Subgroup temperaments: Difference between revisions
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For temperaments that omit various prime harmonics, see: | For temperaments that omit various prime harmonics, see: | ||
* [[No-elevens subgroup temperaments]] | * [[No-elevens subgroup temperaments]] | ||
* [[No-sevens subgroup temperaments]] | * [[No-sevens subgroup temperaments]] | ||
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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]]. | 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]]. | ||
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. | |||
== 2. | |||
== 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. | |||
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. | |||
==== 2.3.25 subgroup ==== | |||
[[Subgroup]]: 2.3.25 | |||
[[Comma list]]: 2048/2025 | |||
{{Mapping|legend=3| 1 0 11 | 0 1 -4 }} | |||
: mapping generators: ~2, ~3 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1198.8214{{c}}, ~3/2 = 704.2059{{c}} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.9585{{c}} | |||
{{Optimal ET sequence|legend=1| 12, 17, 46, 63, 80, 143c′ }} | |||
[[ | [[Badness]] (Sintel): 0.234 | ||
===== 2.3.25.23 subgroup ===== | |||
Subgroup: 2.3.25.23 | |||
Comma list: 576/575, 736/729 | |||
{{ | {{Mapping|legend=2| 1 0 11 -5 | 0 1 -4 6 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1198.9504{{c}}, ~3/2 = 704.6585{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.1888{{c}} | |||
= | {{Optimal ET sequence|legend=0| 12, 17, 63, 80, 97c′, 114c′ }} | ||
Badness (Sintel): 0.223 | |||
==== Sburb ==== | |||
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. | |||
Subgroup: 2.3.25.7 | |||
Comma list: 50/49, 64/63 | |||
{{Mapping|legend=2| 1 0 11 6 | 0 1 -4 -2 }} | |||
== | Optimal tunings: | ||
* WE: ~2 = 1197.6967{{c}}, ~3/2 = 705.6906{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.3438{{c}} | |||
{{Optimal ET sequence|legend=0| 5, 12, 17, 39d }} | |||
Badness (Sintel): 0.253 | |||
===== 2.3.25.7.23 subgroup ===== | |||
Subgroup: 2.3.25.7.23 | |||
Comma list: 50/49, 64/63, 162/161 | |||
{{Mapping|legend=2| 1 0 11 6 -5 | 0 1 -4 -2 6 }} | |||
== | Optimal tunings: | ||
{{ | * WE: ~2 = 1197.5577{{c}}, ~3/2 = 705.3089{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 706.7010{{c}} | |||
{{Optimal ET sequence|legend=0| 5, 12, 17 }} | |||
Badness (Sintel): 0.419 | |||
=== Hypnosis === | |||
Related temperaments: [[swetismic temperaments #Hypnos|hypnos]], [[alphatricot family #Alphatrimot|alphatrimot]]. | |||
[[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 | |||
[[Subgroup]]: | [[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 | {{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]]. | |||
[[Subgroup]]: 2.3.7/5.11/5.13/5 | |||
[[ | [[Comma list]]: [[352/351]] ({{monzo| 5 -3 0 1 -1 }}), [[676/675]] ({{monzo| 2 -3 0 0 2 }}), [[847/845]] ({{monzo| 0 0 1 2 -2 }}) | ||
{{ | {{Mapping|legend=3| 1 0 10 -6 -1 | 0 2 -12 9 3 }} | ||
{{Mapping|legend=5| 1 0 -3/4 37/4 -27/4 -7/4 | 0 2 0 -12 9 3 }} | |||
: mapping generators: ~2, ~26/15 | |||
[[Optimal tuning]]s: | |||
* | * [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.7072{{c}}, ~26/15 = 951.2727{{c}} | ||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.5016{{c}} | |||
= | {{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198 }} | ||
[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents | |||
=== Historical === | |||
{{Distinguish|Historical temperaments}} | |||
{{Distinguish|History (temperament)}} | |||
Historical is essentially an analogue of [[miracle]] that splits [[4/3]] in six rather than [[3/2]]. It tempers out the comma [[4000/3993]] ([[S-expression|S10/S11]]) to set [[11/10]] equal to one-third of 4/3, and [[676/675]] ([[S-expression|S13/S15]]) to equate [[15/13]] to one-half of 4/3, and tempers out [[441/440]] ({{S|21}}) to split 11/10 into two instances of [[22/21]][[~]][[21/20]]. [[Schismatic family #Sextilifourths|Sextilifourths]] adds the [[schismatic family #Schismic|schismic]] mapping of prime 5 (reached by eight fourths) to complete the 13-limit. | |||
[[Subgroup]]: 2.3.7/5.11/5.13/5 | |||
{{ | [[Comma list]]: 364/363 ({{monzo| 2 -1 1 -2 1 }}), 441/440 ({{monzo| -3 2 2 -1 0 }}, 1001/1000 ({{monzo| -3 0 1 1 1 }}) | ||
= | {{Mapping|legend=3| 1 2 0 1 2 | 0 -6 7 2 -9 }} | ||
: mapping generators: ~2, ~21/20 | |||
[[Optimal tuning]]s: | |||
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.0242{{c}}, ~21/20 = 83.0177{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.0144{{c}} | |||
{{Optimal ET sequence|legend=1| 14, 29, 101, 130, 159 }} | |||
[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents | |||
{{ | === Direct breedsmic === | ||
This temperament was first proposed by {{u|Royalmilktea}}. The name was established by [[Lériendil]] in 2024. Related temperaments: [[breedsmic temperaments #Hemififths|hemififths]] and [[septischismic clan #Newt|newt]]. | |||
[[Subgroup]]: 2.3.49/5 | |||
[[Subgroup]]: 2. | |||
[[Comma list]]: | [[Comma list]]: 2401/2400 ({{monzo| -5 -1 2 }}) | ||
{{Mapping|legend= | {{Mapping|legend=3| 1 1 3 | 0 2 1 }} | ||
: mapping generators: ~2, ~49/40 | |||
: | [[Optimal tuning]]: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.0206{{c}}, ~49/40 = 350.9724{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.9743{{c}} | |||
{{ | {{Optimal ET sequence|legend=1| 7, 17, 24, 41, 65, 106, 253, 359, 465, 1036, 1501, 1966*, 5433* }} | ||
: <nowiki/>* Wart for 49/5 | |||
==== Tobago ==== | |||
{{See also| Chromatic pairs #Tobago }} | |||
[[ | Tobago, the 10 & 14 temperament in the 2.3.11.13/5 subgroup, extends [[rastmic clan #Neutral|neutral]] and [[#Barbados|barbados]]. | ||
[[Subgroup]]: 2.3.11.13/5 | |||
[[ | [[Comma list]]: [[243/242]] ({{monzo| -1 5 -2 0 }}), [[676/675]] ({{monzo| 2 -3 0 2 }}) | ||
{{Mapping|legend=3| 2 0 -1 -2 | 0 2 5 3 }} | |||
{{ | |||
{{Mapping|legend=5| 2 0 -5 0 -1 -1 | 0 2 -3/2 0 5 3/2 }} | |||
: mapping generators: ~55/39, ~26/15 | |||
[[Subgroup]]: | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~55/39 = 599.9845{{c}}, ~26/15 = 950.6637{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~55/39 = 600.0000{{c}}, ~26/15 = 950.6776{{c}} | |||
{{Optimal ET sequence|legend=1| 10, 14, 24, 82, 106, 130, 154 }} | |||
[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents | |||
==== Pakkanian hemipyth ==== | |||
[[Subgroup]]: 2.3.11.13/5.17 | |||
[[Comma list]]: 221/220 ({{monzo| -2 0 -1 1 1 }}), 243/242 ({{monzo| -1 5 -2 0 0 }}), 289/288 ({{monzo| -5 -2 0 0 2 }}) | |||
{{Mapping|legend=3| 2 0 -1 -2 5 | 0 2 5 3 2 }} | |||
{{ | [[Optimal tuning]]s: | ||
* [[Tp tuning|subgroup WE]]: ~17/12 = 600.1781{{c}}, ~26/15 = 950.7656{{c}} (~15/13 = 249.2344{{c}}) | |||
* [[Tp tuning|subgroup CWE]]: ~17/12 = 600.0000{{c}}, ~26/15 = 950.6011{{c}} (~15/13 = 249.3989{{c}}) | |||
{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }} | |||
: <nowiki/>* Wart for 13/5 | |||
=== | === Bridgetown === | ||
{{See also| Chromatic pairs #Bridgetown }} | |||
[[ | Bridgetown, the 5 & 24 temperament in the 2.3.11/5.13/5 subgroup, is the no-7/5 restriction of [[#Haumea|haumea]] and the add-11/5 [[extension]] of [[#Barbados|barbados]]. | ||
[[ | [[Subgroup]]: 2.3.11/5.13/5 | ||
{{ | [[Comma list]]: [[352/351]] ({{monzo| 5 -3 1 -1 }}), [[676/675]] ({{monzo| 2 -3 0 2 }}) | ||
{{Mapping|legend=3| 1 0 -6 -1 | 0 2 9 3 }} | |||
{{Mapping|legend= | {{Mapping|legend=5| 1 0 7/3 0 -11/3 4/3 | 0 2 -4 0 5 -1 }} | ||
: mapping generators: ~2, ~26/15 | |||
: [[ | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.6281{{c}}, ~26/15 = 951.3060{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.5580{{c}} | |||
{{Optimal ET sequence|legend=1| 5, 19*, 24, 29, 169, 198, 227, 256, 285, 314, 343 }} | |||
: <nowiki/>* Wart for 11/5 | |||
* | |||
[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents | |||
[[ | === Seventeen-cot === | ||
Seventeen-cot may be described as the 29 & 465 temperament in the 2.3.11/5.13/5 subgroup. It tempers out the [[tendoartisma]] in the 2.3.13/5 subgroup. It can be generated with a ~2/1 octave and a ~169/165 generator which is 1/17 of a ~3/2 perfect fifth. It was named by {{u|FilterNashi}} in 2026. | |||
[[Subgroup]]: 2.3.11/5.13/5 | |||
