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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?

Leonhard Euler
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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? – user1079505 Aug 11 '23 at 18:09
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    pi wouldn't sound good, I don't think. Has to be a whole number and 2 is just the status quo, so 3 is the next best thing. No, please answer the question as stated. – Leonhard Euler Aug 11 '23 at 23:04
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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. – user1079505 Aug 12 '23 at 02:39
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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. Aug 12 '23 at 09:25
  • Also: Wikipedia - Category:Non–octave-repeating scales – Andrew T. Aug 12 '23 at 09:31
  • @user1079505 Do you have any references about "the inner ear structure" relating to the factor of 2 octave? The second part of your comment is the same as my question. – Leonhard Euler Aug 12 '23 at 16:51
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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. – Solomon Slow Aug 12 '23 at 17:37
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    I suggest dividing a "tritave" into 19 equal parts. – mathlander Aug 12 '23 at 21:48
  • See this answer, and the link there https://music.stackexchange.com/a/44792/63781 – user1079505 Aug 13 '23 at 04:31
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    @SolomonSlow Bach has little to do with the ratio between C and G being different from 3:1. In fact, many of the circulating temperaments used in Bach's day did have several justly tuned fifths, in contrast to earlier meantone temperaments where most of the twelfths would have been rather smaller -- in some cases as small as 2.986:1 (a quarter-comma tempered twelfth is 2.991:1). Some circulating temperaments even had wide fifths; for example the fifth from A flat to E flat in Werckmeister IV 3.006:1 -- https://en.wikipedia.org/wiki/Werckmeister_temperament – phoog Aug 13 '23 at 07:01
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    @mathlander interesting. I've thought of dividing a fifth into seven equal parts, but your approach yields somewhat smaller parts. Maybe we should divide 768:5 into 87 equal parts. – phoog Aug 13 '23 at 09:18
  • @RobertM. I don't say "in Western standards". – Leonhard Euler Aug 17 '23 at 15:45
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    @LeonhardEuler True, you said in Western music. Your nitpicking skills are amazing. – Robert M. Aug 19 '23 at 03:34

1 Answers1

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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.

Jeff Zeitlin
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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. Aug 12 '23 at 09:23
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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 Aug 12 '23 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. – Michael Seifert Aug 12 '23 at 13:09
  • The actual scale: https://upload.wikimedia.org/wikipedia/commons/8/82/BP_Just_Lambda_Scale.ogg – OrangeDog Aug 12 '23 at 17:39
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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 Aug 12 '23 at 21:52
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    @Basj https://www.youtube.com/watch?v=60SYLdMYvcE or http://ziaspace.com/ZIA/mp3s/StickMen.mp3 – James K Aug 13 '23 at 11:14
  • @JamesK Nive indeed! Question: would https://www.youtube.com/watch?v=60SYLdMYvcE be playable with a normal 12-semitone tempered scale? It doesn't seem so far from standard chords, and the voice seems not far from a standard scale. – Basj Aug 13 '23 at 14:47
  • I'm pretty sure you need either specialist instruments (the modified keytar she plays) or quite a bit of "unlearning" where to finger the violin. An instrument, like a violin, that can in principle make any pitch could be played. An instrument like a piano couldn't. Nor could a guitar, as the semitone interval is "hard-wired" into the spacing of the frets. – James K Aug 13 '23 at 14:53
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    Years ago, I remapped a midi-capable piano so that semitones became half-semitones, everything centered at middle C. Sure, by that point I'd been heavily conditioned... but I could not figure out any "new" music that I could tolerate listening-to! :) – paul garrett Aug 13 '23 at 17:49
  • @Lazy Octave is such a terrible name for the doubled fundamental, referring to eight steps of different sizes, that there seems little point in trying to salvage it. – Russell Borogove Aug 13 '23 at 18:30
  • @RussellBorogove Look up Gurdjieff's idea of the enneagram. You can safely ignore the silly "enneagram of personality" stuff. – Robert M. Aug 19 '23 at 03:33
  • @Basj Also https://www.youtube.com/watch?v=ImT4XMVr-5s. But you can also search for Bohlen-Pierce or Elaine Walker yourself. ;) – Robert M. Aug 19 '23 at 03:35
  • @RobertM. Yes sure, I already did, and listened to a few songs, but it's always interesting to have other people's recommandations ;) – Basj Aug 19 '23 at 19:29