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Subgroup temperaments: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
A '''subgroup temperament''' is a regular temperament defined on a [[just intonation subgroup]] that is not a full ''p''-limit group.  
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-27 08:37:45 UTC</tt>.<br>
: The original revision id was <tt>258599756</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">Below are some subgroups and temperaments for them. Obviously, no attempt has been made at completeness; attention is focused on subgroups containing interesting chords. The reader may also want to consult the page on [[Chromatic pairs]].


=2.3.7=
For temperaments that omit various prime harmonics, see:
==[[Slendric]]==
* [[No-elevens subgroup temperaments]]
Comma: 1029/1024
* [[No-sevens subgroup temperaments]]
* [[No-fives subgroup temperaments]]
* [[No-threes subgroup temperaments]]
* [[No-twos subgroup temperaments]] (additionally, [[Catalog of 3.5.7 subgroup rank two temperaments]]).


POTE generator: ~8/7 = 233.688
Below are some temperaments for composite subgroups and fractional subgroups. Obviously, no attempt has been made at completeness; attention is focused on subgroups containing interesting chords. The reader may also want to consult the page on [[Chromatic pairs]].


Map: [&lt;1 1 3|, &lt;0 3 -1|]
This page uses an extended [[wart notation]] where squares and cubes of a prime basis element are denoted by ′ and ″, respectively, to reduce the use of temporary warts.  
EDOs: 5, 21, 26, 31, 36, 77, 113, 190
Badness: 0.00461


=2.5.7.11=
== 2.3.… subgroups ==
=== Shrub ===
Named by [[User: VectorGraphics|Vector]] in 2026, shrub is a [[restriction]] of [[diaschismic family #Diaschismic|diaschismic]] which omits the tritone to produce a [[5L 2s|diatonic]] scale. True to its name, it generates a [[neogothic major and minor|shrubmajor]]{{idio}} third (~425{{c}}) in quarter-comma tuning. It has an equal-temperament join of 12 & 17.  


==Larry==
It can also admit a [[41/1|41st harmonic]] by tempering out [[82/81]] (same mapping as [[reversed meantone]]). However, its tuning tends to be flat, and with the rest of the harmonics tending sharp, this will increase the overall error.
Commas: 243/242, 4000/3993
Related temperaments: [[Gravity family#Gravity|gravity]], [[Gravity family#Harry|harry]]


POTE generator: ~40/27 = 683.166
==== 2.3.25 subgroup ====
[[Subgroup]]: 2.3.25


Map: [&lt;1 5 12 12|, &lt;0 -6 -17 -15|]
[[Comma list]]: 2048/2025
EDOs: 7, 58, 65, 137, 202
Badness: 0.0125


=2.3.7.11=
{{Mapping|legend=3| 1 0 11 | 0 1 -4 }}
: mapping generators: ~2, ~3


==Ennea==
[[Optimal tuning]]s:
Commas: 41503/41472, 43923/43904
* [[WE]]: ~2 = 1198.8214{{c}}, ~3/2 = 704.2059{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.9585{{c}}


POTE generator: ~99/98 = 17.6258
{{Optimal ET sequence|legend=1| 12, 17, 46, 63, 80, 143c′ }}


Sval map: [&lt;9 0 11 24|, &lt;0 2 2 1|]
[[Badness]] (Sintel): 0.234
EDOs: 54, 63, 72, 135, 342, 477, 1089, 1566
Badness: 0.00426


==Supra==  
===== 2.3.25.23 subgroup =====
Commas: 64/63, 99/98
Subgroup: 2.3.25.23


[[POTE tuning|POTE generator]]: ~3/2 = 707.192
Comma list: 576/575, 736/729


Sval map: [&lt;1 0 6 13|, &lt;0 1 -2 -6|]
{{Mapping|legend=2| 1 0 11 -5 | 0 1 -4 6 }}
EDOs: 5, 12, 17, 39, 56
Badness: 0.00933


==Skwares==
Optimal tunings:
Commas: 99/98, 243/242
* WE: ~2 = 1198.9504{{c}}, ~3/2 = 704.6585{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.1888{{c}}


POTE generator: ~9/7 = 425.244
{{Optimal ET sequence|legend=0| 12, 17, 63, 80, 97c′, 114c′ }}


Sval map: [&lt;1 3 6 7|, &lt;0 -4 -9 -10|]
Badness (Sintel): 0.223
EDOs: 14, 17, 31, 48, 79, 127, 206
Badness: 0.0107


==Hemif==  
==== Sburb ====
Commas: 243/242, 896/891
Sburb is every other step of [[pajara]], extended to the add-23 add-41 subgroup in the same way as shrub. It can also admit a [[59/1|59th harmonic]] by setting the [[octave reduction|octave-reduced]] 413th harmonic (413/256, 827.998{{c}}) to the diminished seventh.


POTE generator: ~11/9 = 351.535
Subgroup: 2.3.25.7


Sval map: [&lt;1 1 -1 2|, &lt;0 2 13 5|]
Comma list: 50/49, 64/63
EDOs: 7, 17, 41, 58, 99
Badness: 0.0109


=2.9.7.11=
{{Mapping|legend=2| 1 0 11 6 | 0 1 -4 -2 }}


==Machine==
Optimal tunings:
Commas: 64/63, 99/98
* WE: ~2 = 1197.6967{{c}}, ~3/2 = 705.6906{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.3438{{c}}


[[POTE tuning|POTE generator]]: ~8/7 = 214.384
{{Optimal ET sequence|legend=0| 5, 12, 17, 39d }}


[[Smonzos and Svals|Sval map]]: [&lt;1 0 6 13|, &lt;0 1 -1 -3|]
Badness (Sintel): 0.253
EDOs: 5, 6, 11, 17, 28
Badness: 0.00233