[[ | [[Comma basis]]: 43940/43923 ({{monzo| 2 -1 -4 3 }}), 225000/224939 ({{monzo| 3 2 -3 -2 }}) | ||
{{Mapping|legend=3| 1 1 1 1 | 0 17 4 11 }} | |||
: mapping generators: ~2, ~169/165 | |||
{{ | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.993492{{c}}, ~169/165 = 41.291827{{c}} | |||
: [[error map]]: {{val| -0.0065 -0.0004 +0.1566 -0.0104 }} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.000000, ~169/165 = 41.291746{{c}} | |||
: error map: {{val| 0.0000 +0.0047 +0.1628 -0.0047 }} | |||
{{Optimal ET sequence|legend=1| 29, 262, 291, 320, 349, 378, 407, 436, 465, 959, 1424, 6161… }} | |||
[[Badness]] (Sintel): 0.080 | |||
=== 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]]: 2.3.11/7 | ||
{{ | [[Comma list]]: 256/243 ({{monzo| 8 -5 0 }}) | ||
{{Mapping|legend=3| 5 8 0 | 0 0 1 }} | |||
: mapping generators: ~9/8, ~11/7 | |||
[[ | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~8/7 = 238.851{{c}}, ~11/7 = 782.457{{c}} | |||
: [[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 & 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 ({{monzo| 5 -3 1 -1 }}), 364/363 ({{monzo| 2 -1 -2 1 }}) | ||
{{Mapping|legend=3| 1 0 7 12 | 0 1 -4 -7 }} | |||
{{ | {{Mapping|legend=5| 1 0 0 -19/3 2/3 17/3 | 0 1 0 11/3 -1/3 -10/3 }} | ||
: mapping generators: ~2, ~3 | |||
=== | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.3706{{c}}, ~11/7 = 703.4872{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~11/7 = 703.8328{{c}} | |||
{{Optimal ET sequence|legend=1| 12, 17, 29, 75, 104 }} | |||
[[ | [[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents | ||
[[ | === Fendo === | ||
Fendo tempers out [[40/39]] in the 2.3.13/5 subgroup. This equates [[4/3]] with the tendo third [[13/10]]. Fendo can be viewed as an alternative interpretation of scales, chords, etc. commonly associated with [[father family #Father|father]], as it is sometimes perceived that father's characteristics tend to suggest harmonic function more than to actually serve as a temperament. It was named by [[User: VectorGraphics|Vector]] in 2025 as a contraction of ''fourth'' and ''tendo third''. | |||
In fendo, the fourth / tendo third serves the role of both, making the main chords of triadic harmony in fendo essentially suspended chords. | |||
[[ | [[5edo]] and [[13edo]] make good fendo tunings. | ||
[[Subgroup]]: 2.3.13/5 | |||
[[Comma list]]: 40/39 ({{monzo| 3 -1 -1 }}) | |||
{{Mapping|legend=3| 1 0 3 | 0 1 -1 }} | |||
: mapping generators: ~2, ~3 | |||
[[ | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1195.6502{{c}}, ~3/2 = 709.8204{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 710.1529{{c}} | |||
{{ | {{Optimal ET sequence|legend=1| 1, 2, 3, 5, 17*, 22*, 27*, 32* }} | ||
: <nowiki/>* Wart for 13/5 | |||
=== Barbados === | |||
Barbados is the restriction of [[#Bridgetown|bridgetown]] to the 2.3.13/5 subgroup, and tempers out [[676/675]] alone. 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 }}) | |||
{{Mapping|legend=3| 1 0 -1 | 0 2 3 }} | |||
: mapping generators: ~2, ~26/15 | |||
{{ | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9502{{c}}, ~26/15 = 951.0713{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.0933{{c}} | |||
{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 29, 53, 82, 135 }} | |||
[[Badness]] (Smith): 0.002335 | |||
; Music | |||
* ''[[Desert Island Rain]]'' by [[Sevish]] – [https://web.archive.org/web/20201127013711/http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 play] | [https://sevish.bandcamp.com/track/desert-island-rain Bandcamp] | [https://soundcloud.com/sevish/desert-island-rain SoundCloud] | [https://www.youtube.com/watch?v=Gcgawrr2xao YouTube] – in 313edo tuned Barbados[9] | |||
=== 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=3| 1 0 -3 | 0 1 3 }} | |||
: mapping generators: ~2, ~3 | |||
Optimal | [[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 }} | |||
=== | === 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=3| 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 | {{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31*, 50* }} | ||
: <nowiki/>* Wart for 19/7 | |||
[[Badness]] (Sintel): 0.082 | |||
== 2.9.… subgroups == | |||
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]]. | |||
=== Baldy === | |||
{{See also| Chromatic pairs #Baldy }} | |||
Baldy is every other step of [[schismatic family #Garibaldi|garibaldi]]. It is also the add-9 extension of [[no-threes subgroup temperaments #Frostburn|frostburn]]. One of the best extension is to the 2.9.5.7.13 subgroup, mapping 13/8 to +10 whole tones, the same as the cassandra temperament. | |||
[[Subgroup]]: 2.9. | [[Subgroup]]: 2.9.5.7 | ||
[[Comma list]]: | [[Comma list]]: 225/224, 3125/3087 | ||
{{Mapping|legend= | {{Mapping|legend=3| 1 0 15 25 | 0 1 -4 -7 }} | ||
: mapping generators: ~2, ~9 | |||
: | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.1233{{c}}, ~9/8 = 204.1913{{c}} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/8 = 204.1549{{c}} | |||
{{ | {{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47, 194 }} | ||
[[Badness]] (Sintel): 0.274 | |||
==== 2.9.5.7.13 subgroup ==== | |||
Subgroup: 2.9.5.7.13 | |||
Comma list: 225/224, 325/324, 640/637 | |||
{{Mapping|legend=2| 1 0 15 25 -28 | 0 1 -4 -7 10 }} | |||
= | {{Mapping|legend=4| 1 0 15 25 0 28 | 0 1/2 -4 -7 0 10 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1200.0155{{c}}, ~9/8 = 204.0931{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~9/8 = 204.0902{{c}} | |||
{{Optimal ET sequence|legend=0| 6, 35, 41, 47, 100, 147 }} | |||
Badness (Sintel): 0.386 | |||
RMS error: 0.5999 cents | |||
==== Baldanders ==== | |||
Baldanders results from taking every other generator of the [[schismatic family #Andromeda|andromeda]], mapping 11/8 to -9 whole tones. | |||
Subgroup: 2.9.5.7.11 | |||
Comma list: 100/99, 225/224, 245/242 | |||
{{Mapping|legend=2| 1 0 15 25 32 | 0 1 -4 -7 -9 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.1917{{c}}, ~9/8 = 204.7755{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~9/8 = 204.7197{{c}} | |||
{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }} | |||
: | Badness (Sintel): 0.389 | ||
===== 2.9.5.7.11.13 ===== | |||
Subgroup: 2.9.5.7.11.13 | |||
Comma list: 100/99, 144/143, 225/224, 245/242 | |||
= | {{Mapping|legend=2| 1 0 15 25 32 -28 | 0 1 -4 -7 -9 10 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1199.6385{{c}}, ~9/8 = 204.3526{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~9/8 = 204.4234{{c}} | |||
{{Optimal ET sequence|legend=0| 6, 35, 41, 47, 88e }} | |||
Badness (Sintel): 0.653 | |||
=== Daemotertiaschis === | |||
{{See also| Schismatic family #Tertiaschis }} | |||
Daemotertiaschis is produced by taking every other generator of tertiaschis, and the subgroup is chosen so it tempers out exactly the same commas. It is notable due to offering a [[7L 4s|daemotonic 7L 4s]] scale of reasonable hardness (hence the name – daemo- + tertiaschis), which is notoriously difficult to approximate with simple JI or RTT methods. | |||
Subgroup: 2.9.5.7.33.13.17 | |||
Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976 | |||
{{Mapping|legend=2| 1 1 11 -16 13 -18 20 | 0 3 -12 26 -11 30 -22 }} | |||
: mapping generators: ~2, ~33/20 | |||
{{ | Optimal tunings: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.3019{{c}}, 33/20 = 868.1949{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, 33/20 = 867.9773{{c}} | |||
{{Optimal ET sequence|legend=1| 47, 159, 206, 253, 459… }} | |||
[[ | [[Badness]] (Sintel): 0.439 | ||
=== A-team === | |||
{{See also| Dual-fifth temperaments #Dual-3 A-Team }} | |||
A-team is every other step of [[gamelismic clan #Mothra|mothra]]. | |||
[[Subgroup]]: | [[Subgroup]]: 2.9.5.21 | ||
[[Comma list]]: 81/80 | [[Comma list]]: 81/80 ({{monzo| -4 2 -1 0 }}), 1029/1024 ({{monzo| -10 -1 0 3 }}) | ||
{{Mapping|legend= | {{Mapping|legend=3| 1 2 0 4 | 0 3 6 1 }} | ||
: | {{Mapping|legend=5| 1 1 0 3 | 0 3/2 6 -1/2 }} | ||
: mapping generators: ~2, ~21/16 | |||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.9303{{c}}, ~21/16 = 464.7465{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 464.5029{{c}} | |||
{{Optimal ET sequence|legend=1| 13, 18, 31 }} | |||
[[Badness]] (Sintel): 0.177 | |||
===== 2.9.5.21.11 subgroup ===== | |||
