User:Dummy index/Semitritave

Interval information

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{nd}[/math])	~4.22885 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

RMS error:

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

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

RMS error:

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
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

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