #1.]  Locrian b4.  Altered dominant scale.  Super Locrian.

The  Hindu scale  is palindromic.  The other six modes come in three mirror pairs consisting of a bright mode and a dark mode,  increasingly different:

  • Jazz minor scale  and  phrygidorian.
  • Acoustic scale  and  aeolocrian.
  • Lydian augmented  (brightest)  and  altered scale  (darkest mode).

With an extra chromatic  passing tone  between the 5th and 6th degrees ,  mode  I  becomes an  octatonic  scale known as  Bebop melodic minor.

The  other proper Bebop scales  are all modes of two other scales,  unrelated to the melodic minor scale.  Nevertheless,  we may consider Bebop modes obtained from the other heptatonic modes listed above by allowing a passing tone at the  blue-shaded position .  Only mode  I  is widely accepted:

  1. Bebop Melodic Minor.
  2. Half-diminished Bebop.

Altered Dominant Scale :  Futuristic soloing (12:39)  by  Rick Beato   (2016-08-21).
Lydian Augmented:  The dark side of major (11:29)  by  Rick Beato   (2019-05-08).
 
Minor scales   |   Harmonic minor scale   |   Melodic minor scale   |   Jazz minor scale
Jazz scales   |   The melodic minor scale and its modes
Melodic minor scale modes  by  Justin.
Making Music With thw Lydian Dominant Scale (10:22)  Tommaso Zillio  (Music Theory for Guitar, 2020-08-03).

Harmonic Minor Scale :

In the description of scales,  the term  harmonic  indicates the presence of at least one step consisting of an  augmented second  (3 semitones).  Such a step is best abbreviated "A" in the alphabetical version of interval structures  (where H is a half-step of 1 semitone and W is a whole-step of 2 semitones.  The  scientific name  for that is  sesquitone,  but the abbreviation "S" is unused.

The  double-harmonic major scale  has two of those.  The modes of the  harmonic minor scale  only have one.

Dominant  means the 3 is major and the 7 is flat.

2 1 2 2 1 3 1 2 1 2 2 1 3 1 :   Harmonic Minor Scale
All  7  modes ranked by brightness, in 12 keys  (tonic in mode's column "1")
1VI:   Lydian #97 19 
1III:   Ionian Augmented7 18 
1IV:   Romanian Minor7 16 
1I:   Harmonic minor7 15 
1V:   Phrygian Dominant7 14 
1II:   Locrian #67 13 
1VII:   Ultralocrian7 10 
F GAb  Bb  CDb  E F GAb  Bb  CDb  E
CDEbFGAbB CDEbFGAbB
GABbCDEbF# GABbCDEbF#
DEFGABbC# DEFGABbC#
ABCDEFG# ABCDEFG#
EF#GABCD# EF#GABCD#
BC#DEF#GA# BC#DEF#GA#
F#G#ABC#DE# F#G#ABC#DE#
C#D#EF#G#AB# C#D#EF#G#AB#
AbBbCbDbEbFbG AbBbCbDbEbFbG
EbFGbAbBbCbD EbFGbAbBbCbD
BbCDbEbFGbA BbCDbEbFGbA
mmoM+mMMmo mmoM+mMMmo
1st2nd 3rd4th5th 6th7th  <<< mode of the harmonic minor scale.

The 7 modes of the harmonic minor scale are known under these names:

  1. Aeolian #7.  Minor #7.  Harmonic Minor.
  2. Locrian #6. 
  3. Ionian #5.  Ionian augmented.
  4. Dorian #4 (or #11).  Ukrainian Dorian.  Romanian Minor.  Souzinak.
  5. Phrygian #3.  Phrygian Major.  Phrygian dominant.  Hijaz.  Jewish.  Spanish.  Dorico Flamenco.
  6. Lydian #2 (or #9).
  7. [ Mixolydian #1.]   Ultralocrian.
Rick Beato  insists on calling Ultralocrian "Super Locrian bb7" because he always presents modes in the key of C  (although B# is called for here)  which leads to Bbb.  That's a missed opportunity to stress that double alterations are never needed for modes of the four normative scales  (diatonic, melodic minor, harmonic minor and harmonic major).

The harmonic minor scale has no axis of symmetry.  The mirror inverses of its modes are modes of the  harmonic major scale,  described next...

Secrets of A Lydian #9, the undiscovered scale (9:04)  by  Rick Beato   (2016-12-10).
Deconstructing The Harmonic Minor scale (11:01)  by  Rick Beato   (2017-02-01).
Phrygian Major moder (8:38)  by  Rick Beato   (2017-02-18).
Lydian #9 Mode (9:04)  by  Rick Beato   (2016-12-10).
Hidden Sounds of the Harmonic Minor Scale (16:12)  by  Rick Beato   (2017-02-21).
Harmonic Minor Modes for Bass Guitar (16:12)  by  Mark   (TalkingBass, 2017-03-24).
Modes and Scales Music Theory Workshop (53:24)  by  Jamin Coller   (LearnPianoLive, 2018-02-06).
The Mystery at the Heart of Flamenco (9:51)  by  David Bruce   (2018-10-12).


(2020-06-27)   Harmonic Major Scale
The mirror inverse of the  harmonic minor scale.

This is arguably either the most exotic of the normative heptatonic Western scales or the least exotic of the exceptional ones.  It was introduced in 1853 by  Moritz Hauptmann (1792-1868).

