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Final Answers
© 2000-2018   Gérard P. Michon, Ph.D.

Western Art Music
Mathematical Aspects of Music

  • Western musical notation  was born to codify  Gregorian chants.
  • Note durations  follow a binary progression.  Dotting prolongs by  50%.
  • Tempo:  The speed of music.  What the  metronome  measures.
  • Beats of the metronome.  Bar  (measures)  and time signatures.
  • Triplet:  Group of  3  notes equally splitting  twice  their common duration.
  • DynamicsPianissimo  to  fortissimoDiminuendo  or  crescendo.
  • The Frequency Domain:  Sound reduces to a superposition of  tones.
  • Perfect pitch:  Native music speakers can easily name any absolute pitch.
  • Staves and clefs.  Transcribing and reading musical notes.
  • Keyboards  Full pianos (88-key) down to double-octave (25 keys).
  • Musical intervals  are  ratios  of frequencies.
  • The circle of fifths:  Forced wedding of two harmonies  (2:1 and 3:2).
  • Tritone:  Half an octave.  A dissonant interval  (F to B or B to F).
  • Scales.  The key of C major  (or A minor)  only uses  white  piano keys.
  • Chords  are separate notes played together.
  • Cadence:  Resolution of tension.  Musical punctuation and closure.
  • MIDI:  Musical Instrument Digital Interface  (1980).
  • Octave displacement  (Octave dispersion).
  • Jazz:  Lydian Chromatic Concept of Tonal Organization  (George Russell).
  • Microtonal instruments  and exploration of  polychromatic music.
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Related articles:

Related Links (Outside this Site)

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).
Aux marches du palais   |   Le bon roi Dagobert
Au clair de la lune  by  Jean-Baptiste Lully (1632-1687).
Frère Jacques  by  Jean-Philippe Rameau (1683-1764).
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).
When the Saints Go Marching In   |   Louis Armstrong
Max Martin's Melodic Math (11:10)  by  Alastair Errett   (2017-09-01).
How to Write a Melody (33:36)  by  Jamie Henke   (2011-11-10).

Wikipedia :   Music theory   |   Music in Western culture   |   Johann Sebastian Bach (1685-1750)

The  New  Harvard Dictionary of Music,  edited by  Ron Handel  (1986).

 Rowland RD-64

Mathematics of Music

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).

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

For centuries,  chants  were only transmitted orally. 


The  musical staff  was invented 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...

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)

(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 predetermined 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 more 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)
  4Quadruple note
(quadruple rest)
(longa rest)
  2Double note
(double rest)
(breve rest)
(bâton de pause)
  1Whole note
(whole rest)
(semibreve rest)
  1/2Half note
(half rest)
(minim rest)
 1/4Quarter note
(quarter rest)
(crotchet rest)
  1/8Eighth note
(eighth rest)
(quaver rest)
  1/16Sixteenth note
(sixteenth rest)
(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).

(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 rapids clicks morph into a continuous pitch.

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).
Advanced Time Signatures (14:54)  by  Michael New  (2013-11-13).
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 by 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 generally 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).
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)

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
SiB 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.

 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.

 One octave on the piano keyboard

In the  scientific pitch notation  used above  (and elsewhere with growing popularity)  the  ISO  number of each octave is used as a subscript to the name of a pitch to denote a particular tone without any ambiguity.  Other competing systems are still in use,  which are mutually incompatible to some degree.  In all cases,  tones in the same octave  (from C to B)  are denoted alike and we give only the notation corresponding to "C"  (Do, Ut)  in the following table.  Musician will routinely speak a particular tone by identifying the "C" just below is  (e.g., 440 Hz  is "A above middle-C" or concert A, which is rarely called "middle A").

Competing ways of naming an octave 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
LilyPond c,,,c,,c,cc' c''c'''c''''c'''''

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 said  she can sustain  C above high-C  (C,  2093 Hz).

In 2003,  Maria Carey  hit a  G7  (3136 Hz)  during a 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 was just one note  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  A7#  (3729 Hz).  Three semitones above Maria Carey's  Gnot  three octaves above it.  (Did someone confuse semitones and octaves?)

