Research Paper

This paper provides an example of how science and beauty are interrelated through music.

Leyan Lo

Mrs. Marionni

Honors World Literature, period 4

13 June 2003

The Physics of Intonation

            Nearly all of Western music, up until the more recent experimenting with microtones, is based on the diatonic scale, having seven notes in an octave.  The octave is a very fundamental interval for notes, and relates the frequencies of the notes by the ratio of 2:1.  This simple ratio means the notes sound to the ear as if they are the same note.  Virtually all of Western music divides the octave into 12 equal semi-tones on which music is based.  However, the system is a compromise; the octave has not always been divided this way.  Over the ages, there have been four principle tuning systems in wide usage for Western music.

            For pure monophonic (one note at a time) musical melody, it does not matter a great deal what scale or tuning method is used.  However, almost all modern music is polyphonic (many notes sounding simultaneously).  For the music to sound consonant (in tune), the different notes being sounded simultaneously need to be related in a way, which is pleasing to the ear.  The relationship between the frequencies of notes of the important musical intervals is what makes them sound pleasing.  The important ratios are as follows:

6:5       minor third

5:4       major third

4:3       perfect fourth

3:2       perfect fifth

8:5       minor sixth

5:3       major sixth

All four of the main tuning systems considered here were devised around some or most of these intervals.

Western music is generally considered to have started from Pythagoras, the ancient Greek.  Pythagoras devised a system based on mathematical principles.  He defined the scale around the ratios of the fifth, being in the ratio 3:2 exactly, and the fourth being 4:3 exactly.  The difference between these two was then 9:8 (because you need to multiply the fourth ratio by the whole step ratio to end up with the fifth ratio), which he defined as the tone, or whole step.  He then divided the octave so that there were the seven notes, as the diatonic scale has today, but to get the mathematics to add up he was left with two semi-tones, which he defined as 256:243.

Thus the Pythagorean scale has the following intervals:

Cumulative Intervals:    1      9:8    81:64    4:3     3:2    27:16  243:128    2

Note:                    C       D       E       F       G       A       B       C

Incremental Intervals:      9:8     9:8   256:243   9:8     9:8     9:8   256:243

Cents:                      204     204     90      204     204     204     90

(The values given as “cents” represent the intervals where the octave is divided into 1200 cents.  This is the modern way of representing pitch in such a way that each semitone comprises 100 cents.)

It is interesting to note that Pythagoras did not recognize the major third, which is distinctly sharp at 81:64 compared with the ideal of 5:4.

The chromatic Pythagorean scale is formed by inserting semitones equal to 114 cents in such a way as to keep all perfect fifths true, except for the interval G#-Eb, which needs to be adjusted so the intervals add up mathematically.  This difference is known as the “comma of Didymus.”

Ptolemaic Tuning is generally referred to as “just intonation.”  To create perfect major third intervals, this system alters one of the fifths, D-A.  (And thus the major sixths are also perfect, except for F-D.)  This makes the triad D-F-A quite unusable, although the others are perfectly in tune.  The Just Intonation scale employs two different sized tones in the ratios 9:8 and 10:9, and thus it can hardly be considered satisfactory even for purely melodic music.

Cumulative Intervals:    1      9:8     5:4     4:3     3:2     5:3    15:8      2

Note:                    C       D       E       F       G       A       B       C

Incremental Intervals:      9:8    10:9    16:15    9:8    10:9     9:8    16:15

Cents:                      204     182    112      204     182     204    112

The above comments hold true for scales and intervals based solely on the “white notes.”  Adding the “black notes” to give a full chromatic Just Intonation scale creates more perfect fifths, but major thirds and sixths which are not true.  In fact the result is three different sizes of semi-tones.  (16:15, 135:128 and 256:243.)  It is thought that this system, although considerably debated, was not used much.

            In Mean-tone Temperament, the major thirds are made exact.  This results in the fifths becoming slightly flattened but in such a way that the error of the Ptolemaic system is spread out over four fifths.  This reduces the dissonance and makes the fifths more acceptable.  The whole tones are also all equal in size, being half the major third.  Melodically the Mean-tone scale is more acceptable than the Ptolemaic scale.

Cumulative Intervals:    1              5:4                                      2

Note:                    C       D       E       F       G       A       B       C

Cents:                      193     193    117      193     193     193    117

The chromatic Mean-tone scale has semi-tones of two very different sizes: wide 117 cent semi-tones in the diatonic scale, with 76 cent semi-tones balancing the whole-tone interval of 193 cents.  Mean-tone temperament was designed for keyboard instruments and it was an acceptable compromise as long as the “black notes” beyond Eb or G# were not used.  The G#:Eb fifth was so bad as to be unusable-it was often given the name “the wolf.”

In equal temperament, the octave is divided into 12 equal semi-tones each of 100 cents.  The intervals are then built from the semi-tones.  For example a fifth is 7 semi-tones and a third 5.  The Equal temperament scale is universally used today in Western music.

Cumulative Intervals:    1                                                       2

Note:                    C       D       E       F       G       A       B       C

Cents:                      200     200    100      200     200     200    100

The chromatic Equal temperament scale with all semi-tones = 100 cents means that perfectly tuned intervals have been totally eliminated.  However, the mistuning on fifths is only 2 cents and on thirds 14 cents which the ear does not appear to mind.  The big advantage is that all keys are equally usable.

            With the evolution of this more complicated mathematical model for tuning an instrument, and with the increased importance of musicality and performance, music and mathematics in this aspect have lost the close relationship known in ancient Greek times.  As an even-tempered interval could no longer be expressed as a ratio ( is an irrational number), the musicians learnt to tune an instrument by training their ear rather than by applying mathematical principles. Music from this point of view released itself from mathematical domination.

Works Cited

Bloch, Ernst. Essays on the Philosophy of Music. New York: Cambridge University Press, 1985.

Garland, Trudi H., and Charity V. Kahn. Math and music: Harmonious connections. Palo Alto: Dale Seymour Publications, 1995.

Henle, Jim. “Classical Mathematics.” The American Mathematical Monthly. Jan 1996. 18-29. McClain, Ernest G. The Pythagorean Plato. Maine: Nicolas-Hays Inc, 1978.

Reid, Harvey. “On Mathematics and Music.” 1995. 5 June 2002. <>.

Rothwell, James A. The phi factor: mathematical proportions in musical forms. Kansas City: University of Missouri, 1977.

Rusin, Dave. “Mathematics and Music.” 1998. 4 June 2002. <>.