The current name of the scale was popularized by  Nikolai Rimsky-Korsakov (1844-1908)  in his  Practical Manual of Harmony  (1885).

2 2 1 2 1 3 1 2 2 1 2 1 3 1 :   Harmonic Major Scale
All  7  modes ranked by brightness, in 12 keys  (tonic in mode's column "1")
1VI :  Lydian Augmented #2 20 
1IV :  Lydian diminished7 17 
1I:  Harmonic major7 16 
1V :  Mixolydian b27 15 
1II :  Blues heptatonic7 14 
1III :  Phrygian b47 12 
1VII :  Locrian b77 11 
F  G ABb  CDb   E F G  ABb  CDb   E
CDEFG AbBCDEFGAbB
GABCD EbF#GABCDEbF#
DEF#GABbC# DEF#GABbC#
ABC#DEFG# ABC#DEFG#
EF#G#ABCD# EF#G#ABCD#
BC#D#EF#GA# BC#D#EF#GA#
F#G#A#BC#DE# F#G#A#BC#DE#
C#D#E#F#G#AB# C#D#E#F#G#AB#
AbBbCDbEbFbG AbBbCDbEbFbG
EbFGAbBbCbD EbFGAbBbCbD
BbCbDEbFGbA BbCbDEbFGbA
1st2nd3rd 4th5th 6th7th   ...mode of the harmonic major scale.

The 7 modes of the harmonic major scale are called:

  1. Ionian b6.  Major b6.  Harmonic Major.
  2. Dorian b5.  Blues heptatonic.  Kartzihiar.  [1645]
  3. Phrygian b4.
  4. Lydian b3.  Lydian diminished.
  5. Mixolydian b2.  Harmonic Minor Inversed.
  6. [ Aeolian b1.]  Lydian Augmented #2.
  7. Locrian b7.

Harmonic Major Scale (16:24)  by  Rick Beato   (2017-07-17).
he Strangest Normal Scale (6:28)  by  Cory Arnold  (12tone, 2017-10-30).
A Complete Guide to Musical Scales (21:35)  by  Oliver Prehn  (New Jazz, 2019-09-05).


(2020-08-13)   Other heptatonic scales
They can't be given signatures in all 12 keys without double alterations.

The four heptatonic scales given so far  (diatonic, melodic minor, harmonic minor and harmonic major)  are the only ones for which a  simple  key signature can be given in all 12 keys  (without using double alterations or worse).

Seasoned musicians may be unfazed by this fact  (which we prove  elsewhere)  but this is a good pretext to transfer the  discussion of other heptatonic scales  to a dedicated page,  where you'll also find an exhaustive discussion of  Bebop scales,  which have heptatonic and octatonic aspects  (the eighth note is just a  passing tone  but they are usefully classified as modes of just three octatonic scales).

The following articles have been transferred.


(2018-04-13)   Hexatonic Scales
Six notes per octave.

The single mode of the  whole-tone scale  (brightness 15):

The  whole-tone scale  is the only  anhemitonic  hexatonic scale.

The keys of  E# =F, G, A, B=Cb, C# =Db and D# =Eb  are modes of F.  Likewise,  F# =Gb, A# =Bb, B# =C, D, E=Fb and D# =Eb  are modes of C.  Both sets are equimodal,  so the  whole-tone scale  has only one mode.

The 10 key signatures of  Whole-tone :
5 key signatures for the class of  F
E#
F
F
F
F
F#
 
 
 
 
G
G
G
G
G
    A
A
A
A
A
 
 
 
 
Bb
B
B
B
B
Cb
 
 
 
C
 
C#
C#
C#
Db
Db
     
 
D
 
 
D#
D#
Eb
Eb
Eb
     
E
 
 
 
E#
F
F
F
F
F#
 
 
 
 
5 key signatures for the class of  C
 
 
 
 
F
F#
F#
F#
F#
Gb
     
 
 
G
 
G#
G#
G#
Ab
Ab
     
 
A
 
 
A#
A#
Bb
Bb
Bb
 
B
 
 
 
B#
C
C
C
C
C#
 
 
 
 
D
D
D
D
D
 
 
 
 
 
E
E
E
E
E
 
 
 
 
F
F#
F#
F#
F#
Gb

Adding a seventh note to the whole-tone scale always yields one of the modes of Neapolitan Major.

 Come back later, we're
 still working on this one...

The two modes of the  augmented scale :

For either mode  (I or II, corresponding to the column of tonics indicated on the bottom line)  you always have a choice of two differents lines omitting a different letter  (shown on a grey background)  whose line on the  staff  will remain empty.  The two choices are transcribed with totally different  key signatures  but the six tones actually played are the same in the end.

Here are the two  enharmonic  results for the key of  C:

C D# E G Ab B C   or   C Eb Fb G Ab B   for mode I. 
C Db E F G# A   only   for mode II.  (Unless you want to use B#.)

