Quintosec family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
→Decoid: +reduced generators
+CTE tunings. Review optimal ET sequences.
Line 2: Line 2:


== Quintosec ==
== Quintosec ==
Quintosec is naturally a 2.3.5.11 subgroup temperament. It was documented as ''qintosec'' on the temperament finder, which was probably due to a typo<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>. The original, intended spelling is now restored, and the distinctive no-''u'' form is relegated to the lower-accuracy 7-limit extension considered below.  
Quintosec is naturally a 2.3.5.11-subgroup temperament. It was documented as ''qintosec'' on the temperament finder, which was probably due to a typo<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>. The original, intended spelling is now restored, and the distinctive no-''u'' form is relegated to the lower-accuracy 7-limit extension considered below.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 27: Line 27:
Mapping: {{mapping| 5 1 22 45 | 0 2 -3 -8 }}
Mapping: {{mapping| 5 1 22 45 | 0 2 -3 -8 }}


Optimal tuning (CTE): ~1024/891 = 1\5, ~160/99 = 831.0932 (~16/15 = 111.0932)
Optimal tunings:
* CTE: ~1024/891 = 1\5, ~160/99 = 831.0932 (~16/15 = 111.0932)
* CWE: ~1024/891 = 1\5, ~160/99 = 831.0743 (~16/15 = 111.0743)


{{Optimal ET sequence|legend=1| 10e, 55e, 65, 205, 270, 335, 605, 940, 1545c }}
Optimal ET sequence: {{Optimal ET sequence| 10e, 55e, 65, 205, 270, 335, 605, 940, 1545c }}


Badness: 0.0195
Badness: 0.0195
Line 44: Line 46:
{{Multival|legend=1| 10 -15 -10 -47 -44 19 }}
{{Multival|legend=1| 10 -15 -10 -47 -44 19 }}


Optimal tuning (POTE): ~400/343 = 1\5, ~80/49 = 831.553 (~15/14 = 111.553)
Optimal tunings:
* CTE: ~400/343 = 1\5, ~80/49 = 831.7095 (~15/14 = 111.7095)
* POTE: ~400/343 = 1\5, ~80/49 = 831.553 (~15/14 = 111.553)


{{Optimal ET sequence|legend=1| 10, 55d, 65d, 75, 140d, 215dd }}
{{Optimal ET sequence|legend=1| 10, …, 55d, 65d, 75, 140d }}


[[Badness]]: 0.156764
[[Badness]]: 0.156764
Line 57: Line 61:
Mapping: {{mapping| 5 1 22 21 0 | 0 2 -3 -2 5 }}
Mapping: {{mapping| 5 1 22 21 0 | 0 2 -3 -2 5 }}


Optimal tuning (POTE): ~55/48 = 1\5, ~80/49 = 831.311 (~15/14 = 111.311)
Optimal tunings:
* CTE: ~55/48 = 1\5, ~80/49 = 831.1939 (~15/14 = 111.1939)
* POTE: ~55/48 = 1\5, ~80/49 = 831.311 (~15/14 = 111.311)


{{Optimal ET sequence|legend=1| 10, 55d, 65d, 140de, 205dde }}
Optimal ET sequence: {{Optimal ET sequence| 10, 55d, 65d, 140de }}


Badness: 0.077907
Badness: 0.077907
Line 70: Line 76:
Mapping: {{mapping| 5 1 22 21 -7 | 0 2 -3 -2 7 }}
Mapping: {{mapping| 5 1 22 21 -7 | 0 2 -3 -2 7 }}


Optimal tuning (POTE): ~220/189 = 1\5, ~44/27 = 832.448 (~15/14 = 112.448)
Optimal tunings:
* CTE: ~220/189 = 1\5, ~44/27 = 832.4051 (~15/14 = 112.4051)
* POTE: ~220/189 = 1\5, ~44/27 = 832.448 (~15/14 = 112.448)


{{Optimal ET sequence|legend=1| 10, 75, 160cde }}
Optimal ET sequence: {{Optimal ET sequence| 10, 75 }}


Badness: 0.127732
Badness: 0.127732
Line 87: Line 95:
{{Multival|legend=1| 10 -15 0 -47 -28 42 }}
{{Multival|legend=1| 10 -15 0 -47 -28 42 }}


