The New Harvard Dictionary of Music, edited by Ron Handel
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.
(2018-02-16) Duration of notes and rests
Binary progression of standard durations (dotting prolongs by 50%).
A note is an elementary music element with nearly constant pitch and
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)
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.
Beats per Minute
24 bpm and below
24 bpm - 40 bpm
40 bpm - 60 bpm
60 bpm - 66 bpm
66 bpm - 76 bpm
76 bpm - 100 bpm
100 bpm - 120 bpm
120 bpm - 168 bpm
168 bpm - 200 bpm
200 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.
(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.
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).
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).
(2018-02-16) Triplets and offbeat Triplets.
Trained musicians have trouble playing offbeat Triplets at a slow tempo.
A 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
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).
(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.
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
C4 is called
middle C and standard
(A4 , 440 Hz) is dubbed A above middle C.
Each octave starts at a C and ends with the B above it.
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 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 :
The highest note ever sung in a regular performance at New-York's Met Opera
was A above high-C (A6 , 1760 Hz) by soprano
Audrey Luna as the very first note of Leticia in
The Exterminating Angel by Thomas Adès (2017).
said she can sustain C above high-C (C7 , 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
(D7 , 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 G7 ,
not three octaves above it. (Did someone confuse semitones and octaves?)
(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.
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 :
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
Both ends of the 64-key keyboard match the layout of a grand piano.
This pattern helped make allWurlitzer 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.
(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
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 third,
minor third and lesser septimal tritone
(the learned name of an interval which is rarely found outside of
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.
(2018-02-15) The Circle of Fifths was born a spiral...
Two conflicting musical intervals: Perfect fifth (3:2) & octave (2:1).
A 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.
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.
(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.
(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
(A0 , 27.5 Hz). The highest is 108 (C8 , 4186.01 Hz).
The MIDI numbers span almost 11 octaves,
from C-1 (8.176 Hz)
to G9 (12543.854 Hz):