Subgroup temperaments: Difference between revisions

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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]].
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]].


== No-elevens subgroup ==
== No-elevens subgroup ==


=== Sensi (aka Sensation) ===
=== Sensi (aka Sensation) ===
<small>''For full 13-limit extensions, see [[Sensipent family]] or [[Sensi extensions]].''</small>
: <small>''For full 13-limit extensions, see [[Sensipent family]] or [[Sensi extensions]].''</small>


Subgroup: 2.3.5.7.13
Subgroup: 2.3.5.7.13
Line 25: Line 25:


=== [[Tertiaseptal]] ===
=== [[Tertiaseptal]] ===
<small>''For full 13- and 17-limit extensions, see [[Breedsmic temperaments #Tertiaseptal]].''</small>
: <small>''For full 13- and 17-limit extensions, see [[Breedsmic temperaments #Tertiaseptal]].''</small>


Subgroup: 2.3.5.7.13
Subgroup: 2.3.5.7.13
Line 43: Line 43:
[[Tp tuning #T2 tuning|RMS error]]: 0.1383 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.1383 cents


==== 2.3.5.7.13.17 ====
==== 2.3.5.7.13.17 ====
Subgroup: 2.3.5.7.13.17
Subgroup: 2.3.5.7.13.17


Line 60: Line 60:
[[Tp tuning #T2 tuning|RMS error]]: 0.1367 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.1367 cents


== No-sevens subgroup  ==
=== [[Pontiac]] ===
: <small>''For full 13- and 17-limit extensions, see [[Schismatic family #Pontiac]].''</small>


=== Porkypine ===
Subgroup: 2.3.5.7.13
 
[[Comma list]]: 625/624 729/728 4096/4095
 
[[Gencom]]: [2 4/3; 625/624 729/728 4096/4095]
 
[[Gencom|Gencom mapping]]: [{{val|1 2 -1 19 0 -10}}, {{val|0 -1 8 -39 0 33}}]
 
[[Mapping|Sval mapping]]: [{{val|1 2 -1 19 -10}}, {{val|0 -1 8 -39 33}}]
 
[[Tp tuning|POL2 generator]]: ~3/2 = 701.773
 
{{Val list|legend=1| 53, 118, 171, 224, 395 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.1525 cents
 
==== 2.3.5.7.13.17 ====
Subgroup: 2.3.5.7.13.17
 
[[Comma list]]: 625/624, 729/728, 1225/1224, 2880/2873
 
[[Gencom]]: [2 4/3; 625/624 729/728 1225/1224 2880/2873]
 
[[Gencom|Gencom mapping]]: [{{val|1 2 -1 19 0 -10 29}}, {{val|0 -1 8 -39 0 33 -60}}]
 
[[Mapping|Sval mapping]]: [{{val|1 2 -1 19 -10 29}}, {{val|0 -1 8 -39 33 -60}}]
 
[[Tp tuning|POL2 generator]]: ~3/2 = 701.764
 
{{Val list|legend=1| 53, 118, 171, 395, 566… }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.1696 cents
 
=== [[Septidiasemi]] ===
: <small>''For full 13- and 17-limit extensions, see [[Breedsmic temperaments #Septidiasemi]].''</small>
 
Subgroup: 2.3.5.7.13
 
[[Comma list]]: 2205/2197, 2401/2400, 4096/4095
 
[[Gencom]]: [2 15/14; 2205/2197 2401/2400 4096/4095]
 
[[Gencom|Gencom mapping]]: [{{val|1 -1 6 4 0 4}}, {{val|0 26 -37 -12 0 -3}}]
 
[[Mapping|Sval mapping]]: [{{val|1 -1 6 4 4}}, {{val|0 26 -37 -12 -3}}]
 
[[Tp tuning|POL2 generator]]: ~15/14 = 119.297
 
{{Val list|legend=1| 10, 151, 161, 171 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.2002 cents
 
==== 2.3.5.7.13.17 ====
Subgroup: 2.3.5.7.13.17
 
[[Comma list]]: 833/832, 1275/1274, 2025/2023, 2205/2197
 
[[Gencom]]: [2 15/14; 833/832 1275/1274 2025/2023 2205/2197]
 