Optimal tuning (POTE): ~8/7 = 1\5, ~45/28 = 831.113 (~16/15 = 111.113)
[[Optimal tuning]]:
* [[CTE]]: ~8/7 = 1\5, ~45/28 = 831.1061 (~16/15 = 111.1061)
* [[POTE]]: ~8/7 = 1\5, ~45/28 = 831.113 (~16/15 = 111.113)


{{Optimal ET sequence|legend=1| 10, 55, 65, 140dd, 205ddd, 270ddd }}
{{Optimal ET sequence|legend=1| 10, …, 55, 65, 140dd }}


[[Badness]]: 0.156259
[[Badness]]: 0.156259
Line 100: Line 110:
Mapping: {{mapping| 5 1 22 14 0 | 0 2 -3 0 5 }}
Mapping: {{mapping| 5 1 22 14 0 | 0 2 -3 0 5 }}


Optimal tuning (POTE): ~8/7 = 1\5, ~45/28 = 830.659 (~16/15 = 110.659)
Optimal tunings:
* CTE: ~8/7 = 1\5, ~45/28 = 830.7772 (~16/15 = 110.7772)
* POTE: ~8/7 = 1\5, ~45/28 = 830.659 (~16/15 = 110.659)


{{Optimal ET sequence|legend=1| 10, 55, 65 }}
Optimal ET sequence: {{Optimal ET sequence| 10, 55, 65 }}


Badness: 0.081607
Badness: 0.081607
Line 113: Line 125:
Mapping: {{mapping| 5 1 22 14 0 22 | 0 2 -3 0 5 -1 }}
Mapping: {{mapping| 5 1 22 14 0 22 | 0 2 -3 0 5 -1 }}


POTE generator: ~45/28 = 830.910 (~16/15 = 110.910)
Optimal tunings:
* CTE: ~8/7 = 1\5, ~45/28 = 830.8942 (~16/15 = 110.8942)
* POTE: ~8/7 = 1\5, ~45/28 = 830.910 (~16/15 = 110.910)


{{Optimal ET sequence|legend=1| 10, 55, 65 }}
Optimal ET sequence: {{Optimal ET sequence| 10, 55, 65 }}


Badness: 0.056390
Badness: 0.056390
Line 132: Line 146:
{{Multival|legend=1| 20 -30 -10 -94 -72 61 }}
{{Multival|legend=1| 20 -30 -10 -94 -72 61 }}


[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\10, ~8192/4725 = 951.099 (~16/15 = 111.099, or ~225/224 = 8.901)
[[Optimal tuning]]s:
* [[CTE]]: ~15/14 = 1\10, ~8192/4725 = 951.1086 (~16/15 = 111.1086, or ~225/224 = 8.8914)
* [[POTE]]: ~15/14 = 1\10, ~8192/4725 = 951.099 (~16/15 = 111.099, or ~225/224 = 8.901)


{{Optimal ET sequence|legend=1| 10, 120, 130, 270, 2020c, 2290c, 2560c, 2830bc, 3100bcc, 3370bcc, 3640bcc }}
{{Optimal ET sequence|legend=1| 10, …, 130, 270, 2020c, 2290c, 2560c, 2830bc, 3100bcc, 3370bcc, 3640bcc }}


[[Badness]]: 0.033902
[[Badness]]: 0.033902
Line 145: Line 161:
Mapping: {{mapping| 10 0 47 36 98 | 0 2 -3 -1 -8 }}
Mapping: {{mapping| 10 0 47 36 98 | 0 2 -3 -1 -8 }}


Optimal tuning (POTE): ~15/14 = 1\10, ~400/231 = 951.070 (~16/15 = 111.070, or ~225/224 = 8.930)
Optimal tunings:
* CTE: ~15/14 = 1\10, ~400/231 = 951.0943 (~16/15 = 111.0943, or ~225/224 = 8.9057)
* POTE: ~15/14 = 1\10, ~400/231 = 951.070 (~16/15 = 111.070, or ~225/224 = 8.930)


Optimal ET sequence: {{Optimal ET sequence| 10e, 130, 270, 670, 940, 1210, 2150c }}
Optimal ET sequence: {{Optimal ET sequence| 10e, …, 130, 270, 670, 940, 1210, 2150c }}


