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No-fives subgroup temperaments: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings (8/). + infraug ({729/704}) to accommodate the 2.3.11.13-subgroup temp
Switch to Sintel's badness, WE & CWE tunings (9/). - duplicate/dubious entry
Line 892: Line 892:
[[Badness]] (Sintel): 0.185
[[Badness]] (Sintel): 0.185


== Infraug ==
=== Infraug ===
[[Subgroup]]: 2.3.11
[[Subgroup]]: 2.3.11


Line 910: Line 910:
[[Badness]] (Sintel): 0.734
[[Badness]] (Sintel): 0.734


=== 2.3.11.13 ===
==== 2.3.11.13 ====
Subgroup: 2.3.11.13
Subgroup: 2.3.11.13


Line 925: Line 925:
Badness (Sintel): 0.725
Badness (Sintel): 0.725


=== Aerophore ===
==== Aerophore ====
Subgroup: 2.3.11.19
Subgroup: 2.3.11.19


Line 944: Line 944:
[[Subgroup]]: 2.3.11
[[Subgroup]]: 2.3.11


[[Comma list]]: [[4096/3993]]
[[Comma list]]: 4096/3993


{{Mapping|legend=2| 1 0 4 | 0 3 -1 }}
{{Mapping|legend=2| 1 0 4 | 0 3 -1 }}
Line 978: Line 978:


Badness (Sintel): 1.31
Badness (Sintel): 1.31
=== Glaishur ===
This temperament is the no-5 no-7 [[restriction]] of [[#Navy|navy]], as well as the add-11 [[extension]] of [[#Glacier|glacier]].
[[Subgroup]]: 2.3.11
[[Comma list]]: 10554638336/10460353203
{{Mapping|legend=2| 1 1 0 | 0 5 21 }}
: mapping generators: ~2, ~88/81
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0000{{c}}, ~88/81 = 140.537{{c}}
: [[error map]]: {{val| -0.150 +0.493 -0.559 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~88/81 = 140.537{{c}}
: error map: {{val| 0.000 +0.662 -0.326 }}
{{Optimal ET sequence|legend=1| 17, 60e, 77, 94, 111 }}
[[Badness]] (Sintel): 2.27
==== 2.3.11.13 ====
Subgroup: 2.3.11.13
Comma list: 352/351, 531674/531441
Subgroup-val mapping: {{mapping| 1 1 0 3 | 0 5 21 6 }}
Optimal tunings:
* WE: ~2 = 1200.0000{{c}}, ~13/12 = 140.537{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.537{{c}}
{{Optimal ET sequence|legend=0| 17, 60e, 77, 94, 111, 239f, 350f }}
Badness (Sintel): 0.415


== Temperaments with a 2.3.13 gene ==
== Temperaments with a 2.3.13 gene ==
=== Threedic ===
[[Subgroup]]: 2.3.13
[[Comma list]]: 2197/2187
{{Mapping|legend=2| 1 0 0 | 0 3 7 }}
: mapping generators: ~2, ~13/9
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0000, ~13/9 = 634.1729{{c}}
: [[error map]]: {{val| -0.000 +0.563 -1.318 }}
* [[CWE]]: ~2 = 1200.0000, ~13/9 = 634.1729{{c}}
: error map: {{val| 0.000 +0.564 -1.318 }}
{{Optimal ET sequence|legend=1| 15, 17, 36, 53, 70, 123, 193, 316, 755f }}
[[Badness]] (Sintel): 0.160
=== Ultraflat ===
=== Ultraflat ===
Ultraflat is a diatonic-based [[exotemperament]] that makes [[27/26]] vanish, so [[13/8]] is a major sixth.  
Ultraflat is a diatonic-based [[exotemperament]] that makes [[27/26]] vanish, so [[13/8]] is a major sixth.  
Line 1,022: Line 1,075:
[[Badness]] (Sintel): 0.610
[[Badness]] (Sintel): 0.610