Subgroup: 2.9.5.21.11 | |||
Comma list: 81/80 ({{monzo| -4 2 -1 0 }}), 99/98 ({{monzo| -1 2 0 -2 1 }}), 385/384 ({{monzo| -7 -1 1 1 1 }}) | |||
{{Mapping|legend=2| 1 0 | {{Mapping|legend=2| 1 2 0 4 5 | 0 3 6 1 -4 }} | ||
{{Mapping|legend=4| 1 1 0 3 5 | 0 3/2 6 -1/2 -4 }} | |||
Optimal tunings: | |||
* Subgroup WE: ~2 = 1201.3979{{c}}, ~21/16 = 464.6020{{c}} | |||
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~21/16 = 464.1242{{c}} | |||
{{Optimal ET sequence|legend=0| 13, 31, 44, 75b′ }} | |||
Badness (Sintel): 0.213 | |||
=== Glacial === | |||
{{See also| Chromatic pairs #Glacial }} | |||
[[ | [[Subgroup]]: 2.9.5.11.13 | ||
[[Comma list]]: 45/44, 65/64, 81/80 | |||
{{Mapping|legend=3| 1 0 -4 -6 10 | 0 1 2 3 -2 }} | |||
{{Mapping|legend=5| 1 0 -4 0 -6 10 | 0 1/2 2 0 3 -2 }} | |||
: mapping generators: ~2, ~9 | |||
[[ | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1202.6916{{c}}, ~9/8 = 186.1446{{c}} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/8 = 186.1740{{c}} | |||
{{Optimal ET sequence|legend=1| 6, 13, 45b′f }} | |||
{{ | |||
[[ | [[Badness]] (Sintel): 0.281 | ||
[[ | [[Tp tuning #T2 tuning|RMS error]]: 2.887 cents | ||
Music: | |||
* ''[[Thundersnow]]'' by [[Sevish]] (2021) | |||
=== Stacks (a.k.a. 2magic) === | |||
Stacks, the 11 & 30 temperament in the 2.9.15.7.11.13 subgroup, is every other step of [[Magic family #Magic|magic]]. | |||
[[ | [[Subgroup]]: 2.9.15.7 | ||
[[ | [[Comma list]]: 225/224 ({{monzo| -5 0 2 -1 }}), 245/243 ({{monzo| 0 -3 1 2 }}) | ||
{{Mapping|legend=3| 1 0 2 -1 | 0 5 3 6 }} | |||
= | {{Mapping|legend=5| 1 0 2 -1 | 0 5/2 1/2 6 }} | ||
: mapping generators: ~2, ~14/9 | |||
[[ | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1201.0786{{c}}, ~14/9 = 761.3878{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 760.9151{{c}} | |||
{{ | {{Optimal ET sequence|legend=1| 8, 11, 19, 30, 41, 71, 112* }} | ||
: <nowiki/>* Wart for 15 | |||
[[ | [[Badness]] (Sintel): 0.0884 | ||
[[ | [[Tp tuning #T2 tuning|RMS error]]: 1.074 cents | ||
==== 2.9.15.7.11 ==== | |||
Subgroup: 2.9.15.7.11 | |||
Comma list: 100/99 ({{monzo| 2 -2 2 0 -1 }}), 225/224 ({{monzo| -5 0 2 -1 0 }}), 245/243 ({{monzo| 0 -3 1 2 0 }}) | |||
{{Mapping|legend=2| 1 0 2 -1 6 | 0 5 3 6 -4 }} | |||
{{ | {{Mapping|legend=4| 1 0 2 -1 6 | 0 5/2 1/2 6 -4 }} | ||
Optimal tunings: | |||
* Subgroup WE: ~2 = 1200.1372{{c}}, ~14/9 = 761.4799{{c}} | |||
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~14/9 = 761.4017{{c}} | |||
{{Optimal ET sequence|legend=0| 8, 11, 30, 41, 52, 93, 145 }} | |||
Badness (Sintel): 0.166 | |||
RMS error: 1.226 cents | |||
==== | ==== 2.9.15.7.11.13 ==== | ||
Subgroup: 2.9.15.7.11.13 | |||
Comma list: 100/99 ({{monzo| 2 -2 2 0 -1 0 }}), 105/104 ({{monzo| -3 0 1 1 0 -1 }}), 144/143 ({{monzo| 4 1 0 0 -1 -1 }}), 196/195 ({{monzo| 2 0 -1 2 0 -1 }}) | |||
= | {{Mapping|legend=2| 1 0 2 -1 6 -2 | 0 5 3 6 -4 9 }} | ||
{{Mapping|legend=4| 1 0 2 -1 6 -2 | 0 5/2 1/2 6 -4 9 }} | |||
4 | |||
Optimal tunings: | |||
* Subgroup WE: ~2 = 1200.0331{{c}}, ~14/9 = 760.8754{{c}} | |||
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~14/9 = 760.8568{{c}} | |||
= | {{Optimal ET sequence|legend=0| 11, 30, 41 }} | ||
Badness (Sintel): 0.257 | |||
RMS error: 1.540 cents | |||
=== B-team === | |||
B-team may be described as 23 & 41 and is every other step of [[gamelismic clan #Rodan|rodan]]. | |||
[[Subgroup]]: 2.9.15.21 | |||
[ | |||
[[Comma list]]: 245/243 ({{monzo| 0 -4 1 2 }}), 1029/1024 ({{monzo| -10 -1 0 3 }}) | |||
= | {{Mapping|legend=3| 1 2 0 4 | 0 3 10 1 }} | ||
[[ | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.2146{{c}}, ~21/16 = 468.9174{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 468.8518{{c}} | |||
{{ | {{Optimal ET sequence|legend=1| 18, 23, 41, 64, 151 }} | ||
[[Badness]] (Sintel): 0.155 | |||
==== 2.9.15.21.33 subgroup ==== | |||
Subgroup: 2.9.15.21.33 | |||
Comma list: 245/243 ({{monzo| 0 -4 1 2 0 }}), 385/384 ({{monzo| -7 -2 1 1 1 }}), 441/440 ({{monzo| -3 1 -1 2 -1 }}) | |||
= | {{Mapping|legend=2| 1 2 0 4 7 | 0 3 10 1 -5 }} | ||
Optimal tunings: | |||
* Subgroup WE: ~2 = 1200.0553{{c}}, ~21/16 = 468.9391{{c}} | |||
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~21/16 = 468.9189{{c}} | |||
{{ | {{Optimal ET sequence|legend=0| 18, 23, 41, 64, 87, 151, 238b′ }} | ||
Badness (Sintel): 0.160 | |||
[ | === Mabon === | ||
Mabon may be described as the 11 & 62 temperament in the 2.9.7 subgroup. It was derived from a [http://individual.utoronto.ca/kalendis/leap/index.htm#se calendar leap cycle built for the autumn equinox], and named so by {{u|Dummy index}} in 2024. | |||
[[ | [[Subgroup]]: 2.9.7 | ||
[[Comma basis]]: 44957696/43046721 | |||
{{Mapping|legend=3| 1 1 -3 | 0 3 8 }} | |||
: mapping generators: ~2, ~729/448 | |||
[[ | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.0000{{c}}, ~729/448 = 870.4003{{c}} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~729/448 = 871.1047{{c}} | |||
{{ | {{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62, 73b′, 157b′b′ }} | ||
: | [[Badness]] (Sintel): 4.08 | ||
: | ==== 2.9.7.11 subgroup ==== | ||
Subgroup: 2.9.7.11 | |||
Comma basis: 896/891, 1331/1296 | |||
{{Mapping|legend=2| 1 1 -3 2 | 0 3 8 2 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.1182{{c}}, ~18/11 = 870.9904{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 870.9234{{c}} | |||
{{Optimal ET sequence|legend=0| 7d, 11, 40, 51, 62 }} | |||
{{ | |||
Badness (Sintel): 4.08 | |||
[[ | === Machine === | ||
Machine is every other step of [[supra]], most interesting for its scale patterns. | |||
[[ | It was named by [[Mike Battaglia]] in 2011, who spoke highly of this temperament, describing it as "stimulating, like a well-oiled machine".<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_96074.html Yahoo! Tuning Group | ''Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament'']</ref> | ||
[[Subgroup]]: 2.9.7.11 | |||
[[Comma list]]: 64/63, 99/98 | |||
{{Mapping|legend=3| 1 0 6 13 | 0 1 -1 -3 }} | |||
{{ | {{Mapping|legend=5| 1 0 0 6 13 | 0 1/2 0 -1 -3 }} | ||
: | : mapping generators: ~2, ~9 | ||
[[ | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1197.2650{{c}}, ~9/8 = 213.8957{{c}} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/8 = 214.9961{{c}} | |||
{{Optimal ET sequence|legend=1| 5, 6, 11, 17, 28 }} | |||
{{ | |||
[[Badness]] (Sintel): 0.176 | |||
[[ | === Apparatus === | ||
Apparatus is the restriction of [[gamelismic clan #Revelation|revelation]] to the 2.9.7.11 subgroup, and every fourth step of [[breedsmic temperaments #Quasiorwell|quasiorwell]], without prime 5. | |||
[[ | [[Subgroup]]: 2.9.7.11 | ||
[[Comma list]]: 41503/41472, 322102/321489 | |||
{{Mapping|legend=3| 1 -14 1 -11 | 0 19 2 16 }} | |||
{{Mapping|legend= | {{Mapping|legend=5| 1 -7 0 1 -11 | 0 19/2 0 2 16 }} | ||
: | : mapping generators: ~2, ~144/77 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[ | * [[WE]]: ~2 = 1200.0088{{c}}, ~144/77 = 1084.4377{{c}} | ||
* [[ | * [[CWE]]: ~2 = 1200.0000{{c}}, ~144/77 = 1084.4304{{c}} | ||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }} | ||
[[ | [[Badness]] (Sintel): 0.199 | ||
=== | === Penta a.k.a. mechanism === | ||
Penta or mechanism is the 8 & 11 temperament in the 2.9.7.11 subgroup. | |||
[[Subgroup]]: 2.9.7.11 | |||
[[Comma list]]: 896/891, 26411/26244 | |||
[[Comma list]]: | |||
{{Mapping|legend= | {{Mapping|legend=3| 1 0 -1 6 | 0 5 6 -4 }} | ||
{{Mapping|legend= | {{Mapping|legend=5| 1 0 0 -1 6 | 0 5/2 0 6 -4 }} | ||
: | : mapping generators: ~2, ~14/9 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[ | * [[WE]]: ~2 = 1199.4843{{c}}, ~14/9 = 761.2079{{c}} | ||
* [[ | * [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 761.4979{{c}} | ||
{{Optimal ET sequence|legend=1| 8, | {{Optimal ET sequence|legend=1| 8, 11, 30, 41, 52 }} | ||
[[ | [[Badness]] (Sintel): 0.331 | ||
[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents | |||
Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]] | |||
=== Joan === | |||
{{See also| Chromatic pairs #Joan }} | |||
[[ | Joan is the restriction of [[starling temperaments #Casablanca|casablanca]] to the 2.9.7.11 subgroup, as well as every other step of [[orwell]], without prime 5. | ||
[[Subgroup]]: 2.9.7.11 | |||
[[Comma list]]: 99/98, 9317/9216 | |||