[[Gencom|Gencom mapping]]: [{{val|1 -1 6 4 0 4 2}}, {{val|0 26 -37 -12 0 -3 21}}]
 
[[Mapping|Sval mapping]]: [{{val|1 -1 6 4 4 2}}, {{val|0 26 -37 -12 -3 21}}]
 
[[Tp tuning|POL2 generator]]: ~15/14 = 119.297
 
{{Val list|legend=1| 10, 151, 161, 171 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.1867 cents
 
=== [[Mitonic]] ===
: <small>''For full 13- and 17-limit extensions, see [[Minortonic family #Mitonic]].''</small>
 
Subgroup: 2.3.5.7.13
 
[[Comma list]]: 4096/4095, 4375/4374, 13720/13689
 
[[Gencom]]: [2 10/9; 4096/4095 4375/4374 13720/13689]
 
[[Gencom|Gencom mapping]]: [{{val|1 -1 -3 6 0 11}}, {{val|0 17 35 -21 0 -48}}]
 
[[Mapping|Sval mapping]]: [{{val|1 -1 -3 6 11}}, {{val|0 17 35 -21 -48}}]
 
[[Tp tuning|POL2 generator]]: ~10/9 = 182.471
 
{{Val list|legend=1| 46, 125, 171, 388 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.1442 cents
 
==== 2.3.5.7.13.17 ====
Subgroup: 2.3.5.7.13.17
 
[[Comma list]]: 833/832, 1225/1224, 1701/1700, 4096/4095
 
[[Gencom]]: [2 10/9; 833/832 1225/1224 1701/1700 4096/4095]
 
[[Gencom|Gencom mapping]]: [{{val|1 -1 -3 6 0 11 5}}, {{val|0 17 35 -21 0 -48 -6}}]
 
[[Mapping|Sval mapping]]: [{{val|1 -1 -3 6 11 5}}, {{val|0 17 35 -21 -48 -6}}]
 
[[Tp tuning|POL2 generator]]: ~10/9 = 182.471
 
{{Val list|legend=1| 46, 125, 171, 388 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.1341 cents
 
== No-sevens subgroup ==
 
=== Porkypine ===
Related temperament: [[Porcupine]]
Related temperament: [[Porcupine]]


Line 81: Line 189:
[[Tp tuning #T2 tuning|RMS error]]: 2.287 cents
[[Tp tuning #T2 tuning|RMS error]]: 2.287 cents


=== Mohaha ===
=== Mohaha ===
Related temperament: [[mohajira]], [[Meantone family #Migration|migration]]
Related temperament: [[mohajira]], [[Meantone family #Migration|migration]]


Line 100: Line 208:
[[Tp tuning #T2 tuning|RMS error]]: 1.392 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.392 cents


==== Music ====
; Music
[http://micro.soonlabel.com/gene_ward_smith/Others/Stiltner/mohaha10ping2.mp3 Mohaha10ping2] by [[Billy Stiltner]]
[http://micro.soonlabel.com/gene_ward_smith/Others/Stiltner/mohaha10ping2.mp3 Mohaha10ping2] by [[Billy Stiltner]]


=== Larry ===
=== Larry ===
Subgroup: 2.3.5.11
Subgroup: 2.3.5.11


Line 122: Line 230:
[[Tp tuning #T2 tuning|RMS error]]: 0.3025 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.3025 cents


=== Tetracot ===
=== Tetracot ===
<small>''For full 11- and 13-limit extensions, see [[Tetracot family]].''</small>
: <small>''For full 11- and 13-limit extensions, see [[Tetracot family]].''</small>


Subgroup: 2.3.5.11
Subgroup: 2.3.5.11
Line 141: Line 249:
[[Tp tuning #T2 tuning|RMS error]]: 1.182 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.182 cents


==== 2.3.5.11.13 ====
==== 2.3.5.11.13 ====
Subgroup: 2.3.5.11.13
Subgroup: 2.3.5.11.13


Line 158: Line 266:
[[Tp tuning #T2 tuning|RMS error]]: 1.140 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.140 cents


