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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==
Line 58: 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, 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==
Line 67: Line 67:
* 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 micro- meantone region. (for this purpose, 7/5 ≈ 3\5 of ed7/4 and 7/5 ≈ 4\7 of [[ed9/5]] are both extreme...)
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"
{| class="wikitable"
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| 15edo
| 15edo
| hypopent
| hypopent
| simple. "Fifth" is "7\12" ≈ 11/8, off by 3 cents.
| simple. "Fifth" is 7\12edst ≈ 11/8, off by 3 cents.
|-
|-
| 18
| 18
Line 86: Line 88:
| [[23edo and octave stretching|stretched-23edo]]
| [[23edo and octave stretching|stretched-23edo]]
| anpent
| anpent
| This have two "fifth," "11\18" ≈ 7/5 and "10\18" ≈ 19/14. 6/5 and 7/6 are good.
| This have two "fifth," 11\18edst ≈ 7/5 and 10\18edst ≈ 19/14. 6/5 and 7/6 are good.
|-
|-
| 19
| 19
Line 92: Line 94:
| 24edo
| 24edo
| hypopent
| hypopent
| "Fifth" is "11\19" ≈ 11/8. Can convert easily from 19edo.
| "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
| 23
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| 29edo
| 29edo
| anpent
| anpent
| Two "fifth," "14\23" ≈ 7/5, "13\23" ≈ 15/11. 13/11 and 15/13 are precise.
| Two "fifth," 14\23edst ≈ 7/5, 13\23edst ≈ 15/11. 13/11 and 15/13 are precise.
|-
|-
| 26
| 26
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| 34edo
| 34edo
| hyperpent
| 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<nowiki>[12]</nowiki>.
| 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
| 31
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| 39edo
| 39edo
| hypopent
| 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]]).
| "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
| 69
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| 87edo
| 87edo
| amphipent
| amphipent
| "40\69" ≈ 11/8 very precise, and coincidentally contains micro- august. (69=31+19+19=33+12+12+12)
| 40\69edst ≈ 11/8 very precise, and coincidentally contains micro- august. (69=31+19+19=33+12+12+12)
|}
|}


==Rank-2 temperaments==
==Rank-2 temperaments==


===Tribilo as micromeantone===
===Tribilo as a micromeantone===
{{See also|Tribilo family}}
{{See also|Tribilo family}}
====11-limit====
 
Subgroup: 3.2.11
Subgroup: 3.2.11


Line 140: Line 148:
[[POTE generator]]: ~121/96 = 400.0108 (or ~11/8 = 550.9667)
[[POTE generator]]: ~121/96 = 400.0108 (or ~11/8 = 550.9667)


Optimal GPV sequence: b14, b24, b38, b138, b176, b214, b242
[[Optimal ET sequence]]: b14, b24, b38, b138, b176, b214, b252


[[Badness]]: 2.44 × 10<sup>-3</sup>
[[Badness]]: 2.44 × 10<sup>-3</sup>
Line 148: Line 156:


http://x31eq.com/cgi-bin/rt.cgi?limit=3_2_11_13%2F5_17&ets=b38_b62&tuning=po
http://x31eq.com/cgi-bin/rt.cgi?limit=3_2_11_13%2F5_17&ets=b38_b62&tuning=po
===b24 & b66 as microaugust===
===b24 & b66 as a microaugust===
====11-limit====
 
Subgroup: 3.5/2.11/8
Subgroup: 3.5/2.11/8


Line 160: Line 168:
[[Tp tuning|POL2 generator]]: ~288/275 = 82.9018 (or ~11/8 = 551.083)
[[Tp tuning|POL2 generator]]: ~288/275 = 82.9018 (or ~11/8 = 551.083)


Optimal GPV sequence: b24, b66, b90, b114, b138, b252
[[Optimal ET sequence]]: b24, b66, b90, b114, b138, b252


[[Tp tuning #T2 tuning|RMS error]]: 1.252 cents
[[Tp tuning #T2 tuning|RMS error]]:  


====13-limit====
====13-limit====
Subgroup: 3.5/2.11/8.13/2.16/7
Subgroup: 3.5/2.16/7.11/8.13/2


http://x31eq.com/cgi-bin/rt.cgi?limit=3_5%2F2_11%2F8_13%2F2_16%2F7&ets=b24_b66p&tuning=po
http://x31eq.com/cgi-bin/rt.cgi?limit=3_5%2F2_16%2F7_11%2F8_13%2F2&ets=b24_b66p&tuning=po
===b38 & b54 as microsensi===
===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
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
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/>


==Relationship==
2s ≈ 326.324¢ ≈ (6/5),11/9
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==
==Memo==
Retrieved from "https://en.xen.wiki/w/User:Dummy_index/Semitritave"