User:Dummy index/Semitritave
Expression | [math]\sqrt{3/1}[/math] |
Monzo | [0 1/2⟩ |
Size in cents | 950.9775¢ |
Name | semitritave |
Special properties | reduced |
Harmonic entropy (Shannon, [math]\sqrt{n\cdot d}[/math]) |
~4.56202 bits |
open this interval in xen-calc |
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][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][1; 1, 2, 1, 1, 1, ...][/math] - 942.5 cents, between 12/7 and 19/11.
- [math][1; 1, 2, 1, 2, 1, 1, 1, ...][/math] - 950.4 cents, between 45/26 and 71/41.
- [math][1; 1, 2, 1, 3, 1, 1, 1, ...][/math] - 954.6 cents, between 26/15 and 33/19.
False octave
Assuming the 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...)
(more to say, 7/5 results in "5L 3s", micro- oneirotonic.)
N | EDT | Approx. EDO | How "pent" | Comments |
---|---|---|---|---|
12 | 24edt | 15edo | hypopent | simple. "Fifth" is 7\12edst ≈ 11/8, off by 3 cents. |
18 | 36edt | stretched-23edo | anpent | This have two "fifth," 11\18edst ≈ 7/5 and 10\18edst ≈ 19/14. 6/5 and 7/6 are good. |
19 | 38edt | 24edo | hypopent | "Fifth" is 11\19edst ≈ 11/8. Can convert easily from 19edo. "minor tenth" (e.g. (19+3+2)\19edst) ~ 2/1. "Major triad" ≈ 16:19:22. |
23 | 46edt | 29edo | anpent | Two "fifth," 14\23edst ≈ 7/5, 13\23edst ≈ 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\27edst ≈ 18/13 and 15\27edst ≈ 19/14 are precise. Together with 9\27edst ≈ 6/5 and 11\27edst ≈ 5/4, seems good for micro- augene[12]. |
31 | 62edt | 39edo | hypopent | "Fifth" is 18\31edst ≈ 11/8, and "wolf fifth" is 19\31edst ≈ 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\46edst ≈ 3/2, "7/5" ~ 22\46edst ≈ 13/10, "6/5" ~ 12\46edst ≈ 15/13, "10/7" ~ 24\46edst ≈ 4/3, ... |
69 | 138edt | 87edo | amphipent | 40\69edst ≈ 11/8 very precise, and coincidentally contains micro- august. (69=31+19+19=33+12+12+12) |
Rank-2 temperaments
Tribilo as a micromeantone
- See also: Tribilo family
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 ET sequence: b14, b24, b38, b138, b176, b214, b252
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 a microaugust
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 ET sequence: b24, b66, b90, b114, b138, b252
13-limit
Subgroup: 3.5/2.16/7.11/8.13/2
http://x31eq.com/cgi-bin/rt.cgi?limit=3_5%2F2_16%2F7_11%2F8_13%2F2&ets=b24_b66p&tuning=po
b32 & b56 as a microdiminished
Subgroup: 3.16.5.11
http://x31eq.com/cgi-bin/rt.cgi?limit=3_16_5_11&ets=b32_b56&tuning=po
subgroup 3.7.11 seems to be a lot
Vulture (no-fives Buzzard)
Subgroup: 3.2.7
http://x31eq.com/cgi-bin/rt.cgi?limit=3_2_7&ets=b8_b84&tuning=po
b38 & b54 as a microsensi
Subgroup: 3.2.7.11/5
Comma list: 1605632/1594323, 495616/492075
Sval mapping: [⟨2 2 -2 -3], ⟨0 -2 15 12]]
Sval mapping generators: ~704/405, ~896/729
POL2 generator: ~896/729 = 351.4241 (or ~99/70 = 599.5534)
Optimal ET sequence: b38, b54, b92
13-limit
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
b26 & b88 as a microoneirotonic
Subgroup: 3.5.7.26
Comma list: 16875/16807, 676/675
Sval mapping: [⟨2 1 2 4], ⟨0 5 4 5]]
Sval mapping generators: ~26/15, ~26/21
POTE generator: ~26/21 = 367.0018 (or ~7/5 = 583.9757)
Optimal ET sequence: b10, b16, b26, b62, b88, b114
Badness: 1.43 × 10^{-3}
17-limit
Subgroup: 3.4.5.7.11.26.17
http://x31eq.com/cgi-bin/rt.cgi?limit=3_4_5_7_11_26_17&ets=b26_b88&tuning=po
Subgroup: 4.3.5.7.11.26.17
http://x31eq.com/cgi-bin/rt.cgi?limit=4_3_5_7_11_26_17&ets=q33r_q111&tuning=po
Related temperament: mirkat
Another periods
[math]\sqrt{3}^{\sqrt{2}} \approx \varphi^{\varphi}[/math] (off by 3 cents). However, this does not mean that acoustic phi and semitritave should be used together.
Divide or reverse divide by silver Metallic MOS:
[math]\sqrt{3}^{\sqrt{2} - 1}[/math] ≈ 394¢
951¢ => 2 * 394¢ + 1 * 163¢ => 5 * 163¢ + 2 * 68¢
1 * 951¢ + 1 * 394¢ => 3 * 394¢ + 1 * 163¢ => 3 * 231¢ + 4 * 163¢ (3L 4s (1345¢ equivalent))
2 * 951¢ + 1 * 394¢ => 5 * 394¢ + 2 * 163¢ (5L 2s (2296¢ equivalent))
231¢ is near 8/7, 163¢ is near 11/10.
http://x31eq.com/cgi-bin/rt.cgi?limit=3_8_10_7_11_19&ets=b10_b34&tuning=po
3L 4s (1345¢ equivalent)
Cents | In L's and s's | Notation | Approximate ratios^{[1]} | |
---|---|---|---|---|
unison | 0 | 0L + 0s | C | 1/1 |
neutral 2nd | 163.162 | 0L + 1s | vD | 11/10, 10/9, 21/19 |
major 2nd | 230.746 | 1L + 0s | D | 8/7, 9/8 |
neutral 3rd | 393.908 | 1L + 1s | vE | 5/4, 24/19 |
perfect 4th | 557.070 | 1L + 2s | F | 11/8 |
perfect 5th | 787.816 | 2L + 2s | G | 30/19, 11/7 |
neutral 6th | 950.978 | 2L + 3s | vA | 19/11, 33/19 |
neutral 7th | 1181.723 | 3L + 3s | vB | (2/1) |
octave | 1344.885 | 3L + 4s | C | 24/11 |
neutral 9th | 1508.047 | 3L + 5s | vD | 12/5,19/8 |
major 9th | 1575.631 | 4L + 4s | D | (5/2) |
neutral 10th | 1738.793 | 4L + 5s | vE | 30/11,19/7 |
perfect 11th | 1901.955 | 4L + 6s | F | 3/1 |
- ↑ based on treating as a 3.8.10.7.11.19 subgroup; other approaches are possible.
2s ≈ 326.324¢ ≈ (6/5),11/9
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