Xenharmonic Wiki

Subgroup temperaments: Difference between revisions

← Older edit
VisualWikitext
ArrowHead294 (talk | contribs)
15,220 edits
mNo edit summary
Overthink (talk | contribs)
autopatrolled
2,970 edits
→Supramin: add note
 
(62 intermediate revisions by 14 users not shown)
Line 1: Line 1:
{{Technical data page}}
A '''subgroup temperament''' is a regular temperament defined on a [[just intonation subgroup]] that is not a full ''p''-limit group.  
A '''subgroup temperament''' is a regular temperament defined on a [[just intonation subgroup]] that is not a full ''p''-limit group.  


Line 12: Line 13:
= Composite subgroup temperaments =
= Composite subgroup temperaments =
== 2.3.35 subgroup ==
== 2.3.35 subgroup ==
=== Shaka ===
=== Darian calendar ===
{{See also|Kalismic temperaments}}
Darian calendar is described as 24 & 668 temperament in the 2.3.11.19 [[subgroup]] and is named after a certain calendar layout by the same name. The generator is close to the [[36/35]] quartertone, and this allows an extension to the 2.3.35.11.19 subgroup. 5 of them make [[11/8]], 8 of them make [[3/2]], and 6 of them make [[32/19]].


Two commas that split 2/1 in half, corresponding to convergents to sqrt(2), are the [[1682/1681|''sha''ftesburisma]] [[Square superparticular|S29]]/S41 and the [[9801/9800|''ka''lisma S99]], prompting to temper out {S29, S41, S99}, approximating /29 and /41 [[Primodality|primodal]] chords well.
==== 2.3.11.19 subgroup ====
The temperament is simplest in this subgroup, although there is a tradeoff of breaking up the simplicity of the 36/35 quartertone.


Subgroup: 2.3.35.11.29.41
[[Subgroup]]: 2.3.11.19


Comma list: 841/840, 1189/1188, 1681/1680
{{Mapping|legend=2| 4 5 13 18 | 0 8 5 -6 }}


{{Mapping|legend=2|2 2 6 5 7 8|0 1 1 -1 1 1|0 0 2 2 1 1}}
: sval mapping generators: ~6291456/5285401, ~25289/24576


Optimal tuning (CTE): ~41/29 = 1\2, ~3/2 = 702.031, ~41/24 = 926.693
[[Optimal tuning]] ([[CTE]]): ~6291456/5285401 = 1\4, ~25289/24576 = 50.257


[[Support]]ing [[ET]]s: {{EDOs|22, 26, 36, 48, 70, 96, 106, 118, 140, 154, 176, 188, 224, 272, 294, 342}}
[[Support]]ing [[ET]]s: {{EDOs|24, 596, 620, 644, 668, 692, 716}}, ...


Scale: [[Shaka10]]
==== 2.3.35.11.19 subgroup ====
668edo does not map 36/35 consistently, with its own [[direct approximation]] being 27 steps while the direct approximations of its constituent odd harmonics do not sum to that same amount: 3/2, 8/5, and 8/7 are 391, 453, and 129 steps, respectively, and 391 + 391 + 453 + 129 - 668 - 668 = 28, ≠ 27.
 
Subgroup: 2.3.35.11.19
 
Sval mapping: {{mapping| 4 0 5 13 18 | 0 1 8 5 -6 }}
 
: sval mapping generators: ~2240/1881, ~36/35
 
Optimal tuning (CTE): ~2240/1881 = 1\4, ~36/35 = 50.288
 
[[Support]]ing [[ET]]s: {{EDOs|24, 668}}, ...


== 2.9.5.7 subgroup ==
== 2.9.5.7 subgroup ==
Line 89: Line 102:
=== Baldy ===
=== Baldy ===
{{See also|Schismatic family #Garibaldi}}
{{See also|Schismatic family #Garibaldi}}
{{See also|No-threes subgroup temperaments #Frostburn}}


Baldy results from taking every other generator of the [[garibaldi temperament]]. One of the best extension is 2.9.5.7.13 subgroup with mapping 13/8 to +10 whole tones, as well as the cassandra temperament.
Baldy results from taking every other generator of the [[garibaldi]] temperament. One of the best extension is 2.9.5.7.13 subgroup with mapping 13/8 to +10 whole tones, as well as the cassandra temperament.


[[Subgroup]]: 2.9.5.7
[[Subgroup]]: 2.9.5.7
Line 152: Line 166:


{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}
{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}
== 2.3.25 subgroup ==
=== Shrub ===
This is a restriction of diaschismic which omits the tritone to produce a diatonic scale. True to its name, it generates a [[shrubmajor]] third (~425c) in quarter-comma tuning.
Subgroup: 2.3.25
Edo join: 17 & 12
Comma list: [[2048/2025]]
{{Mapping|legend=2| 1 1 7| 0 1 -4}}
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.136
==== 2.3.23.25.41 subgroup ====
''See also: [[Reversed meantone]]''
Edo join: 17 & 12
Comma list: 2048/2025, 576/575, 82/81
{{Mapping|legend=2| 1 1 1 7 3| 0 1 6 -4 4}}
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.264
===== Sburb =====
This temperament sets the [[octave reduction|octave-reduced]] 413th harmonic (413/256, 827.998{{c}}) to the diminished seventh.
Subgroup: 2.3.7.23.25.41.59
Edo join: 17 & 12
Comma list: 64/63, 225/224, 162/161, 82/81, 177/175
{{Mapping|legend=2| 1 1 4 1 7 3 10| 0 1 -2 6 -4 4 -7}}
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 706.387


== 2.9.5.11 subgroup ==
== 2.9.5.11 subgroup ==
Line 290: Line 343:
Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]
Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]


== 2.9.11 subgroup ==
== 2.9.7.13.17 subgroup ==
=== Demon ===
Demon is a temperament which equates 3 [[11/9]] with [[16/9]], or equivalently 3 [[18/11]] with [[9/8]], tempering out [[1331/1296]]. This results in [[11/9]] being tuned flat to a supraminor third, and [[27/22]] being tuned sharp to a submajor third. It was discovered by [[User:CompactStar|CompactStar]] while searching for temperaments assosciated with the [[7L 4s]] ("daemotonic") MOS, known for its lack of representation of simple temperaments. The optimal tuning for demon temperament is near the basic tuning of 7L 4s (13\18), and indeed [[18edo]] supports demon temperament.


[[Subgroup]]: 2.9.11
=== Novisept ===
Novisept is generated by a one-cent-flat 9/7, such that stacking 5 of them gives you 7/4. It can be formed by doubling both generator and period of [[gizzard]].


[[Comma list]]: [[1331/1296]]
[[Subgroup]]: 2.9.7.13.17
 
[[Comma list]]: 729/728, 442/441, 833/832
 
{{Mapping|legend=2| 1 1 1 -1 3| 0 6 5 13 3 }}
 
[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~9/7 = 433.836
 
Badness (Dirichlet): 0.142
 
== 2.9.11 subgroup ==
=== Demon ===
Demon is a temperament which equates 3 [[11/9]] with [[16/9]], or equivalently 3 [[18/11]] with [[9/8]], tempering out [[1331/1296]]. This results in [[11/9]] being tuned flat to a supraminor third, and [[27/22]] being tuned sharp to a submajor third. It was discovered by [[User:CompactStar|CompactStar]] while searching for temperaments assosciated with the [[7L 4s]] ("daemotonic") MOS, known for its lack of representation of simple temperaments. The optimal tuning for demon temperament is near the basic tuning of 7L 4s (13\18), and indeed [[18edo]] supports demon temperament.
 
[[Subgroup]]: 2.9.11
 
[[Comma list]]: [[1331/1296]]


{{Mapping|legend=2|1 1 2|0 3 2}}
{{Mapping|legend=2|1 1 2|0 3 2}}
Line 439: Line 507:
{{Optimal ET sequence|legend=1| 5, 13c, 18, 23, 41, 64, 87, 151 }}
{{Optimal ET sequence|legend=1| 5, 13c, 18, 23, 41, 64, 87, 151 }}


== 2.15.55 subgroup ==
== 2.75.85 subgroup ==
=== Spog ===
=== MVP archagall ===
By tempering out the comma [[24576/24565]] in the 2.75.85 subgroup, we have three [[85/64]]'s up and one octave down as a [[75/64]] and we have two [[128/85]]'s up and one octave down as a [[17/15]] whole tone. It is because of this combination of accuracy, efficiency and mapping-wise simplicity and its corresponding explanatory power in what this comma does that the comma has been named the ''archagallisma''. The ''MVP'' stands for ''minimum viable product'', as this is the core of what the archagall logic achieves, with further extensions adding to the subgroup while avoiding significantly impacting its accuracy. This is a highly accurate temperament that could be considered to be encoding the "high-accuracy logic" of [[superpyth]] and which is inescapably related to the [[17L 5s]] scale form as it is the 17 & 22 temperament (or less accurately, the 5 & 17 temperament) in the 2.75.85 subgroup.


This temperament produces [[Slendro_clan#Superpelog|superpelog]]-like [[5L 4s|semiquartal]] scales while being more accurate ([[Subgroup temperaments#2.15.189.55.325.725.279|see]] rational approximations to their intervals).
[[Subgroup]]: 2.75.85


[[Subgroup]]: 2.15.55
[[Comma list]]: 24576/24565 ({{monzo| 13 1 -3 }})


[[Comma list]]: [[100663296/100656875]]
{{Mapping|legend=2| 1 2 5 | 0 3 1 }}
: mapping generators: ~2, ~85/32


{{Mapping|legend=2| 1 0 5 | 0 5 1 }}
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9692{{c}}, ~85/64 = 491.5853{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~85/64 = 491.5794{{c}}


[[Optimal tuning]] (subgroup [[CTE]]): ~55/32 = 937.655
{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 61, 83, 310, 393, 476, 1345, 1821, 4118*, 5939*, 7760* }}


{{Optimal ET sequence|legend=1|5, 9, 23, 32, 151, 183, 215, 247, 956, 1203, 1450, 3147, 4597 }}
== 2.75.9/7.85 subgroup ==
=== Archagall ===
A fairly natural way to extend [[#MVP archagall|MVP archagall]] is by tempering out [[2025/2023]] ([[S-expression|S15/S17]]), which equates a stack of two [[17/15]]'s with [[9/7]] without much damage. As 9/7 was not previously in the subgroup, this does not decrease the rank of the temperament and qualifies a proper and natural extension. We can equally get the same temperament by tempering out S15/S16 instead (equating a stack of three [[16/15]]'s with [[17/14]]); however, [[16/15]] is not in the subgroup, so it is preferred to think of it as adding 2025/2023.


