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No-fives subgroup temperaments: Difference between revisions

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=== Archy ===
=== Archy ===
See [[Archytas clan #Archy]].  
See [[Archytas clan #Archy]].  
==== Supra ====
See [[Archytas clan #Supra]].
===== Supraphon =====
See [[Archytas clan #Supraphon]].


==== Suhajira ====
==== Suhajira ====
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=== Slendroschismic ===
=== Slendroschismic ===
See [[5th-octave temperaments #Slendroschismic]].  
See [[5th-octave temperaments #Slendroschismic]].  
==== Pentadecoid ====
See [[15th-octave temperaments #Pentadecoid]].


=== Navy ===
=== Navy ===
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=== Bleu ===
=== Bleu ===
Bleu can be described as the {{nowrap| 8d & 9 }} temperament in the no-5 13-limit.  
Bleu can be described as the {{nowrap| 8d & 9 }} temperament in the no-5 13-limit, and is the common [[restriction]] of [[progression]] and [[jerome]].  


[[Subgroup]]: 2.3.7
[[Subgroup]]: 2.3.7
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=== Purpleheart ===
=== Purpleheart ===
{{See also| Whitewood }}
{{Main| Whitewood }}


[[Subgroup]]: 2.3.7
[[Subgroup]]: 2.3.7
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== Temperaments with a 2.3.11 gene ==
== Temperaments with a 2.3.11 gene ==
=== Neutral ===
See [[Rastmic clan #Neutral]].
=== Io ===
=== Io ===
Io is a very low-complexity temperament which tempers out the undecimal quartertone [[33/32]]. This equates very different intervals (for example, the generator itself represents both [[3/2]] and [[16/11]]), and as such some consider it to be an [[exotemperament]]. It has an extremely wide generator range, but the most accurate tunings are generally inside the range of [[flattone]] temperament.
Io is a very low-complexity temperament which tempers out the undecimal quartertone [[33/32]]. This equates very different intervals (for example, the generator itself represents both [[3/2]] and [[16/11]]), and as such some consider it to be an [[exotemperament]]. It has an extremely wide generator range, but the most accurate tunings are generally inside the range of [[flattone]] temperament.


The name Io was coined by [[User:CompactStar|CompactStar]] in 2024 based on the [[color notation|color name]] ilo, prior to which it could only be termed as "undecimal temperament" with 33/32 being known as the undecimal comma.
The name ''io'' was coined by {{u|CompactStar}} in 2024 based on the [[Kite's color notation|color name]] ''ilo'', prior to which it could only be termed as "undecimal temperament" with 33/32 being known as the undecimal comma.


[[Subgroup]]: 2.3.11
[[Subgroup]]: 2.3.11
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{{Mapping|legend=2| 1 0 5 | 0 1 -1 }}
{{Mapping|legend=2| 1 0 5 | 0 1 -1 }}
: mapping generators: ~2, ~3


: mapping generators: ~2, ~3
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1206.6866{{c}}, ~3/2 = 691.7837{{c}}
: [[error map]]: {{val| +6.687 -3.485 -16.355 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 689.2066{{c}}
: error map: {{val| 0.000 -12.748 -40.525 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 692.713
{{Optimal ET sequence|legend=1| 2, 5, 7, 12e, 40ee, 47eee, 54beee, 61beeee }}


{{Optimal ET sequence|legend=1| 2, 5, 7, 12e }}
[[Badness]] (Sintel): 0.185


[[Badness]]: 0.185
=== Alphaxenean ===
Alphaxenean tempers out the [[Alpharabian comma]], and equates a stack of four undecimal quartertones with the [[9/8|Pythagorean whole tone]]. It has a [[semi-octave]] period, and the generator can be taken to be the aforementioned undecimal quartertone or that plus a period, which represents ~[[16/11]]. The interval class of a perfect fifth is found by two generator steps and a period.  


=== Paralimmal ===
[[Subgroup]]: 2.3.11
[[Subgroup]]: 2.3.11


[[Comma list]]: [[4096/3993]]
[[Comma list]]: 131769/131072


{{Mapping|legend=2| 1 0 4 | 0 3 -1 }}
{{Mapping|legend=2| 2 1 8 | 0 2 -1 }}
: mapping generators: ~363/256, ~16/11


[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[16/11]] = 634.320
[[Optimal tuning]]s:
* [[WE]]: ~363/256 = 600.1590{{c}}, ~16/11 = 650.8508{{c}}
: [[error map]]: {{val| +0.318 -0.094 -0.897 }}
* [[CWE]]: ~363/256 = 600.0000{{c}}, ~16/11 = 650.7321{{c}}
: error map: {{val| 0.000 -0.491 -2.050 }}


{{Optimal ET sequence|legend=1| 11b, 13, 15, 17 }}
{{Optimal ET sequence|legend=1| 22, 24, 94, 118, 142, 450e, 592e, 1326beeee }}


[[Tp tuning #T2 tuning|RMS error]]: 1.237 cents
[[Badness]] (Sintel): 0.395


=== Neutral ===
=== Octatonic ===
See [[Rastmic clan #Neutral]]
[[12/11]] is very close to one step of 8edo, and hence this temperament tempers out the [[undecimal octatonic comma]], the difference between a stack of eight 12/11's and the octave.