Scientific pitch notation   |   Helmholtz pitch notation (1863)   |   Hermann von Helmholtz (1821-1894)
C = Do = Ut   |   Sharp (dièse)   |   Flat (bémol)   |   Natural (bécarre)
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).

(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).

(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  was introduced in 1909 spanning eight octaves (97 keys from C0 to C8)  as suggested by composer  Ferruccio Busoni (1866-1924)  to match  Johann Sebastian Bach's  organ works  (the 32-foot registers of some 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 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
32 keys   32-key keyboard
Casio Rapman
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).
Wikipedia :   Musical keyboards   |   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:

Consonant Pure Intervals and Equal-Tempered Approximations Thereof
2 : 1OctaveOctave2.0000000000012
3 : 2FifthQuinte1.498307076887  (-0.01955)
7 : 5Lesser Septimal Tritone1.414213562376  (+0.17488)
4 : 3FourthQuarte1.334839854175  (-0.01955)
5 : 4Major thirdTierce majeure1.259921049894  (+0.13686)
6 : 5Minor thirdTierce mineure1.189207115003  (-0.15641)
9 : 8Major tone1.122462048312  (+0.03910)
10 : 9Minor tone2  (-0.17596)

One percent of a semitone is called a  cent.  The above table shows that the chromatic approximations of the pure musical intervals are very good indeed:  They're less than  2 cents  in the case of the  fourth  and the  fifth.  About  13.6,  15.6  and  17.6 cents,  respectively,  in the cases of the  major thirdminor third  and  lesser septimal tritone  (the learned name of an interval which is rarely found outside of  Blues).

Besides unison and octaves,  there are  11  possible intervals in equal-tempered Western music.  The four most important bear the same name as the  pure  intervals which they approximate:

  • Fifth.  Seven demitones.  (Approx. 3/2 ratio.)
  • Fourth.  Five semitones.  (Approx. 4/3 ratio.)
  • Major third.  Third of an octave.  (Approx. 5/4 ratio.)
  • Minor third.  Quarter of a octave.  (Approx. 6/5 ratio.)

Next in importance are the two intervals which may separate two adjacent  white  notes on a piano keyboard.

  • Major second.  Whole step.  Sixth of an octave.  (Approx. 9/8 ratio).
  • Minor second.  Half step.  Twelfth of an octave.  One semitone.

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...

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).
Wikipedia :   Music   |   Pitch   |   Unison   |   Octave   |   Decade   |   Major third   |   Minor third
Equal temperament   |   Just intonation   |   Perfect fifth   |   Circle of fifths   |   Wolf fifth (quinte du loup)

(2018-02-15)   The  Circle of Fifths  was born a 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.  12  of those intervals is  slightly over  7  octaves  (129.746337890625 vs. 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).
Johann Sebastian Bach (1685-1750)

(2018-02-17)   Tritone:  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.


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...

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,  technically called  lesser septimal tritone.  This unusual interval is actually used in  Blues.

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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.

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.

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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).

(2017-02-26)   Modes

The seven heptatonic modes :

  • Lydian
  • Ionian.  The major scale.
  • Myxolydian
  • Dorian
  • Aeolian
  • Phrygian
  • Lochrian

Why is major  happy ?  (7:35)  by  Adam Neely   (2016-05-30).
Wikipedia :   Mode   |   Degree

(2017-04-09)   Chords
Sets of notes played nearly simultaneously.

The most common chords are  triads  consisting of three distinct notes.  The lowest pitch is called the  root.  In a major chord, the other two notes are one third and one fifth above the root.

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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).
Wikipedia :   Chord

(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 Canad, 2014-03-25).
Wikipedia :   Cadence

(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

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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)

(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)   All but forgotten theory of 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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Original music theory of Jazz (10:46)   Adam Neely  with  Rick Beato   (2017-11-06).
Lydian Chromatic Concept of Tonal Organization for Improvisation (1953)   |   George Russell (1923-2009)

(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).

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