{{Mapping|legend=3| 1 0 1 3 | 0 7 4 1 }} | |||
{{ | {{Mapping|legend=5| 1 0 0 1 3 | 0 7/2 0 4 1 }} | ||
: mapping generators: ~2, ~11/8 | |||
[[ | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.8640{{c}}, ~11/8 = 543.0623{{c}} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 542.8446{{c}} | |||
= | {{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115b′d }} | ||
[[Badness]] (Sintel): 0.407 | |||
[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents | |||
[[ | === Novisept === | ||
Named by {{u|Pailiaq}} in 2026, novisept is generated by a one-cent-flat [[9/7]], such that stacking 5 of them gives [[7/4]]. It can be formed by doubling both generator and period of [[gizzard]]. | |||
[[ | [[Subgroup]]: 2.9.7.13.17 | ||
[[Comma list]]: 442/441, 729/728, 833/832 | |||
{{Mapping|legend=3| 1 | {{Mapping|legend=3| 1 1 1 -1 3 | 0 6 5 13 3 }} | ||
: | : mapping generators: ~2, ~9/7 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[ | * [[WE]]: ~2 = 1200.3044{{c}}, ~9/7 = 433.9023{{c}} | ||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 433.8360{{c}} | |||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 11, 25f, 36, 47, 83, 130, 213g, 296g }} | ||
[[ | [[Badness]] (Sintel): 0.142 | ||
=== | === Genius === | ||
{{ | Genius tempers out the [[Alpharabian comma]] in the 2.9.11 subgroup; it is therefore every other step of [[no-fives subgroup temperaments #Alphaxenean|alphaxenean]]. It was named by {{u|Akselai}} in 2024 after the genius in Roman religion, following the demon (daimon) in Greek mythology. | ||
[[Subgroup]]: 2.9.11 | |||
[[ | [[Comma list]]: 131769/131072 | ||
{{Mapping|legend=2| 1 1 4 | 0 4 -1 }} | |||
: mapping generators: ~2, ~16/11 | |||
{{ | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.3181{{c}}, ~16/11 = 650.8508{{c}} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/11 = 650.7321{{c}} | |||
{{ | {{Optimal ET sequence|legend=1| 11, 24, 59, 83, 142, 225, 367e, 592b′e, 959b′eee, 1326b′b′eeee }} | ||
[[ | [[Badness]] (Sintel): 0.198 | ||
=== Demon === | |||
Demon is a temperament which equates three [[11/9]]'s with [[16/9]], or equivalently three [[18/11]]'s 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 {{u|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 the demon temperament is near the basic tuning of 7L 4s (13\18), and indeed [[18edo]] supports this temperament. | |||
[[ | [[#Mabon|Mabon]] is a strong extension of this temperament to the 2.9.7.11 subgroup. | ||
[[Subgroup]]: 2.9.11 | |||
[[Comma list]]: 1331/1296 | |||
{{Mapping|legend=3| 1 1 2 | 0 3 2 }} | |||
: mapping generators: ~2, ~18/11 | |||
[[ | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1201.1151{{c}}, ~18/11 = 869.4724{{c}} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~18/11 = 869.3196{{c}} | |||
{{ | {{Optimal ET sequence|legend=1| 4, 7, 11, 18, 29, 76e }} | ||
[[Badness]] (Sintel): 0.332 | |||
== 4.3.… subgroups == | |||
=== Tetrameantone === | |||
{{Main| Tetrameantone }} | |||
Named by [[Budjarn Lambeth]] in 2024, tetrameantone is the double-octave restriction of [[meantone family #Meantone|meantone]]. | |||
[[ | [[Subgroup]]: 4.3.5 | ||
[[Comma list]]: 81/80 | |||
{{Mapping|legend=3| 1 0 -2 | 0 1 4 }} | |||
: mapping generators: ~4, ~3 | |||
[[ | [[Optimal tuning]]s: | ||
* [[WE]]: ~4 = 2402.7812{{c}}, ~3 = 1898.4361{{c}} | |||
* [[CWE]]: ~4 = 2400.0000{{c}}, ~3 = 1896.6512{{c}} | |||
[[ | [[Optimal ET sequence]]: [[5ed4|a′5]], [[14ed4|a′14]], [[19ed4|a′19]], [[43ed4|a′43]], [[62ed4|a′62]], [[81ed4|a′81]], [[262ed4|a′262b]], [[343ed4|a′343b]] | ||
[[Badness]] (Sintel): 0.0866 | |||
{{ | === Tetrahanson === | ||
{{Main| Tetrahanson }} | |||
[[ | Named by {{u|CompactStar}} in 2023, tetrahanson is the double-octave restriction of [[kleismic family #Kleismic a.k.a. hanson|hanson]]. | ||
[[Subgroup]]: 4.3.5 | |||
: | |||
[[ | [[Comma list]]: 15625/15552 | ||
{{Mapping|legend=3| 1 -3 -2 | 0 6 5 }} | |||
{{ | : mapping generators: ~4, ~12/5 | ||
[[Optimal tuning]]s: | |||
* [[WE]]: ~4 = 2400.3318{{c}}, ~12/5 = 1517.2163{{c}} | |||
* [[CWE]]: ~4 = 2400.0000{{c}}, ~12/5 = 1517.0308{{c}} | |||
[[ | [[Optimal ET sequence]]: [[8ed4|a′8]], [[11ed4|a′11]], [[19ed4|a′19]], [[68ed4|a′68]], [[87ed4|a′87]], [[106ed4|a′106]], [[511ed4|a′511]], [[617ed4|a′617]], [[723ed4|a′723]], [[829ed4|a′829]], [[935ed4|a′935c]], [[1764ed4|a′1764c]] | ||
[[ | [[Badness]] (Sintel): 0.155 | ||
=== Tetramagic === | |||
Named by [[Budjarn Lambeth]] in 2024, tetramagic is the double-octave restriction of [[magic family #Magic|magic]]. | |||
[[Subgroup]]: 4.3.5 | |||
[[ | [[Comma list]]: 3125/3072 | ||
{{ | {{Mapping|legend=3| 1 0 1 | 0 5 1 }} | ||
: mapping generators: ~4, ~5/4 | |||
[[ | [[Optimal tuning]]s: | ||
* [[WE]]: ~4 = 2402.4898{{c}}, ~5/4 = 380.4527{{c}} | |||
* [[CWE]]: ~4 = 2400.0000{{c}}, ~5/4 = 380.2194{{c}} | |||
[[Optimal ET sequence]]: [[6ed4|a′6]], [[13ed4|a′13]], [[19ed4|a′19]], [[63ed4|a′63]], [[82ed4|a′82]], [[101ed4|a′101]], [[120ed4|a′120]], [[221ed4|a′221c]], [[341ed4|a′341c]] | |||
[[Badness]] (Sintel): 0.459 | |||
[[ | === Blacktetra === | ||
Named by [[Budjarn Lambeth]] in 2024, blacktetra is the double-octave restriction of [[blackwood family #Blackwood|blackwood]]. | |||
[[ | [[Subgroup]]: 4.3.5 | ||
[[Comma list]]: 256/243 | |||
: | |||
{{Mapping|legend=3| | {{Mapping|legend=3| 5 4 0 | 0 0 1 }} | ||
: | : mapping generators: ~4, ~5 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* | * [[WE]]: ~4/3 = 477.7018{{c}}, ~5/4 = 397.6806{{c}} (~16/15 = 80.0212{{c}}) | ||
* | * [[CWE]]: ~4/3 = 480.0000{{c}}, ~5/4 = 395.1255{{c}} (~16/15 = 84.7745{{c}}) | ||
[[Optimal ET sequence]]: [[5ed4|a′5]], [[20ed4|a′20]], [[25ed4|a′25]], [[30ed4|a′30]] | |||
: | |||
[[Badness]] (Sintel): 0.748 | |||
[[ | == 4.6.… subgroups == | ||
=== Meanquad === | |||
Meanquad is the double-octave restriction of [[meantone family #Meantone|meantone]] to the 4.6.5 subgroup. Similarly to meantone, it has a flat [[3/2]] as generator and equates [[5/4]] and [[81/64]]. It possesses [[mos scale]]s of the forms [[3L 4s (4/1-equivalent)|3L 4s<4/1>]], [[7L 3s (4/1-equivalent)|7L 3s<4/1>]], and [[7L 10s (4/1-equivalent)|7L 10s<4/1>]], which could be viewed as equivalents to the pentic, diatonic, and chromatic scales of meantone, respectively. | |||
[[17ed4]] is one of the smallest tunings that [[support]] this temperament and features the decatonic scale. Like [[12edo]], it can be viewed as the boundary between meanquad and [[#quarchy|quarchy]], the 4.6.7-subgroup restriction of [[archy]]. Larger options include 24ed4 (12edo), [[31ed4]] and [[55ed4]]. | |||
[[Subgroup]]: | [[Subgroup]]: 4.6.5 | ||
[[Comma list]]: | [[Comma list]]: 81/80 ({{monzo| -4 4 -1 }}) | ||
{{Mapping|legend= | {{Mapping|legend=3| 1 0 -4 | 0 1 4 }} | ||
: mapping generators: ~ | : mapping generators: ~4, ~6 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* | * [[Tp tuning|Subgroup]] [[WE]]: ~4 = 2402.7812{{c}}, ~3/2 = 697.0455{{c}} | ||
* | * [[Tp tuning|Subgroup]] [[CWE]]: ~4 = 2400.0000{{c}}, ~3/2 = 696.6512{{c}} | ||
[[Optimal ET sequence]]: [[7ed4|a′7]], [[17ed4|a′17]], [[24ed4|a′24]], [[31ed4|a′31]], [[100ed4|a′100]], [[131ed4|a′131]], [[293ed4|a′293*]] | |||
: <nowiki/>* Wart for 6 | |||
[[Badness]] (Sintel): 0.0146 | |||
==== 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 tunings: | ||
{{ | * Subgroup WE: ~4 = 2400.0000{{c}}, ~3/2 = 699.622{{c}} | ||
* Subgroup CWE: ~4 = 2400.0000{{c}}, ~3/2 = 699.622{{c}} | |||
Optimal ET sequence: [[7ed4|a′7]], [[17ed4|a′17]], [[24ed4|a′24]], [[41ed4|a′41]], [[65ed4|a′65cd]] | |||
Badness (Sintel): 0.0974 | |||
[[ | === Quarchy === | ||
[[Subgroup]]: 4.6.7 | |||
{{ | [[Comma list]]: 64/63 ({{monzo| 4 -2 -1 }}) | ||
{{Mapping|legend=3| 1 | {{Mapping|legend=3| 1 0 4 | 0 1 -2 }} | ||
: | : mapping generators: ~4, ~6 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[Tp tuning| | * [[Tp tuning|Subgroup]] [[WE]]: ~4 = 2393.9105{{c}}, ~3/2 = 707.5215{{c}} | ||
* [[Tp tuning| | * [[Tp tuning|Subgroup]] [[CWE]]: ~4 = 2400.0000{{c}}, ~3/2 = 709.3901{{c}} | ||
Optimal ET sequence: [[7ed4|a′7]], [[10ed4|a′10]], [[17ed4|a′17]], [[27ed4|a′27]], [[44ed4|a′44]], [[159ed4|a′159d]], [[203ed4|a′203*dd]] | |||
: <nowiki/>* Wart for 6 | |||
Badness (Sintel): 0.0134 | |||
: | |||