==== Namo ====
[[Subgroup]]: 2.3.11
See [[Rastmic clan #Namo]]


=== Huxley ===
[[Comma list]]: 214990848/214358881
{{Main| Huxley }}


Huxley, the 4 & 13 temperament in the 2.3.11.13 subgroup, extends [[lovecraft]]. Specifically it tunes the ~13/8 to exactly half of ~8/3.
{{Mapping|legend=2| 8 0 15 | 0 1 1 }}
: mapping generators: ~12/11, ~3


[[Subgroup]]: 2.3.11.13
[[Optimal tuning]]s:  
* [[WE]]: ~12/11 = 149.9917{{c}}, ~3/2 = 701.9324{{c}}
: [[error map]]: {{val| -0.066 -0.089 +0.424 }}
* [[CWE]]: ~12/11 = 150.0000{{c}}, ~3/2 = 701.9147{{c}}
: error map: {{val| 0.000 -0.040 +0.597 }}


[[Comma list]]: 512/507, 1352/1331
{{Optimal ET sequence|legend=1| 8, 16, 24, 104, 128, 152, 176, 200, 1024e, 1224e, 1424e, 1624e }}


{{Mapping|legend=2| 1 3 3 3 | 0 -6 2 3 }}
[[Badness]] (Sintel): 0.515


: mapping generators: ~2, ~13/11
=== Betaxenean ===
Betaxenean tempers out the [[Betarabian comma]]. It splits a [[3/1|perfect twelfth]] into two halves of [[512/297]].


[[Optimal tuning]]s:  
[[Subgroup]]: 2.3.11
* [[CTE]]: ~2 = 1\1, ~13/11 = 282.726
* [[CWE]]: ~2 = 1\1, ~13/11 = 282.482


{{Optimal ET sequence|legend=1| 4, 13, 17 }}
[[Comma list]]: 264627/262144


[[Badness]]: 0.0263
{{Mapping|legend=2| 1 0 9 | 0 2 -7 }}
: mapping generators: ~2, ~512/297


=== Aerophore ===
[[Optimal tuning]]s:
[[Subgroup]]: 2.3.11.19
* [[WE]]: ~2 = 1200.5931{{c}}, ~512/297 = 950.6872{{c}}
: [[error map]]: {{val| +0.593 -0.581 -0.791 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~512/297 = 950.1972{{c}}
: error map: {{val| 0.000 -1.561 -2.699 }}


[[Comma list]]: 363/361, 729/704
{{Optimal ET sequence|legend=1| 5, 14e, 19, 24, 125, 149, 173, 197e, 221e, 466bee, 687beee }}


{{Mapping|legend=2| 1 0 -6 -6 | 0 2 12 13 }}
[[Badness]] (Sintel): 0.709


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~19/11 = 945.4
=== Infraug ===
[[Subgroup]]: 2.3.11


{{Optimal ET sequence|legend=1| 9eehh, 14, 19, 33 }}
[[Comma list]]: 729/704


==== Semaerophore ====
{{Mapping|legend=2| 1 0 -6 | 0 1 6 }}
[[Subgroup]]: 2.3.7.11.19
: mapping generators: ~2, ~3


[[Comma list]]: 49/48, 77/76, 729/704
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1202.5969{{c}} ~3/2 = 692.7443{{c}}
: [[error map]]: {{val| +2.597 -6.614 +5.148 }}
* [[CWE]]: ~2 = 1200.0000{{c}} ~3/2 = 692.0116{{c}}
: error map: {{val| 0.000 -9.943 +0.752 }}


{{Mapping|legend=2| 1 0 2 -6 -6 | 0 2 1 12 13 }}
{{Optimal ET sequence|legend=1| 7, 19, 26, 33, 59b, 92b }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/4 = 944.667
[[Badness]] (Sintel): 0.734


{{Optimal ET sequence|legend=1| 9eehh, 14, 33d, 47deh }}
==== 2.3.11.13 subgroup ====
Subgroup: 2.3.11.13


== Temperaments with a 2.3.13 gene ==
Comma list: 144/143, 729/704
=== Superflat ===
Superflat is a diatonic-based temperament that makes [[1053/1024]] vanish, so [[13/8]] is a minor sixth, and [[16/13]] is a major third. The more accurate tunings for this temperament are generated by a fifth at least as flat as those of [[flattone]], although often even flatter (such as [[40edo]]'s fifth). Superflat can be viewed as a 2.3.13 subgroup analogue of [[meantone]] and [[archy]]. Superflat diatonic scales have a character somewhere between neutral third scales (or [[3L 4s|mosh]]) and meantone diatonic scales.


