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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?

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    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? Commented Aug 11, 2023 at 18:09
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    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. Commented Aug 12, 2023 at 2:39
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    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.
    – Robert M.
    Commented Aug 12, 2023 at 9:25
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    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. Commented Aug 12, 2023 at 17:37
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    I suggest dividing a "tritave" into 19 equal parts.
    – mathlander
    Commented Aug 12, 2023 at 21:48

1 Answer 1

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The tripling of the root frequency being considered as the unison is the basis of the Bohlen-Pierce scale. This scale in its equal-tempered chromatic form has thirteen 'semitones' of approximately 146.3 cents between the root and the 'tritave' (semantic equivalent of the octave), and in its diatonic form has ten notes (inclusive) between the root and the tritave. It has 'major' and 'minor' chords, and many of the concepts from standard music theory can be applied to the B-P scale analogously.

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    I had never heard of this, so I went looking and I've been listening to some examples, especially Elaine Walker's stuff. Holy...it's amazing.
    – Robert M.
    Commented Aug 12, 2023 at 9:23
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    The semantic equivalent of octave (eighth step from latin octava) would not be tritave (which is even linguistically wrong, the third step would be the tertia). If it is made up of 10 diatonic steps (inclusive) it would be the decime.
    – Lazy
    Commented Aug 12, 2023 at 11:27
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    To address the "numerical coincidences" in last part of the OP's question: the Bohlen-Pierce scale exploits the fact that 3^(3/13) ≈ 9/7, 3^(4/13) ≈ 7/5, and 3^(6/13) ≈ 5/3. Commented Aug 12, 2023 at 13:09
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    @RobertM. Would you have an example of song (Youtube or other) by Elaine Walker which is representative of this style? I'd love to discover it!
    – Basj
    Commented Aug 12, 2023 at 21:52
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    @Basj youtube.com/watch?v=60SYLdMYvcE or ziaspace.com/ZIA/mp3s/StickMen.mp3
    – James K
    Commented Aug 13, 2023 at 11:14

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