Subgroup temperaments: Difference between revisions
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{{See also|No-threes subgroup temperaments #Frostburn}} | {{See also|No-threes subgroup temperaments #Frostburn}} | ||
Baldy results from taking every other generator of the [[garibaldi | 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 137: | Line 137: | ||
{{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 274: | Line 313: | ||
Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]] | Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]] | ||
== 2.9.7.13.17 subgroup == | |||
=== 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]]. | |||
[[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 == | == 2.9.11 subgroup == | ||
| Line 518: | Line 572: | ||
Fourwar is named after the closely related [[hemiwar]] temperament. | Fourwar is named after the closely related [[hemiwar]] temperament. | ||
{{Todo|inline=1|cleanup}} | |||
<pre> | <pre> | ||
| Line 789: | Line 845: | ||
Mapping generators: ~4, ~9/64 | Mapping generators: ~4, ~9/64 | ||
[[Optimal tuning]] ([[CTE]]): ~ | [[Optimal tuning]] ([[CTE]]): ~9/4 = 1419.190 | ||
[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49 | [[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49 | ||
| Line 807: | Line 863: | ||
: [[gencom]]: [8 9/8; 64/63] | : [[gencom]]: [8 9/8; 64/63] | ||
[[Optimal tuning]] ([[CTE]]): | [[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 814: | Line 870: | ||
= Fractional subgroup temperaments = | = Fractional subgroup temperaments = | ||
== 2.5/ | == 2.5/3.… subgroups == | ||
=== Magicaltet === | === Magicaltet === | ||
{{See also| Chromatic pairs #Magicaltet }} | {{See also| Chromatic pairs #Magicaltet }} | ||
| Line 1,005: | Line 1,061: | ||
{{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 1,017: | Line 1,073: | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = | * [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 882.886 | ||
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = | * [[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 1,024: | Line 1,080: | ||
[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents | [[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents | ||
== 2.….7/ | == 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 the main rank-2 chain of [[guanyin]] and a restriction of [[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 1,034: | Line 1,090: | ||
[[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 | {{Mapping|legend=2| 1 0 1 3 | 0 -3 1 -5 }} | ||
: mapping generators: ~2, ~ | : mapping generators: ~2, ~7/6 | ||
{{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 }} | ||
| Line 1,041: | Line 1,097: | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~ | * ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.455 | ||
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~ | * ([[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* }} | ||
| Line 1,048: | Line 1,104: | ||
[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents | [[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 }}) | |||
{{Mapping|legend=2| 1 0 1 3 1 | 0 -3 1 -5 12 }} | |||
: mapping generators: ~2, ~12/7 | |||
[[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 ==== | ||
| Line 1,142: | Line 1,217: | ||
[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents | [[Tp tuning #T2 tuning|RMS error]]: 1.064 cents | ||
== 2.….9/ | == 2.….9/7.… subgroups == | ||
=== Marveltri === | === Marveltri === | ||
{{See also| Chromatic pairs #Marveltri }} | {{See also| Chromatic pairs #Marveltri }} | ||
| Line 1,184: | Line 1,259: | ||
[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents | [[Tp tuning #T2 tuning|RMS error]]: 1.074 cents | ||
== 2.….7/ | == 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 1,199: | Line 1,273: | ||
[[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}} | ||
=== | === Argentic === | ||
Argentic is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]]. | |||
[[Subgroup]]: 2.3.7/5 | |||
[[Comma list]]: [[5120/5103]] = {{monzo| 10 -6 -1 }} | |||
{{Mapping|legend=2| 1 0 10 | 0 1 -6 }} | |||
: mapping generators: ~2, ~3 | |||
[[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) ==== | ==== Edson (2.3.7/5.11/5.13/5 subgroup) ==== | ||
| Line 1,322: | Line 1,412: | ||
[[Tp tuning #T2 tuning|RMS error]]: 0.7521 cents | [[Tp tuning #T2 tuning|RMS error]]: 0.7521 cents | ||
== 2.….11/ | == 2.….11/5.… subgroups == | ||
=== Petrtri === | === Petrtri === | ||
{{See also| Chromatic pairs #Petrtri }} | {{See also| Chromatic pairs #Petrtri }} | ||
| Line 1,380: | Line 1,469: | ||