[[Subgroup]]: 2.3.13
Subgroup-val mapping: {{mapping| 1 0 -6 10 | 0 1 6 -4 }}


[[Comma list]]: [[1053/1024]]
Optimal tunings:  
* WE: ~2 = 1202.1934{{c}}, ~3/2 = 692.8902{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 691.7585{{c}}


{{Mapping|legend=2|1 1 6|0 1 -4}}
{{Optimal ET sequence|legend=0| 7, 19, 26, 59b }}


[[Optimal tuning]] (CTE): ~2 = 1\1, ~[[3/2]] = 692.939
Badness (Sintel): 0.725


{{Optimal ET sequence|legend=1|5f, 7, 12, 19, 45f, 64f, 147bfff }}
==== Aerophore ====
Subgroup: 2.3.11.19


[[Tp tuning #T2 tuning|RMS error]]: 1.591 cents
Comma list: 363/361, 729/704


==== 2.3.11.13 ====
Subgroup-val mapping: {{mapping| 1 0 -6 -6 | 0 2 12 13 }}
[[Subgroup]]: 2.3.11.13
: mapping generators: ~2, ~19/11


[[Comma list]]: [[144/143]], [[729/704]]
Optimal tunings:  
* WE: ~2 = 1202.6380{{c}}, ~19/11 = 947.4782{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/11 = 945.7779{{c}}


[[Optimal tuning]] (CTE): ~2 = 1\1, ~[[3/2]] = 692.247
{{Optimal ET sequence|legend=0| 14, 19, 33 }}


{{Optimal ET sequence|legend=1|7, 19, 26, 59b }}
Badness (Sintel): 1.59


=== Ultraflat ===
=== Paralimmal ===
Ultraflat is the much more inaccurate cousin of superflat, with even flatter fifths. [[27/26]] is tempered out rather than [[1053/1024]], so [[13/8]] is a major sixth. These temperaments intersect in [[7edo]], where major sixths and minor sixths are not distinguished.
[[Subgroup]]: 2.3.11


[[Subgroup]]: 2.3.13
[[Comma list]]: 4096/3993


[[Comma list]]: [[27/26]]
{{Mapping|legend=2| 1 0 4 | 0 3 -1 }}
: mapping generators: ~2, ~16/11


{{Mapping|legend=2|1 1 2|0 1 3}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1197.9124{{c}}, ~16/11 = 634.1269{{c}}
: [[error map]]: {{val| -2.088 +0.426 +6.205 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/11 = 634.9546{{c}}
: error map: {{val| 0.000 +2.909 +13.727 }}


[[Optimal tuning]] (CTE): ~2 = 1/1, ~[[3/2]] = 688.391
{{Optimal ET sequence|legend=1| 2, 11b, 13, 15, 17 }}


{{Optimal ET sequence|legend=1| 5, 7 }}
[[Badness]] (Sintel): 0.984


[[Tp tuning #T2 tuning|RMS error]]: 4.367 cents
==== Huxley ====
{{Main| Huxley }}


=== Threedic ===
Huxley, the {{nowrap| 4 & 13 }} temperament in the 2.3.11.13 subgroup, extends [[lovecraft]]. Specifically it tunes the ~13/8 to exactly half of ~8/3.  
[[Subgroup]]: 2.3.13


[[Comma list]]: [[2197/2187]]
Subgroup: 2.3.11.13


{{Mapping|legend=2|1 0 0|0 3 7}}
Comma list: 512/507, 1352/1331


[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/9]] = 634.173
Subgroup-val mapping: {{mapping| 1 -3 5 6 | 0 6 -2 -3 }}
: mapping generators: ~2, ~22/13


{{Optimal ET sequence|legend=1|11bff, 13f, 15, 17, 36, 53, 70, 123, 193, 316, 755f }}
Optimal tunings:
* WE: ~2 = 1198.0036{{c}}, ~22/13 = 916.0595{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 917.5184{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 0.2054 cents
{{Optimal ET sequence|legend=0| 4, 13, 17 }}


=== Shoal ===
Badness (Sintel): 1.31
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
=== Glaishur ===
This temperament is the no-5 no-7 [[restriction]] of [[#Navy|navy]], as well as the add-11 [[extension]] of [[#Glacier|glacier]].  


[[Comma list]]: 3888/3887, 12168/12167
[[Subgroup]]: 2.3.11


{{Mapping|legend=2|1 3 6 7|0 -8 -13 -14}}
[[Comma list]]: 10554638336/10460353203


[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~26/23 = 212.261
{{Mapping|legend=2| 1 1 0 | 0 5 21 }}
: mapping generators: ~2, ~88/81


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


Badness (Sintel): 0.021
{{Optimal ET sequence|legend=1| 17, 60e, 77, 94, 111 }}


Scales:
[[Badness]] (Sintel): 2.27
* [[Shoal17]]


Music:
==== 2.3.11.13 ====
* [https://www.youtube.com/watch?v=bwGjN56iMM0 ''Moody Improvisation in the Shoal Temperament''] - [[Budjarn Lambeth]] (2025)
Subgroup: 2.3.11.13


=== Glacier ===
Comma list: 352/351, 531674/531441
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.