Badness: 0.018735
Badness: 0.018735
Line 158: Line 176:
Mapping: {{mapping| 10 0 47 36 98 37 | 0 2 -3 -1 -8 0 }}
Mapping: {{mapping| 10 0 47 36 98 37 | 0 2 -3 -1 -8 0 }}


Optimal tuning (POTE): ~15/14 = 1\10, ~26/15 = 951.083 (~16/15 = 111.083, or ~196/195 = 8.917)
Optimal tunings:
* CTE: ~15/14 = 1\10, ~26/15 = 951.0943 (~16/15 = 111.0943, or ~196/195 = 8.9057)
* POTE: ~15/14 = 1\10, ~26/15 = 951.083 (~16/15 = 111.083, or ~196/195 = 8.917)


Optimal ET sequence: {{Optimal ET sequence| 10e, 130, 270, 940, 1210f, 1480cf }}
Optimal ET sequence: {{Optimal ET sequence| 10e, …, 130, 270, 940, 1210f, 1480cf }}


Badness: 0.013475
Badness: 0.013475
== Notes ==


[[Category:Temperament families]]
[[Category:Temperament families]]

Revision as of 11:43, 2 August 2024

The quintosec family tempers out the quintosec comma, 140737488355328/140126044921875 = [47 -15 -10⟩.

Quintosec

Quintosec is naturally a 2.3.5.11-subgroup temperament. It was documented as qintosec on the temperament finder, which was probably due to a typo[1]. The original, intended spelling is now restored, and the distinctive no-u form is relegated to the lower-accuracy 7-limit extension considered below.

Subgroup: 2.3.5

Comma list: [47 -15 -10⟩

Mapping: [⟨5 1 22], ⟨0 2 -3]]

mapping generators: ~524288/455625, ~16384/10125

Optimal tunings:

  • CTE: ~524288/455625 = 1\5, ~16384/10125 = 831.1061 (~16/15 = 111.1061)
  • POTE: ~524288/455625 = 1\5, ~16384/10125 = 831.105 (~16/15 = 111.105)

Optimal ET sequence: 10, 45c, 55, 65, 140, 205, 270, 1955c, 2225c, 2495bc, 2765bc, 3035bc, 3305bcc, 3575bcc

Badness: 0.139191

2.3.5.11 subgroup

Subgroup: 2.3.5.11

Comma list: 5632/5625, 26214400/26198073

Mapping: [⟨5 1 22 45], ⟨0 2 -3 -8]]

Optimal tunings:

  • CTE: ~1024/891 = 1\5, ~160/99 = 831.0932 (~16/15 = 111.0932)
  • CWE: ~1024/891 = 1\5, ~160/99 = 831.0743 (~16/15 = 111.0743)

Optimal ET sequence: 10e, 55e, 65, 205, 270, 335, 605, 940, 1545c

Badness: 0.0195

Qintosec

Qintosec can be described as the 10 & 65d temperament, tempering out the marvel comma, 225/224 in the 7-limit.

Subgroup: 2.3.5.7

Comma list: 225/224, 2560000/2470629

Mapping: [⟨5 1 22 21], ⟨0 2 -3 -2]]

Wedgie: ⟨⟨ 10 -15 -10 -47 -44 19 ]]

Optimal tunings:

  • CTE: ~400/343 = 1\5, ~80/49 = 831.7095 (~15/14 = 111.7095)
  • POTE: ~400/343 = 1\5, ~80/49 = 831.553 (~15/14 = 111.553)

Optimal ET sequence: 10, …, 55d, 65d, 75, 140d

Badness: 0.156764

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 3840/3773

Mapping: [⟨5 1 22 21 0], ⟨0 2 -3 -2 5]]

Optimal tunings:

  • CTE: ~55/48 = 1\5, ~80/49 = 831.1939 (~15/14 = 111.1939)
  • POTE: ~55/48 = 1\5, ~80/49 = 831.311 (~15/14 = 111.311)

Optimal ET sequence: 10, 55d, 65d, 140de

Badness: 0.077907

Qinto

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 332750/321489

Mapping: [⟨5 1 22 21 -7], ⟨0 2 -3 -2 7]]

Optimal tunings:

  • CTE: ~220/189 = 1\5, ~44/27 = 832.4051 (~15/14 = 112.4051)
  • POTE: ~220/189 = 1\5, ~44/27 = 832.448 (~15/14 = 112.448)

Optimal ET sequence: 10, 75

Badness: 0.127732

Sengasec

Sengasec can be described as the 10 & 55 temperament, tempering out the cloudy comma, 16807/16384 and the sengic comma, 686/675 in the 7-limit.