[[ | == 4.9.… subgroups == | ||
=== Meansquared === | |||
Named by {{u|CompactStar}} in 2023, meansquared tempers out the double syntonic comma, [[6561/6400]], in the 4.9.25 subgroup. | |||
[[Subgroup]]: 4.9.25 | |||
[[Comma list]]: 6561/6400 | |||
{{Mapping|legend=3| 1 0 -4 | 0 1 4 }} | |||
: mapping generators: ~4, ~9 | |||
[[ | [[Optimal tuning]]s: | ||
* [[WE]]: ~4 = 2402.7812{{c}}, ~9/4 = 1394.0910{{c}} | |||
* [[CWE]]: ~4 = 2400.0000{{c}}, ~9/4 = 1393.3025{{c}} | |||
[[ | [[Optimal ET sequence]]: [[5ed4|a′5]], [[7ed4|a′7]], [[12ed4|a′12]], [[19ed4|a′19]], [[31ed4|a′31]], [[50ed4|a′50]], [[81ed4|a′81]], [[131ed4|a′131b′]] | ||
{{ | [[Badness]] (Sintel): 0.173 | ||
=== Archsquared === | |||
Named by {{u|CompactStar}} in 2023, archsquared tempers out the double archytas comma, [[4096/3969]], in the 4.9.49 subgroup. | |||
[[Subgroup]]: 4.9.49 | |||
[[ | [[Comma list]]: 4096/3969 | ||
{{ | {{Mapping|legend=3| 1 0 6 | 0 1 -2 }} | ||
: mapping generators: ~4, ~9 | |||
[[ | [[Optimal tuning]]s: | ||
* [[WE]]: ~4 = 2393.9105{{c}}, ~9/4 = 1415.0430{{c}} | |||
* [[CWE]]: ~4 = 2400.0000{{c}}, ~9/4 = 1418.7801{{c}} | |||
[[Optimal ET sequence]]: [[5ed4|a′5]], [[12ed4|a′12]], [[17ed4|a′17]], [[22ed4|a′22]], [[137ed4|a′137b′d′d′]], [[159ed4|a′159b′d′d′d′]], [[181ed4|a′181b′b′d′d′d′]] | |||
[[Badness]] (Sintel): 0.159 | |||
[[ | == 8.9.… subgroups == | ||
=== Sixscared === | |||
Sixscared is a tuning which still maintains some consonance, while eviscerating the rules of conventional 12-tone harmony. The familiar major, minor and perfect intervals are nowhere to be found, and octaves are far and few between, so the seventh harmonic becomes the backbone of harmony. Approximating the harmonics 7, 8, 9, sixscared was named by [[Budjarn Lambeth]] in 2023 for the classic dad joke: "Why was six scared? Because seven ate nine." | |||
[[ | [[Subgroup]]: 8.9.7 | ||
[[Comma list]]: 64/63 | |||
: | |||
{{Mapping|legend=3| 1 | {{Mapping|legend=3| 1 0 2 | 0 1 -1 }} | ||
: | : mapping generators: ~8, ~9 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[ | * [[WE]]: ~8 = 3590.8657{{c}}, ~9/8 = 219.0878{{c}} | ||
* [[ | * [[CWE]]: ~8 = 3600.0000{{c}}, ~9/8 = 218.7801{{c}} | ||
[[Optimal ET sequence]]: [[16ed8|a″16]], [[33ed8|a″33]], [[148ed8|a″148b′d]], [[181ed8|a″181b′d]], [[214ed8|a″214b′d]], [[247ed8|a″247b′b′d]] | |||
[[ | [[Badness]] (Sintel): 0.0265 | ||
=== | == 2.5.… subgroups == | ||
{{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 [[orwellismic family #Guanyin|guanyin]] and a restriction of [[semicomma family #Orwell|orwell]]. It is defined by tempering out [[176/175]] and [[540/539]], which imply [[1728/1715]] as well as [[3388/3375]] are tempered out. The tonic and the first three generator steps make a [[guanyin tetrad]], hence the name. | |||
[[Subgroup]]: 2.7/3.11 | [[Subgroup]]: 2.5.7/3.11/3 | ||
[[Comma list]]: | [[Comma list]]: 176/175 ({{monzo| 4 -2 -1 1 }}), 540/539 ({{monzo| 2 1 -2 -1 }}) | ||
{{Mapping|legend= | {{Mapping|legend=3| 1 0 1 3 | 0 3 -1 5 }} | ||
{{Mapping|legend= | {{Mapping|legend=5| 1 0 0 2 -2 | 0 -4/3 3 -7/3 11/3 }} | ||
: | : mapping generators: ~2, ~12/7 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[Tp tuning| | * [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.2336{{c}}, ~12/7 = 929.3128{{c}} (~7/6 = 269.9208{{c}}) | ||
* [[Tp tuning| | * [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~12/7 = 929.7608{{c}} (~7/6 = 270.2392{{c}}) | ||
{{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 add-13 subgroup by the equivalences ([[12/11]])<sup>3</sup>~[[13/10]] and ([[15/14]])<sup>3</sup>~[[16/13]]. However, note that it is not supported by tridecimal orwell (31 & 53), but instead the less accurate [[semicomma family #Winston|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 }} | |||
Optimal tunings: | |||
* Subgroup WE: ~2 = 1199.3604{{c}}, ~12/7 = 929.2866{{c}} (~7/6 = 270.0738{{c}}) | |||
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~12/7 = 929.7980{{c}} (~7/6 = 270.2020{{c}}) | |||
{{Optimal ET sequence|legend=0| 9, 22f, 31, 40, 71, 111, 151, 262c* }} | |||
: <nowiki/>* Wart for 7/3 | |||
{{ | |||
Badness (Sintel): 0.329 | |||
==== Laz ==== | |||
{{See also | Chromatic pairs #Laz }} | |||
[[ | Laz is related to [[orwellismic family #Avalokita|avalokita]] as well as to [[semicomma family #Winston|winston]]. | ||
Subgroup: 2.5.7/3.11/3.13/3 | |||
{{ | Comma list: 144/143 ({{monzo| 4 0 0 -1 -1 }}), 176/175 ({{monzo| 4 -2 -1 1 }}), 196/195 ({{monzo| 2 -1 2 0 -1 }} | ||
{{Mapping|legend=2| 1 0 2 -2 6 | 0 3 -1 5 -5 }} | |||
{{ | {{Mapping|legend=4| 1 -3/2 0 1/2 -7/2 9/2 | 0 1/4 3 -3/4 21/4 -19/4 }} | ||
Optimal tunings: | |||
* Subgroup WE: ~2 = 1198.4320{{c}}, ~12/7 = 929.4834{{c}} (~7/6 = 268.9486{{c}}) | |||
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~12/7 = 930.5797{{c}} (~7/6 = 269.4203{{c}}) | |||
= | {{Optimal ET sequence|legend=0| 9, 31, 40, 49, 156c*†, 205c*† }} | ||
: <nowiki/>* Wart for 7/3 | |||
: † Wart for 11/3 | |||
RMS error: 0.8790 cents | |||
=== Kryptonite === | |||
{{See also| Chromatic pairs #Kryptonite }} | |||
Kryptonite is related to [[sengic family #Krypton|krypton]]. | |||
Subgroup: 2.5.7/3.11/3.13/3 | |||
{{ | Comma list: 56/55 ({{monzo| 3 -1 1 -1 }}), 78/77 ({{monzo| 1 0 -1 -1 1 }}), 91/90 ({{monzo| -1 -2 1 0 1 }}) | ||
{{Mapping|legend=2| 1 2 1 2 2 | 0 3 2 -1 1 }} | |||
= | {{Mapping|legend=4| 1 -5/4 2 -1/4 3/4 3/4 | 0 -1/2 3 3/2 -3/2 1/2 }} | ||
: mapping generators: ~2, ~13/12 | |||
Optimal tunings: | |||
* Subgroup WE: ~2 = 1197.6484{{c}}, ~13/12 = 132.1686{{c}} | |||
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~13/12 = 131.8445{{c}} | |||
{{Optimal ET sequence|legend=0| 1, …, 8, 9 }} | |||
RMS error: 2.545 cents | |||
=== Marveltri === | |||
{{See also| Chromatic pairs #Marveltri }} | |||
[[ | Marveltri, the {{nowrap| 3 & 13 }} temperament in the 2.5.9/7 subgroup, is related to [[marvel family #Marvel|marvel]], [[magic family #Magic|magic]], and the unnamed {{nowrap| 22 & 47 }} temperament. The tonic and the first two generator steps make a [[marvel triad]], hence the name. | ||
[[Subgroup]]: 2.5.9/7 | |||
[[ | [[Comma list]]: 225/224 ({{monzo| -5 2 1 }}) | ||
{{Mapping|legend=3| 1 0 5 | 0 1 -2 }} | |||
{{Mapping|legend= | {{Mapping|legend=5| 1 2 0 -1 | 0 -4/5 1 2/5 }} | ||
: mapping generators: ~2, ~ | : mapping generators: ~2, ~5 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[Tp tuning| | * [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.5971{{c}}, ~5/4 = 383.8285{{c}} | ||
* [[Tp tuning| | * [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 383.8540{{c}} | ||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }} | ||
< | : <nowiki/>* Wart for 9/7 | ||
[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents | |||
==== | ==== Sulis ==== | ||
Sulis is related to [[marvel family #Minerva|minerva]] and [[würschmidt family #Septimal würschmidt|würschmidt]]. | |||
Subgroup: 2.5.9/7.11/9 | |||
Comma list: 99/98 ({{monzo| -1 0 2 1 }}), 176/175 ({{monzo| 4 -2 1 1 }}) | |||
{{Mapping|legend=2| 1 0 5 -9 | 0 1 -2 4 }} | |||
{{ | Optimal tunings: | ||
* Subgroup WE: ~2 = 1200.1086{{c}}, ~5/4 = 386.5931{{c}} | |||
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.5785{{c}} | |||
{{ | {{Optimal ET sequence|legend=0| 3, …, 22, 25, 28, 31, 59 }} | ||
RMS error: 1.074 cents | |||
{{ | == 2.5/3.… subgroups == | ||
=== Magicaltet === | |||
{{See also| Chromatic pairs #Magicaltet }} | |||
[[ | Magicaltet is related to [[keemic family #Keemic|keemic]], [[gamelismic clan #Superkleismic|superkleismic]], and [[magic family #Magic|magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name. | ||
[[Subgroup]]: 2.5/3.7.11 | |||
[[Comma list]]: 100/99 ({{monzo| 2 2 0 -1 }}), 385/384 ({{monzo| -7 1 1 1 }}) | |||
{{Mapping|legend=3| 1 0 5 2 | 0 1 -3 2 }} | |||
{{Mapping|legend=5| 1 0 0 5 2 | 0 -1/2 1/2 -3 2 }} | |||
: mapping generators: ~2, ~5/3 | |||
{{ | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9868{{c}}, ~5/3 = 877.3312{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 877.3417{{c}} | |||
{{ | {{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }} | ||
: | : <nowiki/>* Wart for 5/3 | ||
[[ | [[Tp tuning #T2 tuning|RMS error]]: 1.206 cents | ||
{{ | === Starlingtet === | ||