[[Subgroup]]: 2.3.13
Subgroup-val mapping: {{mapping| 1 1 0 3 | 0 5 21 6 }}


[[Comma list]]: [[373248/371293]]
Optimal tunings:  
* WE: ~2 = 1200.0000{{c}}, ~13/12 = 140.537{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.537{{c}}


{{Mapping|legend=2|1 1 3|0 5 6}}
{{Optimal ET sequence|legend=0| 17, 60e, 77, 94, 111, 239f, 350f }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/12]] = 140.360
Badness (Sintel): 0.415


{{Optimal ET sequence|legend=1|9, 17, 26, 43, 60, 77, 94, 111, 137, 171}}
=== Profanity ===
Profanity identifies [[11/9]] with 2\7.


Badness (Sintel): 0.383
[[Subgroup]]: 2.3.11


==== Glaishur ====
[[Comma list]]: 19487171/19131876
[[Subgroup]]: 2.3.11.13


[[Comma list]]: [[352/351]], [[531674/531441]]
{{Mapping|legend=2| 7 0 2 | 0 1 2 }}
: mapping generators: ~1458/1331, ~3


{{Mapping|legend=2|1 1 0 3|0 5 21 6}}
[[Optimal tuning]]s:
* [[WE]]: ~1458/1331 = 171.4369{{c}}, ~3/2 = 702.9304{{c}}
: [[error map]]: {{val| +0.058 +1.033 -2.467 }}
* [[CWE]]: ~1458/1331 = 171.4286{{c}}, ~3/2 = 702.9442{{c}}
: error map: {{val| 0.000 -0.989 -2.572 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/12]] = 140.537
{{Optimal ET sequence|legend=1| 7, … 49, 56, 63, 70 }}


{{Optimal ET sequence|legend=1|17, 77, 94, 111, 128, 145, 205}}
[[Badness]] (Sintel): 3.03


Badness (Sintel): 0.400
== Temperaments with a 2.3.13 gene ==
=== Threedic ===
[[Subgroup]]: 2.3.13


===== 2.3.7.11.13 =====
[[Comma list]]: 2197/2187
[[Subgroup]]: 2.3.7.11.13


[[Comma list]]: [[352/351]], [[729/728]], [[1573/1568]]
{{Mapping|legend=2| 1 0 0 | 0 3 7 }}
: mapping generators: ~2, ~13/9


{{Mapping|legend=2|1 1 0 1 3|0 5 24 21 6}}
[[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 tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/12]] = 140.429
{{Optimal ET sequence|legend=1| 15, 17, 36, 53, 70, 123, 193, 316, 755f }}


{{Optimal ET sequence|legend=1|17, 77, 94, 111, 128, 145, 205}}
[[Badness]] (Sintel): 0.160


Badness (Sintel): 0.415
=== Ultraflat ===
Ultraflat is a diatonic-based [[exotemperament]] that makes [[27/26]] vanish, so [[13/8]] is a major sixth.  


===== 2.3.7.11.13.23 =====
[[Subgroup]]: 2.3.13
[[Subgroup]]: 2.3.7.11.13.23


[[Comma list]]: [[352/351]], [[729/728]], [[253/252]], [[1288/1287]]
[[Comma list]]: 27/26


{{Mapping|legend=2|1 1 0 1 3 3|0 5 24 21 6 13}}
{{Mapping|legend=2| 1 0 -1 | 0 1 3 }}
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/12]] = 140.426
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.6561{{c}}, ~3/2 = 686.9485{{c}}
: [[error map]]: {{val| +1.656 -13.350 +23.630 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 687.1143{{c}}
: error map: {{val| 0.000 -14.841 +20.815 }}


{{Optimal ET sequence|legend=1|17, 77, 94, 111, 205}}
{{Optimal ET sequence|legend=1| 2, 5, 7 }}


Badness (Sintel): 0.452
[[Badness]] (Sintel): 0.200


===== 2.3.7.11.13.23.29 =====
=== Tridecapyth ===
[[Subgroup]]: 2.3.7.11.13.23.29
Tridecapyth sets a stack of ten [[9/8]]'s equal to [[13/4]]. It shows a way of giving a mapping of prime 13 to any temperament with an extremely accurate tuning of its fifth (like the [[53edo]] tuning of [[schismic]]). Interestingly, the mapping is ''so'' accurate that more optimized tunings of schismic that use a flatter fifth are not accurate enough to preserve the mapping – for 118edo we get the 118f [[val]] that takes the second-closest, flat mapping of prime 13, and the same is true for [[171edo]] where we get the 171f val. However, [[41edo]] uses this mapping, and so does [[94edo]], the val sum of 41 and 53. Thus it is of interest to flatter tunings of [[garibaldi]]/[[cassandra]] with fifths tending close to pure. If we add this mapping to 5-limit schismic instead we get [[tridecaschismic]].