Both choices of key signatures yield the same tones and  brightness.
1I:   Augmented Scale6 18
 1II:   Inverted Augmented6 12
F
F
   G#
G#
A
A
  
B
B#
C
C#
Db
D
 
 E
E
F
F
   G#
G#
A
A
  
B
C
C
  
D
D#
Eb
E
Fb
F
 
  G
G
Ab
Ab
  B
B
C
C
  
D
D#
Eb
E
Fb
F
 
 
G
G
  
A
A#
Bb
B
Cb
C
 
  D
D
Eb
Eb
  F#
F#
G
G
  
A
A#
Bb
B
Cb
C
 
 
D
D
  
E
E#
F
F#
Gb
G
 
  A
A
Bb
Bb
  C#
C#
D
D
  
E
E#
F
F#
Gb
G
 
 
A
A
  
B
B#
C
C#
Db
D
 
  E
E
F
F
  G#
G#
A
A
  
B
B#
C
C#
Db
D
 
 
E
Fb
F
 
 G
G
Ab
Ab
   B
B
C
C
  
D
D#
Eb
E
Fb
F
 
 G
G
Ab
Ab
  
B
Cb
C
 
 D
D
Eb
Eb
   F#
F#
G
G
  
A
A#
Bb
B
Cb
C
 
 D
D
Eb
Eb
  
F#
Gb
G
 
 A
A
Bb
Bb
   C#
C#
D
D
  
E
E#
F
F#
Gb
G
 
 A
A
Bb
Bb
  
C#
Db
D
 
 E
E
F
F
   G#
G#
A
A
  
B
B#
C
C#
Db
D
 
 E
E
F
F
  
Ab
Ab
  B
B
C
C
  
D
D#
Eb
E
Fb
F
 
 G
G
Ab
Ab
  B
B
C
C
  
D
Eb
Eb
   F#
F#
G
G
  
A
A#
Bb
B
Cb
C
 
 D
D
Eb
Eb
   F#
F#
G
G
  
A
Bb
Bb
   C#
C#
D
D
  
E
E#
F
F#
Gb
G
 
 A
A
Bb
Bb
   C#
C#
D
D
  
E
I  II  <<<  mode of the augmented scale.

 Come back later, we're
 still working on this one...

The Whole Tone Scale (10:40)  by  Rick Beato   (2016-11-17).
[The augmented scale] is the pros' secret weapon (31:17)  by  Rick Beato   (2019-01-17).
 
Hexatonic scales   |   Augmented scale (pianoscales.org)


(2018-04-19)   Diminished Scales.  Double-diminished chords.
Two scales:  Whole-half diminished scale and Half-whole diminished.

Octatonic symmetrical scale,  where the 8 notes are obtained by repeating four times the same two-note progression spanning three semitones  (either  semitone-tone  or  tone-semitone).  four times.  That gives two modes:

  • HWHWHWHW   Half-whole diminished.  Also called  dominant.
  • WHWHWHWH   Whole-half diminished.

There are only three distinct keys:

  • C  =  Eb  =  Gb  =  A
  • B  =  D  =  F  =  Ab
  • Bb  =  Db  =  E  =  G

No avoid notes.  All the chords are interchangeable...

 Come back later, we're
 still working on this one...

Diminished Scale & Double-Diminished Chord (14:21)  by  Anton Petrov   (Walk That Bass, 2016-03-19).
 
Advanced Jazz Improvisation:  Diminished-Scale Patterns (16:07)  by  Kent Hewitt   (2016-03-09).
 
The Diminished Scale (16:05)  by  Rick Beato   (2016-08-25).
 
Diminished Scale Theory (17:27)  by  Rick Beato   (2017-01-30).
 
Analyzing the Octatonic Scales (5:36)  by  Kari Juusela   (Berklee Online, 2017-05-09).
 
Octatonic scale   |   Nikolai Rimsky-Korsakov (1844-1908)   |   Edmond de Polignac (1834-1901)


(2017-04-09)   Chords and tertian harmony  (stacking thirds).
The way simultaneous sets of notes are organized in Western music.

The tightest chords are  triads  consisting of three distinct notes,  characterized be by the two intervals which separate the root from the other two notes.  The first of those is a  third;  either a major third (M3; worth 3 semitones) or a minor third (m3; worth 2 semitones).  The second one can be a perfect fifth (P5; worth 7 semitones), a diminished fifth (d5; worth 6 semitones) or an augmented fifth (A5; worth 8 semitones). That yields four different  qualities  of triads:

The four different qualities of triads (in closed position)
ValidityQualityInterval 1Interval 2 Interval Structure
Consonant MajorM3P5 1   3 5
Minorm3P5 1 35
Dissonant AugmentedM3A5 1 35
Diminishedm3d5 1 35
Invalid M3d5 1 35
m3A5 1 35

A triad can be denoted by the roman numeral corresponding to its  root;

  • UPPER-case for a major triad.
  • Lower-case for a minor triad.
  • UPPER-case with a "+" superscript for an augmented triad.
  • Lower-case with a "o" superscript for a diminished triad.

The  seventh chords  are four-note chords  (tetrads)  derived from the above triads.

Spread Triads, Chord Inversions and Figured Bass :

The above describes the most basic form of a triad, called close-position, root-position.  Other versions of the same chord are obtained by a so-called   inversion  which consists in displacing one of the three notes by a full octave.

 Come back later, we're
 still working on this one...

Secondary Chords :

 Come back later, we're
 still working on this one...