==== 2.15.55.325 ====
[[Subgroup]]: 2.75.9/7.85
[[Subgroup]]: 2.15.55.325


[[Comma list]]: [[4225/4224]], [[6656/6655]]
[[Comma list]]: 2025/2023 ({{monzo| 2 -2 1 0 }}), 24576/24565 ({{monzo| 13 1 0 -3 }})


{{Mapping|legend=2| 1 0 5 6 | 0 5 1 3 }}
{{Mapping|legend=2| 1 2 6 5 | 0 3 -4 1 }}
: mapping generators: ~2, ~85/32


[[Optimal tuning]] (subgroup [[CTE]]): ~55/32 = 937.647
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.0241{{c}}, ~85/64 = 491.3358{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~85/64 = 491.3290{{c}}


[[Support]]ing [[ET]]s: 5, 9, 13[-15], 14, 23, 32, 37, 41, 50, 55, 64, 73, 78, 87, 96, 101, 105, 119, 128, 151, 183, 206, 311
{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 83, 105, 127, 403, 530, 657, 784, 1441* }}


==== 2.15.189.55.325 ====
== 4.3.5 subgroup ==
Related temperament: [[Lehmerismic_temperaments#Lux|lux]]
=== Tetrahanson ===
{{Main| Tetrahanson }}


[[Subgroup]]: 2.15.189.55.325
[[Subgroup]]: 4.3.5


[[Comma list]]: [[2080/2079]], [[3025/3024]], [[4096/4095]]
[[Comma list]]: 15625/15552


{{Mapping|legend=2| 1 0 6 5 6 | 0 5 2 1 3 }}
{{Mapping|legend=2| 1 3 3 | 0 -6 -5 }}


[[Optimal tuning]] (subgroup [[CTE]]): ~55/32 = 937.677
: Mapping generators: ~4, ~5/3


[[Support]]ing [[ET]]s: 5, 9, 14, 23, 32, 37, 41, 50, 55, 64, 73, 78, 87, 96, 101, 105, 119, 128, 151, 183, 206, 311
[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~5/3 = 882.941


==== 2.15.189.55.325.725 ====
[[Support]]ing [[ET]]s: {{EDs|19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79|equave=4}}


[[Subgroup]]: 2.15.189.55.325.725
=== Tetrameantone ===
{{Main| Tetrameantone }}


[[Comma list]]: [[1625/1624]], [[2080/2079]], [[3025/3024]], [[4096/4095]]
[[Subgroup]]: 4.3.5


{{Mapping|legend=2| 1 0 6 5 6 -3 | 0 5 2 1 3 16 }}
[[Comma list]]: 81/80


[[Optimal tuning]] (subgroup [[CTE]]): ~55/32 = 937.649
{{Mapping|legend=2| 1 1 2 | 0 -1 -4 }}


[[Support]]ing [[ET]]s: 9[-725], 14[+725], 23, 32, 41[-725], 55, 73[-725], 87, 105[-725], 119,  142[+725], 151, 183, 206[+725], 311
: Mapping generators: ~4, ~4/3


==== 2.15.189.55.325.725.279 ====
[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~4/3 = 503.761


Here are rational approximations to the intervals of the semiquartal scale.
[[Support]]ing [[ET]]s: {{EDs|5, 9, 14, 19, 24, 43, 62, 81, 100|equave=4}}


Sharp: 12/11, 25/21, 33/26, 18/13, 31/21 ~ 65/44 ~ 96/65, 50/31 ~ 29/18, 55/32, 15/8.
=== Tetramagic ===


Flat: 16/15, 64/55, 31/25 ~ 36/29, 42/31 ~ 65/48 ~ 88/65, 13/9, 52/33, 42/25, 11/6.
[[Subgroup]]: 4.3.5
 
[[Subgroup]]: 2.15.189.55.325.725.279
[[Comma list]]: 3125/3072


[[Comma list]]: [[1625/1624]], [[2016/2015]], [[2080/2079]], [[3025/3024]], [[4096/4095]]
{{Mapping|legend=2| 1 0 1 | 0 5 1 }}


{{Mapping|legend=2| 1 0 6 5 6 -3 5 | 0 5 2 1 3 16 4 }}
: Mapping generators: ~4, ~5/4


[[Optimal tuning]] (subgroup [[CTE]]): ~55/32 = 937.638
[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~5/4 = 380.059


[[Support]]ing [[ET]]s: 9[-725], 14[+725], 23, 32, 41[-725], 55, 73[-725], 87, 105[-725], 119, 151, 183, 206[+725], 311
[[Support]]ing [[ET]]s: {{EDs|6, 13, 19, 25, 38, 44, 63, 82|equave=4}}


== 4.3.5 subgroup ==
=== Blacktetra ===
=== Tetrahanson ===
{{Main| Tetrahanson }}


[[Subgroup]]: 4.3.5
[[Subgroup]]: 4.3.5


[[Comma list]]: 15625/15552
[[Comma list]]: 256/243


{{Mapping|legend=2| 1 3 3 | 0 -6 -5 }}
{{Mapping|legend=2| 5 4 6 | 0 0 -1 }}


: Mapping generators: ~4, ~5/3
: Mapping generators: ~4, ~16/15


[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~5/3 = 882.941
[[Optimal tuning]] ([[POTE]]): 1\5ed4 = 480.0, ~16/15 = 80.4062


[[Support]]ing [[ET]]s: {{EDs|19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79|equave=4}}
[[Support]]ing [[ET]]s: {{EDs|5, 10, 15, 20, 25, 30, 55, 85, 115|equave=4}}


=== Tetrameantone ===
== 4.6.5 subgroup ==
{{Main| Tetrameantone }}
=== Meanquad ===
{{Main| Meanquad }}


[[Subgroup]]: 4.3.5
[[Subgroup]]: 4.6.5


[[Comma list]]: 81/80
[[Comma list]]: [[81/80]] = {{monzo| -4 4 -1 }}


{{Mapping|legend=2| 1 1 2 | 0 -1 -4 }}
{{Mapping|legend=2| 1 0 -4| 0 1 4 }}


: Mapping generators: ~4, ~4/3
: mapping generators: ~4, ~6


[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~4/3 = 503.761
[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 697.214


[[Support]]ing [[ET]]s: {{EDs|5, 9, 14, 19, 24, 43, 62, 81, 100|equave=4}}
[[Support]]ing [[ET]]s: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69


=== Tetramagic ===
<nowiki />* Wart for 4


[[Subgroup]]: 4.3.5
==== 4.6.5.7 subgroup (tetrominant) ====
[[Subgroup]]: 4.6.5.7


[[Comma list]]: 3125/3072
[[Comma list]]: [[36/35]] = {{monzo| 0 2 -1 -1 }}, [[64/63]] = {{monzo| 4 -2 0 -1 }}


{{Mapping|legend=2| 1 0 1 | 0 5 1 }}
{{Mapping|legend=2| 1 0 -4 4 | 0 1 4 -2 }}


: Mapping generators: ~4, ~5/4
[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 699.622


[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~5/4 = 380.059
[[Support]]ing [[ET]]s: *7, *10, *17, *24, *27[+5], *31, *38[+7], *41, *44[+5], *55[+7], *58[+5, +7], *65[+5, +7], *75[+5, +7]


[[Support]]ing [[ET]]s: {{EDs|6, 13, 19, 25, 38, 44, 63, 82|equave=4}}
<nowiki />* Wart for 4


=== Blacktetra ===
=== Fourwar ===
The 23-limit version of Fourwar was created first, as an attempt to approximate subgroup 4.6.5.7.11.13.17.19.23 as accurately as possible using 25 to 35 notes per equave. Then the lower limit versions were created by simply extrapolating the temperament downwards.


[[Subgroup]]: 4.3.5
Fourwar is named after the closely related [[hemiwar]] temperament.


[[Comma list]]: 256/243
{{Todo|inline=1|cleanup}}


{{Mapping|legend=2| 5 4 6 | 0 0 -1 }}
<pre>
 
Reduced Mapping
: Mapping generators: ~4, ~16/15
4 6 5
 
[ ⟨ 1 0 1 ]
[[Optimal tuning]] ([[POTE]]): 1\5ed4 = 480.0, ~16/15 = 80.4062
⟨ 0 16 2 ] ⟩
 
[[Support]]ing [[ET]]s: {{EDs|5, 10, 15, 20, 25, 30, 55, 85, 115|equave=4}}
TE Generator Tunings (cents)
 
⟨2399.3973, 193.8643]
== 4.6.5 subgroup ==
=== Meanquad ===
TE Step Tunings (cents)
{{Main| Meanquad }}
⟨25.21211, 47.81337]
TE Tuning Map (cents)
⟨2399.397, 3101.829, 2787.126]
TE Mistunings (cents)
⟨-0.603, -0.126, 0.812]
Complexity 1.369085
Adjusted Error 0.692892 cents
TE Error 0.268047 cents/octave
Unison Vector
[8, 1, -8⟩ (393216:390625)


[[Subgroup]]: 4.6.5
Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>


[[Comma list]]: [[81/80]] = {{monzo| -4 4 -1 }}
==== 4.6.5.7 ====
 
<pre>
{{Mapping|legend=2| 1 0 -4| 0 1 4 }}
Reduced Mapping
 
4 6 5 7
: mapping generators: ~4, ~6
[ ⟨ 1 0 1 1 ]
⟨ 0 16 2 5 ] ⟩
TE Generator Tunings (cents)
⟨2399.4195, 193.8654]
TE Step Tunings (cents)
⟨25.23883, 47.79592]
TE Tuning Map (cents)
⟨2399.420, 3101.846, 2787.150, 3368.747]
TE Mistunings (cents)
⟨-0.580, -0.109, 0.837, -0.079]
Complexity 1.192044
Adjusted Error 0.653313 cents
TE Error 0.232715 cents/octave
Unison Vectors
[-2, -1, -2, 4⟩ (2401:2400)
[3, 0, -5, 2⟩ (3136:3125)
[5, 1, -3, -2⟩ (6144:6125)
[8, 1, -8, 0⟩ (393216:390625)


[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 697.214
Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>


[[Support]]ing [[ET]]s: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69
==== 4.6.5.7.11 ====
 
<pre>
<nowiki />* Wart for 4
 
==== 4.6.5.7 subgroup (tetrominant) ====
[[Subgroup]]: 4.6.5.7
 
[[Comma list]]: [[36/35]] = {{monzo| 0 2 -1 -1 }}, [[64/63]] = {{monzo| 4 -2 0 -1 }}
 
{{Mapping|legend=2| 1 0 -4 4 | 0 1 4 -2 }}
 
[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 699.622
 
[[Support]]ing [[ET]]s: *7, *10, *17, *24, *27[+5], *31, *38[+7], *41, *44[+5], *55[+7], *58[+5, +7], *65[+5, +7], *75[+5, +7]
 
<nowiki />* Wart for 4
 
=== Fourwar ===
The 23-limit version of Fourwar was created first, as an attempt to approximate subgroup 4.6.5.7.11.13.17.19.23 as accurately as possible using 25 to 35 notes per equave. Then the lower limit versions were created by simply extrapolating the temperament downwards.
 