[[Comma list]]: [[352/351]], [[729/728]], [[253/252]], [[1288/1287]], [[378/377]]
[[Subgroup]]: 2.3.13


{{Mapping|legend=2|1 1 0 1 3 3 1|0 5 24 21 6 13 33}}
{{mapping|legend=2| 1 0 -28 | 0 1 20 }}
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/12]] = 140.426
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9778{{c}}, ~3/2 = 702.0170{{c}}
: [[error map]]: {{val| -0.022 +0.040 -0.011 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0283{{c}}
: error map: {{val| 0.000 +0.070 -0.019 }}


{{Optimal ET sequence|legend=1|17, 77, 94, 111j}}
{{Optimal ET sequence|legend=1| 12, 29f, 41, 53, 94, 147, 494, 641, 788, 2217, 3005 }}


Badness (Sintel): 0.511
[[Badness]] (Sintel): 0.211


===== 2.3.7.11.13.19.23.29 =====
=== Superflat ===
[[Subgroup]]: 2.3.7.11.13.19.23.29
Superflat is a less inaccurate cousin of ultraflat, with less flat fifths. It tempers out [[1053/1024]], so [[13/8]] is a minor sixth, and [[16/13]] is a major third. Superflat and ultraflat intersect in [[7edo]], where major sixths and minor sixths are not distinguished.  


[[Comma list]]: [[352/351]], [[729/728]], [[209/208]], [[253/252]], [[1288/1287]], [[378/377]]
The more accurate tunings for this temperament are generated by a fifth at least as flat as those of [[flattone]], although often even flatter (such as [[40edo]]'s fifth). Superflat can be viewed as a 2.3.13 subgroup analogue of [[meantone]] and [[archy]]. Superflat diatonic scales have a character somewhere between neutral third scales (or [[3L 4s|mosh]]) and meantone diatonic scales.


{{Mapping|legend=2|1 1 0 1 3 -6 3 1|0 5 24 21 6 -15 13 33}}
[[Subgroup]]: 2.3.13


[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/12]] = 140.426
[[Comma list]]: 1053/1024


{{Optimal ET sequence|legend=1|17, 77, 94, 111j}}
{{Mapping|legend=2| 1 0 10 | 0 1 -4 }}
: mapping generators: ~2, ~3


Badness (Sintel): 0.699
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1203.1291{{c}} ~3/2 = 695.6489{{c}}
: [[error map]]: {{val| +3.129 -3.177 -4.349 }}
* [[CWE]]: ~2 = 1200.0000{{c}} ~3/2 = 693.6081{{c}}
: error map: {{val| 0.000 -8.347 -14.960 }}


== Temperaments with a higher-limit gene ==
{{Optimal ET sequence|legend=1| 5f, 7, 12, 19, 45f, 64f, 147bfff }}
=== Semitonic ===
[[Subgroup]]: 2.3.17


[[Comma list]]: [[289/288]]
[[Badness]] (Sintel): 0.610


{{Mapping|legend=2| 2 0 5 | 0 1 1 }}
=== 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.


: sval mapping generators: ~17/12, ~3
[[Subgroup]]: 2.3.13


: [[gencom]]: [17/12 3; 289/288]
[[Comma list]]: 816293376/815730721


[[Optimal tuning]] ([[CTE]]): ~17/12 = 1\2, ~3/2 = 702.3472 (~17/16 = 102.3472)
{{Mapping|legend=2| 1 -5 -7 | 0 8 13 }}
: mapping generators: ~2, ~3888/2197


{{Optimal ET sequence|legend=1| 12, 58, 70, 82, 94, 106, 118, 224g }}
[[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 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.2247 cents
{{Optimal ET sequence|legend=1| 17, 79, 96, 113, 130, 147, 424, 571, 1289, 10883ff, 12172ff }}


=== Gigapyth ===
Badness (Sintel): 0.135
[[Subgroup]]: 2.3.85


[[Comma list]]: 2.3.85 {{val|-40 1 6 }}
==== 2.3.13.23 subgroup ====
Subgroup: 2.3.13.23


{{mapping|legend=2| 1 4 6 | 0 -6 1 }}
Comma list: 3888/3887, 12168/12167


: mapping generators: ~2, ~85/64
Subgroup-val mapping: {{mapping| 1 -5 -7 -7 | 0 8 13 14 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~85/64 = 483.034
Optimal tunings:
* WE: ~2 = 1199.9883{{c}}, ~23/13 = 987.7325{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~23/13 = 987.7408{{c}}


[[Support]]ing [[ET]]s: 5, 47, 52, 57, 62, 67, 72, 77*, 82*, 87*, 92*, 139*, 149*, 159**
{{Optimal ET sequence|legend=0| 17, 79, 96, 113, 130, 147, 424, 571, 1289, 1860, 3149 }}


<nowiki>*</nowiki>Wart for 85
Badness (Sintel): 0.0213


==== 2.3.7.85 subgroup ====
Scales:
[[Subgroup]]: 2.3.7.85
* [[Shoal17]]


[[Comma list]]: 1029/1024, 7225/7203
; Music
* [https://www.youtube.com/watch?v=bwGjN56iMM0 ''Moody Improvisation in the Shoal Temperament''] by [[Budjarn Lambeth]] (2025)


{{mapping|legend=2| 1 4 2 6 | 0 -6 2 1}}
=== Glacier ===
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 nearly pure 13/12 can serve as the generator, but [[94edo]] provides a much better tuning.


