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In musical tuning, the Pythagorean comma (or ditonic comma^[1]), named after the ancient mathematician and philosopher Pythagoras, is the small interval (or comma) existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B♯ (Play^ⓘ), or D♭ and C♯.^[2] It is equal to the frequency ratio 531441:524288, or approximately 23.46 cents, roughly a quarter of a semitone (in between 75:74 and 74:73^[3]). The comma which musical temperaments often refer to tempering is the Pythagorean comma.^[4]

The Pythagorean comma can be also defined as the difference between a Pythagorean apotome and a Pythagorean limma^[5] (i.e., between a chromatic and a diatonic semitone, as determined in Pythagorean tuning), or the difference between twelve just perfect fifths and seven octaves, or the difference between three Pythagorean ditones and one octave (this is the reason why the Pythagorean comma is also called ditonic comma).

The diminished second, 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 notes in a Pythagorean scale, such as C and B♯ (Play), or D♭ and C♯ (see below).
Difference between Pythagorean apotome and Pythagorean limma.
Difference between twelve just perfect fifths and seven octaves.
Difference between three Pythagorean ditones (major thirds) and one octave.

A just perfect fifth has a frequency ratio 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 semitones 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) is an interval formed by two major tones. 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

{\hbox{apotome}}-{\hbox{limma}}\approx 113.69-90.23\approx 23.46~{\hbox{cents}}\!

or more exactly, in terms of frequency ratios:

{\frac {\hbox{apotome}}{\hbox{limma}}}={\frac {3^{7}/2^{11}}{2^{8}/3^{5}}}={\frac {3^{12}}{2^{19}}}={\frac {531441}{524288}}=1.0136432647705078125\!

Circle of fifths and enharmonic change

The Pythagorean comma can also be thought of as the discrepancy between twelve justly tuned perfect fifths (ratio 3:2) (play^ⓘ) and seven octaves (ratio 2:1):

{\frac {\hbox{twelve fifths}}{\hbox{seven octaves}}}=\left({\tfrac {3}{2}}\right)^{12}\!\!{\Big /}\,2^{7}={\frac {3^{12}}{2^{19}}}={\frac {531441}{524288}}=1.0136432647705078125\!

Ascending by perfect fifths
Note	Fifth	Frequency ratio	Decimal ratio
C	0	1 : 1	1
G	1	3 : 2	1.5
D	2	9 : 4	2.25
A	3	27 : 8	3.375
E	4	81 : 16	5.0625
B	5	243 : 32	7.59375
F♯	6	729 : 64	11.390625
C♯	7	2187 : 128	17.0859375
G♯	8	6561 : 256	25.62890625
D♯	9	19683 : 512	38.443359375
A♯	10	59049 : 1024	57.6650390625
E♯	11	177147 : 2048	86.49755859375
B♯ (≈ C)	12	531441 : 4096	129.746337890625

Ascending by octaves
Note	Octave	Frequency ratio
C	0	1 : 1
C	1	2 : 1
C	2	4 : 1
C	3	8 : 1
C	4	16 : 1
C	5	32 : 1
C	6	64 : 1
C	7	128 : 1

In the following table of musical scales in the circle of fifths, the Pythagorean comma is visible as the small interval between e.g. F♯ and G♭.

The 6♭ and the 6♯ scales* are not identical - even though they are on the piano keyboard - but the ♭ scales are one Pythagorean comma lower. Disregarding this difference leads to enharmonic change.

Template:Circle of fifths unrolled * The 7♭ and 5♯, respectively 5♭ and 7♯ scales differ in the same way by one Pythagorean comma. Scales with 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 schemes of the chromatic scale, because in Western music, 12 perfect fifths and seven octaves are treated as the same interval. Equal temperament, 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 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

Chinese mathematicians had been aware of the Pythagorean comma as early as 122 BC (its calculation is detailed in the Huainanzi), 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).

The first to mention the comma's proportion of 531441:524288 was Euclid, 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.^[6] The necessary calculations read:

Calculation of G:

262144\cdot \left(\textstyle {\frac {9}{8}}\right)^{6}=531441

Calculation of the double of A:

262144\cdot \left(\textstyle {\frac {2}{1}}\right)^{1}=524288

Sound Recording

The Pythagorean Comma; A Play

First broadcast:Saturday 22 December 2012 Loosely based on Jules Verne's story "Mr Ray Sharp and Miss Me Flat", "The Pythagorean Comma" is a music drama with text by Blake Morrison and music by Gavin Bryars. It's about one of the oldest mysteries in the science of sound. The story says Bryars, "has wit, whimsy, fantasy and magic and is also about scientific experiment".

