# 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 fi, then
fi+1 = (3/2) fi if that result is less than 2 f0
fi+1 = (3/4) fi 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 1.00000 1.00000
265.20531441/524288 = 1.013643
279.392187/2048 = 1.067871 1.0417-6.9 1.0595-2.2
294.339/8 = 1.125000 1.12500 1.1225-0.7
298.354782969/4194304 = 1.140349
314.3119683/16384 = 1.201355 1.2000-0.4 1.1892-3.2
331.1381/64 = 1.265625 1.2500-4.1 1.2599-1.5
353.60177147/131072 = 1.351524 1.3333-4.8 1.3348-4.4
372.52729/512 = 1.423828 1.4063-4.6 1.4142-2.5
392.453/2 = 1.500000 1.50000 1.4983-0.4
397.801594323/1048576 = 1.520465
419.086561/4096 = 1.601807 1.6000-0.47 1.5874-3.8
441.5027/16 = 1.687500 1.66675.4 1.6818-1.5
447.5214348907/8388608 = 1.710523
471.4759049/32768 = 1.802032 1.8000-0.5 1.7818-5.3
496.69243/128 = 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.

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