: mapping generators: ~2, ~85/64
[[Subgroup]]: 2.3.13


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~85/64 = 483.031
[[Comma list]]: 373248/371293


[[Support]]ing [[ET]]s: 5, 47, 52, 57, 62, 67, 72, 77*, 82*, 87*, 92*, 139*, 149*, 159**
{{Mapping|legend=2| 1 1 3 | 0 5 6 }}
: mapping generators: ~2, ~13/12


<nowiki>*</nowiki>Wart for 85
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8406{{c}}, ~13/12 = 140.3695{{c}}
: [[error map]]: {{val| -0.159 -0.267 +1.211 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~13/12 = 140.3605{{c}}
: error map: {{val| 0.000 -0.153 +1.635 }}


=== Dog ===
{{Optimal ET sequence|legend=1| 8, 9, 17, 60, 77, 94, 171, 265, 359f, 983ff }}
The dog temperament is based by [[2L 5s]] or [[7L 2s]] scale that makes [[81/76]] vanish, so [[19/16]] is a major third. It can be viewed as a 2.3.19 subgroup analogue of [[Pelogic family|mavila]].


Subgroup: 2.3.19
[[Badness]] (Sintel): 0.383


[[Comma list]]: 81/76
== Temperaments with a higher-limit gene ==
=== Semitonic ===
[[Subgroup]]: 2.3.17


[[Gencom]]: [2 4/3; 81/76]
[[Comma list]]: [[289/288]]


[[Mapping|Sval mapping]]: [{{val|1 2 6}}, {{val|0 -1 -4}}]
{{Mapping|legend=2| 2 0 5 | 0 1 1 }}
: mapping generators: ~17/12, ~3


[[Tp tuning|POL2 generator]]: ~4/3 = 521.403
[[Optimal tuning]]s:
* [[WE]]: ~17/12 = 600.1471{{c}}, ~3/2 = 701.9563{{c}} (~17/16 = 101.8091{{c}})
: [[error map]]: {{val| +0.294 +0.295 -1.969 }}
* [[CWE]]: ~17/12 = 600.0000{{c}}, ~3/2 = 702.0260{{c}} (~17/16 = 102.0260{{c}})
: error map: {{val| 0.000 +0.071 -2.929 }}


{{Optimal ET sequence|legend=1| 5h, 7, 16, 23 }}
{{Optimal ET sequence|legend=1| 10, 12, 58, 70, 82, 94, 106, 118, 224g }}


[[Tp tuning #T2 tuning|RMS error]]: 4.943 cents
[[Badness]] (Sintel): 0.0454


=== Boethian ===
=== Boethian ===
Line 1,169: Line 1,246:
[[Subgroup]]: 2.3.19
[[Subgroup]]: 2.3.19


[[Comma list]]: [[513/512]]
[[Comma list]]: 513/512


{{Mapping|legend=2| 1 0 9 | 0 1 -3 }}
{{Mapping|legend=2| 1 0 9 | 0 1 -3 }}
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


[[Optimal tuning]] (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 701.3288{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2498{{c}}, ~3/2 = 701.4958{{c}}
: [[error map]]: {{val| +0.250 -0.209 -0.501 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.3445{{c}}
: error map: {{val| 0.000 -0.610 -1.547 }}


{{Optimal ET sequence|legend=1| 5, 7, 12, 41, 53, 65, 77, 219, 296 }}
{{Optimal ET sequence|legend=1| 5, 7, 12, 41, 53, 65, 77, 219, 296, 1557bhhhh, 1853bhhhh }}


[[Badness]] (Smith): 0.000374
[[Badness]] (Sintel): 0.0294
 
=== Deviaug ===
Deviaug tempers out [[6912/6859]] in the 2.3.19 subgroup, setting [[24/19]] to 1/3 of an octave.
 