Verne's story takes place in a 19th century Swiss village. This contemporary take on the original is set on a remote fictional Scottish island but the essential story is unchanged.

A village organist gets old and deaf and stops playing and the organ falls silent. A mysterious stranger arrives who not only plays the organ beautifully but also declares that he will develop a new organ registration with the voices of the children in the school. Each will have his or her own note that has a special resonance.

Though the children are musically untrained, the stranger rehearses them with an iron discipline and prepares them for a Christmas concert. It's at this concert that he demonstrates his phenomenon of a "human organ". He tells the children that he will make them famous and that they are a choir like no other choir.

A boy and girl who are arch rivals are given their special notes. They're angry because this strange music maestro seems to have given them the same note. However he explains that there is a tiny beating sound between them - and this difference is the Pythagorean Comma. The two children are relieved that they have their own notes but strangely, once they start to sing, their old rivalry disappears and it is as if a new harmony has come to them and to the village in general.

The stranger seems to have a power over the choir and they outperform everyone's expectations in a Christmas concert for the island community.

Composer Gavin Bryars and author Blake Morrison have collaborated before on a Jules Verne story, 'Doctor Ox's Experiment' - also about Verne's interest in music and science.

Gerda Stevenson stars as the narrator, Anna. She's the church warden and mother of a child she christened Ian but who now has the new name of Ray because his special note is Ray sharp. She sees at first hand how the stranger brings his gift of music.

References

^ not to be confused with the diatonic comma, better known as syntonic comma, equal to the frequency ratio 81:80, or around 21.51 cents. See: Johnston B. (2006). "Maximum Clarity" and Other Writings on Music, edited by Bob Gilmore. Urbana: University of Illinois Press. ISBN 0-252-03098-2.
^ Apel, Willi (1969). Harvard Dictionary of Music, p.188. ISBN 978-0-674-37501-7. "...the difference between the two semitones of the Pythagorean scale..."
^ Ginsburg, Jekuthiel (2003). Scripta Mathematica, p.287. ISBN 978-0-7661-3835-3.
^ Coyne, Richard (2010). The Tuning of Place: Sociable Spaces and Pervasive Digital Media, p.45. ISBN 978-0-262-01391-8.
^ Kottick, Edward L. (1992). The Harpsichord Owner's Guide, p.151. ISBN 0-8078-4388-1.
^ Euclid: Katatome kanonos (lat. Sectio canonis). Engl. transl. in: Andrew Barker (Ed.): Greek Musical Writings. Vol. 2: Harmonic and Acoustic Theory, Cambridge Mass.: Cambridge University Press, 2004, pp. 190–208, here: p. 199.

[1] t to be confused with the diatonic comma, better known as syntonic comma, equal to the frequency ratio 81:80, or around 21.51 cents. See: Johnston B. (2006). "Maximum Clarity" and Other Writings on Music, edited by Bob Gilmore. Urbana: University of Illinois Press. ISBN 0-252-03098-2.

[2] Apel, Willi (1969). Harvard Dictionary of Music, p.188. ISBN 978-0-674-37501-7. "...the difference between the two semitones of the Pythagorean scale..."

[3] Ginsburg, Jekuthiel (2003). Scripta Mathematica, p.287. ISBN 978-0-7661-3835-3.

[4] Coyne, Richard (2010). The Tuning of Place: Sociable Spaces and Pervasive Digital Media, p.45. ISBN 978-0-262-01391-8.

[5] Kottick, Edward L. (1992). The Harpsichord Owner's Guide, p.151. ISBN 0-8078-4388-1.

[6] Euclid: Katatome kanonos (lat. Sectio canonis). Engl. transl. in: Andrew Barker (Ed.): Greek Musical Writings. Vol. 2: Harmonic and Acoustic Theory, Cambridge Mass.: Cambridge University Press, 2004, pp. 190–208, here: p. 199.

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+{{merge to|"Sound Recording" section is *totally* off-topic and should be a separate article.  |date=January 2013}}
 [[Image:Pythagorean comma (difference A1-m2).PNG|thumb|right|400px|Pythagorean comma ('''PC''') defined in [[Pythagorean tuning]] as difference between semitones (A1&ndash;m2), or interval between [[enharmonic|enharmonically equivalent]] notes (from D{{Music|b}} to C{{Music|#}}). The [[diminished second]] has the same width but an opposite direction (from  to C{{Music|#}} to D{{Music|b}}).]]
 [[Image:Pythagorean comma on C.png|thumb|right|Pythagorean comma on C. {{audio|Pythagorean comma on C.mid|Play}}. The note depicted as lower on the staff (B[[semitone#Just intonation|{{music|#}}]][[syntonic comma|+++]]) is slightly higher in pitch (than C{{music|natural}}).]]