Triads and seventh chords   |   Chord-scale system   |   Secondary chords
Jazz chords   |   Jazz scales   |   Side-slipping
 
Understanding The Chord/Scale Relationship (35:06)  by  Rick Beato   ().
Major and minor triads (10:12)  by  Scott Murphy   (2015-07-11).
Clock arithmetic (10:12)  by  Scott Murphy   (2015-07-15).
Vocabulary of Chords (14:06)  by  Rick Beato   (2016-06-08).
Chord Families and Their Modes (34:13)  by  Rick Beato   (2016-12-23).
Basics of Music Theory:  2016-12-23 (6:01)  &  2016-15-29 (16:13)  by  Rick Beato.
 
A fully-extended chord is a scale (18:12)  Anton Petrov   (Walk That Bass, 2015-12-25).
Why you love suspended chords (12:26)  by  Rick Beato   (2018-09-21).
Suspended chords (13:09)  by  Michael New   (2015-03-30).
Augmented 6th Chords in Common Practice  by  Jesse Strickland  (Two-Minute Music Theory, 2017-10-04).
Why are Minor Chord Songs so rare? (12:44)  by  David Bennett  (2019-12-16).
Altered dominant chords (12:29) Tommaso Zillio  (Music Theory for Guitar, 2020-03-02).
Negative dominant & subdominant chords (9:54) Tommaso Zillio  (Music Theory for Guitar, 2020-06-29).
 
(Tonal) Major Scale vs. Ionian Mode  by  Jesse Strickland  (Two-Minute Music Theory #29, 2017-11-13).
Modal Chord Progressions (26:22)  by  Brian Kelly  (Zombie Guitar, 2018-02-09).
How to Learn Songs by Ear (30:03)  by  Brian Kelly  (Zombie Guitar, 2019-09-10).
 
The 7 Levels of Jazz Harmony (13:11)  by  Adam Neely  (2019-08-30).
 
The 7 Levels of Gospell Harmony (13:11)  by  Corey Taylor  (Skilled Musician, 2019-10-30).
 
My New Favorite Chord (9:33)  by  Cory Arnold  (12tone, 2019-06-14).
 
Triads, Inversions & Figured Bass (1:17:38)  by  Declan Plummer  (Decky Pipes #2, 2015).
 
Aeolian & Natural Minor aren't the same thing (10:30) Tommaso Zillio  (Music Theory for Guitar, 2020-06-22).
 
Wikipedia :   Chord   |   Roman numeral analysis   |   Figured bass (thoroughbass)
Nashville number system   |   Neal Matthews (1929-2000)  of  The Jordanaires   |   Charlie McCoy (1941-)
 
Inversions (12:22) & Figured Bass (6:58)  by  Stephen Wiles  (musictheoryguy, August 2012).


(2020-08-17)   Harmonization  &  Functional Harmony

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Close and open harmony   |   Voice leading   |   Counterpoint   |   Guerino Mazzola (1947-)
 
Voice Leading, Terminology, Basic Harmonisation (1:17:38)  by  Declan Plummer  (Decky Pipes #3, 2015).
 
How To Analyze Songs (9:06)  by  Cory Arnold  (12tone, 2018-11-30).
 
Picardy Third: Minor Resolving to Major Chord (9:35)  by  David Bennett  (2019-12-23).
 
Beginner's Guide to Voicing and Voiceleading (16:34)  by  8-bit Music Theory  (2021-04-23)   |   Chiptune


(2018-02-25)   Cadence:  Resolution of tension and closure.
Musical punctuation.  Dominant and tonic.

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V7 to I :  Dominant Seventh to Tonic (11:03)  by  Rhythmic Canada   (2014-03-25).
GOD CHORDS - Writing Epic Changes (6:59)  by  Jake Lizzio  (Signals Music Studio, 2018-02-10).
 
Cadences & Pre-Dominants  (1:12:21 & 47:40)  by  Declan Plummer  (Decky Pipes #4, 2015).
 
Wikipedia :   Cadence


(2021-04-25)   Modulation and Key Changes

Modulation is the essential part of the art.  Without it there is little music,
for a piece derives its true beauty not from the large number of fixed modes 
which it embraces but rather from the subtle fabric of its modulation.

  Charles-Henri de Blainville (1767)

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The Greatest Key Change in Pop Music (27:01)  by  Adam Neely   (2021-04-16).
 
Wikipedia :   Cadence


(2018-03-20)   Idiot's Guide to Composition:
A few basic principles composers have been using.

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 still working on this one...

Why Modern Music is Awful (20:00)  by  Thoughty2  (2017-08-05).
Geometry of Consonance:  Music and Mathematics (1:27:33)  by  Dmitri Tymoczko   (Santa Fe, 2009-08-27).
Max Martin's Melodic Math (11:10)  by  Alastair Errett   (The Art of Songs, 2017-09-01).
How to Write a Melody (33:36)  by  Jamie Henke   (fastforwardable, 2011-11-10).
Basic Music Theory for Producers and Songwriters (20:18, 15:45, 13:09)  by  Dezz Asante   (May 2016).
 