Fourwar is named after the closely related [[hemiwar]] temperament.
 
<pre>  
Reduced Mapping
Reduced Mapping
4 6 5
4 6 5 7 11
[ ⟨ 1 0 1 ]
[ ⟨ 1 0 1 1 1 ]
⟨ 0 16 2 ] ⟩
⟨ 0 16 2 5 9 ] ⟩
   
   
TE Generator Tunings (cents)
TE Generator Tunings (cents)
⟨2399.3973, 193.8643]
⟨2400.1097, 193.9498]
   
   
TE Step Tunings (cents)
TE Step Tunings (cents)
⟨25.21211, 47.81337]
⟨24.18752, 48.52491]
   
   
TE Tuning Map (cents)
TE Tuning Map (cents)
⟨2399.397, 3101.829, 2787.126]
⟨2400.110, 3103.196, 2788.009, 3369.859, 4145.658]
   
   
TE Mistunings (cents)
TE Mistunings (cents)
⟨-0.603, -0.126, 0.812]
⟨0.110, 1.241, 1.696, 1.033, -5.660]
   
   
Complexity 1.369085
Complexity 1.068792
Adjusted Error 0.692892 cents
Adjusted Error 2.926965 cents
TE Error 0.268047 cents/octave
TE Error 0.846083 cents/octave
   
   
Unison Vector
Unison Vectors
[8, 1, -8⟩ (393216:390625)
[-1, -1, -1, 0, 2⟩ (121:120)
[2, 0, -2, -1, 1⟩ (176:175)
[-3, -1, 1, 1, 1⟩ (385:384)
[-1, 0, 3, -3, 1⟩ (1375:1372)
[-2, -1, -2, 4, 0⟩ (2401:2400)
[1, 0, 1, -4, 2⟩ (2420:2401)


Subsets
Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
</pre>
</pre>


==== 4.6.5.7 ====
==== 4.6.5.7.11.13 ====
 
<pre>
<pre>
Reduced Mapping
Reduced Mapping
4 6 5 7
4 6 5 7 11 13
[ ⟨ 1 0 1 1 ]
[ ⟨ 1 0 1 1 1 0 ]
⟨ 0 16 2 5 ] ⟩
⟨ 0 16 2 5 9 23 ] ⟩
   
   
TE Generator Tunings (cents)
TE Generator Tunings (cents)
⟨2399.4195, 193.8654]
⟨2401.2305, 193.5378]
   
   
TE Step Tunings (cents)
TE Step Tunings (cents)
⟨25.23883, 47.79592]
⟨42.79107, 35.98524]
   
   
TE Tuning Map (cents)
TE Tuning Map (cents)
⟨2399.420, 3101.846, 2787.150, 3368.747]
⟨2401.230, 3096.606, 2788.306, 3368.920, 4143.071, 4451.371]
   
   
TE Mistunings (cents)
TE Mistunings (cents)
⟨-0.580, -0.109, 0.837, -0.079]
⟨1.230, -5.349, 1.992, 0.094, -8.247, 10.843]
   
   
Complexity 1.192044
Complexity 1.219191
Adjusted Error 0.653313 cents
Adjusted Error 6.699599 cents
TE Error 0.232715 cents/octave
TE Error 1.810487 cents/octave
   
   
Unison Vectors
Unison Vectors
[-2, -1, -2, 4⟩ (2401:2400)
[0, 1, -1, 0, 1, -1⟩ (66:65)
[3, 0, -5, 2⟩ (3136:3125)
[-1, -1, -1, 0, 2, 0⟩ (121:120)
[5, 1, -3, -2⟩ (6144:6125)
[1, 2, 0, 0, -1, -1⟩ (144:143)
[8, 1, -8, 0⟩ (393216:390625)
[2, 0, -2, -1, 1, 0⟩ (176:175)
[-2, 1, 1, 1, 0, -1⟩ (105:104)
[-3, -1, 1, 1, 1, 0⟩ (385:384)
[-3, 0, 0, 1, 2, -1⟩ (847:832)
[1, 3, -1, 0, 0, -2⟩ (864:845)
[-1, 0, 3, -3, 1, 0⟩ (1375:1372)


Subsets
Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
</pre>
</pre>


==== 4.6.5.7.11 ====
==== 4.6.5.7.11.13.17 ====
<pre>
<pre>
Reduced Mapping
Reduced Mapping
4 6 5 7 11
4 6 5 7 11 13 17
[ ⟨ 1 0 1 1 1 ]
[ ⟨ 1 0 1 1 1 0 1 ]
⟨ 0 16 2 5 9 ] ⟩
⟨ 0 16 2 5 9 23 13 ] ⟩
   
   
TE Generator Tunings (cents)
TE Generator Tunings (cents)
⟨2400.1097, 193.9498]
⟨2400.4701, 193.4599]
   
   
TE Step Tunings (cents)
TE Step Tunings (cents)
⟨24.18752, 48.52491]
⟨43.39350, 35.55764]
   
   
TE Tuning Map (cents)
TE Tuning Map (cents)
⟨2400.110, 3103.196, 2788.009, 3369.859, 4145.658]
⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449]
   
   
TE Mistunings (cents)
TE Mistunings (cents)
⟨0.110, 1.241, 1.696, 1.033, -5.660]
⟨0.470, -6.596, 1.076, -1.056, -9.709, 9.050, 10.494]
   
   
Complexity 1.068792
Complexity 1.129881
Adjusted Error 2.926965 cents
Adjusted Error 8.082725 cents
TE Error 0.846083 cents/octave
TE Error 1.977443 cents/octave
   
   
Unison Vectors
Unison Vectors
[-1, -1, -1, 0, 2⟩ (121:120)
[0, 1, -1, 0, 1, -1, 0⟩ (66:65)
[2, 0, -2, -1, 1⟩ (176:175)
[1, 1, 1, -1, 0, 0, -1⟩ (120:119)
[-3, -1, 1, 1, 1⟩ (385:384)
[1, 2, 0, 0, -1, -1, 0⟩ (144:143)
[-1, 0, 3, -3, 1⟩ (1375:1372)
[-2, 1, 1, 1, 0, -1, 0⟩ (105:104)
[-2, -1, -2, 4, 0⟩ (2401:2400)
[-1, 2, 2, 0, 0, -1, -1⟩ (225:221)
[1, 0, 1, -4, 2⟩ (2420:2401)
[-1, 1, 2, -2, 0, -1, 1⟩ (1275:1274)


Subsets
Subsets
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg
</pre>
</pre>


==== 4.6.5.7.11.13 ====
==== 4.6.5.7.11.13.17.19 ====
 
<pre>
<pre>
Reduced Mapping
Reduced Mapping
4 6 5 7 11 13
4 6 5 7 11 13 17 19
[ ⟨ 1 0 1 1 1 0 ]
[ ⟨ 1 0 1 1 1 0 1 1 ]
⟨ 0 16 2 5 9 23 ] ⟩
⟨ 0 16 2 5 9 23 13 14 ] ⟩
   
   
TE Generator Tunings (cents)
TE Generator Tunings (cents)
⟨2401.2305, 193.5378]
⟨2399.9219, 193.3952]
   
   
TE Step Tunings (cents)
TE Step Tunings (cents)
⟨42.79107, 35.98524]
⟨44.14256, 35.03670]
   
   
TE Tuning Map (cents)
TE Tuning Map (cents)
⟨2401.230, 3096.606, 2788.306, 3368.920, 4143.071, 4451.371]
⟨2399.922, 3094.324, 2786.712, 3366.898, 4140.479, 4448.090, 4914.060, 5107.455]
   
   
TE Mistunings (cents)
TE Mistunings (cents)
⟨1.230, -5.349, 1.992, 0.094, -8.247, 10.843]
⟨-0.078, -7.631, 0.399, -1.928, -10.839, 7.562, 9.104, 9.942]
   
   
Complexity 1.219191
Complexity 1.058472
Adjusted Error 6.699599 cents
Adjusted Error 8.712222 cents
TE Error 1.810487 cents/octave
TE Error 2.050935 cents/octave
   
   
Unison Vectors
Unison Vectors
[0, 1, -1, 0, 1, -1⟩ (66:65)
[0, 1, -1, 0, 1, -1, 0, 0⟩ (66:65)
[-1, -1, -1, 0, 2, 0⟩ (121:120)
[-1, 0, 0, 1, 1, 0, 0, -1⟩ (77:76)
[1, 2, 0, 0, -1, -1⟩ (144:143)
[2, 1, -1, 0, 0, 0, 0, -1⟩ (96:95)
[2, 0, -2, -1, 1, 0⟩ (176:175)
[1, 1, 1, -1, 0, 0, -1, 0⟩ (120:119)
[-2, 1, 1, 1, 0, -1⟩ (105:104)
[0, 1, 1, 1, -1, 0, 0, -1⟩ (210:209)
[-3, -1, 1, 1, 1, 0⟩ (385:384)
[0, 0, 1, -2, 1, 0, 1, -1⟩ (935:931)
[-3, 0, 0, 1, 2, -1⟩ (847:832)
[2, 0, -3, 1, 0, 0, -1, 1⟩ (2128:2125)
[1, 3, -1, 0, 0, -2⟩ (864:845)
 
[-1, 0, 3, -3, 1, 0⟩ (1375:1372)
 
Subsets
Subsets
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh
</pre>
</pre>


