User:Dummy index/Semitritave: Difference between revisions
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| "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]]). | | "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]]). | ||
|- | |||
| 46 | |||
| 92edt | |||
| 58edo | |||
| hyperpent | |||
| Good for micro- sensi. "5/3 ~ 34\46" ≈ 3/2, "7/5 ~ 22\46" ≈ 13/10, "6/5 ~ 12\46" ≈ 15/13, "10/7 ~ 24\46" ≈ 4/3, ... | |||
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| 69 | | 69 | ||
Revision as of 13:57, 6 March 2022
| Interval information |
Interval
Semitritave, square root of 3:1, is an interseptimal interval. It divide 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.
The following table compares selected JI semitwelfth pairs:
| Ratios | prime limit | distance from 950.9775c |
|---|---|---|
| 125/72, 216/125 | 5 | 4.054 |
| 7/4, 12/7 | 7 | 17.848 |
| 140/81, 243/140 | 7 | 3.658 |
| 512/297, 891/512 | 11 | 8.160 |
| 1331/768, 2304/1331 | 11 | 1.021 |
| 26/15, 45/26 | 13 | 1.281 |
| 85/49, 147/85 | 17 | 2.640 |
| 19/11, 33/19 | 19 | 4.782 |
Merciful intonation
Semitritave is an candidate for "practically merciful intonation", because it is [math]\displaystyle{ [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:
- [math]\displaystyle{ [1; 1, 2, 1, 1, 1, ...] }[/math] - 942.5 cents, between 12/7 and 19/11.
- [math]\displaystyle{ [1; 1, 2, 1, 2, 1, 1, 1, ...] }[/math] - 950.4 cents, between 45/26 and 71/41.
- [math]\displaystyle{ [1; 1, 2, 1, 3, 1, 1, 1, ...] }[/math] - 954.6 cents, between 26/15 and 33/19.
False octave
Semitritave is available for false octave. Differ from acoustic phi or ed7/4, two equave makes 3:1, well-known equave.
2*N-edt
Every even-numbered EDT has semitritave interval. Treating it as equave. Another preferable intervals...
- 5edt - 380 cents major third
- 6edt - 317 cents minor third
- 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 micro- meantone region. (for this purpose, 7/5 ≈ 3\5 of ed7/4 and 7/5 ≈ 4\7 of ed9/5 are both extreme...)
| N | EDT | Approx. EDO | How "pent" | Comments |
|---|---|---|---|---|
| 12 | 24edt | 15edo | hypopent | simple. "Fifth" is "7\12" ≈ 11/8, off by 3 cents. |
| 18 | 36edt | stretched-23edo | anpent | This have two "fifth," "11\18" ≈ 7/5 and "10\18" ≈ 19/14. 6/5 and 7/6 are good. |
| 19 | 38edt | 24edo | hypopent | "Fifth" is "11\19" ≈ 11/8. Can convert easily from 19edo. |
| 23 | 46edt | 29edo | anpent | Two "fifth," "14\23" ≈ 7/5, "13\23" ≈ 15/11. 13/11 and 15/13 are precise. |
| 26 | 52edt | 33edo | hypopent | Quadruple BP. Micro- flattone (4434443) can't put to use BP intervals. How is 5424542? |
| 27 | 54edt | 34edo | hyperpent | 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[12]. |
| 31 | 62edt | 39edo | hypopent | "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). |
| 46 | 92edt | 58edo | hyperpent | Good for micro- sensi. "5/3 ~ 34\46" ≈ 3/2, "7/5 ~ 22\46" ≈ 13/10, "6/5 ~ 12\46" ≈ 15/13, "10/7 ~ 24\46" ≈ 4/3, ... |
| 69 | 138edt | 87edo | amphipent | "40\69" ≈ 11/8 very precise, and coincidentally contains micro- august. (69=31+19+19=33+12+12+12) |
Rank-2 temperaments
Tribilo as micromeantone
11-limit
Subgroup: 3.2.11
Comma list: 1771561/1769472
Sval mapping: [⟨2 0 1], ⟨0 3 8]]
Sval mapping generators: ~1331/768, ~121/96
POTE generator: ~121/96 = 400.0108 (or ~11/8 = 550.9667)
Optimal GPV sequence: b14, b24, b38, b138, b176, b214, b242
Badness: 2.44 × 10-3
17-limit
Subgroup: 3.2.11.13/5.17
http://x31eq.com/cgi-bin/rt.cgi?limit=3_2_11_13%2F5_17&ets=b38_b62&tuning=po
b24 & b66 as microaugust
11-limit
Subgroup: 3.5/2.11/8
Comma list: 15625/15552
Sval mapping: [⟨6 5 2], ⟨0 0 -1]]
Sval mapping generators: ~6/5, ~288/275
POL2 generator: ~288/275 = 82.9018 (or ~11/8 = 551.083)
Optimal GPV sequence: b24, b66, b90, b114, b138, b252
RMS error: 1.252 cents
13-limit
Subgroup: 3.5/2.11/8.13/2.16/7
http://x31eq.com/cgi-bin/rt.cgi?limit=3_5%2F2_11%2F8_13%2F2_16%2F7&ets=b24_b66p&tuning=po
b38 & b54 as microsensi
Subgroup: 3.2.7.11/5.13/5
http://x31eq.com/cgi-bin/rt.cgi?limit=3_2_7_11%2F5_13%2F5&ets=b38_b54&tuning=po
Relationship
On this micromeantone, "minor tenth" (e.g. "(19+3+2)\19") ~ 2/1. Re-breaking by real octave, results in 6L 3s e. g. LsLLLsLLs, tribilo or triforce.
Memo
3.5/2.11/8 => 24edt, 3.5.7.13 => 30edt, 3.5/2.7/2 => 36edt, 3.2.11.17 => 38edt, 3.2.11/5.13/5 => 46edt, 3.10.14.13/8.34 => 52edt, 3.2.5.13.17 => 54edt, 3.10.14.17.11/8 => 62edt