Three Tips for Writing a Great Melody (5:46)  by  Cory Arnold  (12tone, 2017-05-19).
Surprising Enemy of Good Songwriting (8:15)  by  Cory Arnold  (12tone, 2020-01-31).
How to Write a Melody  (19:51, 27:01)  by  JJay Berthume   (2017-01-19).
How to Create Chord Melodies (48:51)  by  Rick Beato   (2017-06-15).
Composing a Melody from a Chord Scheme (14:06)  by  Gareth Green  (Music Matters, 2019-01-03).
Developing a Melody Line (16:17)  by  Gareth Green  (Music Matters, 2019-01-17).
Classical Melody Writing Tricks (19:15)  by  Gareth Green  (Music Matters, 2020-04-30).
Composing a Motif (15:03)  by  Gareth Green  (Music Matters, 2020-06-04).
How to Write a Theme (11:12)  Sideways  (2018-05-17).
Modulation Using the Circle of Fifths (12:08)  by  Gareth Green  (Music Matters, 2019-03-21).
How to use modulation  (23:2520:54)  by  Rick Beato   (March 2017).
 
5 composers, 1 theme (27:39)  Adam Neely, Nahre Sol, Ben Levin, Martin Keary  (David Bruce, 2019-05-17).
5 composers, 1 orchestra (33:45)  Adam Neely, Brechtje van Dijk, Ben Levin, Martin Keary  (David Bruce, 2019-11-01).
5 composers, 1 idea (26:28)  June Lee, David Bennett, Ben Levin, Allison Wright  (David Bruce, 2020-06-09).
The "Tonight" Hook (8:20)  by  David Bennett  (2019-06-17).
Best & Worst Endings in Classical Music (18:57)  David Bruce  (2019-06-28).
 
Counterpoint   |   Gradus ad Parnassum (1725)   |   Cantus firmus
Johann Joseph Fux (c.1660-1741)   |   Alfred Mann (1917-2006)


(2020-08-05)   Musical Dice
Composing a musical piece without human intervention.

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Musikalisches Würfelspiel (1757)   |   Johann Philipp Kirnberger (1721-1883)
Algorithmic composition   |   Aleatoric music
The Glass Bead Game (1943)   |   Hermann Hesse (1877-1962)
Doodle (2019-03-21)  celenrating  Johann Sebastian Bach (1685-1750).
 
The Dice Game That Lets Anyone be a Composer (12:26)  by  Cory Arnold  (12tone, 2019-09-20).


(2020-07-23)   Ornaments and Embellishments
Changing the sound of a tune without affecting its structure.

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Variationx: 14 Ways to Compose with One Idea (9:30)  by  David Bruce   (2020-02-03).
 
Ornaments   |   Mordent   |   Trill (formerly: shake)   |   Turn
Grace note   |   Acciaccatura   |   Appoggiatura   |   Glissando   |   Portamento   |   Portato


(2018-02-16)   Octave Displacement  (Octave Dispersion)
Transforming a melody by moving one or more note to a different octave.

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What Bach and Charlie Parker Had In Common (17:33)  by  Rick Beato   (2017-02-25).
Cantus Firmus for Jazz Soloing (9:30)  by  Adam Neely   (2016-02-21).
 
Counterpoint   |   Gradus ad Parnassum (1725)   |   Cantus firmus
Johann Joseph Fux (c.1660-1741)   |   Alfred Mann (1917-2006)


(2018-02-12)   Seminal theory of modal Jazz   (George Russell, 1953)
The Lydian Chromatic Concept of Tonal Organization  (LCCTO).

George Russell published the first edition of LCCTO in 1953 (age 30) and spent nearly fifty years refining it.  He published the final edition in 2001:

The Lydian Chromatic Concept of Tonal Organization:
The Art and Science of Tonal Gravity
  (George Russell, 2001)

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Jazz Theory with  Barry Harris  (1, 234567Jazz Academy   (2014).
The Elements of Jazz Explained (19:21)  by  Rick Beato   (2017-03-11).
Original music theory of Jazz (10:46)   Adam Neely  with  Rick Beato   (2017-11-06).
Kind of Blue: How Miles Davis Changed Jazz (10:40)   Polyphonic   (2018-04-19).
Miles Davis:  Two days that changed music (18:15)  by  Rick Beato   (2020-09-10).
 
Lydian Chromatic Concept of Tonal Organization for Improvisation (1953)   |   George Russell (1923-2009)


(2020-08-08)   Telescoping Scales

According to  Andy Chamberlain,  that's what  Jacob Collier  has been promoting using a silly compound prefix.  What we'll call here  telescoping Lydian  is what Collier dubs  super-ultra-hyper-mega-meta Lydian.

The idea hinted at by Collier and articulated by Chamberlain is to consider the first 7 notes of some heptatonic mode  (e.g., Lydian)  and prolong that with the notes corresponding to the same mode rooted at the n-th degree,  provided the initial seven notes are not affected by the switch.  This works with either n=4 or n=5 (not both) for all diatonic modes.

Telescoping F-Lydian
FGABCDE
CDEFGAB
GABCDEF#
DEF#GABC#
ABC#DEF#G#
EF#G#ABC#D#
B
Cb
C#
Db
D#
Eb
E
Fb
F#
Gb
G#
Ab
A#
Bb
F#
Gb
G#
Ab
A#
Bb
B
Cb
C#
Db
D#
Eb
E#
F
C#
Db
D#
Eb
E#
F
F#
Gb
G#
Ab
A#
Bb
B#
C
AbBbCDbEbFG
EbFGAbBbCD
BbCDEbFGA
FGABCDE

The highlighted last line is identical to the first,  which corresponds to the repetition of the 48-note pattern  (not 49)  with a period of 7 octaves.  By starting the full pattern at the beginning of the relevant line,  you obtain telescoping Lydian in any of the twelve possible keys.

