# What if an octave was three times the frequency of the root?

In Western music, probably others, a root note, say middle C, and the octave above are differ by an order of two in regards to frequency of the note.

When played together, they sound completely harmonious, if a little boring. And there are 11 notes between them, each jumping up about 2^1/12 between them, with some corrections made for Western music that I don't care about here.

What if instead of an octave frequency differing by a factor of 2, it differed by a factor of 3?

Part of what makes a factor of 2 in 12 semitones work is that 2^(7/12) ~ 3/2 and 2^(5/12) ~ 4/3 for the fifth and fourth respectively. Would a different number of intervals work for an 3x octave? Has anyone tried this and did it work or are our ears too attuned to the Western scales?

• What is the motivation of this question? Why the number 3, rather than 5, 7 or π? You seem to neither know how it would sound, nor know how to calculate the interval sizes... perhaps asking about how to do it would be a better direction for exploration? Aug 11, 2023 at 18:09
• No, 2 is not just a status quo. For most people, sounds an octave apart sound very similar, which is perhaps due to the inner ear structure, which gives the factor of 2 a very special place. Factor of 3, while consonant, doesn't have the same property. With various equal divisions of 3 you will get some intervals which sound ok (in Western standards), and some which sound bad. I expect the same for mostly any number. Aug 12, 2023 at 2:39
• You say "in Western standards", and that's a key point right there. What sounds similar or consonant is at least in part a cultural phenomenon. In other words, very much the status quo. Aug 12, 2023 at 9:25
• FYI: Anybody who wants to hear a 3:1 interval, Just play a "C" on your piano, and then play the "G" an octave and a half higher (or, play any note, and then play whatever note is 19 semi-tones higher.) It's not exactly 3:1 (thanks, J.S. Bach!) but it's really close: Approx 2.997:1. Aug 12, 2023 at 17:37
• I suggest dividing a "tritave" into 19 equal parts. Aug 12, 2023 at 21:48