==Apparatus==  
===== 2.3.25.7.23 subgroup =====
Commas: 41503/41472, 322102/321489
Subgroup: 2.3.25.7.23


[[POTE tuning|POTE generator]]: ~77/72 = 115.570
Comma list: 50/49, 64/63, 162/161


[[Smonzos and Svals|Sval map]]: [&lt;1 5 3 5|, &lt;0 -19 -2 -16|]
{{Mapping|legend=2| 1 0 11 6 -5 | 0 1 -4 -2 6 }}
EDOs: 10, 21, 31, 52, 83, 135, 353, 488, 623
Badness: 0.00263


==Mechanism==
Optimal tunings:
Commas: 896/891, 26411/26244
* WE: ~2 = 1197.5577{{c}}, ~3/2 = 705.3089{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 706.7010{{c}}


[[POTE tuning|POTE generator]]: ~9/7 = 438.465
{{Optimal ET sequence|legend=0| 5, 12, 17 }}


[[Smonzos and Svals|Sval map]]: [&lt;1 5 5 2|, &lt;0 -5 -6 4|]
Badness (Sintel): 0.419
EDOs: 8, 11, 30, 41, 52
 
Badness: 0.00439</pre></div>
=== Hypnosis ===
<h4>Original HTML content:</h4>
Related temperaments: [[swetismic temperaments #Hypnos|hypnos]], [[alphatricot family #Alphatrimot|alphatrimot]].
<div style="width:100%; max-height:400pt; overflow:auto; background-color
 
[[Subgroup]]: 2.3.7.11/5.13
 
[[Comma list]]: 169/168 ({{monzo| -3 -1 -1 0 2 }}), 540/539 ({{monzo| 2 3 -2 -1 0 }}), 729/728 ({{monzo| -3 6 -1 0 -1 }})
 
{{Mapping|legend=3| 1 0 -3 8 0 | 0 3 11 -13 7 }}
: mapping generators: ~2, ~13/9
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.6306{{c}}, ~13/9 = 633.8505{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~13/9 = 633.5546{{c}}
 
{{Optimal ET sequence|legend=1| 17, 36, 125f, 161f, 197f }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents
 
=== Darian calendar ===
Darian calendar is described as the 24 & 668 temperament in the 2.3.35.11.19 subgroup. The generator is close to [[36/35]]. Five of them make [[11/8]], six of them make [[32/19]], and eight of them make [[3/2]].
 
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.
 
Darian calendar was named by [[Eliora]] in 2023 after a certain calendar layout by the same name.
 
[[Subgroup]]: 2.3.35.11.19
 
[[Comma list]]: 29282/29241 ({{monzo| 1 -4 0 4 -2 }}), 42875/42768 ({{monzo| -4 -5 3 -1 0 }}), 885115/884736 ({{monzo| -15 -3 1  3 1 }})
 
{{Mapping|legend=3| 4 5 18 13 18 | 0 8 15 5 -6 }}
: mapping generators: ~2240/1881, ~36/35
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2240/1881 = 299.9912{{c}}, ~36/35 = 50.2951{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2240/1881 = 300.0000{{c}}, ~36/35 = 50.2947{{c}}
 
{{Optimal ET sequence|legend=1| 24, …, 524, 548, 572, 596, 620, 644, 668 }}
 
=== Hydrothermal ===
Named by [[Budjarn Lambeth]] in 2024, hydrothermal tempers out 50/49, the [[jubilisma]], in the 2.3.7/5 subgroup. A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, will lend it a mysterious, heavy atmosphere. The 6-tone [[mos]] is melodically interesting and flavorful. The 18-tone mos is a useful "chromatic" scale for taking subsets of.
 
[[Subgroup]]: 2.3.7/5
 
[[Comma list]]: 50/49 ({{monzo| 1 0 -2 }})
 
{{Mapping|legend=3| 2 0 1 | 0 1 0 }}
: mapping generators: ~7/5, ~3
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 599.6673{{c}}, ~3/2 = 702.5906{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 600.0000{{c}}, ~3/2 = 702.4574{{c}}
 
{{Optimal ET sequence|legend=1| 2, 6, 8, 10, 12, 34, 46* }}
: <nowiki/>* Wart for 7/5
 
=== Argentic ===
Argentic is the 2.3.7/5-subgroup temperament tempering out 5120/5103, the [[aberschisma]].
 
[[Subgroup]]: 2.3.7/5
 
[[Comma list]]: 5120/5103 ({{monzo| 10 -6 -1 }})
 
{{Mapping|legend=3| 1 0 10 | 0 1 -6 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.7172{{c}}, ~3/2 = 702.6636{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8166{{c}}
 
{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}
 
[[Badness]] (Sintel): 0.119
 
==== Edson ====
{{See also| Chromatic pairs #Edson }}
 
Edson is related to [[aberschismic family #Pele|pele]] and [[schismatic family #Andromeda|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=3| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
 
{{Mapping|legend=5| 1 0 -49/4 -9/4 19/4 39/4 | 0 1 29/4 5/4 -11/4 -23/4 }}
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.4965{{c}}, ~3/2 = 703.1192{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.4225{{c}}
 
{{Optimal ET sequence|legend=1| 12, 17, 29 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents
 
==== Haumea ====
{{See also| Chromatic pairs #Haumea }}
 
Haumea is the add-7/5 extension of [[#Bridgetown|bridgetown]], which is in turn the add-11/5 extension of [[#Barbados|barbados]]. Other related temperamnets include [[schismatic family #Hemigari|hemigari]], [[the Archipelago #Parizekmic|parizekmic]], and [[aberschismic family #Namaka|namaka]
Retrieved from "https://en.xen.wiki/w/Subgroup_temperaments"