{{See also | Chromatic pairs #Starlingtet }} | |||
[[ | Starlingtet, the {{nowrap| 4 & 15 }} temperament in the 2.5/3.7/3 subgroup, is related to [[starling family #Starling|starling]] as well as to [[starling temperaments #Myna|myna]]. The tonic and the first three generator steps make a [[starling tetrad]], hence the name. | ||
[[Subgroup]]: 2.5/3.7/3 | |||
[[Comma list]]: 126/125 ({{monzo| 1 -3 1 }}) | |||
{{Mapping|legend=3| 1 0 -1 | 0 1 3 }} | |||
{{Mapping|legend=5| 1 1/3 1/3 -2/3 | 0 -4/3 -1/3 5/3 }} | |||
: mapping generators: ~2, ~5/3 | |||
{{ | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.7931{{c}}, ~5/3 = 888.6923{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 888.8022{{c}} | |||
{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }} | |||
[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents | |||
==== Skateboard ==== | |||
{{See also| Chromatic pairs #Skateboard }} | |||
Skateboard is related to [[starling family #Thrasher|thrasher]]. | |||
Subgroup: 2.5/3.7/3.11.13/9 | |||
Comma list: 56/55 ({{monzo| 3 -1 1 -1 }}), 91/90 ({{monzo| -1 -1 1 0 1 }}), 100/99 ({{monzo| 2 2 0 -1 }}) | |||
{{Mapping|legend=2| 1 0 -1 2 2 | 0 1 3 2 -2 }} | |||
{{Mapping|legend= | {{Mapping|legend=4| 1 -3/7 -3/7 -10/7 2 8/7 | 0 0 1 3 2 -2 }} | ||
{{ | Optimal tunings: | ||
* Subgroup WE: ~2 = 1197.4723{{c}}, ~5/3 = 884.2917{{c}} | |||
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.6469{{c}} | |||
{{Optimal ET sequence|legend=0| 4, 11, 15, 19, 23, 42e }} | |||
RMS error: 2.396 cents | |||
==== Greeley ==== | |||
{{See also| Chromatic pairs #Greeley }} | |||
Greeley is related to [[porcupine family #Opossum|opossum]] as well as to [[starling temperaments #Nusecond|nusecond]]. | |||
Subgroup: 2.5/3.7/3.11/3 | |||
Comma list: 121/120 ({{monzo| -3 -1 0 2 }}), 126/125 ({{monzo| 1 -3 1 }}) | |||
{{Mapping|legend=2| 1 1 2 2 | 0 -2 -6 -1 }} | |||
{{Mapping|legend= | {{Mapping|legend=4| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }} | ||
: mapping generators: ~2, ~11/10 | |||
{{ | Optimal tunings: | ||
: | * Subgroup WE: ~2 = 1200.4124{{c}}, ~11/10 = 155.8292{{c}} | ||
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~11/10 = 155.7399{{c}} | |||
{{Optimal ET sequence|legend=0| 8, 15, 23, 54, 77, 131* }} | |||
: <nowiki/>* Wart for 11/3 | |||
RMS error: 1.034 cents | |||
[[ | === Gariberttet === | ||
Gariberttet is the 2.5/3.7/3 altergene{{idio}}{{clarify}} of [[no-twos subgroup temperaments #Sirius|sirius]], tempering out [[3125/3087]]. A notable tuning besides WE and CWE given below is [[CTE]]/[[POTE]] at 293.6785 [[cent]]s. | |||
[[Subgroup]]: 2.5/3.7/3 | |||
[[Subgroup]]: 2. | |||
[[Comma list]]: | [[Comma list]]: 3125/3087 | ||
{{Mapping|legend= | {{Mapping|legend=3| 1 0 0 | 0 3 5 }} | ||
: mapping generators: ~2, ~25/21 | |||
{{ | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9983{{c}}, ~25/21 = 293.7393{{c}} | |||
: [[error map]]: {{val| -0.002 -3.141 1.826 }} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~25/21 = 293.7395{{c}} | |||
: error map: {{val| 0.000 -3.140 1.827 }} | |||
{{Optimal ET sequence|legend=0| 4, 25*, 29, 33, 37, 41, 45, 49, 94, 143, 192 }} | |||
: <nowiki/>* Wart for 7/3 | |||
==== 2.5/3.7/3.13/11 subgroup ==== | |||
{{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. Extensions to the full 7-, 11-, and 13-limits include [[quasitemp]]. | ||
Subgroup: 2.5/3.7/3.13/11 | |||
Comma list: 275/273 ({{monzo| 0 2 -1 -1 }}), 847/845 ({{monzo| 0 -1 1 -2 }}) | |||
{{Mapping|legend=2| 1 0 0 0 | 0 3 5 1 }} | |||
{{Mapping|legend=4| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }} | |||
{{ | Optimal tunings: | ||
* Subgroup WE: ~2 = 1199.9976{{c}}, ~13/11 = 293.6780{{c}} | |||
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~13/11 = 293.6782{{c}} | |||
{{ | {{Optimal ET sequence|legend=0| 4, 25*, 29, 33, 37, 41, 45, 49, 94, 143† }} | ||
: | : <nowiki/>* Wart for 7/3 | ||
: † Wart for 13/11 | |||
RMS error: 0.6914 cents | |||
{{ | ==== Indium ==== | ||
{{See also| Chromatic pairs #Indium }} | |||
Indium can be described as the {{nowrap| 8 & 33 }} temperament in the 2.5/3.7/3.11/3 subgroup. | |||
Subgroup: 2.5/3.7/3.11/3 | |||
Comma list: 3025/3024 ({{monzo| -4 2 -1 2 }}), 3125/3087 ({{monzo| 0 5 -3 }}) | |||
{{Mapping|legend=2| 1 0 0 2 | 0 6 10 -1 }} | |||
{{Mapping|legend=4| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }} | |||
: mapping generators: ~2, ~12/11 | |||
{{ | Optimal tunings: | ||
* Subgroup WE: ~2 = 1199.5499{{c}}, ~12/11 = 146.9551{{c}} | |||
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~12/11 = 146.9960{{c}} | |||
{{ | {{Optimal ET sequence|legend=0| 8, 33, 41, 49, 204*† }} | ||
: | : <nowiki/>* Wart for 7/3 | ||
: † Wart for 11/3 | |||
RMS error: 0.7788 cents | |||
{{ | ==== Ammon ==== | ||
{{See also| Chromatic pairs #Ammon }} | |||
[[ | 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|tridec]], and is related to [[porcupine family #Ammonite|ammonite]]. It is generated by ~20/13, whose octave complement, ~13/10, may be considered 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 | |||
Comma list: 121/120 ({{monzo| -3 -1 0 2 }}), 169/168 ({{monzo| -3 0 -1 0 2 }}), 275/273 ({{monzo| 0 2 -1 1 -1 }}) | |||
{{Mapping|legend=2| 1 -3 -5 0 -1 | 0 6 10 3 5 }} | |||
{{Mapping|legend= | {{Mapping|legend=4| 1 9/5 -6/5 -16/5 9/5 4/5 | 0 -24/5 6/5 26/5 -9/5 1/5 }} | ||
: mapping generators: ~2, ~20/13 | |||
Optimal tunings: | |||
* Subgroup WE: ~2 = 1200.9125{{c}}, ~20/13 = 747.3256{{c}} | |||
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~20/13 = 746.8104{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 8, 29, 37, 45 }} | ||
RMS error: 1.052 cents | |||
=== | === Sentry === | ||
{{See also | Chromatic pairs #Sentry }} | |||
Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, tempers out [[245/243]] and shares the same generator with [[sensipent family #Sensi|sensi]]. A notable tuning besides WE and CWE given below is [[CTE]]/[[POTE]] at 440.9020 [[cent]]s. | |||
[[Subgroup]]: 2.3.7 | [[Subgroup]]: 2.5/3.9/7 | ||
[[Comma list]]: | [[Comma list]]: 245/243 ({{monzo| 0 1 -2 }}) | ||
{{Mapping|legend= | {{Mapping|legend=3| 1 0 0 | 0 2 1 }} | ||
{{Mapping|legend=5| 1 0 0 0 | 0 0 2 -1 }} | |||
: mapping generators: ~2, ~9/7 | |||
{{ | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9983{{c}}, ~9/7 = 440.9014{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 440.9017{{c}} | |||
{{Optimal ET sequence|legend=1| 3, 5, 8, 11, 19, 30, 41, 49 }} | |||
[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents | |||
[[ | |||
=== 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. [[Kleismic family #2.3.5.13 subgroup (cata)|Cata]] is a very natural extension of this temperament to the [[2.3.5.13 subgroup]]. | ||
[[ | [[Subgroup]]: 2.5/3.13/9 | ||
{{ | [[Comma list]]: 325/324 ({{monzo| -2 2 1 }}) | ||
{{Mapping|legend=3| 1 0 2 | 0 1 -2 }} | |||
= | {{Mapping|legend=5| 1 -2/3 -7/3 0 0 2/3 | 0 1/2 3/2 0 0 -1 }} | ||
: mapping generators: ~2, ~5/3 | |||
[[ | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.1340{{c}}, ~5/3 = 882.9597{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 882.8731{{c}} | |||
{{ | {{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }} | ||
[[ | [[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents | ||
{{ | == 2.9/5.… subgroups == | ||
=== Kiribati === | |||
{{See also| Chromatic pairs #Kiribati }} | |||
[[ | Kiribati is related to [[octagar family #Nakika|nakika]] as well as to [[tetracot family #Octacot|octacot]]. | ||
[[Subgroup]]: 2.9/5.7/3.11/9 | |||
[[Subgroup]]: 2. | |||
[[Comma list]]: | [[Comma list]]: 100/99 ({{monzo| 2 -2 0 -1 }}), 245/242 ({{monzo| -1 -1 2 -2 }}) | ||
{{Mapping|legend= | {{Mapping|legend=3| 1 1 1 0 | 0 -2 3 4 }} | ||
{{Mapping|legend=5| 1 1/10 -4/5 11/10 1/5 | 0 -3/2 -1 3/2 1 }} | |||
: mapping generators: ~2, ~21/20 | |||
{{ | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.6137{{c}}, ~21/20 = 87.8636{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 87.8385{{c}} | |||
{{Optimal ET sequence|legend=1| 13, 14, 27, 41 }} | |||
[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents | |||
[[ | |||
=== Terrain === | |||
{{Redirect|Terrain|the scale|Terrain (scale)}} | |||
{{See also| Chromatic pairs #Terrain }} | |||