==== 4.6.5.7.11.13.17 ====
==== 4.6.5.7.11.13.17.19.23 ====
<pre>
<pre>
Reduced Mapping
Reduced Mapping
4 6 5 7 11 13 17
4 6 5 7 11 13 17 19 23
[ ⟨ 1 0 1 1 1 0 1 ]
[ ⟨ 1 0 1 1 1 0 1 1 0 ]
⟨ 0 16 2 5 9 23 13 ] ⟩
⟨ 0 16 2 5 9 23 13 14 28 ] ⟩
   
   
TE Generator Tunings (cents)
TE Generator Tunings (cents)
⟨2400.4701, 193.4599]
⟨2399.3286, 193.5316]
   
   
TE Step Tunings (cents)
TE Step Tunings (cents)
⟨43.39350, 35.55764]
⟨37.31613, 39.63311]
   
   
TE Tuning Map (cents)
TE Tuning Map (cents)
⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449]
⟨2399.329, 3096.506, 2786.392, 3366.987, 4141.113, 4451.227, 4915.240, 5108.771, 5418.885]
   
   
TE Mistunings (cents)
TE Mistunings (cents)
⟨0.470, -6.596, 1.076, -1.056, -9.709, 9.050, 10.494]
⟨-0.671, -5.449, 0.078, -1.839, -10.205, 10.699, 10.284, 11.258, -9.389]
   
   
Complexity 1.129881
Complexity 1.115920
Adjusted Error 8.082725 cents
Adjusted Error 9.502017 cents
TE Error 1.977443 cents/octave
TE Error 2.100561 cents/octave
   
   
Unison Vectors
Unison Vectors
[0, 1, -1, 0, 1, -1, 0⟩ (66:65)
[0, 1, -1, 0, 1, -1, 0, 0, 0⟩ (66:65)
[1, 1, 1, -1, 0, 0, -1⟩ (120:119)
[1, 0, 0, -1, 0, -1, 0, 0, 1⟩ (92:91)
[1, 2, 0, 0, -1, -1, 0⟩ (144:143)
[0, -1, 1, 0, 0, 0, 0, -1, 1⟩ (115:114)
[-2, 1, 1, 1, 0, -1, 0⟩ (105:104)
[1, 1, 1, -1, 0, 0, -1, 0, 0⟩ (120:119)
[-1, 2, 2, 0, 0, -1, -1⟩ (225:221)
[2, 0, -2, -1, 1, 0, 0, 0, 0⟩ (176:175)
[-1, 1, 2, -2, 0, -1, 1⟩ (1275:1274)
[-3, -1, 1, 1, 1, 0, 0, 0, 0⟩ (385:384)
[1, 0, -2, 1, 0, 0, 1, -1, 0⟩ (476:475)
[1, 0, 0, -2, 1, 0, -1, 1, 0⟩ (836:833)
[0, 0, 1, -2, 1, 0, 1, -1, 0⟩ (935:931)
[1, -1, 0, 0, 0, 0, -2, 1, 1⟩ (874:867)


Subsets
Subsets
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii
</pre>
</pre>


==== 4.6.5.7.11.13.17.19 ====
== 4.9.25 subgroup ==
<pre>
=== Meansquared ===
Reduced Mapping
[[Subgroup]]: 4.9.25
4 6 5 7 11 13 17 19
 
[ ⟨ 1 0 1 1 1 0 1 1 ]
[[Comma list]]: [[6561/6400]]
⟨ 0 16 2 5 9 23 13 14 ] ⟩
 
{{Mapping|legend=2| 1 3 4 | 0 1 4 }}
TE Generator Tunings (cents)
⟨2399.9219, 193.3952]
TE Step Tunings (cents)
⟨44.14256, 35.03670]
TE Tuning Map (cents)
⟨2399.922, 3094.324, 2786.712, 3366.898, 4140.479, 4448.090, 4914.060, 5107.455]
TE Mistunings (cents)
⟨-0.078, -7.631, 0.399, -1.928, -10.839, 7.562, 9.104, 9.942]
Complexity 1.058472
Adjusted Error 8.712222 cents
TE Error 2.050935 cents/octave
Unison Vectors
[0, 1, -1, 0, 1, -1, 0, 0⟩ (66:65)
[-1, 0, 0, 1, 1, 0, 0, -1⟩ (77:76)
[2, 1, -1, 0, 0, 0, 0, -1⟩ (96:95)
[1, 1, 1, -1, 0, 0, -1, 0⟩ (120:119)
[0, 1, 1, 1, -1, 0, 0, -1⟩ (210:209)
[0, 0, 1, -2, 1, 0, 1, -1⟩ (935:931)
[2, 0, -3, 1, 0, 0, -1, 1⟩ (2128:2125)


Subsets
Mapping generators: ~4, ~9/64
q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh
 
</pre>
[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429
 
[[Support]]ing [[ET]]s: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]
 
== 4.9.49 subgroup ==
=== Archsquared ===
[[Subgroup]]: 4.9.49
 
[[Comma list]]: 4096/3969
 
{{Mapping|legend=2| 1 3 0 | 0 1 -2 }}


==== 4.6.5.7.11.13.17.19.23 ====
Mapping generators: ~4, ~9/64
<pre>
 
Reduced Mapping
[[Optimal tuning]] ([[CTE]]): ~9/4 = 1419.190
4 6 5 7 11 13 17 19 23
[ ⟨ 1 0 1 1 1 0 1 1 0 ]
⟨ 0 16 2 5 9 23 13 14 28 ] ⟩
TE Generator Tunings (cents)
⟨2399.3286, 193.5316]
TE Step Tunings (cents)
⟨37.31613, 39.63311]
TE Tuning Map (cents)
⟨2399.329, 3096.506, 2786.392, 3366.987, 4141.113, 4451.227, 4915.240, 5108.771, 5418.885]
TE Mistunings (cents)
⟨-0.671, -5.449, 0.078, -1.839, -10.205, 10.699, 10.284, 11.258, -9.389]
Complexity 1.115920
Adjusted Error 9.502017 cents
TE Error 2.100561 cents/octave
Unison Vectors
[0, 1, -1, 0, 1, -1, 0, 0, 0⟩ (66:65)
[1, 0, 0, -1, 0, -1, 0, 0, 1⟩ (92:91)
[0, -1, 1, 0, 0, 0, 0, -1, 1⟩ (115:114)
[1, 1, 1, -1, 0, 0, -1, 0, 0⟩ (120:119)
[2, 0, -2, -1, 1, 0, 0, 0, 0⟩ (176:175)
[-3, -1, 1, 1, 1, 0, 0, 0, 0⟩ (385:384)
[1, 0, -2, 1, 0, 0, 1, -1, 0⟩ (476:475)
[1, 0, 0, -2, 1, 0, -1, 1, 0⟩ (836:833)
[0, 0, 1, -2, 1, 0, 1, -1, 0⟩ (935:931)
[1, -1, 0, 0, 0, 0, -2, 1, 1⟩ (874:867)


Subsets
[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii
</pre>
 
== 4.9.25 subgroup ==
=== Meansquared ===
[[Subgroup]]: 4.9.25
 
[[Comma list]]: [[6561/6400]]
 
{{Mapping|legend=2| 1 3 4 | 0 1 4 }}
 
Mapping generators: ~4, ~9/64
 
[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429
 
[[Support]]ing [[ET]]s: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]
 
== 4.9.49 subgroup ==
=== Archsquared ===
[[Subgroup]]: 4.9.49
 
[[Comma list]]: 4096/3969
 
{{Mapping|legend=2| 1 3 0 | 0 1 -2 }}
 
Mapping generators: ~4, ~9/64
 
[[Optimal tuning]] ([[CTE]]): ~[[9/8]] = 219.190
 
[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49


== 8.9.7 subgroup ==
== 8.9.7 subgroup ==
Line 891: Line 926:
: [[gencom]]: [8 9/8; 64/63]
: [[gencom]]: [8 9/8; 64/63]


[[Optimal tuning]] ([[CTE]]): 1\[[3ed8]] = 1600.0, ~9/8 = 219.1898
[[Optimal tuning]] ([[CTE]]): ~9/8 = 219.1898


[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}
[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}
Line 898: Line 933:


= Fractional subgroup temperaments =
= Fractional subgroup temperaments =
== 2.5/3… subgroups ==
== 2.5/3.… subgroups ==
=== Magicaltet ===
=== Magicaltet ===
{{See also| Chromatic pairs #Magicaltet }}
{{See also| Chromatic pairs #Magicaltet }}


Magicaltet is related to [[supermagic]], [[superkleismic]], and [[magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.  
Magicaltet is related to [[keemic]], [[superkleismic]], and [[magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.  


[[Subgroup]]: 2.5/3.7.11
[[Subgroup]]: 2.5/3.7.11
Line 909: Line 944:


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


{{Mapping|legend=3| 1 -1/2 1/2 2 4 | 0 1/2 -1/2 3 -2 }}
{{Mapping|legend=3| 1 -1/2 1/2 2 4 | 0 1/2 -1/2 3 -2 }}
: [[gencom]]: [2 6/5; 100/99 385/384]
: [[gencom]]: [2 6/5; 100/99 385/384]


Line 921: Line 954:


{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
 
: <nowiki/>* wart for 5/3
<nowiki/>* Wart for 5/3


[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents
Line 940: Line 972:


{{Mapping|legend=3| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
{{Mapping|legend=3| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
: [[gencom]]: [2 6/5; 126/125]
: [[gencom]]: [2 6/5; 126/125]


Line 963: Line 994:


{{Mapping|legend=3| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}
{{Mapping|legend=3| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}
: [[gencom]]: [2 11/10; 121/120 126/125]
: [[gencom]]: [2 11/10; 121/120 126/125]


Line 971: Line 1,001:


{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }}
{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }}
 
: <nowiki/>* wart for 11/3
<nowiki/>* Wart for 11/3


[[Tp tuning #T2 tuning|RMS error]]: 1.034 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.034 cents
Line 988: Line 1,017:


{{Mapping|legend=3| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}
{{Mapping|legend=3| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}
: [[gencom]]: [2 6/5; 56/55 91/90 100/99]
: [[gencom]]: [2 6/5; 56/55 91/90 100/99]


Line 1,005: Line 1,033:
{{See also | Chromatic pairs #Gariberttet }}
{{See also | Chromatic pairs #Gariberttet }}


Gariberttet can be described as the {{nowrap| 4 & 29 }} temperament in the 2.5/3.7/3.13/11 subgroup.  
Gariberttet can be described as the {{nowrap| 4 & 29 }} temperament in the 2.5/3.7/3.13/11 subgroup. Extensions to the full 7-, 11-, and 13-limits include [[quasitemp]].