=== Emka ===
=== Emka ===
<small>''For full 11- and 13-limit extensions, see [[Hemimean clan #Emka]] or [[Horwell temperaments #Emkay]].''</small>
: <small>''For full 11- and 13-limit extensions, see [[Hemimean clan #Emka]] or [[Horwell temperaments #Emkay]].''</small>


Subgroup: 2.3.5.11
Subgroup: 2.3.5.11
Line 177: Line 285:
[[Tp tuning #T2 tuning|RMS error]]: 0.1188 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.1188 cents


==== 2.3.5.11.13 ====
==== 2.3.5.11.13 ====
Subgroup: 2.3.5.11.13
Subgroup: 2.3.5.11.13


Line 194: Line 302:
[[Tp tuning #T2 tuning|RMS error]]: 0.1250 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.1250 cents


== No-fives subgroup ==
== No-fives subgroup ==


=== [[Semaphore and Godzilla|Semaphore]] ===
=== [[Semaphore and Godzilla|Semaphore]] ===
Line 213: Line 321:
[[Tp tuning# T2 tuning|RMS error]]: 2.523 cents
[[Tp tuning# T2 tuning|RMS error]]: 2.523 cents


=== Archy ===
=== Archy ===
Archy (properly pronounced "arky", after the Greek theorist Archytas) can be thought of as "no-fives [[dominant]]" or "no-fives [[superpyth]]". The name comes from the fact that it tempers out 64/63, the Archytas comma.
Archy (properly pronounced "arky", after the Greek theorist Archytas) can be thought of as "no-fives [[dominant]]" or "no-fives [[superpyth]]". The name comes from the fact that it tempers out 64/63, the Archytas comma.


Line 232: Line 340:
[[Tp tuning #T2 tuning|RMS error]]: 1.856 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.856 cents


==== Supra ====
==== Supra ====
Subgroup: 2.3.7.11
Subgroup: 2.3.7.11


Line 249: Line 357:
[[Tp tuning #T2 tuning|RMS error]]: 1.977 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.977 cents


===== Supraphon =====
===== Supraphon =====
Subgroup: 2.3.7.11.13
Subgroup: 2.3.7.11.13


Line 266: Line 374:
[[Tp tuning #T2 tuning|RMS error]]: 2.095 cents
[[Tp tuning #T2 tuning|RMS error]]: 2.095 cents


==== Suhajira ====
==== Suhajira ====
Subgroup: 2.3.7.11
Subgroup: 2.3.7.11


Line 283: Line 391:
[[Tp tuning #T2 tuning|RMS error]]: 1.968 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.968 cents


===== 2.3.7.11.13 =====
===== 2.3.7.11.13 =====
Subgroup: 2.3.7.11.13
Subgroup: 2.3.7.11.13


Line 332: Line 440:
[[Tp tuning #T2 tuning|RMS error]]: 0.5452 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.5452 cents


==== 2.3.7.13.17 ====
==== 2.3.7.13.17 ====
Subgroup: 2.3.7.13.17
Subgroup: 2.3.7.13.17


Line 347: Line 455:
[[Tp tuning #T2 tuning|RMS error]]: 0.5073 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.5073 cents


=== Lee ===
=== Lee ===
Subgroup: 2.3.7
Subgroup: 2.3.7


Line 364: Line 472:
[[Tp tuning #T2 tuning|RMS error]]: 0.3519 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.3519 cents


=== Skwares ===
=== Skwares ===
Related temperament: [[Meantone family #Squares|squares]]
Related temperament: [[Meantone family #Squares|squares]]


Line 383: Line 491:
[[Tp tuning #T2 tuning|RMS error]]: 1.149 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.149 cents


==== 2.3.7.11 ====
==== 2.3.7.11 ====
Subgroup: 2.3.7.11
Subgroup: 2.3.7.11


Line 400: Line 508:
[[Tp tuning #T2 tuning|RMS error]]: 1.099 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.099 cents


==== 2.3.7.11.13 ====
==== 2.3.7.11.13 ====
Subgroup: 2.3.7.11.13
Subgroup: 2.3.7.11.13


Line 417: Line 525:
[[Tp tuning #T2 tuning|RMS error]]: 1.769 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.769 cents


==== Skwairs ====
==== Skwairs ====
Subgroup: 2.3.7.11.13
Subgroup: 2.3.7.11.13


Line 434: Line 542:
[[Tp tuning #T2 tuning|RMS error]]: 1.290 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.290 cents


