Set 56 Problem number 14

Problem

Show that if we start at any key on a piano, which is tuned so that each key gives a frequency 2^(1/12) = 1.0594 (approx) times that of the previous, and count up 7 keys we will achieve a ratio of pitches very nearly equal to 3/2 (the ratio of a 'fifth').

Show that if we start at any key and count up 12 keys we will achieve a pitch ratio of 2/1 (the ratio of an 'octave').

Determine how many keys we must count up from a given key to come as close as possible to the ratios 4/3 (called a 'fourth'), 5/4 (called a 'third), and 6/5 (called a 'minor third').

How close do the ratios based on the ratio 2^(1/12) come to mimicking the natural ratios of the harmonics of a string?

If 12 divisions of a 2/1 ratio come this close, it seems like a greater number of divisions, for example 21, might do an even better job.  Use your calculator to find the ratio that would correspond to  21 subdivisions, and determine how nearly the natural ratios could be mimicked using multiples of this ratio.

It turns out that the next number of divisions that actually improves on 12 is 43. This ratio is used in some Eastern music.  How much better can we do with a division of the ratio 2 into  43 equal subdivisions, as compared to 12 subdivisions?

Solution

If we count up 7 keys,

• the first will have ratio 2 ^(1/12),
• the second a ratio of [2^(1/12)] ^2 = 2^(2/12) = 2^(1/6),
• the third a ratio of [2^(1/12)] ^3 = 2^(3/12) = 2^(1/4),
• the fourth a ratio of [2^(1/12)] ^4 = 2^(4/12) = 2^(1/3),
• ....
• and the seventh a ratio of [2^(1/12)] ^7 = 2^(7/12).

If you key this into your calculator you will see that it is very close to 3/2, or 1.5.

The other ratios are similarly calculated:

• Counting up 12 keys we would obtain a ratio of [2^(1/12)] ^12 = 2^(12/12) = 2^(1) = 2.
• The ratio obtained for 5 keys is 2^(5/12), close to the ratio 4/3 = 1.333....
• The ratios for 4 and 3 keys are, as indicated above, 2^(1/3) and 2^(1/4). A calculator will verify that these ratios are close to 5/4 = 1.20 and 6/5 = 1.166....

The ratios obtained for the 21-note division of the 2/1 ratio are

• 2^(1/ 21), 2^(2/ 21), 2^(3/ 21), ..., 2^(n/ 21), where n can be any counting number.

You can figure these ratios out with your calculator. You will not find good matches for most of the naturally occurring ratios 3/2, 4/3, 5/4, ....  In any case the matches are not as good as for a division into 12 equal ratios.

The 43-note division is done in a very similar manner, and does result in an improvement over the 12-note scale. However, an instrument with 43 notes to a doubling (and 'octave') is for the most part impractical (imagine a piano with 88 * (43/12) keys).  12 notes does the job well enough that very few people can even learn to distinguish the difference between the musical intervals based on this scale and the natural harmonics.