[[Subgroup]]: 2.5/3.7/3.13/11
[[Subgroup]]: 2.5/3.7/3.13/11
Line 1,014: Line 1,042:


{{Mapping|legend=3| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}
{{Mapping|legend=3| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}
: [[gencom]]: [2 13/11; 275/273 847/845]
: [[gencom]]: [2 13/11; 275/273 847/845]


Line 1,021: Line 1,048:


{{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143* }}
{{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143* }}
 
: <nowiki/>* wart for 13/11
<nowiki/>* Wart for 13/11


[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents
Line 1,038: Line 1,064:


{{Mapping|legend=3| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}
{{Mapping|legend=3| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}
: [[gencom]]: [2 12/11; 3025/3024 3125/3087]
: [[gencom]]: [2 12/11; 3025/3024 3125/3087]


Line 1,046: Line 1,071:


{{Optimal ET sequence|legend=1| 8, 33, 41, 49, 204*<sup>†</sup> }}
{{Optimal ET sequence|legend=1| 8, 33, 41, 49, 204*<sup>†</sup> }}
 
: <nowiki/>* wart for 7/3
<nowiki/>* Wart for 7/3
: <sup>†</sup> wart for 11/3
 
<sup>†</sup> Wart for 11/3


[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents


==== Semidim ====
==== Ammon ====
{{See also| Chromatic pairs #Semidim }}
{{See also| Chromatic pairs #Ammon }}


Semidim can be described as the {{nowrap| 8 & 29 }} temperament in the 2.5/3.7/3.11/3.13/3 subgroup. It extends [[tridec]], and is related to [[ammonite]]. It is generated by a semidiminished fourth, hence the name.  
Ammon can be described as the {{nowrap| 8 & 29 }} temperament in the 2.5/3.7/3.11/3.13/3 subgroup. It extends [[tridec]], and is related to [[ammonite]]. It is generated by a semidiminished fourth, hence the old name ''semidim'', which has been rejected since 2025 to avoid confusion with another temperament of the same name.


[[Subgroup]]: 2.5/3.7/3.11/3.13/3
[[Subgroup]]: 2.5/3.7/3.11/3.13/3
Line 1,065: Line 1,088:


{{Mapping|legend=3| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}
{{Mapping|legend=3| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}
: [[gencom]]: [2 13/10; 121/120 169/168 275/273]
: [[gencom]]: [2 13/10; 121/120 169/168 275/273]


Line 1,088: Line 1,110:


{{Mapping|legend=3| 1 0 0 0 | 0 0 2 -1 }}
{{Mapping|legend=3| 1 0 0 0 | 0 0 2 -1 }}
: [[gencom]]: [2 9/7; 245/243]
: [[gencom]]: [2 9/7; 245/243]


Line 1,095: Line 1,116:


{{Optimal ET sequence|legend=1| 8, 11, 19, 30, 41, 49, 52, 145*, 166<sup>†</sup>, 197*<sup>†</sup>, 215<sup>†</sup>, 264*<sup>†</sup> }}
{{Optimal ET sequence|legend=1| 8, 11, 19, 30, 41, 49, 52, 145*, 166<sup>†</sup>, 197*<sup>†</sup>, 215<sup>†</sup>, 264*<sup>†</sup> }}
 
: <nowiki/>* wart for 5/3
<nowiki/>* Wart for 5/3
: <sup>†</sup> wart for 9/7
 
<sup>†</sup> Wart for 9/7


[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents
Line 1,105: Line 1,124:
{{See also| Chromatic pairs #Marveltwintri }}
{{See also| Chromatic pairs #Marveltwintri }}


Marveltwintri can be described as the {{nowrap| 3 & 4 }} temperament in the 2.5/3.13/9 subgroup. The tonic and the first two generator steps make a [[marveltwin triad]], hence the name.  
Marveltwintri can be described as the {{nowrap| 3 & 4 }} temperament in the 2.5/3.13/9 subgroup. The tonic and the first two generator steps make a [[marveltwin triad]], hence the name. [[Cata]] is a very natural extension of this temperament to the [[2.3.5.13 subgroup|2.3.5.13-subgroup]].


[[Subgroup]]: 2.5/3.13/9
[[Subgroup]]: 2.5/3.13/9
Line 1,114: Line 1,133:


{{Mapping|legend=3| 1 -1/6 5/6 0 0 -1/3 | 0 -1/2 -3/2 0 0 1 }}
{{Mapping|legend=3| 1 -1/6 5/6 0 0 -1/3 | 0 -1/2 -3/2 0 0 1 }}
: [[gencom]]: [2 6/5; 325/324]
: [[gencom]]: [2 6/5; 325/324]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.886
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 882.886
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.861
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 882.861


{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}
{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}
Line 1,125: Line 1,143:
[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents


== 2.….7/3… subgroups ==
== 2.….7/3.… subgroups ==
=== Guanyintet ===
=== Guanyintet ===
{{See also | Chromatic pairs #Guanyintet }}
{{See also | Chromatic pairs #Guanyintet }}


Guanyintet, the {{nowrap| 4 & 9 }} temperament in the 2.5.7/3.11/3 subgroup, is related to [[guanyin]] as well as to [[orwell]]. The tonic and the first three generator steps make a [[guanyin tetrad]], hence the name.  
Guanyintet, the {{nowrap| 4 & 9 }} temperament in the 2.5.7/3.11/3 subgroup, is the main rank-2 chain of [[guanyin]] and a restriction of [[orwell]]. It is defined by tempering out [[1728/1715]] ({{S|6/S7}}) and [[540/539]] (S12/S14), which imply [[176/175]] (S8/S10) as well as S11/S15 being tempered out. The tonic and the first three generator steps make a [[guanyin tetrad]], hence the name.  


[[Subgroup]]: 2.5.7/3.11/3
[[Subgroup]]: 2.5.7/3.11/3
Line 1,135: Line 1,153:
[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[540/539]] ({{monzo| 2 1 -2 -1 }})
[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[540/539]] ({{monzo| 2 1 -2 -1 }})


{{Mapping|legend=2| 1 0 2 -2 | 0 3 -1 5 }}
{{Mapping|legend=2| 1 0 1 3 | 0 -3 1 -5 }}
 
: mapping generators: ~2, ~7/6
: mapping generators: ~2, ~12/7


{{Mapping|legend=3| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}
{{Mapping|legend=3| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}
: [[gencom]]: [2 7/6; 176/175 540/539]
: [[gencom]]: [2 7/6; 176/175 540/539]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~12/7 = 929.545
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.455
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~12/7 = 929.907
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.093


{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
: <nowiki/>* wart for 7/3
[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents
==== Tridecimal guanyintet ====
Guanyintet can extend to the 13th harmonic by the equivalences ([[12/11]])<sup>3</sup> = [[13/10]] and ([[15/14]])<sup>3</sup> = [[16/13]], therefore tempering out {S11/S12/S14/S15}. However, note that it is not supported by the 31 & 53 orwell extension dubbed "tridecimal orwell", but instead the less accurate [[winston]] (22f & 31), as orwell prefers slightly sharper tunings than guanyintet. [[40edo]] remains an excellent tuning.
[[Subgroup]]: 2.5.7/3.11/3.13
[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 0 }}), [[540/539]] ({{monzo| 2 1 -2 -1 0 }}), [[1573/1568]] ({{monzo| -5 0 -2 2 1 }})


<nowiki/>* wart for 7/3
{{Mapping|legend=2| 1 0 1 3 1 | 0 -3 1 -5 12 }}
: mapping generators: ~2, ~12/7


[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents
[[Optimal tuning]]s:
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.152
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.218
 
{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 71, 111, 151, 262c*}} <small> using subgroup TE </small>
: <nowiki/>* wart for 7/3
 
Badness (Sintel): 0.329


==== Laz ====
==== Laz ====
{{See also | Chromatic pairs #Laz }}
{{See also | Chromatic pairs #Laz }}


Laz is related to [[georgian]] as well as to [[winston]].  
Laz is related to [[avalokita]] as well as to [[winston]].  


[[Subgroup]]: 2.5.7/3.11/3.13/3
[[Subgroup]]: 2.5.7/3.11/3.13/3
Line 1,165: Line 1,199:


{{Mapping|legend=3| 1 -5/4 3 -1/4 7/4 -1/4 | 0 -1/4 -3 3/4 -21/4 19/4 }}
{{Mapping|legend=3| 1 -5/4 3 -1/4 7/4 -1/4 | 0 -1/4 -3 3/4 -21/4 19/4 }}
: [[gencom]]: [2 7/6; 144/143 176/175 196/195]
: [[gencom]]: [2 7/6; 144/143 176/175 196/195]


Line 1,173: Line 1,206:


{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 156c*†, 205c*† }}
{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 156c*†, 205c*† }}
 
: <nowiki/>* wart for 7/3
<nowiki/>* wart for 7/3
: † wart for 11/3
 
† wart for 11/3


[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents
Line 1,190: Line 1,221:


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


{{Mapping|legend=3| 1 -5/4 2 -1/4 3/4 3/4 | 0 -1/2 3 3/2 -3/2 1/2 }}
{{Mapping|legend=3| 1 -5/4 2 -1/4 3/4 3/4 | 0 -1/2 3 3/2 -3/2 1/2 }}
: [[gencom]]: [2 13/12; 56/55 78/77 91/90]
: [[gencom]]: [2 13/12; 56/55 78/77 91/90]


Line 1,215: Line 1,244:


{{Mapping|legend=2| 1 1 1 0 | 0 -2 3 4 }}
{{Mapping|legend=2| 1 1 1 0 | 0 -2 3 4 }}
: mapping generators: ~2, ~21/20
: mapping generators: ~2, ~21/20


{{Mapping|legend=3| 1 1/10 -4/5 11/10 1/5 | 0 -3/2 -1 3/2 1 }}
{{Mapping|legend=3| 1 1/10 -4/5 11/10 1/5 | 0 -3/2 -1 3/2 1 }}
: [[gencom]]: [2 21/20; 100/99 245/242]
: [[gencom]]: [2 21/20; 100/99 245/242]


Line 1,240: Line 1,267:


{{Mapping|legend=2| 1 0 1 | 0 1 2 }}
{{Mapping|legend=2| 1 0 1 | 0 1 2 }}
: mapping generators: ~2, ~7/3
: mapping generators: ~2, ~7/3