=== Bleu ===
=== Bleu ===
Subgroup: 2.3.7
Subgroup: 2.3.7


Line 451: Line 559:
[[Tp tuning #T2 tuning|RMS error]]: 1.917 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.917 cents


==== 2.3.7.11 ====
==== 2.3.7.11 ====
Subgroup: 2.3.7.11
Subgroup: 2.3.7.11


Line 468: Line 576:
[[Tp tuning #T2 tuning|RMS error]]: 1.829 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.829 cents


==== 2.3.7.11.13 ====
==== 2.3.7.11.13 ====
Subgroup: 2.3.7.11.13
Subgroup: 2.3.7.11.13


Line 485: Line 593:
[[Tp tuning #T2 tuning|RMS error]]: 1.752 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.752 cents


=== Hemif ===
=== Hemif ===
Related temperament: [[Breedsmic temperaments #Hemififths|hemififths]], namo
Related temperament: [[Breedsmic temperaments #Hemififths|hemififths]], namo


Line 504: Line 612:
[[Tp tuning #T2 tuning|RMS error]]: 0.2344 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2344 cents


==== 2.3.7.11 ====
==== 2.3.7.11 ====
Subgroup: 2.3.7.11
Subgroup: 2.3.7.11


Line 521: Line 629:
[[Tp tuning #T2 tuning|RMS error]]: 0.6108 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.6108 cents


==== 2.3.7.11.13 ====
==== 2.3.7.11.13 ====
Subgroup: 2.3.7.11.13
Subgroup: 2.3.7.11.13


Line 538: Line 646:
[[Tp tuning #T2 tuning|RMS error]]: 0.7167 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.7167 cents


=== Ennea ===
=== Ennea ===
Subgroup: 2.3.7.11
Subgroup: 2.3.7.11


Line 555: Line 663:
[[Tp tuning #T2 tuning|RMS error]]: 0.0383 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.0383 cents


=== Leapfrog ===
=== Leapfrog ===
{{see also|Gentle region}}
{{see also|Gentle region}}


Line 574: Line 682:
[[Tp tuning #T2 tuning|RMS error]]: 0.6202 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.6202 cents


==== 2.3.7.11 ====
==== 2.3.7.11 ====
Subgroup: 2.3.7.11
Subgroup: 2.3.7.11


Line 591: Line 699:
[[Tp tuning #T2 tuning|RMS error]]: 0.6047 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.6047 cents


==== 2.3.7.11.13 ====
==== 2.3.7.11.13 ====
Subgroup: 2.3.7.11.13
Subgroup: 2.3.7.11.13


Line 608: Line 716:
[[Tp tuning #T2 tuning|RMS error]]: 0.7541 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.7541 cents


==== Music ====
; Music
*Suite for Harpsichord in A Locrian, tuning: Eb-G# in [[46edo]] by [[User:IlL|IlL]] (in progress):
* Suite for Harpsichord in A Locrian, tuning: Eb-G# in [[46edo|46EDO]] by [[User:IlL|IlL]] (in progress):
**I. Prelude
** I. Prelude
**II. Allemande
** II. Allemande
**III. Courante
** III. Courante
**[[:File:Locrian Suite Sarabande.mp3|IV. Sarabande]] ([[:File:Locrian Suite Sarabande Score.pdf|score]], [[:File:Locrian Suite Sarabande 17edo.mp3|17edo version]])
** [[:File:Locrian Suite Sarabande.mp3|IV. Sarabande]] ([[:File:Locrian Suite Sarabande Score.pdf|score]], [[:File:Locrian Suite Sarabande 17edo.mp3|17EDO version]])
**[[:File:Locrian Suite Menuet.mp3|V. Menuet and Trio]]
** [[:File:Locrian Suite Menuet.mp3|V. Menuet and Trio]]
**[[:File:Locrian Suite Gavotte.mp3|VI. Gavotte I and II]]
** [[:File:Locrian Suite Gavotte.mp3|VI. Gavotte I and II]]
**[[:File:Locrian Suite Gigue.mp3|VII. Gigue]]
** [[:File:Locrian Suite Gigue.mp3|VII. Gigue]]


