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It is ever possible to have an equal temperament without using the pure octave. In this case we can produce equal temperament where an universal half tone is used. It will permit to stay nearest possible to the just intonation and in the same time to modulate in all tonalities. This temperament exist and is named TEDJ. French name where TemperamentEquale with DifferencialeJust. Who know more about this temperament?

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    I don't know about TEDj (nor can I find much online), but why would you want a nearly just tuning system that doesn't match octaves? (I'm not saying it's a bad question, I'm legitimately curious...it seems to defeat the purpose!) – Richard Sep 3 '17 at 19:33
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    @Richard It's a sensible idea on piano, and any other instrument with "stretched" tuning because of unavoidable inharmonicity which makes the harmonics "sharp" compared with the fundamental. Somebody came up with a piano tuning system based on just intonation 12ths (slightly wider than conventional ET 12ths) and wide octaves - sorry, I can't remember what it was called, but I'm pretty sure it wasn't "TEDj". – user19146 Sep 3 '17 at 19:57
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    Stretch tuning is designed to give the illusion of pure octaves. – Laurence Payne Sep 3 '17 at 20:20
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    Ah - I found the reference: Bernhard Stopper proposed it. And this is what it sounds like: youtube.com/watch?v=5BrcWplvGJY. Stopper's own website seems to be down right now, but Google will find it for you. – user19146 Sep 3 '17 at 20:36
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    @LaurencePayne the usual definition of "stretch tuning" is an empirical system - it's different for every piano, and different over different parts of the pitch range. Stopper's tuning system is mathematically consistent, and it's designed to give the illusion of pure diatonic chords, not just octaves. – user19146 Sep 3 '17 at 20:37
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31 tone et gets very close to providing all of the just intervals that can be produced from voices and fretless instruments. But it is unwieldy to play a keyboard with 31 keys to the octave. I would suggest getting a piano tuned to well temperament. While some of the keys contain wildly sharp Pythagorean thirds they at least want to go outward and suggest movement to the fifth versus the sickly sounding thirds of et. Also, keys have different feeling in well temperament due to the different intonation they possess. Of course this can be grouped between closely related keys. Also c major and it's closely related keys contain practically perfect triads that sound sweet and glorious. I think overall the temperament that satisfied Bach's disappointment with mean tone is the best solution for keyboard instruments.

  • It's also possible to use a subset of 31-edo as a kind of well temperament. That gives all the nice intervals within one key without impeding playability, but also allows modulations without wolves quite easily by picking a different subset. – leftaroundabout Sep 30 '17 at 21:14

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