[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents | [[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents | ||
=== Trisect === | |||
=== | Trisect divides every Pythagorean interval into three, and is the much more accurate subgroup restriction of [[Augmented family #Trisected|trisected]]. | ||
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. | [[Subgroup]]: 2.3.7.11/5 | ||
[[Comma list]]: | [[Comma list]]: 1029/1024, 4000/3993 | ||
{{Mapping|legend=2| | {{Mapping|legend=2| 3 0 10 5 | 0 3 -1 -1 }} | ||
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.742 | |||
{{Optimal ET sequence|legend=1| 15, 21, 36, 123, 159, 195, 231 }} | |||
[[Tp tuning #T2 tuning|RMS error]]: ??? | |||
: | |||
[[ | ==== 2.3.7.11/5.13 subgroup ==== | ||
[[Subgroup]]: 2.3.7.11/5.13 | |||
[[Comma list]]: 1029/1024, 1575/1573, 2080/2079 | |||
{{Mapping|legend=2| 3 0 10 5 0 | 0 3 -1 -1 7 }} | |||
[[ | [[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.918 | ||
{{Optimal ET sequence|legend=1| 15, 21f, 36, 87, 123, 159 }} | |||
[[ | [[Tp tuning #T2 tuning|RMS error]]: ??? | ||
==== 2.3.7.11/5.13.17 subgroup ==== | |||
[[Subgroup]]: 2.3.7.11/5.13.17 | |||
[[ | [[Comma list]]: 273/272, 833/832, 1575/1573, 2080/2079 | ||
{{Mapping|legend=2| 3 0 10 5 0 -2 | 0 3 -1 -1 7 9 }} | |||
== | [[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.820 | ||
{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123, 159 }} | |||
[[ | [[Tp tuning #T2 tuning|RMS error]]: ??? | ||
[[ | ===== Trisector ===== | ||
[[Subgroup]]: 2.3.7.11/5.13.17.19 | |||
[[Comma list]]: 210/209, 273/272, 286/285, 595/594, 2080/2079 | |||
{{Mapping|legend=3 | {{Mapping|legend=2| 3 0 10 5 0 -2 8 | 0 3 -1 -1 7 9 3 }} | ||
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~ | [[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.894 | ||
{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123h, 159h }} | |||
[[Tp tuning #T2 tuning|RMS error]]: ??? | |||
===== 2.3.7.11/5.13.17.19.23 subgroup ===== | |||
[[Subgroup]]: 2.3.7.11/5.13.17.19.23 | |||
[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 595/594, 2080/2079 | |||
{{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]]: ??? | |||
== 2.….11/7.… subgroups == | |||
=== Pepperoni === | |||
{{Main| Parapyth }} | |||
{{See also| Chromatic pairs #Pepperoni }} | |||
Pepperoni is generated by a fifth and can be described as the 5 & 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 | ||
==== Pakkanian hemipyth ==== | [[Comma list]]: 352/351, 364/363 | ||
[[Subgroup]]: 2.3.11.13/5.17 | |||
{{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 & 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 | [[Comma list]]: 221/220, 243/242, 289/288 | ||
{{Mapping|legend=2| 2 0 -1 -2 5 | 0 2 5 3 2 }} | {{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]] | |||
== 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]]. | |||
[[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}} | |||
{{Optimal ET sequence|legend=1| 7, 18, 25 }} | |||
[[ | [[Badness]] (Sintel): 0.005 | ||
==== 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. | |||
Subgroup: 2.17/7.19/7.23/7 | |||
[[ | Comma list: [[323/322]], [[392/391]] | ||
Subgroup-val mapping: {{Mapping| 1 0 4 3 | 0 1 -2 -1 }} | |||
{{ | Optimal tunings: | ||
* Subgroup WE: ~2 = 1199.871{{c}}, ~17/14 = 336.243{{c}} | |||
* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 336.296{{c}} | |||
{{Optimal ET sequence|legend=0| 7, 18, 25 }} | |||
Badness (Sintel): 0.029 | |||
==== 2.25/7.17/7.19/7.23/7 subgroup ==== | |||
Subgroup: 2.25/7.17/7.19/7.23/7 | |||
Comma list: [[323/322]], [[392/391]], [[476/475]] | |||
Subgroup-val mapping: {{Mapping| 1 -2 0 4 3 | 0 3 1 -2 -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}} | |||
{{ | {{Optimal ET sequence|legend=0| 7, 18, 25 }} | ||
Badness (Sintel): 0.053 | |||
== 3/2.5/2.… subgroups == | |||
== 3/2.5/ | |||
{{Main|Half-prime subgroup}} | {{Main|Half-prime subgroup}} | ||
| Line 1,579: | Line 1,839: | ||
[[Optimal ET sequence]]: [[8edf]], [[11edf]] | [[Optimal ET sequence]]: [[8edf]], [[11edf]] | ||
== 3/2.5/ | == 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 1,625: | Line 1,885: | ||
== 5/2-equave subgroups == | == 5/2-equave subgroups == | ||
=== Hyperion === | === Hyperion === | ||
[[Subgroup]]: 5/2.7.11 | [[Subgroup]]: 5/2.7.11 | ||
| Line 1,645: | Line 1,904: | ||
* [[Substitute harmonic]] temperaments | * [[Substitute harmonic]] temperaments | ||
<!-- main article --> | [[Category:Subgroup temperaments| ]] <!-- main article --> | ||
[[Category:Temperament collections]] | |||
[[Category:Temperament collections]] | {{Todo| review | cleanup }} | ||