=== Parapyth (Rank 3) ===
=== Parapyth (Rank 3) ===
Subgroup: 2.3.7.11
Subgroup: 2.3.7.11


Line 635: Line 743:
[[Tp tuning #T2 tuning|RMS error]]: 0.4149 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.4149 cents


==== 2.3.7.11.13 ====
==== 2.3.7.11.13 ====
Subgroup: 2.3.7.11.13
Subgroup: 2.3.7.11.13


Line 654: Line 762:
[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents


=== Neutral ===
=== Neutral ===
Neutral can be thought of as the 2.3.11 version of either [[mohajira]] or [[maqamic]], as well as suhajira and ringo. Among other things, it is the temperament optimizing the [[neutral tetrad]].
Neutral can be thought of as the 2.3.11 version of either [[mohajira]] or [[maqamic]], as well as suhajira and ringo. Among other things, it is the temperament optimizing the [[neutral tetrad]].


Line 679: Line 787:
[[neutraltet17|Seventeen note mega chromatic scale]]
[[neutraltet17|Seventeen note mega chromatic scale]]


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


Line 696: Line 804:
[[Tp tuning #T2 tuning|RMS error]]: 0.7038 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.7038 cents


== No-threes subgroup ==
== No-threes subgroup ==


=== Didacus ===
=== Didacus ===
Related temperaments: [[Chromatic pairs #Roulette|roulette]], [[Gamelismic clan #Hemithirds|hemithirds]]
Related temperaments: [[Chromatic pairs #Roulette|roulette]], [[Gamelismic clan #Hemithirds|hemithirds]]


Line 717: Line 825:
[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents


=== Llywelyn ===
=== Llywelyn ===
Subgroup: 2.5.7
Subgroup: 2.5.7


Line 734: Line 842:
[[Tp tuning #T2 tuning|RMS error]]: 0.5391 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.5391 cents


=== Rainy ===
=== Rainy ===
Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]].
Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]].


Line 753: Line 861:
[[Tp tuning #T2 tuning|RMS error]]: 0.058596 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.058596 cents


== 2.9.7.11 subgroup ==
== 2.9.7.11 subgroup ==


=== Machine ===
=== Machine ===
Subgroup: 2.9.7.11
Subgroup: 2.9.7.11


Line 772: Line 880:
[[Tp tuning #T2 tuning|RMS error]]: 1.977 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.977 cents


=== Apparatus ===
=== Apparatus ===
Subgroup: 2.9.7.11
Subgroup: 2.9.7.11


Line 789: Line 897:
[[Tp tuning #T2 tuning|RMS error]]: 0.0673 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.0673 cents


=== Mechanism ===
=== Mechanism ===
Subgroup: 2.9.7.11
Subgroup: 2.9.7.11


Line 806: Line 914:
[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents


== 2.9.15.7 subgroup ==
== 2.9.15.7 subgroup ==


=== Stacks (aka 2magic) ===
=== Stacks (aka 2magic) ===
Subgroup: 2.9.15.7
Subgroup: 2.9.15.7


Line 825: Line 933:
[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents


==== 2.9.15.7.11 ====
==== 2.9.15.7.11 ====
Subgroup: 2.9.15.7.11
Subgroup: 2.9.15.7.11


Line 842: Line 950:
RMS error: 1.226 cents
RMS error: 1.226 cents


==== 2.9.15.7.11.13 ====
==== 2.9.15.7.11.13 ====
Subgroup: 2.9.15.7.11.13
Subgroup: 2.9.15.7.11.13


Line 859: Line 967:
RMS error: 1.540 cents
RMS error: 1.540 cents


== 2.9.21 subgroup ==
== 2.9.21 subgroup ==


=== A-team ===
=== A-team ===
Subgroup: 2.9.21
Subgroup: 2.9.21


Line 878: Line 986:
[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents


== Miscellaneous subgroup temperaments ==
== Miscellaneous subgroup temperaments ==


=== Historical ===
=== Historical ===
Subgroup: 2.3.7/5.11/5.13/5
Subgroup: 2.3.7/5.11/5.13/5


Line 893: Line 1,001:
[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents


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