User:Dummy index/Semitritave: Difference between revisions
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==Interval== | ==Interval== | ||
Semitritave, square [[root]] of 3:1, is an [[interseptimal]] interval. It divide pure [[tritave]] into two equal parts. Every even-numbered [[EDT]] has this interval. | Semitritave, square [[root]] of 3:1, is an [[interseptimal]] interval. It divide pure [[tritave]] into two equal parts. Every even-numbered [[EDT]] has this interval. It is strongly related to island comma, [[676/675]], via 13-limit approximant 26/15 and 45/26. | ||
==Merciful intonation== | ==Merciful intonation== | ||
Semitritave is an candidate for "practically [[merciful intonation]]", because it is <math>[1; 1, 2, 1, 2, ...]</math> in continued fraction, have many gradually proximal ratios | Semitritave is an candidate for "practically [[merciful intonation]]", because it is <math>[1; 1, 2, 1, 2, ...]</math> in continued fraction, have many gradually proximal ratios, 7/4, 19/11, 26/15, 71/41, ..., makes rich dissonance. | ||
Approximating it by noble number: | Approximating it by noble number: | ||
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* so 30edt? | * so 30edt? | ||
To do mechanical translation from diatonic scores, "fifth" sound is preferred to be consonance. 7/5 is better, but it makes [[3L 2s]]. 11/8 corresponds to meantone region. (for this purpose, [[ed9/5]] | To do mechanical translation from diatonic scores, "fifth" sound is preferred to be consonance. 7/5 is better, but it makes [[3L 2s]]. 11/8 corresponds to meantone region. (for this purpose, 7/5 ≈ 3\5 of ed7/4 and 7/5 ≈ 4\7 of [[ed9/5]] are both extreme...) | ||
* 24edt - simple. "Fifth" is "7\12" ≈ 11/8, off by 3 cents. | * 24edt - simple. "Fifth" is "7\12" ≈ 11/8, off by 3 cents. | ||
* | * 36edt - approximately [[23edo and octave stretching|stretched-23edo]]. This have two "fifth," "11\18" ≈ 7/5 and "10\18" ≈ 19/14. 6/5 and 7/6 are good. | ||
* 38edt - approximately 24edo. "Fifth" is "11\19" ≈ 11/8. Can convert easily from 19edo. | * 38edt - approximately 24edo. "Fifth" is "11\19" ≈ 11/8. Can convert easily from 19edo. | ||
* | * 46edt - approximately 29edo. Two "fifth," "14\23" ≈ 7/5, "13\23" ≈ 15/11. 13/11 and 15/13 are precise. | ||
* 54edt - approximately 34edo. Two "fifth," "16\27" ≈ 18/13 and "15\27" ≈ 19/14 are precise. Together with "9\27" ≈ 6/5 and "11\27" ≈ 5/4, seems good for micro- augene<nowiki>[12]</nowiki>. | * 54edt - approximately 34edo. Two "fifth," "16\27" ≈ 18/13 and "15\27" ≈ 19/14 are precise. Together with "9\27" ≈ 6/5 and "11\27" ≈ 5/4, seems good for micro- augene<nowiki>[12]</nowiki>. | ||
* 62edt - approximately 39edo. "Fifth" is "18\31" ≈ 11/8, and "wolf fifth" is "19\31" ≈ 7/5. | * 62edt - approximately 39edo. "Fifth" is "18\31" ≈ 11/8, and "wolf fifth" is "19\31" ≈ 7/5. By the way, "upmajor 3rd" and "downminor 3rd" approximate 17/14 and 17/15, where (17/14)*(17/15) ≈ (11/8)*([[1156/1155]]). | ||
==Memo== | ==Memo== | ||
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[http://x31eq.com/cgi-bin/pregular.cgi?limit=3.2.11.17&error=5.0 3.2.11.17 => 38edt], | [http://x31eq.com/cgi-bin/pregular.cgi?limit=3.2.11.17&error=5.0 3.2.11.17 => 38edt], | ||
[http://x31eq.com/cgi-bin/pregular.cgi?limit=3.2.11%2F5.13%2F5&error=5.0 3.2.11/5.13/5 => 46edt], | [http://x31eq.com/cgi-bin/pregular.cgi?limit=3.2.11%2F5.13%2F5&error=5.0 3.2.11/5.13/5 => 46edt], | ||
[http://x31eq.com/cgi-bin/pregular.cgi?limit=3.2.5. | [http://x31eq.com/cgi-bin/pregular.cgi?limit=3.2.5.13.17&error=5.0 3.2.5.13.17 => 54edt], | ||
[http://x31eq.com/cgi-bin/pregular.cgi?limit=3.10.14.17. | [http://x31eq.com/cgi-bin/pregular.cgi?limit=3.10.14.17.11%2F8&error=5.0 3.10.14.17.11/8 => 62edt] | ||