[[Subgroup]]: 2.3.19
 
[[Comma list]]: 6912/6859
 
{{Mapping|legend=2| 3 0 8 | 0 1 1 }}
: mapping generators: ~24/19, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~24/19 = 399.8571{{c}}, ~3/2 = 701.9795{{c}}
: [[error map]]: {{val| -0.429 -0.404 +2.894 }}
* [[CWE]]: ~24/19 = 400.0000{{c}}, ~3/2 = 701.8810{{c}}
: error map: {{val| 0.000 -0.074 +4.368 }}
 
{{Optimal ET sequence|legend=1| 9, 12, 75, 87, 99, 111, 123h, 135h }}
 
[[Badness]] (Sintel): 0.237


=== Lipsett ===
=== Lipsett ===
Lipsett temperament is a pleasantly melodic little temperament with a highly useable 5-tone and 9-tone mos. It is audibly similar to [[semaphore]] temperament, so it could be thought of as semaphore but for the 23rd harmonic instead of the 7th. It is named for Arthur Lipsett, the director off the Canadian short film ’21-87’. Leia’s prison cell in Star Wars is numbered ‘2187’, as a nod to the influence the film had on George Lucas.
Lipsett is a pleasantly melodic little temperament with a highly usable 5-tone and 9-tone mos. It is audibly similar to [[semaphore]] temperament, so it could be thought of as semaphore but for the 23rd harmonic instead of the 7th. It is named for Arthur Lipsett, the director of the Canadian short film ''21-87''. Leia's prison cell in ''Star Wars'' is numbered 2187, as a nod to the influence the film had on George Lucas.


[[Subgroup]]: 2.3.23
[[Subgroup]]: 2.3.23


[[Comma list]]: [[2187/2116]]
[[Comma list]]: 2187/2116


{{Mapping|legend=2| 1 0 -1 | 0 2 7}}
{{Mapping|legend=2| 1 0 -1 | 0 2 7 }}
: mapping generators: ~2, ~46/27


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~46/27 = 948.526
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.5339{{c}}, ~46/27 = 948.5629{{c}}
: [[error map]]: {{val| +0.534 -4.829 +11.132 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~46/27 = 948.3272{{c}}
: error map: {{val| 0.000 -5.301 +10.016 }}


{{Optimal ET sequence|legend=1| 5, 14, 19, 43, 62i, 81i }}
{{Optimal ET sequence|legend=1| 5, 14, 19, 43, 62i, 81i }}


[[Badness]] (Smith): 8.998 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.801


=== Porpoise ===
=== Porpoise ===
Line 1,200: Line 1,306:
[[Comma list]]: 24576/24389
[[Comma list]]: 24576/24389


[[Mapping]]: [{{val|1 2 5}}, {{val|0 3 -1}}]
{{Mapping|legend=2| 1 2 5 | 0 -3 -1 }}
: mapping generators: ~2, ~32/29


[[CTE tuning|CTE generator]]: ~32/29 = 166.067
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5519{{c}}, ~32/29 = 165.7453{{c}}
: [[error map]]: {{val| -0.448 -0.087 +2.437 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~32/29 = 165.9004{{c}}
: error map: {{val| 0.000 +0.344 +4.522 }}


{{Optimal ET sequence|legend=1| 7, 22, 29, 94, 123, 152j, 275jj, 427jjj }}
{{Optimal ET sequence|legend=1| 7, 22, 29, 94, 123, 152j, 275jj, 427jjj }}


=== Sematology ===
[[Badness]] (Sintel): 0.317
This temperament tempers out 4107/4096 and thus equates 2 [[37/32]]'s with [[4/3]].


[[Subgroup]]: 2.3.37
=== Jackpot ===
Named by {{u|CompactStar}} in 2024, jackpot tempers out the [[7-29-comma]] in 2.3.29, setting 32/29 to one step of [[7edo]] with an independent generator for the [[3/1|3rd harmonic]]. The name refers to the [[Temperament merging|EDO join]] of 7 & 77.


[[Comma list]]: 4107/4096
[[Subgroup]]: 2.3.29


[[Gencom]]: [2 37/32; 4107/4096]
[[Comma list]]: 17249876309/17179869184


[[Mapping]]: [{{val|1 1 5}}, {{val|0 -2 1}}]
{{Mapping|legend=2| 7 0 34 | 0 1 0 }}
: mapping generators: ~32/29, ~3


[[POTE generator]]: ~[[37/32]] = 249.075
[[Optimal tuning]]s:
* [[WE]]: ~32/29 = 171.4434{{c}}, ~3/2 = 701.8515{{c}}
: [[error map]]: {{val| +0.103 -0.000 -0.503 }}
* [[CWE]]: ~32/29 = 171.4286{{c}}, ~3/2 = 701.8456{{c}}
: error map: {{val| 0.000 -0.109 -1.006 }}


{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 53, 77, 130 }}
{{Optimal ET sequence|legend=1| 7, 42, 49, 56, 63, 70, 77, 147, 224 }}


==== 2.3.7.37 subgroup ====
[[Badness]] (Sintel): 1.02
[[Subgroup]]: 2.3.7.37


[[Comma list]]: 4107/4096, 259/256
=== Sematology ===
This temperament tempers out [[4107/4096]] and thus equates a stack of two [[37/32]]'s with [[4/3]].


