Musical scale based on fifths

One can create a musical scale based solely on the "fifth" and the octave. First, pick a starting pitch, now go up a fifth (multiply the frequency by 3/2), then go up another fifth and convert this back down an octave, go up a fifth from that - if the result is beyond the octave, go back down an octave.

Mathematically, starting with a pitch f₀, the next pitch is f₁ = 3f₀/2, and f₂ = (3/2)f₁/2. More generally, given the pitch f_i, then
f_i+1 = (3/2) f_i if that result is less than 2 f₀
f_i+1 = (3/4) f_i if the previous result was not less.

Of course, this process can be repeated indefinately and one will stop after a while to keep the number of notes in the scale reasonable.

Here is a table which results from that procedure. I have included more notes than we usually use for the sake of illustration. Here f₀ = 261.63 Hz was used as an example and corresponds to "middle C." Frequency differences (in Hz) are based on this f₀.

Freq (Hz)	Ratio to Fundamental	Closest Ratio in Just Scale	Freq. difference of Just Scale (Hz)	Closest Ratio in Equal Tempered	Freq. Difference of Equal Tempered
261.63	1 = 1.0000	1.0000	0	1.0000	0
265.20	531441/524288 = 1.013643
279.39	2187/2048 = 1.067871	1.0417	-6.9	1.0595	-2.2
294.33	9/8 = 1.125000	1.1250	0	1.1225	-0.7
298.35	4782969/4194304 = 1.140349
314.31	19683/16384 = 1.201355	1.2000	-0.4	1.1892	-3.2
331.13	81/64 = 1.265625	1.2500	-4.1	1.2599	-1.5
353.60	177147/131072 = 1.351524	1.3333	-4.8	1.3348	-4.4
372.52	729/512 = 1.423828	1.4063	-4.6	1.4142	-2.5
392.45	3/2 = 1.500000	1.5000	0	1.4983	-0.4
397.80	1594323/1048576 = 1.520465
419.08	6561/4096 = 1.601807	1.6000	-0.47	1.5874	-3.8
441.50	27/16 = 1.687500	1.6667	5.4	1.6818	-1.5
447.52	14348907/8388608 = 1.710523
471.47	59049/32768 = 1.802032	1.8000	-0.5	1.7818	-5.3
496.69	243/128 = 1.898438	1.8750	-6.1	1.8878	-2.8

Note that the "octave" for this scale, the eighth note of the scale, should be a fifth above one of these notes, and not the usual octave. The closest would be a frequency ratio of 2.027286, slightly larger than our normal octave. Various schemes have been introduced to try to "fix" the octave for such a scale.

Scales based at least in part on this procedure were introduced by Pythagoras (the Pythagorean Scale) and can also be found in Chinese history. The ratio 81/64 is known as the Pythagorean third, for example, which is quite high compared to many other tuning schemes.