{{Mapping|legend=3| 1 -1/2 0 1/2 3 | 0 -1/2 0 1/2 2 }}
{{Mapping|legend=3| 1 -1/2 0 1/2 3 | 0 -1/2 0 1/2 2 }}
: [[gencom]]: [2 7/6; 99/98]
: [[gencom]]: [2 7/6; 99/98]


Line 1,255: Line 1,280:
[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents


== 2.….9/7… subgroups ==
== 2.….9/7.… subgroups ==
=== Marveltri ===
=== Marveltri ===
{{See also| Chromatic pairs #Marveltri }}
{{See also| Chromatic pairs #Marveltri }}
Line 1,266: Line 1,291:


{{Mapping|legend=2| 1 0 5 | 0 1 -2 }}
{{Mapping|legend=2| 1 0 5 | 0 1 -2 }}
: mapping generators: ~2, ~5
: mapping generators: ~2, ~5


{{Mapping|legend=3| 1 2 0 -1 | 0 -4/5 1 2/5 }}
{{Mapping|legend=3| 1 2 0 -1 | 0 -4/5 1 2/5 }}
: [[gencom]]: [2 5; 225/224]
: [[gencom]]: [2 5; 225/224]


Line 1,278: Line 1,301:


{{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }}
{{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }}
 
: <nowiki/>* wart for 9/7
<nowiki/>* Wart for 9/7


[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents
Line 1,300: Line 1,322:
[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents


== 2.….15/11… subgroups ==
== 2.….7/5.… subgroups ==
 
=== Poggers ===
Related temperaments: [[Stearnsmic_clan#Pogo|pogo]], [[Stearnsmic_clan#Supers|supers]]
 
[[Subgroup]]: 2.9.7.15/11.13
 
[[Comma list]]: [[540/539]], [[1716/1715]], [[2080/2079]]
 
{{Mapping|legend=2| 1 1 1 -1 -1 | 0 6 5 4 13 }}
 
[[Optimal tuning]] (subgroup [[CTE]]): ~9/7 = 433.888
 
[[Support]]ing [[ET]]s: 8[+9, +7, +13], 11, 14[-13], 19[+9, +7, ++13], 25[-13], 36, 47, 58, 61[-13], 69[+13], 80[+13], 83, 91[+9, +7, +13], 105
 
== 2.….7/5… subgroups ==
 
=== Hydrothermal ===
=== Hydrothermal ===
A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, lend it a mysterious, heavy atmosphere. The 6-tone (hexatonic) MOS is melodically interesting and flavorful. The 18-tone MOS is a useful 'chromatic' scale for taking subsets of.
A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, lend it a mysterious, heavy atmosphere. The 6-tone (hexatonic) MOS is melodically interesting and flavorful. The 18-tone MOS is a useful 'chromatic' scale for taking subsets of.
Line 1,330: Line 1,336:
[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}
[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}


=== Edson ===
=== Argentic ===
Edson is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].
Argentic is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].  


==== Edson (2.3.7/5.11/5.13/5 subgroup) ====
[[Subgroup]]: 2.3.7/5
{{See also| Chromatic pairs #Edson }}


Edson is related to [[pele]] and [[andromeda]].
[[Comma list]]: [[5120/5103]] = {{monzo| 10 -6 -1 }}


[[Subgroup]]: 2.3.7/5.11/5.13/5
{{Mapping|legend=2| 1 0 10 | 0 1 -6 }}
: mapping generators: ~2, ~3


[[Comma list]]: [[196/195]] = {{monzo| 2 -1 2 0 -1 }}, [[352/351]] = {{monzo| 5 -3 0 1 -1 }}, [[364/363]] = {{monzo| 2 -1 1 -2 1 }}
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 702.792
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 702.830
 
{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}
<small> based on subgroup TE </small>
 
Badness (Sintel): 0.119
 
==== Edson (2.3.7/5.11/5.13/5 subgroup) ====
{{See also| Chromatic pairs #Edson }}
 
Edson is related to [[pele]] and [[andromeda]].
 
[[Subgroup]]: 2.3.7/5.11/5.13/5
 
[[Comma list]]: [[196/195]] = {{monzo| 2 -1 2 0 -1 }}, [[352/351]] = {{monzo| 5 -3 0 1 -1 }}, [[364/363]] = {{monzo| 2 -1 1 -2 1 }}


{{Mapping|legend=2| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
{{Mapping|legend=2| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


{{Mapping|legend=3| 1 1 -5 -1 2 4 | 0 1 29/4 5/4 -11/4 -23/4 }}
{{Mapping|legend=3| 1 1 -5 -1 2 4 | 0 1 29/4 5/4 -11/4 -23/4 }}
: [[gencom]]: [2 3/2; 196/195, 352/351, 364/363]
: [[gencom]]: [2 3/2; 196/195, 352/351, 364/363]


Line 1,370: Line 1,390:


{{Mapping|legend=3| 1 2 -3/4 -11/4 9/4 5/4 | 0 -2 0 12 -9 -3 }}
{{Mapping|legend=3| 1 2 -3/4 -11/4 9/4 5/4 | 0 -2 0 12 -9 -3 }}
: [[gencom]]: [2 15/13; 352/351 676/675 847/845]
: [[gencom]]: [2 15/13; 352/351 676/675 847/845]


Line 1,383: Line 1,402:
{{distinguish|History (temperament)}}, which is the rank-3 version of this temperament in the full 13-limit.
{{distinguish|History (temperament)}}, which is the rank-3 version of this temperament in the full 13-limit.


Historical is essentially an analogue of [[miracle]] that splits [[4/3]] in six rather than [[3/2]]. It tempers out the comma S10/S11 = [[4000/3993]] to set [[11/10]] equal to one-third of 4/3, and S13/S15 = [[676/675]] to equate [[15/13]] to one-half of 4/3, and tempers out S21 = [[441/440]] to split 11/10 into two instances of [[22/21]]~[[21/20]].
Historical is essentially an analogue of [[miracle]] that splits [[4/3]] in six rather than [[3/2]]. It tempers out the comma S10/S11 = [[4000/3993]] to set [[11/10]] equal to one-third of 4/3, and S13/S15 = [[676/675]] to equate [[15/13]] to one-half of 4/3, and tempers out S21 = [[441/440]] to split 11/10 into two instances of [[22/21]]~[[21/20]]. [[Sextilifourths]] adds the [[schismic]] mapping of prime 5 (reached by eight fourths) to complete the 13-limit.


[[Subgroup]]: 2.3.7/5.11/5.13/5
[[Subgroup]]: 2.3.7/5.11/5.13/5
Line 1,410: Line 1,429:


{{Mapping|legend=3| 3 10/9 -7/9 2/9 | 0 -2/3 -1/3 2/3 }}
{{Mapping|legend=3| 3 10/9 -7/9 2/9 | 0 -2/3 -1/3 2/3 }}
: [[gencom]]: [63/50 10/9; 250047/250000]
: [[gencom]]: [63/50 10/9; 250047/250000]


Line 1,432: Line 1,450:


{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 | 0 0 0 -4 3 1 }}
{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 | 0 0 0 -4 3 1 }}
: [[gencom]]: [2 13/10; 847/845 1001/1000]
: [[gencom]]: [2 13/10; 847/845 1001/1000]


Line 1,441: Line 1,458:
[[Tp tuning #T2 tuning|RMS error]]: 0.1613 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.1613 cents


== 2.….11/5… subgroups ==
==== Naiadec ====
[[Subgroup]]: 2.7/5.11/5.13/5.17/5
 
[[Comma list]]: [[170/169]], [[221/220]], [[847/845]]
 
{{Mapping|legend=2| 1 2 0 1 1 | 0 -4 3 1 2 }}
 
{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 1/4 | 0 0 0 -4 3 1 2 }}
: [[gencom]]: [2 13/10; 170/169 221/220 847/845]
 
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.882
 
{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 95<sup>t</sup>, 124<sup>t</sup> }}
: <sup>t</sup> wart for 17/5
 
[[Tp tuning #T2 tuning|RMS error]]: 0.7521 cents


== 2.….11/5.… subgroups ==
=== Petrtri ===
=== Petrtri ===
{{See also| Chromatic pairs #Petrtri }}
{{See also| Chromatic pairs #Petrtri }}
Line 1,456: Line 1,489:


{{Mapping|legend=3| 1 0 -1/3 0 -1/3 2/3 | 0 0 -4/3 0 5/3 -1/3 }}
{{Mapping|legend=3| 1 0 -1/3 0 -1/3 2/3 | 0 0 -4/3 0 5/3 -1/3 }}
: [[gencom]]: [2 13/10; 2200/2197]
: [[gencom]]: [2 13/10; 2200/2197]


Line 1,477: Line 1,509:


{{Mapping|legend=3| 1 2 -5/3 0 4/3 1/3 | 0 -2 4 0 -5 1 }}
{{Mapping|legend=3| 1 2 -5/3 0 4/3 1/3 | 0 -2 4 0 -5 1 }}
: [[gencom]]: [2 15/13; 352/351 676/675]
: [[gencom]]: [2 15/13; 352/351 676/675]


Line 1,487: Line 1,518:


=== Hypnosis ===
=== Hypnosis ===
Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Hemifamity temperaments #Tricot|tricot]]
Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Alphatricot family #Alphatricot|alphatricot]]


[[Subgroup]]: 2.3.7.11/5.13
[[Subgroup]]: 2.3.7.11/5.13
Line 1,501: Line 1,532:
[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents


== 2.….11/7… subgroups ==
=== Trisect ===
=== Pepperoni ===
Trisect divides every Pythagorean interval into three, and is the much more accurate subgroup restriction of [[Augmented family #Trisected|trisected]].
{{Main| Parapyth }}
{{See also| Chromatic pairs #Pepperoni }}


Pepperoni is generated by a fifth and can be described as the 5 &amp; 12 temperament in the 2.3.11/7.13/7 subgroup. It is the single-chain retraction of [[parapyth]]. The [[Peppermint-24|Pepper fifth]], which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.
Extending this temperament to the full [[11-limit|11-]], [[13-limit|13-]], or [[17-limit]] through [[portent]] or [[landscape]] results in the [[weak extension]] known as [[tritikleismic]].


