MT 007.02 / SL 266.01
Ideas in Mathematics: The Grammar of Numbers

Musical Intervals: The harmonic series and a 'pure' scale

We refer to the proportional distance between two musical notes (roughly within an acoustical range from 0030 Hz to 4000 Kz) as an interval.
This presentation discusses, along Pythagorean lines, the derivation of the intervals used in traditional Western music and their numerical interrelationships.


If a string (or other ideal vibrating medium) of a given length sounds at the pitch of (capital) C, i.e. the note on the second line below the bass clef or the key that is two octaves below middle c (c') on a standard keyboard, then a string of half that length will produce a note that sounds one octave higher, namely (small) c, i.e. the note in the second space (from the bottom) on the bass clef or the key that is one octave below middle c (c') on a standard keyboard.
The two notes C-c are in a proportion (higher to lower) of 2:1, and this proportion characterizes the musical interval called the octave.

Note that discussions of musical intervals nearly always choose a c as the starting note because the major scale of c on standard keyboard instruments uses only the 'white' keys, i.e. contains no sharps or flats.
Note also that some discussions characterize interval proportions from bottom to top (1:2) and/or as fractions (2/1 or 1/2). As long as one maintains consistency in one's own discussion, these differences are distracting but not fatal.

Subsequent divisions of the string halves into further halves will produce notes at

one fourth of the original string length,
one line above the bass clef or one line below the treble clef,
middle c
one eighth of the original string length,
third space in the treble clef,
one octave above middle c
one sixteenth of the original string length,
two lines above the treble clef,
two octaves above middle c
For purposes of general musical discussion this four-octave range (C-c''') suffices, since it covers a full singing range from the basso profundo's low C to the operatic soprano's 'high c' (c''').


A trisecting of the string produces sections which sound at a pitch roughly halfway between those for one half and one fourth of the original string length. (On a fractional ruler 1/3 falls roughly half way between 1/2 and 1/4).
The proportion 3:2 (and its integral multiples such as 6:4, 12:8) characterizes the musical interval of the fifth (do-sol), or in our plan c-g (also c'-g' and c''-g'').

Note that if a string were vibrating at a frequency of 0100 Hz (cycles/second), its half sections (i.e. octaves) would be vibrating at 0200 Hz, its quarter sections (i.e. two octaves above the fundamental string length) would vibrate at 0400 Hz, a note three octaves above the fundamental would be at 0800 Hz, and a note four octaves above the fundamental would sound at 1600 Hz.
The first trisection would be at 0300 Hz, and the distance between 0200 Hz and 0300 Hz would be heard as a musical fifth (do-sol). Likewise the proportional distances 0400 and 0600 Hz, 0800 and 1200 Hz, etc. would be heard as fifths.
In increasingly higher octaves the absolute 'size' of the fifth increases (in our example 0100 Hz, 0200 Hz, 0400 Hz respectively) because the ear perceives geometrically, not linearly, but the proportion of a fifth always remains 3:2.


In order to add intervals we multiply the proportions (see Exercise 24),
and in order to subtract one interval from another, we divide their proportions (larger by smaller).
Thus, by subtracting a fifth from an octave, we derive a new interval, the fourth (g-c', sol-do', transposable to c-f), with a proportion 4:3 because
octave minus fifth
= 2:1 - 3:2
=> 2:1 x 2:3
= 4:3.

Thirds (major and minor)

Dividing our sample string into five parts will give us a note e' which sounds at a musical major third (do-mi) in a proportion of 5:4 over c'.
By proportion the interval c'' to e'' (10:8) is also a major third, one octave higher.

If we subtract a major third (5:4) from a fifth (3:2), the remainder gives us a new interval called the minor third (mi-sol) with a proportion of 6:5 because
fifth minus major third
= 3:2 - 5:4
=> 3:2 x 4:5
= 12:10 = 6:5

Seconds (major and minor)

One ninth of a string against one eighth of a string gives a note roughly halfway between c'' and e''. This is the (major) second (do-re), proportion 9:8.

Using fractional proportions this could also be transposed down into other octaves:
c' to d' would be 4.5:5,
c to d would be 2.25 to 2,
C to D would be 1.125 to 1.

And, by derivation, if we subtract a (major) second from a (major) third, this will give an interval narrower than a major second, namely the minor second, proportion 10:9, because:
major third minus (major) second
= 10:8 - 9:8
=> 10:8 x 8:9
= 80:72 = 10:9


The interval heard in the proportion 16:15 corresponds to the semitone between b'' and c''' (si-do', by transposition mi-fa).

The German musical tradition uses the letter b to denote what is b-flat elsewhere. What we call b (natural) is for them h. Disambiguating the situation would require that, for clarity's sake, we always speak of b-flat and b-natural.
What we call Bach's B-minor Mass was for him the _Messe in h-moll_.
This letter/scale-note h also made it possible for Bach to compose fugues and themes which incorporate his name:

Ekmelic notes

The note for the 7th vibration mode (i') (and its multiple 14 (i'')), as well as the 11th (k''), and the 13th (l'') --all non-low primes!-- have no part in traditional Western music and are called 'ekmelic' (non-melodic) notes. Coach drivers once used i'(a very flat b-flat) in blowing alarm signals on a natural horn, and k'' (an out-of-tune f-sharp) occurs in Swiss Alphorn melodies and yodels.

The harmonic tone series

Taking all the integral vibration notes within our compass C to c''' we get the full set of the first sixteen members of the harmonic tone series:

The pure-tempered scale

The melodic (emmelic) intervals obtained thus far can combine to form a diatonic scale in c with pure or 'natural' temperament:

We shall be using this natural scale as the basis of our work in discussing the mathematics of various temperament (tuning) systems, including the famous equal temperament, whose ultimate working out was one of the great moments of applied mathematics.
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