In
musical tuning
In music, there are two common meanings for tuning:
* Tuning practice, the act of tuning an instrument or voice.
* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.
Tuning practice
Tun ...
, the Pythagorean comma (or ditonic comma), named after the ancient mathematician and philosopher
Pythagoras
Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samos, Samian, or simply ; in Ionian Greek; ) was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymou ...
, is the small
interval (or
comma
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
) existing in
Pythagorean tuning
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2.Bruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh edition, 2 vols. (Boston: Mc ...
between two
enharmonically equivalent
In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. The enharmonic spelling of a written no ...
notes such as C and B (), or D and C. It is equal to the
frequency ratio
In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth (for example C to G) is 3:2 (), 1.5, and may be approximated by an equal tempered perfect fifth () which is 27/ ...
=
≈ 1.01364, or about 23.46
cents, roughly a quarter of a
semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.
It is defined as the interval between two adjacent no ...
(in between 75:74 and 74:73). The comma that
musical temperament
In musical tuning, a temperament is a tuning system that slightly compromises the pure intervals of just intonation to meet other requirements. Most modern Western musical instruments are tuned in the equal temperament system. Tempering is the ...
s often refer to tempering is the Pythagorean comma.
The Pythagorean comma can be also defined as the difference between a
Pythagorean apotome
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.
It is defined as the interval between two adjacent no ...
and a
Pythagorean limma
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.
It is defined as the interval between two adjacent no ...
(i.e., between a chromatic and a diatonic
semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.
It is defined as the interval between two adjacent no ...
, as determined in Pythagorean tuning), or the difference between twelve
just
Just or JUST may refer to:
__NOTOC__ People
* Just (surname)
* Just (given name)
Arts and entertainment
* ''Just'', a 1998 album by Dave Lindholm
* "Just" (song), a song by Radiohead
* "Just", a song from the album ''Lost and Found'' by Mudvayne ...
perfect fifth
In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so.
In classical music from Western culture, a fifth is the interval fro ...
s and seven
octave
In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
s, or the difference between three Pythagorean
ditone
In music, a ditone (, from , "of two tones") is the interval of a major third. The size of a ditone varies according to the sizes of the two tones of which it is compounded. The largest is the Pythagorean ditone, with a ratio of 81:64, also cal ...
s and one octave (this is the reason why the Pythagorean comma is also called a ''ditonic comma'').
The
diminished second
In modern Western tonal music theory, a diminished second is the interval produced by narrowing a minor second by one chromatic semitone.Bruce Benward and Marilyn Saker (2003). ''Music: In Theory and Practice, Vol. I'', p. 54. . Specific example ...
, in Pythagorean tuning, is defined as the difference between limma and apotome. It coincides, therefore, with the opposite of a Pythagorean comma, and can be viewed as a ''descending'' Pythagorean comma (e.g. from C to D), equal to about −23.46 cents.
Derivation
As described in the introduction, the Pythagorean comma may be derived in multiple ways:
* Difference between two
enharmonically equivalent
In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. The enharmonic spelling of a written no ...
notes in a Pythagorean scale, such as C and B (), or D and C (see
below).
* Difference between
Pythagorean apotome
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.
It is defined as the interval between two adjacent no ...
and
Pythagorean limma
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.
It is defined as the interval between two adjacent no ...
.
* Difference between twelve just
perfect fifth
In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so.
In classical music from Western culture, a fifth is the interval fro ...
s and seven
octaves.
* Difference between three Pythagorean
ditone
In music, a ditone (, from , "of two tones") is the interval of a major third. The size of a ditone varies according to the sizes of the two tones of which it is compounded. The largest is the Pythagorean ditone, with a ratio of 81:64, also cal ...
s (
major third
In classical music, a third is a musical interval encompassing three staff positions (see Interval number for more details), and the major third () is a third spanning four semitones. Forte, Allen (1979). ''Tonal Harmony in Concept and P ...
s) and one octave.
A just perfect fifth has a
frequency ratio
In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth (for example C to G) is 3:2 (), 1.5, and may be approximated by an equal tempered perfect fifth () which is 27/ ...
of 3:2. It is used in Pythagorean tuning, together with the octave, as a yardstick to define, with respect to a given initial note, the frequency ratio of any other note.
Apotome and limma are the two kinds of
semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.
It is defined as the interval between two adjacent no ...
s defined in Pythagorean tuning. Namely, the apotome (about 113.69 cents, e.g. from C to C) is the chromatic semitone, or augmented unison (A1), while the limma (about 90.23 cents, e.g. from C to D) is the diatonic semitone, or minor second (m2).
A ditone (or
major third
In classical music, a third is a musical interval encompassing three staff positions (see Interval number for more details), and the major third () is a third spanning four semitones. Forte, Allen (1979). ''Tonal Harmony in Concept and P ...
) is an interval formed by two
major tone
In Western music theory, a major second (sometimes also called whole tone or a whole step) is a second spanning two semitones (). A second is a musical interval encompassing two adjacent staff positions (see Interval number for more deta ...
s. In Pythagorean tuning, a major tone has a size of about 203.9 cents (frequency ratio 9:8), thus a Pythagorean ditone is about 407.8 cents.
Size
The size of a Pythagorean comma, measured in
cents, is
:
or more exactly, in terms of
frequency ratios:
:
Circle of fifths and enharmonic change
The Pythagorean comma can also be thought of as the discrepancy between twelve
justly tuned perfect fifth
In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so.