Now,  if we impose the additional requirement of a perfect naming of the notes  (repeating the letters A,B,C,D,E,F,G in sequence)  the 7-octave pattern can't be repeated indefinitely  (that would require 49 notes instead of 48).  The maximal perfect pattern contains just  67  tones,  over  9½  octaves:

Telescoping Diatonics
EF#G#ABC#D#
CbDbEbFb GbAbBb
GbAbBbCb DbEbF
DbEbFGb AbBbC
AbBbCDbEbFG
EbFGAbBbCD
BbCDEbFGA
FGABCDE
CDEFGAB
GABCDEF#
DEF#GABC#
ABC#DEF#G#
EF#G#ABC#D#
BC#D#E F#G#A#
F#G#A#B C#D#E#
C#D#E#F# G#A#B#
G#A#B#C# D# E# G

The musical explorations of Jacob Collier (b. 1994)

Jacob Collier is from a musical family and has perfect pitch.  He is an alumnus from the  Purcell School for Young Musicians.  He dropped out of the Jazz piano class at the  Royal Academy of Music (where his mother is a professor).  Describing him as an  autodidact  is a  misleading  term,  possibly damaging for young people who are still unsure about the benefits of a formal education.  When successful,  education may seem unnecessary  a posteriori,  but this ain't so.

Collier has been sharing his creation on YouTube since 2012  (at age 18).  He has received accolades from the music industry  (including several  Grammys).

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 still working on this one...

Jacob Collier (1994-)   |   Jazz prodigy Jacob Collier (The Guardian, )
What does Jacob Collier really mean? (10:36)  by  Andy Chamberlain  (2018-12-15).
Explaining harmony (15:41)   Jacob Collier  &  Herbie Hancock  (WIRED, 2018-01-08).
Why Musicians Love Jacob Collier (16:39)   by  Charles Cornell  (2020-06-01).


(2018-02-15)   Musical Instrument Digital Interface  (MIDI, 1980)
Protocol for transmitting and recording keyboard performances.

A time-stamped MIDI event correspond to depressing a certain key at a certain velocity and for a certain duration.

A 7-bit MIDI note number  n  (between 0 and 127)  corresponds to the following frequency:

f   =   2 (n-69) / 12  ×  440 Hz

This is to say that note  69  is  concert-A  (440 Hz; A above middle-C)  by definition.  Middle-C is 60.  The lowest note on the piano is number  21  (A,  27.5 Hz).  The highest is  108  (C,  4186.01 Hz).  The MIDI numbers span almost  11  octaves,  from  C-1  (8.176 Hz)  to  G (12543.854 Hz):

The 128  MIDI  note numbers :
Organ Register   32' 16'8'4'2'1'6'' 3'' 
ISO Octave  -1  0  123456789
DoC 012243648607284 96108120
 C# 11325374961738597109121
D 21426385062748698110122
 D# 31527395163758799111123
MiE 416284052647688100112124
FaF 517294153657789101113125
 F# 618304254667890102114126
SolG 719314355677991103115127
 G# 820324456688092104116  
LaA 921 33455769 8193105117
 A# 1022344658708294106118
SiB 1123354759718395107119

Typically,  very low or very high codes are silenced for different ranges in voices which are intended to represent actual musical instruments.  For example,  with the built-in  Grand Piano  of  Ableton Live,  no sound corresponds to codes below  21  (A0, the lowest key on the piano)  or above  108  (C8,  the highest key on the piano).  That means silencing  most  of the keys in both of the extreme octave shiftts provided by some small keyboards  (like the M-Audio Keystation Mini 32).

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MIDI  (1980)   |   MIDI tuning standard  (MRS, 1992)
ABC music notation :    |   Home   |   Primer   |   Chris Walshaw
GUIDO music notation   |   Guido of Arezzo (AD 992, fl. 1033)
 
Setting Up Your Midi Controller in Ableton Live (9:30)  by  Anthony Thogmartin   (Seed to Stage,  2017-05-15).
ProTools for Beginners:  MIDI Setup (8:22)  by  Brian Boyle   (2012-07-17).


(2018-02-26)   Microtonal and Polychromatic Music
Beyond twelve tones per octave and back to the origins of music.

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Polychromatic music and modern microtonal instruments (14:07)  by  Dolores Catherino  (2015-10-28).
Polychromatic music (13:50)  by  Dolores Catherino  (TEDx Sacramento, 2016-02).
How To Sing Microtonal Pitches Like Jacob Collier (13:29)  by  Aimee Nolte  (2017-06-05).
History of the Microtonal Guitar (9:13)  by  Tolgahan Çogulu  (2018-08-06).


(2020-09-04)   Lyres and Harps

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The Ancient Greek Kithara (9:03)  by  Peter Pringle  (2015-10-26).


(2018-02-26)   Fretted String Instruments
Guitars and others.

The  standard tuning  for the open tones of a six-string guitar is:

E2   A2   D3   G3   B3   E4  

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 still working on this one...

The lute family   |   Tanbur (Mesopotamia, c.3100 BC)   |   Barbat   |   Arabic oud
 
Standard guitar tuning   |   Regular tunings   |   All-fourths tuning   |   All-fifths tuning
 
History of the Guitar (55:11)  by  Rob Scalon & Brandon Acker   (2020-07-25).
 