Terrain, the 6 & 21 temperament in the 2.9/5.9/7 subgroup, tempers out the [[landscape comma]] and is related to [[minortone family #Domain|domain]]. It is a remarkable temperament, in that while its complexity is low, it has no discernible error. The 1–7/5–9/5 and 1–9/7–9/5 chords are characteristic. | |||
[[ | [[Subgroup]]: 2.9/5.9/7 | ||
{{ | [[Comma list]]: 250047/250000 ({{monzo| -4 6 -3 }}) | ||
{{Mapping|legend=3| 3 0 -4 | 0 1 2 }} | |||
: mapping generators: ~63/50, ~9/5 | |||
= | [[Optimal tuning]]s: | ||
[[Subgroup]]: | * [[Tp tuning|Subgroup]] [[WE]]: ~63/50 = 400.0012{{c}}, ~9/5 = 1017.5415{{c}} (~10/9 = 182.4620{{c}}) | ||
* [[Tp tuning|Subgroup]] [[CWE]]: ~63/50 = 400.0000{{c}}, ~9/5 = 1017.5388{{c}} (~10/9 = 182.4612{{c}}) | |||
{{Optimal ET sequence|legend=1| 6, 21, 27, 33, 105, 138, 171, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558, 3729, 3900, 7971 }} | |||
[[Tp tuning #T2 tuning|RMS error]]: 0.00844 cents | |||
== 2.75.… subgroups == | |||
=== Archagall === | |||
{{See also| Fifthplus }} | |||
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. | |||
[[ | It is perhaps worth noting that [[83edo]] is approximately an optimal tuning for getting all of [[75/64]], [[85/64]], and the interval between them, [[17/15]], close to just, in that larger tunings compromise on tone-efficiency. | ||
[[Subgroup]]: 2.75.85 | |||
[[Comma list]]: 24576/24565 ({{monzo| 13 1 -3 }}) | |||
{{Mapping|legend=3| 1 2 5 | 0 3 1 }} | |||
: mapping generators: ~2, ~85/32 | |||
[[ | [[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 ET sequence|legend=1| 5, 12, 17, 22, 61, 83, 310, 393, 476, 1345, 1821, 4118…, 5939…, 7760… }} | ||
==== 2.75.9/7.85 subgroup ==== | |||
A fairly natural way to extend 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. | |||
Subgroup: 2.75.9/7.85 | |||
{{ | Comma list: 2025/2023 ({{monzo| 2 -2 1 0 }}), 24576/24565 ({{monzo| 13 1 0 -3 }}) | ||
{{Mapping|legend=2| 1 2 6 5 | 0 3 -4 1 }} | |||
= | Optimal tunings: | ||
= | * Subgroup WE: ~2 = 1200.0241{{c}}, ~85/64 = 491.3358{{c}} | ||
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~85/64 = 491.3290{{c}} | |||
{{Optimal ET sequence|legend=0| 5, 12, 17, 22, 83, 105, 127, 403, 530, 657, 784, 1441… }} | |||
== 2.7/3.… subgroups == | |||
=== Mothwelltri === | |||
{{See also| Chromatic pairs #Mothwelltri }} | |||
[[ | Mothwelltri, the {{nowrap| 4 & 5 }} temperament in the 2.7/3.11 subgroup, tempers out [[99/98]] and is related to [[semicomma family #Orwell|orwell]]. The tonic and the first two generator steps make a [[mothwellsmic triad]], hence the name. | ||
[[ | [[Subgroup]]: 2.7/3.11 | ||
{{ | [[Comma list]]: 99/98 ({{monzo| -1 -2 1 }}) | ||
{{Mapping|legend=3| 1 0 1 | 0 1 2 }} | |||
{{Mapping|legend=5| 1 0 0 0 1 | 0 -1/2 0 1/2 2 }} | |||
: mapping generators: ~2, ~7/3 | |||
==== | [[Optimal tuning]]s: | ||
{{ | * [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.3175{{c}}, ~7/6 = 273.2462{{c}} | ||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~7/6 = 273.4280{{c}} | |||
{{Optimal ET sequence|legend=1| 4, 9, 13, 22, 79 }} | |||
[[ | [[Tp tuning #T2 tuning|RMS error]]: 1.064 cents | ||
== 2.7/5.… subgroups == | |||
=== Tridec === | |||
{{See also| Chromatic pairs #Tridec }} | |||
{{See also| Non-over-1 temperament #Tridec }} | |||
Tridec, the 5 & 8 temperament in the 2.7/5.11/5.13/5 subgroup, extends [[#Petrtri]]. | |||
[[Subgroup]]: 2.7/5.11/5.13/5 | |||
[[ | [[Comma list]]: 847/845 ({{monzo| 0 1 2 -2 }}), 1001/1000 ({{monzo| -3 1 1 1 }}) | ||
{{ | {{Mapping|legend=3| 1 -2 3 2 | 0 4 -3 -1 }} | ||
{{Mapping|legend=5| 1 0 -3/4 -11/4 9/4 5/4 | 0 0 0 4 -3 -1 }} | |||
: mapping generators: ~2, ~20/13 | |||
== | [[Optimal tuning]]s: | ||
[[ | * [[Tp tuning|subgroup]] [[WE]]: ~2 = 1200.0343{{c}}, ~20/13 = 745.4658{{c}} | ||
* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~20/13 = 745.4431{{c}} | |||
{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 37, 66 }} | |||
[[Tp tuning #T2 tuning|RMS error]]: 0.1613 cents | |||
==== Naiadec ==== | |||
Subgroup: 2.7/5.11/5.13/5.17/5 | |||
{{ | Comma list: 170/169 ({{monzo| 1 0 0 -2 1 }}), 221/220 ({{monzo| -2 0 -1 1 1 }}), 847/845 ({{monzo| 0 1 2 -2 0 }}) | ||
= | {{Mapping|legend=2| 1 -2 3 2 3 | 0 4 -3 -1 -2 }} | ||
{{Mapping|legend=4| 1 0 -3/4 -11/4 9/4 5/4 9/4 | 0 0 0 4 -3 -1 -2 }} | |||
: mapping generators: ~2, ~20/13 | |||
Optimal tunings: | |||
* Subgroup WE: ~2 = 1200.4068{{c}}, ~20/13 = 745.3706{{c}} | |||
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~20/13 = 745.0940{{c}} | |||
{{ | {{Optimal ET sequence|legend=1| 5, 8, 21, 29, 95*, 124* }} | ||
: <nowiki/>* Wart for 17/5 | |||
RMS error: 0.7521 cents | |||
== 2.25/7.… subgroups == | |||
=== Supramine === | |||
Supramine extends [[#Supramin|supramin]] and 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]]: 2.25/7.17/7.19/7.23/7 | ||
[[Comma list]]: 323/322 ({{monzo| -1 0 1 1 -1 }}), 392/391 ({{monzo| 3 0 -1 0 -1 }}), 476/475 ({{monzo| 2 -1 1 -1 0 }}) | |||
= | {{Mapping|legend=3| 1 -2 0 4 3 | 0 3 1 -2 -1 }} | ||
: mapping generators: ~2, ~17/14 | |||
[[Subgroup]]: 2. | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.7558{{c}}, ~17/14 = 335.4278{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~17/14 = 335.4885{{c}} | |||
{{Optimal ET sequence|legend=1| 7, 18, 25, 68, 93 }} | |||
[[Badness]] (Sintel): 0.053 | |||
== 2.11/5.… subgroups == | |||
=== Petrtri === | |||
{{See also| Chromatic pairs #Petrtri }} | |||
{{See also| 5L 3s/Temperaments #Petrtri }} | |||
Petrtri can be described as 3 & 5 temperament in the 2.11/5.13/5 subgroup and tempers out [[2200/2197]]. It restricts [[#Tridec|tridec]]. The tonic and the first two generator steps make a [[petrmic triad]], hence the name. | |||
[[ | [[Subgroup]]: 2.11/5.13/5 | ||
[[Comma list]]: 2200/2197 ({{monzo| 3 1 -3 }}) | |||
{{Mapping|legend=3| 1 0 1 | 0 3 1 }} | |||
{{Mapping|legend=5| 1 0 -1/3 0 -1/3 2/3 | 0 0 -4/3 0 5/3 -1/3 }} | |||
: mapping generators: ~2, ~13/10 | |||
{{ | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9572{{c}}, ~13/10 = 454.9958{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~13/10 = 455.0039{{c}} | |||
{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 153, 182, 211, 240, 269, 509, 778…, 1287… }} | |||
[[Tp tuning #T2 tuning|RMS error]]: 0.0749 cents | |||
== 2 | == Other 2-repeating subgroups == | ||
=== | === Supramin === | ||
Supramin restricts [[#Supramine|supramine]] and 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 {{u|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 [[Mabila family #Cohemimabila|cohemimabila]]. | |||
[[Subgroup]]: 2. | [[Subgroup]]: 2.17/7.19/7 | ||
[[Comma list]]: | [[Comma list]]: 5491/5488 ({{monzo| -4 2 1 }}) | ||
{{Mapping|legend= | {{Mapping|legend=3| 1 0 4 | 0 1 -2 }} | ||
: mapping generators: ~2, ~ | : mapping generators: ~2, ~17/7 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = | * [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.0220{{c}}, ~17/14 = 335.7933{{c}} | ||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~ | * [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~17/14 = 335.7852{{c}} | ||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 7, 18, 25, 118, 143, 168, 193, 218, 461, 679, 1576, 2255… }} | ||
[[Badness]] (Sintel): 0.005 | |||
[[Badness]] (Sintel): 0. | |||
== 3/2.5/2.… subgroups == | == 3/2.5/2.… subgroups == | ||
{{Main|Half-prime subgroup}} | {{Main| Half-prime subgroup }} | ||
=== Hemihemi === | === Hemihemi === | ||
Named by {{u|CompactStar}} in 2023, hemihemi tempers out 10976/10935, the [[hemimage comma]], in the 3/2.5/2.7/2 subgroup. | |||
[[Subgroup]]: 3/2.5/2.7/2 | [[Subgroup]]: 3/2.5/2.7/2 | ||
[[Comma list]]: | [[Comma list]]: 10976/10935 ({{monzo| -7 -1 3 }}) | ||
{{Mapping|legend= | {{Mapping|legend=3| 1 2 3 | 0 3 1 }} | ||
: mapping generators: ~3/2, ~28/27 | |||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~3/2 = 702.6069{{c}}, ~28/27 = 60.4665{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~3/2 = 701.9550{{c}}, ~28/27 = 60.8011{{c}} | |||
[[ | [[Optimal ET sequence]]: [[11edf|*11]], [[12edf|*12]], [[23edf|*23]], [[35edf|*35]], [[58edf|*58]], [[93edf|*93]], [[151edf|*151†]], [[244edf|*244†]] | ||
: <nowiki/>* Wart for 3/2 | |||
: † Wart for 7/2 | |||
=== Halftone === | === Halftone === | ||