=== Threedic ===
=== Shoal ===
The 2.3.13.23-subgroup [[microtemperament]] is remarkable for containing not one but two [[superparticular]] intervals as small as [[3888/3887]] and [[12168/12167]]. Tempering out both of them gives us this rank-2 temperament where a sharp whole tone of [[26/23]] is the generator, two of which stack to a [[23/18]] supermajor third, and eight of which stack to a [[8/3]] perfect eleventh. [[17edo]] is a trivial tuning where 26/23 is equated to [[9/8]], tempering out the comma [[208/207]]. More accurate tunings of shoal create a 17-note [[mos]] scale, serving as a [[circulating temperament]] of 17edo, where 208/207 is the [[chroma]] between large and small steps.
 
[[Subgroup]]: 2.3.13
[[Subgroup]]: 2.3.13


[[Comma list]]: [[2197/2187]]
[[Comma list]]: 816293376/815730721


{{Mapping|legend=2|1 0 0|0 3 7}}
{{Mapping|legend=2| 1 -5 -7 | 0 8 13 }}
: mapping generators: ~2, ~3888/2197


[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/9]] = 634.173
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9922{{c}}, ~3888/2197 = 987.7360{{c}}
: [[error map]]: {{val| -0.008 -0.028 +0.095 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3888/2197 = 987.7415{{c}}
: error map: {{val| 0.000 -0.023 +0.112 }}


{{Optimal ET sequence|legend=1| 11bff, 13f, 15, 17, 36, 53, 70, 123, 193, 316, 755f }}
{{Optimal ET sequence|legend=1| 17, 79, 96, 113, 130, 147, 424, 571, 1289, 10883ff, 12172ff }}


[[Tp tuning #T2 tuning|RMS error]]: 0.2054 cents
Badness (Sintel): 0.135


=== Shoal ===
==== 2.3.13.23 ====
The 2.3.13.23 subgroup is remarkable for containing not one but two [[superparticular]] intervals as small as [[3888/3887]] and [[12168/12167]]. Tempering out both of them gives us a rank-2 temperament where a sharp whole tone of [[26/23]] is the generator, two of which stack to a [[23/18]] supermajor third, and eight of which stack to a [[8/3]] perfect eleventh. [[17edo]] is a trivial tuning where 26/23 is equated to [[9/8]], tempering out the comma [[208/207]]. More accurate tunings of shoal create a 17-note [[MOS]] scale, serving as a [[circulating temperament]] of 17edo, where 208/207 is the [[chroma]] between large and small steps.
Subgroup: 2.3.13.23


[[Subgroup]]: 2.3.13.23
Comma list: 3888/3887, 12168/12167


[[Comma list]]: 3888/3887, 12168/12167
Subgroup-val mapping: {{mapping| 1 -5 -7 -7 | 0 8 13 14 }}


{{Mapping|legend=2|1 3 6 7|0 -8 -13 -14}}
Optimal tunings:
 
* WE: ~2 = 1199.9883{{c}}, ~23/13 = 987.7325{{c}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~26/23 = 212.261
* CWE: ~2 = 1200.0000{{c}}, ~23/13 = 987.7408{{c}}


{{Optimal ET sequence|legend=1|11fi, 17, 79, 96, 113, 130, 147, 424, 571, 1289, 1860, 3149 }}
{{Optimal ET sequence|legend=0| 17, 79, 96, 113, 130, 147, 424, 571, 1289, 1860, 3149 }}


Badness (Sintel): 0.021
Badness (Sintel): 0.0213


Scales:
Scales:
Line 1,054: Line 1,114:


Music:
Music:
* [https://www.youtube.com/watch?v=bwGjN56iMM0 ''Moody Improvisation in the Shoal Temperament''] - [[Budjarn Lambeth]] (2025)
* [https://www.youtube.com/watch?v=bwGjN56iMM0 ''Moody Improvisation in the Shoal Temperament''] by [[Budjarn Lambeth]] (2025)


=== Glacier ===
=== Glacier ===
The 2.3.13 gene subgroup is not nearly as good as Shoal, but it can extend extremely well to other no-5 subgroups. It is very well represented in [[26edo]], where a pure 13/12 can serve as the generator, but [[94edo]] provides a much better tuning in higher subgroups.
This 2.3.13-subgroup gene is not nearly as good as shoal, but it can extend extremely well to other no-5 subgroups. It is the common [[restriction]] of [[#Bleu|bleu]] and [[#Navy|navy]]. It is very well represented in [[26edo]], where a pure 13/12 can serve as the generator, but [[94edo]] provides a much better tuning.