Subgroup: 2.3.5.7

Comma list: 686/675, 16807/16384

Mapping: [⟨5 1 22 14], ⟨0 2 -3 0]]

Wedgie: ⟨⟨ 10 -15 0 -47 -28 42 ]]

Optimal tuning:

  • CTE: ~8/7 = 1\5, ~45/28 = 831.1061 (~16/15 = 111.1061)
  • POTE: ~8/7 = 1\5, ~45/28 = 831.113 (~16/15 = 111.113)

Optimal ET sequence: 10, …, 55, 65, 140dd

Badness: 0.156259

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 385/384, 686/675

Mapping: [⟨5 1 22 14 0], ⟨0 2 -3 0 5]]

Optimal tunings:

  • CTE: ~8/7 = 1\5, ~45/28 = 830.7772 (~16/15 = 110.7772)
  • POTE: ~8/7 = 1\5, ~45/28 = 830.659 (~16/15 = 110.659)

Optimal ET sequence: 10, 55, 65

Badness: 0.081607

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 243/242, 686/675

Mapping: [⟨5 1 22 14 0 22], ⟨0 2 -3 0 5 -1]]

Optimal tunings:

  • CTE: ~8/7 = 1\5, ~45/28 = 830.8942 (~16/15 = 110.8942)
  • POTE: ~8/7 = 1\5, ~45/28 = 830.910 (~16/15 = 110.910)

Optimal ET sequence: 10, 55, 65

Badness: 0.056390

Decoid

See also: Breedsmic temperaments § Decoid

Subgroup: 2.3.5.7

Comma list: 2401/2400, 67108864/66976875

Mapping: [⟨10 0 47 36], ⟨0 2 -3 -1]]

mapping generators: ~15/14, ~8192/4725

Wedgie: ⟨⟨ 20 -30 -10 -94 -72 61 ]]

Optimal tunings:

  • CTE: ~15/14 = 1\10, ~8192/4725 = 951.1086 (~16/15 = 111.1086, or ~225/224 = 8.8914)
  • POTE: ~15/14 = 1\10, ~8192/4725 = 951.099 (~16/15 = 111.099, or ~225/224 = 8.901)

Optimal ET sequence: 10, …, 130, 270, 2020c, 2290c, 2560c, 2830bc, 3100bcc, 3370bcc, 3640bcc

Badness: 0.033902

11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 5632/5625, 9801/9800

Mapping: [⟨10 0 47 36 98], ⟨0 2 -3 -1 -8]]

Optimal tunings:

  • CTE: ~15/14 = 1\10, ~400/231 = 951.0943 (~16/15 = 111.0943, or ~225/224 = 8.9057)
  • POTE: ~15/14 = 1\10, ~400/231 = 951.070 (~16/15 = 111.070, or ~225/224 = 8.930)

Optimal ET sequence: 10e, …, 130, 270, 670, 940, 1210, 2150c

Badness: 0.018735

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 1716/1715, 4096/4095

Mapping: [⟨10 0 47 36 98 37], ⟨0 2 -3 -1 -8 0]]

Optimal tunings:

  • CTE: ~15/14 = 1\10, ~26/15 = 951.0943 (~16/15 = 111.0943, or ~196/195 = 8.9057)
  • POTE: ~15/14 = 1\10, ~26/15 = 951.083 (~16/15 = 111.083, or ~196/195 = 8.917)

Optimal ET sequence: 10e, …, 130, 270, 940, 1210f, 1480cf

Badness: 0.013475

Notes

  1. ↑ Yahoo Tuning Group | 2D temperament names, part I -- reclassified temperaments from message #101780
Retrieved from "https://en.xen.wiki/index.php?title=Quintosec_family&oldid=149343"