[[Subgroup]]: 2.3.11/7.13/7
[[Subgroup]]: 2.3.7.11/5


[[Comma list]]: 352/351, 364/363
[[Comma list]]: 1029/1024, 4000/3993


{{Mapping|legend=2| 1 0 7 12 | 0 1 -4 -7 }}
{{Mapping|legend=2| 3 0 10 5 | 0 3 -1 -1 }}


{{Mapping|legend=3| 1 1 0 -8/3 1/3 7/3 | 0 1 0 11/3 -1/3 -10/3 }}
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.742


: [[gencom]]: [2 3/2; 352/351 364/363]
{{Optimal ET sequence|legend=1| 15, 21, 36, 123, 159, 195, 231 }}


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 703.856
[[Tp tuning #T2 tuning|RMS error]]: ???


{{Optimal ET sequence|legend=1| 5, 7, 12f, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595b*<sup>†</sup> }}
==== 2.3.7.11/5.13 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13


<nowiki />* Wart for 11/7
[[Comma list]]: 1029/1024, 1575/1573, 2080/2079


<sup>†</sup> Wart for 13/7
{{Mapping|legend=2| 3 0 10 5 0 | 0 3 -1 -1 7 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.918


== 2.….13/5… subgroups ==
{{Optimal ET sequence|legend=1| 15, 21f, 36, 87, 123, 159 }}
=== Barbados ===
The [[minimax tuning]] for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.


[[Subgroup]]: 2.3.13/5
[[Tp tuning #T2 tuning|RMS error]]: ???


[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }}
==== 2.3.7.11/5.13.17 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13.17


[[Sval]] [[mapping]]: [{{val| 1 0 -1 }}, {{val| 0 2 3 }}]
[[Comma list]]: 273/272, 833/832, 1575/1573, 2080/2079


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621
{{Mapping|legend=2| 3 0 10 5 0 -2 | 0 3 -1 -1 7 9 }}


{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.820


[[Badness]]: 0.002335
{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123, 159 }}


; Music
[[Tp tuning #T2 tuning|RMS error]]: ???
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 ''Desert Island Rain''] in 313edo tuned Barbados[9], by [https://soundcloud.com/sevish/desert-island-rain Sevish]


==== Tobago ====
===== Trisector =====
{{See also| Chromatic pairs #Tobago }}
[[Subgroup]]: 2.3.7.11/5.13.17.19


Tobago, the 10 &amp; 14 temperament in the 2.3.11.13/5 subgroup, extends [[neutral]] and [[barbados]].
[[Comma list]]: 210/209, 273/272, 286/285, 595/594, 2080/2079


[[Subgroup]]: 2.3.11.13/5
{{Mapping|legend=2| 3 0 10 5 0 -2 8 | 0 3 -1 -1 7 9 3 }}


[[Comma list]]: [[243/242]], [[676/675]]
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.894


{{Mapping|legend=2| 2 0 -1 -2 | 0 2 5 3 }}
{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123h, 159h }}


{{Mapping|legend=3| 2 4 -2 0 9 2 | 0 -2 3/2 0 -5 -3/2 }}
[[Tp tuning #T2 tuning|RMS error]]: ???


: [[gencom]]: [55/39 15/13; 243/242 676/675]
===== 2.3.7.11/5.13.17.19.23 subgroup =====
[[Subgroup]]: 2.3.7.11/5.13.17.19.23


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~55/39 = 1\2, ~15/13 = 249.312
[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 595/594, 2080/2079


{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }}
{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 | 0 3 -1 -1 7 9 3 1 }}
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 634.038
 
{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}
 
[[Tp tuning #T2 tuning|RMS error]]: ???
 
===== 2.3.7.11/5.13.17.19.23.29 subgroup =====
[[Subgroup]]: 2.3.7.11/5.13.17.19.23.29
 
[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 320/319, 595/594, 2080/2079
 
{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 13 | 0 3 -1 -1 7 9 3 1 1 }}
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~29/23 = 1\3, ~13/9 = 634.102
 
{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents
[[Tp tuning #T2 tuning|RMS error]]: ???


==== Pakkanian hemipyth ====
== 2.….11/7.… subgroups ==
=== Blackweed ===
Blackweed is a [[restriction]] of undecimal [[blackwood]] as it tempers out 256/243 alike but in the 2.3.11/7 subgroup. 20edo is close to the optimum, which has 4\20 as the period and 420{{c}} as the generator.


[[Subgroup]]: 2.3.11.13/5.17
[[Subgroup]]: 2.3.11/7


[[Comma list]]: 221/220, 243/242, 289/288
[[Comma list]]: {{monzo| 8 -5 }} (256/243)


{{Mapping|legend=2| 2 0 -1 -2 5 | 0 2 5 3 2 }}
{{Mapping|legend=2| 5 8 0 | 0 0 1 }}
: mapping generators: ~9/8, ~11/7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup CTE]]: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344)
* [[Tp tuning|subgroup]] [[WE]]: ~8/7 = 238.851{{c}}, ~11/7 = 782.457{{c}}
* [[Tp tuning|subgroup CWE]]: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989)
: [[error map]]: {{val| -5.746 +8.852 -0.035 }}
* [[Tp tuning|subgroup]] [[CWE]]: ~8/7 = 240.000{{c}}, ~11/7 = 784.967{{c}}
: error map: {{val| 0.000 +18.045 +2.475 }}
 
{{Optimal ET sequence|legend=1| 15, 20, 35b, 55b }}
 
=== Pepperoni ===
{{Main| Parapyth }}
{{See also| Chromatic pairs #Pepperoni }}
 
Pepperoni is generated by a fifth and can be described as the 5 &amp; 12 temperament in the 2.3.11/7.13/7 subgroup. It is the single-chain retraction of [[parapyth]]. The [[Peppermint-24|Pepper fifth]], which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.
 
[[Subgroup]]: 2.3.11/7.13/7
 
[[Comma list]]: 352/351, 364/363


{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
{{Mapping|legend=2| 1 0 7 12 | 0 1 -4 -7 }}
 
{{Mapping|legend=3| 1 1 0 -8/3 1/3 7/3 | 0 1 0 11/3 -1/3 -10/3 }}
: [[gencom]]: [2 3/2; 352/351 364/363]
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 703.856
 
{{Optimal ET sequence|legend=1| 5, 7, 12f, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595b*<sup>†</sup> }}
: <nowiki />* wart for 11/7
: <sup>†</sup> wart for 13/7
 
[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
 
== 2.….13/5.… subgroups ==
=== Barbados ===
The [[minimax tuning]] for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.
 
[[Subgroup]]: 2.3.13/5
 
[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }}
 
[[Sval]] [[mapping]]: [{{val| 1 0 -1 }}, {{val| 0 2 3 }}]
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621
 
{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}
 
[[Badness]]: 0.002335
 
; Music
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 ''Desert Island Rain''] in 313edo tuned Barbados[9], by [https://soundcloud.com/sevish/desert-island-rain Sevish]
 
==== Tobago ====
{{See also| Chromatic pairs #Tobago }}
 
Tobago, the 10 &amp; 14 temperament in the 2.3.11.13/5 subgroup, extends [[neutral]] and [[barbados]].
 
[[Subgroup]]: 2.3.11.13/5
 
[[Comma list]]: [[243/242]], [[676/675]]
 
{{Mapping|legend=2| 2 0 -1 -2 | 0 2 5 3 }}
 
{{Mapping|legend=3| 2 4 -2 0 9 2 | 0 -2 3/2 0 -5 -3/2 }}
: [[gencom]]: [55/39 15/13; 243/242 676/675]
 
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~55/39 = 1\2, ~15/13 = 249.312
 
{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents
 
==== Pakkanian hemipyth ====
[[Subgroup]]: 2.3.11.13/5.17
 
[[Comma list]]: 221/220, 243/242, 289/288
 
{{Mapping|legend=2| 2 0 -1 -2 5 | 0 2 5 3 2 }}
 
[[Optimal tuning]]s:
* [[Tp tuning|subgroup CTE]]: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344)
* [[Tp tuning|subgroup CWE]]: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989)
 
{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
: <nowiki />* wart for 13/5
 
=== Oceanfront ===
Related temperaments: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]]
 
[[Subgroup]]: 2.3.7.13/5
 
[[Comma list]]: 64/63, 91/90
 
{{Mapping|legend=2| 1 0 6 -5 | 0 1 -2 4 }}
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 713.910
 
{{Optimal ET sequence|legend=1| 5, 22, 27, 32, 37 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents
 
Scales: [[Oceanfront scales]]
 
=== Seventeen-cot ===
 
Seventeen-cot is a rank-2 temperament in the 2.3.13/5 and 2.3.11/5.13/5 subgroups. It tempers out the [[Tendoartisma]] in the 2.3.13/5 subgroup. It can be generated with a ~2/1 octave and a ~2250/2197 or ~169/165 generator which is a 17th of a ~3/2 perfect fifth. It can be described as the 29 & 146 temperament in these subgroups.
 
====2.3.13/5 subgroup====
 
Comma basis: {{monzo| -6 -11 17 }} (2.3.13/5)
 
edo join: 29 & 146
 
{{Mapping|legend=2| 1 1 1 | 0 17 11 }}
: mapping generators: ~2, ~2250/2197
 
{{Todo|inline=1|correct maths|comment=Optimal tunings and error maps showed below is not yet precise enough.}}
 
Optimal tunings:
* WE: ~2 = 1200.000{{c}}, ~2250/2197 = 41.291{{c}}
: error map: {{val| +0.000 +0.001 -0.007}}
* CWE: ~2 = 1200.000{{c}}, ~2250/2197 = 41.292{{c}}
: error map: {{val| +0.000 +0.001 -0.007}}
 
edos: 29, 465, 494, 436, 523, 407, 378, 349, 30[-3], 28[+3], 320, 291, 59[-3], 262
 
Badness (Sintel): 0.064
 
====2.3.11/5.13/5 subgroup====
 
Comma basis: 225000/224939, 43940/43923
 
edo join: 29 & 146
 
{{Mapping|legend=2| 1 1 1 1 | 0 17 4 11 }}
: mapping generators: ~2, ~169/165
 
Optimal tunings:
* WE: ~2 = 1199.993{{c}}, ~169/165 = 41.292{{c}}
: error map: {{val| -0.007 -0.000 +0.157 -0.010}}
* CWE: ~2 = 1200.000{{c}}, ~169/165 = 41.292{{c}}
: error map: {{val| +0.000 +0.005 +0.163 -0.005}}
 
edos: 29, 465, 494, 436, 523, 407, 378, 349, 320, 291, 30[-3], 262, 28[+3], 233
 
Badness (Sintel): 0.080
 
== 2.….49/5.… subgroups ==
=== Direct breedsmic ===
Related temperament: [[hemithirds]], [[newt]]
 
[[Subgroup]]: 2.3.49/5
 
[[Comma list]]: 2401/2400
 
{{Mapping|legend=2| 1 1 3 | 0 2 1 }}
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~49/40 = 350.966
 
{{Optimal ET sequence|legend=1|7, 10, 17}}
 
[[Tp tuning #T2 tuning|RMS error]]: ?
 