[[Subgroup]]: 2.3.13
[[Subgroup]]: 2.3.13


[[Comma list]]: [[373248/371293]]
[[Comma list]]: 373248/371293


{{Mapping|legend=2|1 1 3|0 5 6}}
{{Mapping|legend=2| 1 1 3 | 0 5 6 }}
: mapping generators: ~2, ~13/12


[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/12]] = 140.360
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.8406{{c}}, ~13/12 = 140.3695{{c}}
{{Optimal ET sequence|legend=1|9, 17, 26, 43, 60, 77, 94, 111, 137, 171}}
: [[error map]]: {{val| -0.159 -0.267 +1.211 }}
 
* [[CWE]]: ~2 = 1200.0000{{c}}, ~13/12 = 140.3605{{c}}
Badness (Sintel): 0.383
: error map: {{val| 0.000 -0.153 +1.635 }}
 
==== Glaishur ====
[[Subgroup]]: 2.3.11.13
 
[[Comma list]]: [[352/351]], [[531674/531441]]
 
{{Mapping|legend=2|1 1 0 3|0 5 21 6}}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/12]] = 140.537
 
{{Optimal ET sequence|legend=1|17, 77, 94, 111, 128, 145, 205}}
 
Badness (Sintel): 0.400
 
===== 2.3.7.11.13 =====
[[Subgroup]]: 2.3.7.11.13
 
[[Comma list]]: [[352/351]], [[729/728]], [[1573/1568]]
 
{{Mapping|legend=2|1 1 0 1 3|0 5 24 21 6}}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/12]] = 140.429
 
{{Optimal ET sequence|legend=1|17, 77, 94, 111, 128, 145, 205}}
 
Badness (Sintel): 0.415
 
===== 2.3.7.11.13.23 =====
[[Subgroup]]: 2.3.7.11.13.23
 
[[Comma list]]: [[352/351]], [[729/728]], [[253/252]], [[1288/1287]]
 
{{Mapping|legend=2|1 1 0 1 3 3|0 5 24 21 6 13}}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/12]] = 140.426
 
{{Optimal ET sequence|legend=1|17, 77, 94, 111, 205}}
 
Badness (Sintel): 0.452
 
===== 2.3.7.11.13.23.29 =====
[[Subgroup]]: 2.3.7.11.13.23.29
 
[[Comma list]]: [[352/351]], [[729/728]], [[253/252]], [[1288/1287]], [[378/377]]
 
{{Mapping|legend=2|1 1 0 1 3 3 1|0 5 24 21 6 13 33}}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/12]] = 140.426
 
{{Optimal ET sequence|legend=1|17, 77, 94, 111j}}
 
Badness (Sintel): 0.511
 
===== 2.3.7.11.13.19.23.29 =====
[[Subgroup]]: 2.3.7.11.13.19.23.29
 
[[Comma list]]: [[352/351]], [[729/728]], [[209/208]], [[253/252]], [[1288/1287]], [[378/377]]
 
{{Mapping|legend=2|1 1 0 1 3 -6 3 1|0 5 24 21 6 -15 13 33}}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/12]] = 140.426


{{Optimal ET sequence|legend=1|17, 77, 94, 111j}}
{{Optimal ET sequence|legend=1| 8, 9, 17, 60, 77, 94, 171, 265, 359f, 983ff }}


Badness (Sintel): 0.699
[[Badness]] (Sintel): 0.383


== Temperaments with a higher-limit gene ==
== Temperaments with a higher-limit gene ==
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