User:PrySigneToFexia/Quasi-12edo

Quasi-12edo (or just-intoned well tuning-system) is a concept that has existed for a long time. It involves achieving harmonious and easily transposable intervals by approximating just intervals as closely as possible with the equal-tempered intervals divided into hundreds of steps.

Throughout history, many people have proposed the concept of Quasi-12edo. Here, I will introduce my recently conceived Quasi-12edo—an 'excellent' tuning system.

Content

My 'excellent' musical tuning borrows BMTC’s just-intoned well tuning-system. It originates from 8-dimensional and 7-dimensional(19 and 17) pure tunings, but I think it’s possible to get even closer by methods that directly extract the square root of 2 as the boundary for interval inversion (even though that thing sounds terrible, and is even considered the 'Devil’s interval').

In order to achieve an effect closer to the 12-tone equal temperament, I used other dimensions.

We use the C of the twelve-tone equal temperament as the reference, and every modulation is done in whole cents, but all intervals within the key are based on just intonation, effectively masking the dissonance of the twelve-tone equal temperament (even though this can create some very unpleasant pitch ','[sic]-s).

So, now we have this thing:

1. C: +0c
2. C#/Db: +4.955410c
3. D: +3.910002c
4. D#/Eb: -2.486984c
5. E: +0.108480c
6. F: -1.955001c
7. F#/Gb: +0c
8. G: +1.955001c
9. G#/Ab: +0.108480c
10. A: +2.486984c
11. A#/Bb: -3.910002c
12. B: -4.955410c

Extension to more equal temperaments

We can also extend my 'excellent' tuning system(hereinafter shortened as ETS) to more edos and even maybe ed3s, ed5s, and so on.

First, we need to determine how many notes we want to divide an octave (or other prime harmonics) into. If the number of notes we want to divide is even, then we must set √2 ≈ 1.41421356 as the dividing line.

For example, for 24-tone equal temperament, we have the following method for ETS:

1. C: +0c
2. C: +3.272943c
3. C#/Db: +4.955410c
4. D: +0.637059c
5. D: +3.910002c
6. D: +2.268038c
7. D#/Eb: -2.486984c
8. E: +0.616902c
9. E: +0.108480c
10. E/F: -0.725382c
11. F: -1.955001c
12. F: +1.317942c
13. F#/Gb: +0c
14. G/F: -1.317942c
15. G: +1.955001c
16. G: +0.725382c
17. G#/Ab: +0.108480c
18. A: -0.616902c
19. A: +2.486984c
20. A/B: -2.268038c
21. A#/Bb: -3.910002c
22. B: -0.637059c
23. B: -4.955410c
24. C: -3.272943c

Really 'harmonic', isn't it?

Examples

Prorogato

https://www.bilibili.com/video/BV1Qt4y1n751