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{{Infobox Interval
{{Infobox Interval
| Icon =
| Ratio = \sqrt{3/1}
| Ratio = (3/1)^(1/2)
| Monzo = 0 1/2
| Monzo = 0 1/2
| Cents = 950.97750
| Cents = 950.97750
| Name = semitritave
| Name = semitritave
| Calc = sqrt(3/1)
}}
}}
==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 [[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:
 
{| class="wikitable"
! | 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==
==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, 5/3, 7/4, 19/11, 26/15, 71/41, ..., makes rich dissonance.
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:
Line 18: Line 58:


==False octave==
==False octave==
Semitritave is available for [[Ed5/3 through ed7/3|false octave]]. Differ from [[acoustic phi]] or [[ed7/4]], two equave makes [[3/1|3:1]], well-known equave.
Assuming the semitritave is available for [[Ed5/3 through ed7/3|false octave]]. Differ from [[acoustic phi]] or [[ed7/4]], two equave makes 3:1, well-known equave.


==2*N-edt==
==2*N-edt==
Every even-numbered [[EDT]] has semitritave interval. Treating it as equave. Another preferrable intervals...
Every even-numbered EDT has semitritave interval. Treating it as equave. Another preferable intervals...


* 5edt - 380 cents major third
* 5edt - 380 cents major third
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* so 30edt?
* so 30edt?


To do mechanical translation from diatonic scores, "fifth" sound want to be consonance. 7/5 is better, but it makes [[3L 2s]]. 11/8 corresponds meantone region. (for this purpose, [[ed9/5]] is good answer.)
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.)
 
{| class="wikitable"
! | N
! | EDT
! | Approx. EDO
! | How "pent"
! | Comments
|-
| 12
| 24edt
| 15edo
| hypopent
| simple. "Fifth" is 7\12edst ≈ 11/8, off by 3 cents.
|-
| 18
| 36edt
| [[23edo and octave stretching|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<nowiki>[12]</nowiki>.
|-
| 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]]: [{{val| 2 0 1 }}, {{val| 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<sup>-3</sup>
 
====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]]: [{{val| 6 5 2 }}, {{val| 0 0 -1 }}]
 
Sval mapping generators: ~6/5, ~288/275
 
[[Tp tuning|POL2 generator]]: ~288/275 = 82.9018 (or ~11/8 = 551.083)
 
[[Optimal ET sequence]]: b24, b66, b90, b114, b138, b252
 
[[Tp tuning #T2 tuning|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
===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]]: [{{val| 2 2 -2 -3 }}, {{val| 0 -2 15 12 }}]
 
Sval mapping generators: ~704/405, ~896/729
 
[[Tp tuning|POL2 generator]]: ~896/729 = 351.4241 (or ~99/70 = 599.5534)
 
[[Optimal ET sequence]]: b38, b54, b92
 
[[Tp tuning #T2 tuning|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]]: [{{val| 2 1 2 4 }}, {{val| 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<sup>-3</sup>
 
====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]]:<br />
<math>\sqrt{3}^{\sqrt{2} - 1}</math> ≈ 394¢<br />
951¢ => 2 * 394¢ + 1 * 163¢ => 5 * 163¢ + 2 * 68¢<br />
1 * 951¢ + 1 * 394¢ => 3 * 394¢ + 1 * 163¢ => 3 * 231¢ + 4 * 163¢ (3L 4s (1345¢ equivalent))<br />
2 * 951¢ + 1 * 394¢ => 5 * 394¢ + 2 * 163¢ (5L 2s (2296¢ equivalent))<br />
231¢ is near 8/7, 163¢ is near 11/10.<br />
http://x31eq.com/cgi-bin/rt.cgi?limit=3_8_10_7_11_19&ets=b10_b34&tuning=po
 
===== 3L 4s (1345¢ equivalent) =====
{| class="wikitable"
!
!Cents
!In L's and s's
!Notation
!Approximate ratios<ref>based on treating as a 3.8.10.7.11.19 subgroup; other approaches are possible.</ref>
|-
|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
|}
<references/>
 
2s ≈ 326.324¢ ≈ (6/5),11/9


* 46edt - approximately 29edo. This have two "fifth," 14\23 ≈ 7/5, 13\23 ≈ 15/11.
==Memo==
* 38edt - approximately 24edo. "Fifth" is 11\19 ≈ 11/8. Can convert easily from 19edo.
[http://x31eq.com/cgi-bin/pregular.cgi?limit=3.5%2F2.11%2F8&error=5.0 3.5/2.11/8 => 24edt],
* 36edt - approximately [[23edo and octave stretching|stretched 23edo]]. Another candidate with "fifth" 7/5...
[http://x31eq.com/cgi-bin/pregular.cgi?limit=3.5.7.13&error=5.0 3.5.7.13 => 30edt],
[http://x31eq.com/cgi-bin/pregular.cgi?limit=3.5%2F2.7%2F2&error=5.0 3.5/2.7/2 => 36edt],
[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=3.10.14.13%2F8.34&error=5.0 3.10.14.13/8.34 => 52edt],
[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=3.10.14.17.11%2F8&error=5.0 3.10.14.17.11/8 => 62edt]
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