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I am trying to understand how to construct the pythagorean scale. I have gathered, that the basic idea is to use only the 3/2 ratio starting from some base frequency. As this takes you out of the octave quickly, using the 1/2 ratio (=one octave) you move back into the original octave. The video

explains this quite well. However for the last step he suddly starts from the base frequency with a 4/3 ratio and calls this "the oddball one". Using the same approach as all notes before and starting from the previous note would take you (in this example) to 372.525 Hz. This is not the same but within the octave so i do not understand why the approach is suddenly different for the "last" note (This is another question: where to stop).

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    Maybe the wiki article on this, which is pretty complete could also help? – Tom Feb 10 at 8:53
  • My problem with the wiki article is twofold: it starts out with already naming notes ("A is tuned such that its frequency equals 3/2 times the frequency of D") while my understanding is that Pytagoras had no knowlege of note names so would not be able to say "ok, now i have all 12 notes and i am done". Also, the only part where the article actually talks about scale construction using frequencies, they also "suddenly" change the approach without explaination just suddenly "Starting from the same point working the other way" – Joe Feb 10 at 10:45
  • This tuning tries to stay as tuned as possible compared to the base note. Starting from there, you can see it this way: pick one step up, one step down, and so on. There is no sudden change, you go back and forth above and under the base note (3/2 then (2/3)^-1 and scale back to one octave). At some point, you get two frequencies that are very close (G# and Ab on the wiki) which is a good point to stop. The first time such a close set happens is after 12 iterations. You can obviously go higher, that's why they are usually denoted pythagorean scaleS. – Tom Feb 10 at 12:41
  • The fact that, in the video he is going several steps one way, before getting the other way is a bit confusing, and I agree sounds a bit "out of the hat". The good way of constructing is to alternate. I hope I did clarify things a bit… If you want I can make it an answer… – Tom Feb 10 at 12:44
  • The answers firm here : music.stackexchange.com/questions/110699/… might also be of help! – Tom Feb 10 at 12:53
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The combination of Information from the comments made it much clearer. I actually used this method to calculate the resulting frequencies starting with 440Hz and you can see quite nicely how two frequencies (618.05Hz and 626.48Hz) are really close to each other and "mess up" the otherwise almost equal distribution.

Plot of Pythagorean scale

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  • Neat! I guess the key point was to see that you need to alternate up and down around the base note. – Tom Feb 10 at 17:01
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This approach is suddenly different for the last note, because the last note, the subdominant of the scale, does not exist in the harmonic series of the tonic. 4/3 is not a harmonic of 1. You could also construct the scale only using 3/2 ratios (and bringing down into one octave) by starting on the subdominant (say, F in a C major scale) and going up from there. This is because the subdominant is the "flattest" note in a major scale.

And do check out the wiki that Tom suggested.

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