In classical music from Western culture, a fifth is the interval fro ...
s (ratio 3:2) () and seven octaves (ratio 2:1):
:
In the following table of
musical scale
In music theory, a scale is any set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is an ascending scale, and a scale ordered by decreasing pitch is a descending scale.
Often, especially in the ...
s in the
circle of fifths
In music theory, the circle of fifths is a way of organizing the 12 chromatic pitches as a sequence of perfect fifths. (This is strictly true in the standard 12-tone equal temperament system — using a different system requires one interval ...
, the Pythagorean comma is visible as the small interval between e.g. F and G. Going around the circle of fifths with just intervals results in a
comma pump In music theory, a comma pump (or comma drift) is a sequence of notes, often a chord progression, where the pitch shifts up or down by a comma (a small interval) every time the sequence is traversed. Comma pumps often arise from a sequence of just ...
by the Pythagorean comma.
The 6 and the 6 scales* are not identical - even though they are on the
piano keyboard
A musical keyboard is the set of adjacent depressible levers or keys on a musical instrument. Keyboards typically contain keys for playing the twelve notes of the Western musical scale, with a combination of larger, longer keys and smaller, sh ...
- but the scales are one Pythagorean comma lower. Disregarding this difference leads to
enharmonic change.
* The 7 and 5, respectively 5 and 7 scales differ in the same way by one Pythagorean comma. Scales wit
seven accidentals
are seldom used, because the enharmonic scales with five accidentals are treated as equivalent.
This interval has serious implications for the various
tuning
Tuning can refer to:
Common uses
* Tuning, the process of tuning a tuned amplifier or other electronic component
* Musical tuning, musical systems of tuning, and the act of tuning an instrument or voice
** Guitar tunings
** Piano tuning, adjusti ...
schemes of the
chromatic scale
The chromatic scale (or twelve-tone scale) is a set of twelve pitches (more completely, pitch classes) used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce the ...
, because in Western music,
12 perfect fifths and seven octaves are treated as the same interval.
Equal temperament
An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, wh ...
, today the most common tuning system used in the West, reconciled this by flattening each fifth by a twelfth of a Pythagorean comma (approximately 2 cents), thus producing perfect octaves.
Another way to express this is that the just fifth has a frequency ratio (compared to the tonic) of 3:2 or 1.5 to 1, whereas the seventh semitone (based on 12 equal logarithmic divisions of an octave) is the seventh power of the
twelfth root of two
The twelfth root of two or \sqrt 2/math> (or equivalently 2^) is an algebraic irrational number, approximately equal to 1.0594631. It is most important in Western music theory, where it represents the frequency ratio (musical interval) of a semi ...
or 1.4983... to 1, which is not quite the same (out by about 0.1%). Take the just fifth to the twelfth power, then subtract seven octaves, and you get the Pythagorean comma (about 1.4% difference).
History
The first to mention the comma's proportion of 531441:524288 was
Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
, who takes as a basis the whole tone of Pythagorean tuning with the ratio of 9:8, the octave with the ratio of 2:1, and a number A = 262144. He concludes that raising this number by six whole tones yields a value G which is larger than that yielded by raising it by an octave (two times A). He gives G to be 531441. The necessary calculations read:
Calculation of G:
::
Calculation of the double of A:
::
Chinese mathematicians had been aware of the Pythagorean comma as early as 122 BC (its calculation is detailed in the ''
Huainanzi
The ''Huainanzi'' is an ancient Chinese text that consists of a collection of essays that resulted from a series of scholarly debates held at the court of Liu An, Prince of Huainan, sometime before 139. The ''Huainanzi'' blends Daoist, Confuci ...
''), and circa 50 BC,
Ching Fang discovered that if the cycle of perfect fifths were continued beyond 12 all the way to 53, the difference between this 53rd pitch and the starting pitch would be much smaller than the Pythagorean comma. This much smaller interval was later named
Mercator's comma (''see:
history of 53 equal temperament'').
In George Russell's ''
Lydian Chromatic Concept of Tonal Organization
The ''Lydian Chromatic Concept of Tonal Organization'' is a 1953 jazz music theory book written by George Russell. The book is the founding text of the Lydian Chromatic Concept (LCC), or Lydian Chromatic Theory (LCT). Russell's work postulates ...
'' (1953) the half step between the Lydian Tonic and 2 in his Altered Major and Minor Auxiliary Diminished Blues scales is theoretically based on the interval of a Pythagorean comma.
[Russell, George (2001) ]953
Year 953 ( CMLIII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar.
Events
By place
Byzantine Empire
* Battle of Marash: Emir Sayf al-Dawla marches north into the Byzantine Empire an ...
George Russell's ''Lydian chromatic concept of tonal organization''. Volume One: The art and science of tonal gravity (Fourth (Second printing, corrected, 2008) ed.). Brookline, Massachusetts: Concept Publishing Company. pp. 17, 57-59. .
See also
*
Syntonic comma
In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125) ...
*
Schisma
In music, the schisma (also spelled ''skhisma'') is the interval between a Pythagorean comma (531441:524288) and a syntonic comma (81:80) and equals or 32805:32768 = 1.00113, which is 1.9537 cents (). It may also be defined as:
* the differ ...
*
Holdrian comma
In music, 53 equal temperament, called 53 TET, 53 EDO, or 53 ET, is the tempered scale derived by dividing the octave into 53 equal steps (equal frequency ratios). Each step represents a frequency ratio of 2, or 22.6415& ...
Notes
References
{{Intervals, state=expanded
Commas (music)
3-limit tuning and intervals