The CAGED system for guitar and beyond (27:32)  by  Rick Beato   (2019-03-03).


(2018-02-26)   Classical Fretless String Instruments
Violinviolavioloncello (or cello)  and  double-bass  (or  bass).

The smallest related instruments  (violino piccolo & pochette)  are all but forgotten.  The standard tuning of the 4 strings in modern instruments is:

  • Violin  (all fifths tuning):  G3,  D4,  A4,  E5.
  • Viola  (tuned in fifths):  C3,  G3,  D4,  A4.
  • Violoncello  (tuned in fifths):  C2,  G2,  D3,  A3.
  • Double Bass  (all fourths tuning) :  E1,  A1,  D2,  G2.
    • 5-String Double Bass:  B0,  E1,  A1,  D2,  G2    (most often).
    • 6-String Double Bass:  B0,  E1,  A1,  D2,  G2,   C3.

Violin SizesAdults normally play 7/8 or 4/4  (full size).
Size :1/161/101/81/41/22/47/84/4
Body'' 89101112½1313½14
cm 2336
Total'' 14½161718½20½2122½23½
cm
Neck to
Wrist
'' 141516½182021½22≥ 23
cm

Famous violin makers of the past include:

The only Stradivarius violin still in  as new  condition is the  Messiah (1716)  which has almost never been played.  It was the favorite of the Mirecourt-born Parisian luthier Jean-Baptiste Vuillaume (1798-1875) who bought it in 1854 from the heirs of the violin dealer  Luigi Tarisio (1796-1854).  Vuillaume once owned 24 other Stradivari.  He made an ornate tailpiece and carved pegs to upgrade the  Messiah,  which is now kept at the  Ashmolean_Museum  in  Oxford.  Vuillaume made about 3000 instruments, including direct copies of his revered  Messiah.

Some makers have achieved a great reputation for specific violin parts:

  • Bridges :  Aubert Lutherie,  in  Mirecourt,  France.

Byzantine lyra (3-5 strings)   |   Lira da braccio (7 strings)   |   Fiddle   |   Violin family   |   Violone (3-6 strings)
Violin construction   |   Purfling   |   Tailpiece   |   Fingerboard   |   Nut   |   Scroll   |   Pegs   |   Bridge   |   Soundpost
 
The Basics of Strings (12:47)  by  Rick Beato   (2016-10-08).
 
: Violin vs. Viola: What's the difference? (8:20)  by  Amber Reed   (Violin Viola Masterclass, 2020-03-25).
 
Adjusting the Bridge and Sound Post (13:55)  by  Edgar E. Russ   (Luthier in Cremona, 2018-03-19).
 
6 things you need to start playing violin (14:51)  by  Olaf Grawert   (Violin Studio in Brisbane, 2020-07-29).


(2020-06-18)   Recorder  (French:  flûte à bec )
Sopranino, soprano, alto, tenor, basset, great bass, contrabass, etc.

Sometimes just called  flutes  (of which they are now the most common kind)  recorders  have been known by that name since  1388,  or earlier.

Many languages which don't use a straight adaptation of the word  recorder  employ a locution which translates as  block flute,  refering to the internal construction of the mouthpiece,  featuring a solid wooden  fipple plug.

That's often called a  cedar block,  because it's almost exclusively made from  pencil cedar,  the moth-repellent wood famously used for  cedar chests.  That  wood  is actually best called  junniper  (French: genévrier)  because it's not technically a cedar at all.  Juniperus virginiana,  commonly goes by many other names:  eastern redcedar, Virginian juniper, eastern juniper, red juniper, pencil-cedar, aromatic-cedar, etc.  It's used for recorder blocks because of its excellent resistance to rot in humid enviroments.

Modern recorders come in the sizes listed below from smallest to largest, usually with  baroque-english fingering  (as opposed to German fingering).  The quoted tuning refers to the lowest note the instrument can produce  (always using  scientific notation).

One of those is dubbed  Big babe  on account of its size  (about 2.8 m tall)  and as a mnemonic  for its low note  (Bb).  It can play  more than one octave lower  than the cello but  several people  who play it keep misrepresenting that as just "one semitone lower than the lowest tone of a cello" (which is C2, not C1 !).  This goes to show that they know more about recorders than  string instruments... Big Babe  belongs to the  Dutch  Royal Wind Music,  a 13-player  ensemble  featuring that sub-contrabass and two  sextets  which are playing one octave apart.

The above is mostly the streamlined modern lineup,  which allows recorder players to play any instrument by mastering only two fingering systems  (C and F).  Historically however,  there are traces of a traditional nomenclature where successive members of the recorder family were uniformly separated by a perfect fifth  (incidentally,  this makes the nominal key sufficient to identify the octave, with room to spare for an unplayable instrument  in Eb = D#  at either end of the spectrum).  This scheme,  which was probably never really enforced,  applies to the three deepest recorders in use.  It drifts from there but names remain almost recognizable for smaller instruments...