[[Gencom]]: [2 37/32; 4107/4096 259/256]
[[Subgroup]]: 2.3.37


[[Mapping]]: [{{val|1 1 1 5}}, {{val|0 -2 -1 1}}]
[[Comma list]]: 4107/4096


[[POTE generator]]: ~[[37/32]] = 247.782
{{Mapping|legend=2| 1 0 6 | 0 2 -1 }}
: mapping generators: ~2, ~64/37


{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 53d }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2184{{c}}, ~64/37 = 950.9546{{c}}
: [[error map]]: {{val| +0.218 -0.046 -0.988 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/37 = 950.8250{{c}}
: error map: {{val| 0.000 -0.305 -2.169 }}


==== 2.3.5.37 subgroup ====
{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 53, 77, 130, 443l, 573ll, 703ll, 1536blllll }}
It is difficult to extend sematology to include 5, due the 5th harmonic being quite high-complexity.


[[Subgroup]]: 2.3.5.37
[[Badness]] (Sintel): 0.0690
 
[[Comma list]]: 4107/4096, 17592186044416/17562397269605
 
[[Gencom]]: [2 37/32; 4107/4096 17592186044416/17562397269605]
 
[[Mapping]]: [{{val|1 1 4 5}}, {{val|0 -2 -8 1}}]
 
[[POTE generator]]: ~[[37/32]] = 251.393
 
{{Optimal ET sequence|legend=1| 5, 14c, 19, 43, 62 }}
 
===== 2.3.5.7.37 subgroup =====
[[Subgroup]]: 2.3.5.7.37
 
[[Comma list]]: 4107/4096, 17592186044416/17562397269605, 259/256
 
[[Gencom]]: [2 37/32; 4107/4096 17592186044416/17562397269605 259/256]
 
[[Mapping]]: [{{val|1 1 4 1 5}}, {{val|0 -2 -8 -1 1}}]
 
[[POTE generator]]: ~[[37/32]] = 251.204
 
{{Optimal ET sequence|legend=1| 5, 14c, 19 }}


=== Reversed mavila ===
=== Reversed mavila ===
Line 1,267: Line 1,364:
[[Comma list]]: 81/74
[[Comma list]]: 81/74


[[Gencom]]: [2 4/3; 81/74]
{{Mapping|legend=2| 1 0 -1 | 0 1 4 }}
: mapping generators: ~2, ~3


[[Mapping]]: [{{val|1 1 0}}, {{val|0 -1 12}}]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1201.9908{{c}}, ~3/2 = 676.4865{{c}}
: [[error map]]: {{val| +1.991 -23.478 +60.575 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 676.7603{{c}}
: error map: {{val| 0.000 -25.195 +55.697 }}


[[POTE generator]]: ~4/3 = 521.397
{{Optimal ET sequence|legend=1| 2, 5l, 7l, 9, 16l }}


{{Optimal ET sequence|legend=1| 5l, 7l, 9, 16l }}
[[Badness]] (Sintel): 0.623


=== Reversed meantone ===
=== Reversed meantone ===
{{Main| Reversed meantone }}
{{Main| Reversed meantone }}


Subgroup: 2.3.41
[[Subgroup]]: 2.3.41


[[Comma list]]: 82/81
[[Comma list]]: 82/81


[[Gencom]]: [2 4/3; 82/81]
{{Mapping|legend=2| 1 0 -1 | 0 1 4 }}
: mapping generators: ~2, ~3


[[Mapping|Sval mapping]]: [{{val|1 2 7}}, {{val|0 -1 -4}}]
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1199.6907{{c}}, ~3/2 = 705.3096{{c}}
[[Tp tuning|POL2 generator]]: ~4/3 = 494.509
: [[error map]]: {{val| -0.309 +3.045 -8.752 }}
 
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 705.2699{{c}}
{{Optimal ET sequence|legend=1| 5, 12, 17 }}
: error map: {{val| 0.000 +3.315 -7.983 }}
 
==== 2.3.7.41 subgroup ====
Subgroup: 2.3.7.41
 
[[Comma list]]: 64/63, 82/81
 
[[Gencom]]: [2 4/3; 64/63 82/81]
 
[[Mapping|Sval mapping]]: [{{val| 1 2 2 7 }}, {{val| 0 -1 2 -4 }}]
 
[[POTE generator]]: ~4/3 = 490.0323
 
[[TOP tuning|TOP generator]]s: ~2 = 1197.2342, ~4/3 = 488.9029
 
{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 49 }}
 
==== 2.3.7.11.41 subgroup ====
Subgroup: 2.3.7.11.41
 
[[Comma list]]: 64/63, 82/81, 99/98
 
[[Gencom]]: [2 4/3; 64/63 82/81 99/98]
 
[[Mapping|Sval mapping]]: [{{val| 1 2 2 1 7 }}, {{val| 0 -1 2 6 -4 }}]
 
[[POTE generator]]: ~4/3 = 492.1787


[[TOP tuning|TOP generator]]s: ~2 = 1197.9683, ~4/3 = 491.3454
{{Optimal ET sequence|legend=1| 5, 12, 17, 97m, 114m, 131m }}


{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 39d }}
[[Badness]] (Sintel): 0.0841


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Subgroup temperaments]]
[[Category:Subgroup temperaments]]
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/No-fives_subgroup_temperaments"