== 2.….17/5.… subgroups ==
=== Fiventeen ===
Fiventeen tempers out [[136/135]] ({{monzo| 3 -3 1 }}) in 2.3.17/5. It equates [[17/15]] with [[9/8]], so it implies a [[supersoft]] [[pentic]] [[pentad]] of [[~]]30:34:40:45:51. [[17edo]] makes a good tuning especially for its size, which gives a [[supersoft]] pentic scale corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and [[97edo]] (= 80 + 17) and  [[114edo]] (= 97 + 17) do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, finding [[34edo]], then [[80edo]], then [[114edo]] (= 34 + 80) and even [[194edo|194bc-edo]] (= 80 + 114), though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and [[143edo]] (= 63 + 80) tunings are found in the optimal ET sequence for fiventeen.
 
[[Subgroup]]: 2.3.17/5
 
[[Comma list]]: 136/135 ({{monzo| 3 -3 1 }})
 
{{Mapping|legend=2| 1 0 -3 | 0 1 3 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}
 
{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}
 
== 2.….19/7.… subgroups ==
=== Surprise ===
This temperament was named by [[User:VectorGraphics|Vector]] in 2025, as he was surprised that the temperament of [[57/56]] did not have a name. This is the [[rank-2 temperament|rank-2]] version of the temperament; Vector surmises that the name ''hendrix'' would be more thoughtfully given to the [[rank-3]] version.
 
[[Subgroup]]: 2.3.19/7
 
[[Comma list]]: [[57/56]] ({{Monzo| -3 1 1 }})
 
{{Mapping|legend=2| 1 0 3 | 0 1 -1 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1202.4345{{c}}, ~3/2 = 697.4314{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.3981{{c}}
 
{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31*, 50* }}
 
<nowiki/>* wart for 19/7
 
[[Badness]] (Sintel): 0.082
 
=== Supramin ===
This is a remarkable low-complexity microtemperament that contains the 14:17:19 triad within just four generator steps. An excellent tuning is [[25edo]], which provides an accurate yet tone-efficient tuning of this temperament. It was named by [[User:Overthink|Overthink]] in 2026 after the fact that the generator is a [[17/14]] supraminor third, two of which reach [[28/19]]. It is related to [[cohemimabila]].
 
[[Subgroup]]: 2.17/7.19/7
 
[[Comma list]]: [[5491/5488]] ({{Monzo| -4 2 1 }})
 
{{Mapping|legend=2| 1 0 4 | 0 1 -2 }}
: mapping generators: ~2, ~17/7
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.022{{c}}, ~17/14 = 335.793{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.000{{c}}, ~17/14 = 335.785{{c}}


<nowiki />* Wart for 13/5
{{Optimal ET sequence|legend=1| 7, 18, 25 }}


=== Oceanfront ===
[[Badness]] (Sintel): 0.005
Related temperaments: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]]


[[Subgroup]]: 2.3.7.13/5
==== Supramine ====
This extension approximates the 14:17:19:23:25 pentad in just six generator steps, at the cost of some accuracy. 25edo remains a strong tuning.


[[Comma list]]: 64/63, 91/90
Subgroup: 2.17/7.19/7.23/7
 
{{Mapping|legend=2| 1 0 6 -5 | 0 1 -2 4 }}


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 713.910
Comma list: [[323/322]], [[392/391]]


{{Optimal ET sequence|legend=1| 5, 22, 27, 32, 37 }}
Subgroup-val mapping: {{Mapping| 1 0 4 3 | 0 1 -2 -1 }}


[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents
Optimal tunings:
* Subgroup WE: ~2 = 1199.871{{c}}, ~17/14 = 336.243{{c}}
* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 336.296{{c}}


Scales: [[Oceanfront scales]]
{{Optimal ET sequence|legend=0| 7, 18, 25 }}


== 2.….49/5… subgroups ==
Badness (Sintel): 0.029
=== Direct breedsmic ===
Related temperament: [[hemithirds]], [[newt]]


[[Subgroup]]: 2.3.49/5
==== 2.25/7.17/7.19/7.23/7 subgroup ====


[[Comma list]]: 2401/2400
Subgroup: 2.25/7.17/7.19/7.23/7


{{Mapping|legend=2| 1 1 3 | 0 2 1 }}
Comma list: [[323/322]], [[392/391]], [[476/475]]


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~49/40 = 350.966
Subgroup-val mapping: {{Mapping| 1 -2 0 4 3 | 0 3 1 -2 -1 }}


{{Optimal ET sequence|legend=1|?}}
Optimal tunings:
* Subgroup WE: ~2 = 1199.757{{c}}, ~17/14 = 335.428{{c}}
* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 335.479{{c}}


[[Tp tuning #T2 tuning|RMS error]]: ?
{{Optimal ET sequence|legend=0| 7, 18, 25 }}


Badness (Sintel): 0.053


== 3/2.5/2… subgroups ==
== 3/2.5/2.… subgroups ==
{{Main|Half-prime subgroup}}
{{Main|Half-prime subgroup}}


Line 1,640: Line 1,897:


{{Mapping|legend=2| 1 3 4 | 0 -4 -5 }}
{{Mapping|legend=2| 1 3 4 | 0 -4 -5 }}
: sval mapping generators: ~3/2, ~15/14
: sval mapping generators: ~3/2, ~15/14


Line 1,646: Line 1,902:


Supporting ETs: *5, *6, *7[+5/2, +7/2], *9[-5/2, --7/2], *11, *16, *17[+5/2], *23[+5/2, +7/2], *21[-7/2], *27, *28[+5/2], *38, *43[-7/2], *49
Supporting ETs: *5, *6, *7[+5/2, +7/2], *9[-5/2, --7/2], *11, *16, *17[+5/2], *23[+5/2, +7/2], *21[-7/2], *27, *28[+5/2], *38, *43[-7/2], *49
 
: <nowiki />* wart for 3/2
<nowiki />* Wart for 3/2


==== 3/2.5/2.7/2.11/2 ====
==== 3/2.5/2.7/2.11/2 ====
Line 1,655: Line 1,910:


{{Mapping|legend=2| 1 3 4 4 | 0 -4 -5 1 }}
{{Mapping|legend=2| 1 3 4 4 | 0 -4 -5 1 }}
: sval mapping generators: ~3/2, ~15/14
: sval mapping generators: ~3/2, ~15/14


Line 1,661: Line 1,915:


[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2], *21[-7/2], *38[-11/2], *43[-7/2, -11/2], *59[-7/2, -11/2], *70[-7/2, -11/2], *75[--7/2, -11/2]
[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2], *21[-7/2], *38[-11/2], *43[-7/2, -11/2], *59[-7/2, -11/2], *70[-7/2, -11/2], *75[--7/2, -11/2]
 
: <nowiki />* wart for 3/2
<nowiki />* Wart for 3/2


==== 3/2.5/2.7/2.11/2.13/2 ====
==== 3/2.5/2.7/2.11/2.13/2 ====
Line 1,674: Line 1,927:


[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2]
[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2]
 
: <nowiki />* wart for 3/2
<nowiki />* Wart for 3/2


=== Semiwolf ===
=== Semiwolf ===
Line 1,712: Line 1,964:
[[Optimal ET sequence]]: [[8edf]], [[11edf]]
[[Optimal ET sequence]]: [[8edf]], [[11edf]]


== 3/2.5/4… subgroups ==
== 3/2.5/4.… subgroups ==
=== Poseidon ===
=== Poseidon ===
'''This temperament will be subjected to renaming due to a conflict.'''
'''This temperament will be subjected to renaming due to a conflict.'''
Line 1,741: Line 1,993:


Supporting ETs: *5, *6[+13], *7[-7, -13], *9, *11[+13], *13, *14, *17[-7, -13], *19[+13], *21[-7, -13], *22[-7], *23[+13], *25[-7, -13], *31[-7]
Supporting ETs: *5, *6[+13], *7[-7, -13], *9, *11[+13], *13, *14, *17[-7, -13], *19[+13], *21[-7, -13], *22[-7], *23[+13], *25[-7, -13], *31[-7]
 
: <nowiki />* wart for 3/2
<nowiki />* Wart for 3/2


=== Doubleton ===
=== Doubleton ===
Line 1,756: Line 2,007:


Supporting ETs: *6, *10, *16, *14[-13], *8[+7], *22, *18[-13], *26, *24[-13], *28[+7], *20[+7], *36[-13], *12[+7, +13], *34[-13]
Supporting ETs: *6, *10, *16, *14[-13], *8[+7], *22, *18[-13], *26, *24[-13], *28[+7], *20[+7], *36[-13], *12[+7, +13], *34[-13]
 
: <nowiki />* wart for 3/2
<nowiki />* Wart for 3/2


== 5/2-equave subgroups ==
== 5/2-equave subgroups ==
=== Hyperion ===
=== Hyperion ===
[[Subgroup]]: 5/2.7.11
[[Subgroup]]: 5/2.7.11
Line 1,773: Line 2,022:


Supporting ETs: *5[-7], *8, *19[+7], *21[-7], *27[+7], *29[-7], *35[+7], *43[+7], *37[-7], *51[+7, +11], *45[-7], *59[+7, +11]
Supporting ETs: *5[-7], *8, *19[+7], *21[-7], *27[+7], *29[-7], *35[+7], *43[+7], *37[-7], *51[+7, +11], *45[-7], *59[+7, +11]
 
: <nowiki />* wart for 5/2
<nowiki />* Wart for 5/2


= Related temperament collections =
= Related temperament collections =
Line 1,781: Line 2,029:
* [[Substitute harmonic]] temperaments
* [[Substitute harmonic]] temperaments


<!-- main article -->
[[Category:Subgroup temperaments| ]] <!-- main article -->
 
[[Category:Temperament collections]]
[[Category:Temperament collections]][[Category:Subgroup]]
[[Category:Rank 2]]
{{Todo| review | cleanup }}
Retrieved from "https://en.xen.wiki/w/Subgroup_temperaments"