  1. Unused.  Would be  way  too small for a real chromatic instrument.
  2. Sopranissimo in Ab = G#6  (four tones above  C Garklein).
  3. Sopranino in Db = C#6  (four tones above F5 ).
  4. Soprano in Gb = F#5  (three tones above  C5 ).
  5. Alto in B4  (three tones above  F4 ).
  6. Tenor in E4  (two tones above  C4 ).  One tone above  D voice flute.
  7. Basset in A3  (two tones above  F3 ).
  8. Greatbass in D3  (one tone above  C3 ).
  9. Contrabass in G2  (one tone above  F2 ).
  10. Sub-greatbass in C2.
  11. Contrabass in F1.
  12. Sub-contrabass in Bb0.  (Modern creation by  Adriana Breukink.)
  13. Unused.  Would require extreme lung capacity.

In a pinch,  professional recorder players can play one semitone lower than the nominal pitch of their instrument by closing all finger holes  (including "hole 0" for the left thumb on the underside)  and partially blocking the end of the pipe "hole 8"; that's what the first line of the table below is intended to convey.  The other lines indicate how each note of the chromatic scale  can  be played  (there are often other solutions)  to cover   2¾  octaves.

Garklein
Soprano
Tenor
Great Bass
Baroque  Fingering Sopranino
Alto
Basset
Contrabass
01234 5678
B 012345678- E
C 01234567  F
C# / Db 01234567-  F# / Gb
D 0123456   G
D# / Eb 0123456-   G# / Ab
E 012345    A
F 01234 67  A# / Bb
F# / Gb 0123 56   B
G 0123      C
G# / Ab 012 456-   C# / Db
A 012       D
A# / Bb 01 3 56   D# / Eb
B 01        E
C 0 2       F
C# / Db  12       F# / Gb
D   2       G
D# / Eb   23456   G# / Ab
E 0-12345    A
F 0-1234 6   A# / Bb
F# / Gb 0-123 5    B
G 0-123      C
G# / Ab 0-12 4     C# / Db
A 0-12       D
A# / Bb 0-12  56 (7-)  D# / Eb
B 0-12 45    E
C 0-1  45    F
C# / Db 0-1 34 67-8 F# / Gb
D 0-1 34 67-  G
D# / Eb 0- 23 ( 5 6 7-)  G# / Ab
E 0- 23456   A
F 0-12 45  8 A# / Bb
F# / Gb 0-12 45    B
G 0-1  4     C

The  open fingerings  are defined to be the patterns where the first holes are closed and the last ones are open.  They tend to be easier, more reliable and more stable than the other patterns,  which are called  forked fingerings  (where open and closed holes alternate).

In 1926,  the German instrument maker  Peter Harlan (1898-1966)  introduced a modified version of the recorder,  where the fourth fingerhole is larger than the fifth, in an effort to make the instrument faster to learn.  That modification yields an  open fingering  for the first F in a C-descant  (so that all simple 1-octave tunes in C-Major can be played with open fingerings only).  That so-called  German fingering  is still used in the educational market.  However, only the simplest songs are easier to play on a German instrument and the early habits so acquired are a hindrance to the use of better recorders,  which  all  use Baroque fingering.

Recorder   |   Fipple flute (duct flute)   |   Flutes   |   Flue pipe   |   Woodwind   |   Wind instruments
 
Peter Bressan, Pierre Jaillard "le Bressan" (1663-1732)
 
Georg Philipp Telemann (1681-1767)
 
Fred Morgan (1940-1999)
 
The Royal Wind Music (1997-)   |   Paul Leenhouts (1957-)   |   Adriana Breukink (c.1958-)
 
Making a new block for a recorder (1983)  by  Philippe Bolton.
 
Resin recorders  by  Vincent Bernolin.
 
German vs. Baroque fingering (9:35) by  Sarah Jeffery  (Team Recorder, 2020-02-20).
Which recorder should you buy (15:05) by  Sarah Jeffery  (Team Recorder, 2017-05-18).
Choosing a PLASTIC recorder (19:19) by  Sarah Jeffery  (Team Recorder, 2018-04-12).
Soprano, Alto, Tenor, Bass: Yamaha YRA-302B III (15:05) by  Florentin Tise  (2018-08-29).
 
Playing ALL the notes (12:02) by  Sarah Jeffery  (Team Recorder, 2017-09-28).
Sounding good on a soprano (21:45) by  Sarah Jeffery  (Team Recorder, 2016-04-28).
Hitting the high notes (12:36) by  Sarah Jeffery  (Team Recorder, 2016-06-29).
Hand position (15:30) by  Sarah Jeffery  (Team Recorder, 2016-08-09).
Trills Tutorial (15:05) by  Sarah Jeffery  (Team Recorder, 2016-09-30).
Alternative Fingerings (16:50) by  Sarah Jeffery  (Team Recorder, 2016-10-20).
Tuning (16:50) by  Sarah Jeffery  (Team Recorder, 2017-10-05).
 
How wooden recorders are made at AAFAB (8:56) by  Anton Cornelissen  (Team Recorder, 2017-06-01).
 
Greensleves - Fingering tutorial (5:33) by  You Can Play It  (2013-08-19).
Yesterday - The beattles (20:20) by  You Can Play It  (2017-11-12).
Hey Jude - The beattles (20:20) by  You Can Play It  (2017-02-13).
Ennio Morricone: The Good, the Bad and the Ugly (5:53) by  You Can Play It  (2020-07-09).
El Condor Pasa (a - minor) - recorder tutorial (3:58) by  Iryna Flute  (2019-05-03).

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