What is the Comma, and where does it come from?neither harmonious nor dissonant. But when two...

© Kevin Vincent Halpin

kevinvincenthalpin.com

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This text is based on excerpts from

Chapters 1 to 4 of Euphony.

What is the Comma, and

where does it come from?

In the sixth century BC Pythagoras discovered that the harmonious relationships

between sounds lie in the fact that they are based on ratios of simple whole

numbers. The same single note from two identical instruments (unison, 1:1) is

neither harmonious nor dissonant. But when two different notes are sounded

together only certain notes achieve harmony.

It is said that Pythagoras discovered this while walking past a blacksmith’s

workshop. He noticed that the ringing of the different hammers on the anvils was

producing simple harmonies, and the differences in sound were owing to the

different sizes of the hammers. A hammer weighing half the weight of another rang

at a frequency (or pitch) twice as high, which is an interval of an octave (ratio 2:1).

A pair with weights in the ratio 3:2 sounded nice as well, which is an interval of a

perfect fifth. Others say the differences in sound were because sheets of metal of

different sizes were being hammered, and the hammers were the same. Others say

the blacksmith story is a myth. Even if a myth the physics involved is true and such

a discovery could have been made that way.

Pythagoras then experimented. He observed that the same ratios exist in stringed

instruments, pipes and bells. It is thus a principle which may be quantified and

proven true. From this the physics was quantified: strings at the same tension but

whose lengths were different in these simple whole numbered ratios, or strings of

the same length but whose tension was different by the same ratios, produced the

notes which are naturally related to each other harmonically. This is Pythagorean

harmony, and the behaviour of a vibrating string will present these numbers

phenomenally.

A vibrating string is a ground note. In music theory today it would be called a

tonic, and in harmonic analysis it would be called the fundamental (or prime).

When a duplicate string is stretched tighter its tension is greater. It will then

produce a higher frequency. When the duplicate string with that higher frequency

is an octave above the first string it will vibrate in two equal parts (the ratio of the

higher frequency to the lower is 2:1). A still point then appears on that higher tuned

string—at the middle—where the vibration is displaced. Such a point of vibrational

displacement is called a node. Tune another identical string even higher so that

there are two still points and its pitch is then a perfect fifth above that octave (3:2

higher than the octave string); tuned at a quarter, there are three points of

displacement and the sound is then a perfect fourth above the fifth (4:3 higher than

the octave + fifth), making its tone a full two octaves above the ground note. It was

discovered after Pythagoras’ lifetime that five vibrating parts is a major third again

above the fundamental (5:4 higher again).



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Pythagoras was also a mathematician. Besides experimenting with strings, bells

and pipes, he progressed the perfect fifth twelve times and identified all the unique

perfect fifths. He also discovered that these notes would define a total interval of

seven octaves, which is a compound perfect interval. Another discovery was that

the notes identified by the continued progression are also the very notes in any 12-

tone octave, though in a different sequential order. A third discovery was that in

each arrangement, seven octaves or one, a comma manifested. The comma was a

difference between what a sound should be by these whole-numbered ratios, and

the sound actually produced when tuned by them.

Mathematically, progressing 3:2 to the twelfth power (3:212

) is 129.7463379 (1.5

x 1.5 x 1.5 and so on twelve times). But, the mathematical expression of an interval

of seven octaves is 27, 128 exactly. This difference of 1.7463379 between

129.7463379 and 128 is the Pythagorean comma.

The mathematics of the Pythagorean comma has a corresponding physical and

aural expression. Tune all the Cs on a piano by 27 then play C and its octave notes



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over seven octaves. You arrive at a sound for the seventh octave of C (its name is

C8 or B#7). Now start from C again by fifths, and tune them 3:2 higher than the

other. The final note, the final fifth, must be the same note, C8 (enharmonically

B#7), but now it sounds noticeably higher than C8 when tuned by 27. Two identical

pianos, one tuned by fifths and another by octaves, would then not be in tune

together. Commas would exist between the notes throughout both instruments.

The comma is a serious problem because the differences in sound cause

dissonance, an unpleasant-sounding clash between sounds which should be

harmonious. Commas prohibited the development of tonal harmony for almost two

thousand years after Pythagoras.

The comma has similar manifestations across shorter intervals. When Pythagoras

used his notes in a single scale they produced sounds which were not in tune the

way they were expected to be, similarly. This is the ditonic comma, the same

comma in seven octaves but within the interval of an octave.

Name of comma Where it resides Mathematical

expression

Size as a ratio

Pythagorean

comma

Between C1 and

B#7 by octaves

and C1 and B#7 by

fifths

3:212

:27 R = 3:2

12 ÷ 2

7

=

1.013643265

ditonic comma Between any two

notes that should

be the same

531441:524288 (≈

74:73)

R =

1.013643265,

≈ 74:73

(1.013698630)

In the second century AD Claudius Ptolemy (c.90–168) devised just intonation,

a tuning system for the diatonic scale (Pythagoras’ scale, the doe–rei–me–fa–sol–

la–tee scale). It too could not avoid manifesting the comma in any octave of any

key, and the size of the comma was almost twice as great. It was almost half a

semitone out. Not only was just intonation worse but now commas could be

defined between the two systems. Not good when different instruments playing

together could be tuned one by Pythagorean tuning and another by just intonation.

Name of comma Where it resides Mathematical

expression

Size

diesis Between any two

notes that should

be the same

128:125 1.024

syntonic comma Between a 5:4

major third and the

Pythagorean major

third (81:64)

81:80 1.0125 = 81 ÷ 80 =

(81:64) ÷ (5:4)

schisma Between syntonic

comma and ditonic

comma

32805:32768 1.00112915

There are many other commas, some named after those in the ancient world who

like Ptolemy tried to fix things, could not, and the remaining comma was named

after them. There is for instance the comma of Didymus (or Didymic comma),

which is found as a difference between A tuned by a 5:4 major third from F and the

same A if tuned by a perfect fifth above D by 3:2. Today some people refer to any

difference between any two notes which should be the same as a diesis.



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Additionally, a syntonic comma can be the difference between six tones and an

octave, which should be the same but is separated by a comma. The difference

between three major thirds at 5:4 and an octave (which is the same thing as six

tones anyway) can be referred to as being either a syntonic comma or a ditonic

comma.

Since the Renaissance the problem of the comma has been addressed by

temperament, methods of adjusting sounds to remove the unexpected differences.

In the eighteenth century, ‘equal temperament’ was the method finally discovered

to temper the comma effectively. Music written in the centuries since then relies on

equal temperament to sound right, including jazz and rock music.

But today there still remains a comma in any enharmonic note by equal

temperament (for example C# can require a different fingering or breath than is

required for Db even though they are enharmonically the same vibrating string or

note from a wind instrument). Mathematically, 216

:310

= 1.10985715 and 9:8 =

1.125. The difference between the two ratios is only 0.01514285. But this is a

noticeable comma aurally.

Euphony tells the story of the comma and its impact on cultural and scientific

history, outlines its solution, and surveys how nature is saturated with the

principles underlying music. The book can be viewed or downloaded as a PDF

from the link on the Comma page of kevinvincenthalpin.com.

What is the Comma, and where does it come from?neither harmonious nor dissonant. But when two...

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