{{Main| Halftone }} | {{Main| Halftone }} | ||
Named by {{u|CompactStar}} in 2023, halftone tempers out 9604/9375 in the 3/2.5/2.7/2 subgroup. | |||
[[Subgroup]]: 3/2.5/2.7/2 | [[Subgroup]]: 3/2.5/2.7/2 | ||
[[Comma list]]: 9604/9375 | [[Comma list]]: 9604/9375 ({{monzo| -1 -5 4 }}) | ||
{{Mapping|legend= | {{Mapping|legend=3| 1 3 4 | 0 -4 -5 }} | ||
: | : mapping generators: ~3/2, ~15/14 | ||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~3/2 = 702.6568{{c}}, ~15/14 = 128.2849{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~3/2 = 701.9550{{c}}, ~15/14 = 128.8183{{c}} | |||
[[Optimal ET sequence]]: [[5edf|*5]], [[11edf|*11]], [[27edf|*27]], [[38edf|*38]] | |||
: <nowiki />* | : <nowiki/>* Wart for 3/2 | ||
==== 3/2.5/2.7/2.11/2 ==== | ==== 3/2.5/2.7/2.11/2 subgroup ==== | ||
Subgroup: 3/2.5/2.7/2.11/2 | |||
Comma list: 1232/1215 ({{monzo| -5 -1 1 1 }}), 9604/9375 ({{monzo| -1 -5 4 0 }}) | |||
{{Mapping|legend=2| 1 3 4 4 | 0 -4 -5 1 }} | {{Mapping|legend=2| 1 3 4 4 | 0 -4 -5 1 }} | ||
Optimal tunings: | |||
* WE: ~3/2 = 704.8651{{c}}, ~15/14 = 130.9114{{c}} | |||
* CWE: ~3/2 = 701.9550{{c}}, ~15/14 = 129.6376{{c}} | |||
[[ | Optimal ET sequence: [[5edf|*5]], [[11edf|*11]], [[16edf|*16]], [[27edf|*27…]], [[43edf|*43…]], [[70edf|*70…]] | ||
: <nowiki />* | : <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 subgroup ==== | ||
Subgroup: 3/2.5/2.7/2.11/2.13/2 | |||
Comma list: 275/273 ({{monzo| -1 2 -1 1 -1 }}), 1232/1215 ({{monzo| -5 -1 1 1 0 }}), 1323/1300 ({{monzo| 3 -2 2 0 -1 }}) | |||
{{Mapping|legend=2| 1 3 4 4 5 | 0 -4 -5 1 -2 }} | {{Mapping|legend=2| 1 3 4 4 5 | 0 -4 -5 1 -2 }} | ||
Optimal tunings: | |||
* WE: ~3/2 = 702.9683{{c}}, ~15/14 = 130.0424{{c}} | |||
* CWE: ~3/2 = 701.9550{{c}}, ~15/14 = 129.6034{{c}} | |||
[[ | Optimal ET sequence: [[5edf|*5]], [[11edf|*11]], [[16edf|*16]], [[27edf|*27†]] | ||
: <nowiki />* | : <nowiki/>* Wart for 3/2 | ||
: † Wart for 11/2 | |||
=== Semiwolf === | === Semiwolf === | ||
Semiwolf provides an interpretation for the {{MOS scalesig|link=true|3L 2s<3/2>}} [[mos scale]]. The name ''semiwolf'' comes from two [[~]][[7/6]] generators approximating a [[27/20]] wolf fourth. This is further extended to the 11-limit in two interpretations: semilupine, where 2 major 2-mossteps (LL) equal 11/9, and hemilycan. where 1 major and 2 minor 2-mossteps (sLs) equal 11/9. Basic [[8edf]] fits both extensions. | |||
[[Subgroup]]: 3/2.5/2.7/4 | [[Subgroup]]: 3/2.5/2.7/4 | ||
[[Comma list]]: 245/243 | [[Comma list]]: 245/243 ({{monzo| -5 1 2 }}) | ||
{{Mapping|legend= | {{Mapping|legend=3| 1 1 2 | 0 2 -1 }} | ||
: mapping generators: ~3/2, ~9/7 | |||
: | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~3/2 = 703.7414{{c}}, ~9/7 = 440.9014{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~3/2 = 701.9550{{c}}, ~9/7 = 440.8307{{c}} | |||
[[Optimal ET sequence]]: [[8edf|*8]], [[43edf|*43]], [[51edf|*51]], [[59edf|*59]], [[67edf|*67]], [[75edf|*75…]], [[83edf|*83…]] | |||
: <nowiki/>* Wart for 3/2 | |||
==== Hemilycan ==== | |||
Subgroup: 3/2.5/2.7/4.11/4 | |||
Comma list: 245/243 ({{monzo| -5 1 2 0 }}), 441/440 ({{monzo| 2 -1 2 -1 }}) | |||
{{Mapping|legend=2| 1 1 2 5 | 0 2 -1 -4 }} | |||
Optimal tunings: | |||
* Subgroup WE: ~3/2 = 703.2870{{c}}, ~9/7 = 441.1967{{c}} | |||
* Subgroup CWE: ~3/2 = 701.9550{{c}}, ~9/7 = 440.4501{{c}} | |||
Optimal ET sequence: [[8edf|*8]], [[27edf|*28]], [[35edf|*35]], [[43edf|*43]], [[51edf|*51]], [[110edf|110…]] | |||
: <nowiki/>* Wart for 3/2 | |||
==== Semilupine ==== | ==== Semilupine ==== | ||
Subgroup: 3/2.5/2.7/4.11/4 | |||
Comma list: 100/99 ({{monzo| -3 2 0 -1 }}), 245/243 ({{monzo| -5 1 2 0 }}) | |||
{{Mapping|legend=2| 1 1 2 0 | 0 2 -1 4 }} | {{Mapping|legend=2| 1 1 2 0 | 0 2 -1 4 }} | ||
[[ | Optimal tunings: | ||
* Subgroup WE: ~3/2 = 704.6266{{c}}, ~9/7 = 439.2433{{c}} | |||
* Subgroup CWE: ~3/2 = 701.9550{{c}}, ~9/7 = 438.6989{{c}} | |||
Optimal ET sequence: [[5edf|*5]], [[8edf|*8]], [[45edf|*45…]], [[53edf|*53…]], [[61edf|*61…]], [[69edf|*69…]], [[77edf|*77…]] | |||
: <nowiki/>* Wart for 3/2 | |||
[[ | == 3/2.5/4.… subgroups == | ||
=== Ambulatory === | |||
Ambulatory tempers out [[686/675]] in the 3/2.5/4.7/4 subgroup and is generated by a sharp [[~]][[15/14]]. This implies a stack of two generators represents [[7/6]], and a stack of three generators represents [[5/4]]. | |||
Ambulatory can be considered the 3/2.5/4.7/4 analog of [[no-twos subgroup temperaments #BPS|BPS]] or [[meantone family #Meantone|meantone]], using [[5:6:7]] as its fundamental consonant chord in the place of [[3:5:7]] or of [[4:5:6]]. Ambulatory is actually similar to meantone in several ways: | |||
[[ | * They are both of low [[badness]].{{dubious}} | ||
* In most [[optimal tuning]]s, meantone tunes [[3/2]] flat and [[5/4]] slightly sharp, while ambulatory tunes [[7/5]] flat and [[6/5]] slightly sharp. | |||
* Meantone[7] has [[5/4]] and [[6/5]] share a scale degree. Ambulatory[5] has [[6/5]] and [[7/6]] share a scale degree. | |||
* Meantone[7] is a tempering of the 7-note [[AGS]](5/4, 6/5) (Ptolemy's intense diatonic) with the large and medium steps equated. Ambulatory[5] is a tempering of the 5-note AGS(6/5, 7/6) (assuming an equivalence of 3/2) with the medium and small steps equated. | |||
Ambulatory was first documented by {{u|CompactStar}} who originally proposed the name ''emergency'', which refers to the North American emergency telephone number 911, because [[11edf|9\11edf]] is a possible generator. This was deemed as too dark so the name ''ambulatory''" was suggested by {{u|Frostburn}} to more subtly reference the same thing. | |||
[[Subgroup]]: 3/2.5/4.7/4 | |||
[[Comma list]]: | [[Comma list]]: 686/675 ({{monzo| -3 -2 3 }}) | ||
{{Mapping|legend= | {{Mapping|legend=3| 1 0 1 | 0 3 2 }}] | ||
: mapping generators: ~3/2, ~15/14 | |||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~3/2 = 701.8452{{c}}, ~15/14 = 129.7976{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~3/2 = 701.9550{{c}}, ~15/14 = 129.5324{{c}} | |||
[[Optimal ET sequence]]: [[ | [[Optimal ET sequence]]: [[5edf|*5]], [[11edf|*11]], [[27edf|*27]], [[38edf|*38]] | ||
: <nowiki/>* Wart for 3/2 | |||
=== Poseidon === | === Poseidon === | ||
{{Todo|inline=1|rename|resolve temperament name conflict}} | |||
[[Subgroup]]: 3/2.5/4.11/8 | [[Subgroup]]: 3/2.5/4.11/8 | ||
[[Comma list]]: 121/120 | [[Comma list]]: 121/120 ({{monzo| -1 -1 2 }}) | ||
{{Mapping|legend= | {{Mapping|legend=3| 1 1 1 | 0 -2 -1 }}] | ||
: mapping generators: ~3/2, ~12/11 | |||
: [[ | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~3/2 = 701.8452{{c}}, ~12/11 = 157.8337{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~3/2 = 701.9550{{c}}, ~12/11 = 157.8574{{c}} | |||
[[Optimal | [[Optimal ET sequence]]: [[4edf|*4]], [[5edf|*5]], [[9edf|*9]], [[22edf|*22]], [[31edf|*31]], [[40edf|*40]] | ||
: <nowiki/>* Wart for 3/2 | |||
== Other 3/2-repeating subgroups == | |||
=== Auk === | |||
Auk tempers out the comma [[87808/85293]] in the 3/2.7.13 subgroup, equating three instances of [[7/1]] (fifth-reduced to [[112/81]]) with [[13/1]] (fifth-reduced to 832/729). The generator when reduced is a superfourth in the neighborhood of 550 cents, representing 112/81. It possesses nontrivial mos of the families {{MOS scalesig|1L 3s<3/2>}} ("neptunian"), {{MOS scalesig|4L 1s<3/2>}}, {{MOS scalesig|5L 4s<3/2>}}, and {{MOS scalesig|9L 5s<3/2>}}. [[9edf]] is a particularly good equal temperament that [[support]]s it. It contains tempered versions of the chords 1–832/729–112/81 (fifth-reduction of 1:7:13) and 1–63/52–112/81 (fifth-reduction of 1/(13:7:1)). | |||
[[Subgroup]]: 3/2.7.13 | [[Subgroup]]: 3/2.7.13 | ||
[[Comma list]]: 87808/85293 | [[Comma list]]: 87808/85293 | ||
{{Mapping|legend= | {{Mapping|legend=3| 1 0 -8 | 0 1 3 }} | ||
: mapping generators: ~3/2, ~7 | |||
: | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~3/2 = 706.5197{{c}}, ~112/81 = 538.8834{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~3/2 = 701.9550{{c}}, ~112/81 = 543.8799{{c}} | |||
[[Optimal | [[Optimal ET sequence]]: [[4edf|*4]], [[13edf|*13]], [[17edf|*17df]], [[21edf|*21df]], [[38edf|*38ddff]], [[59edf|*59ddffff]] | ||
: <nowiki/>* Wart for 3/2 | |||
: <nowiki />* | |||
=== Doubleton === | === Doubleton === | ||
[[Subgroup]]: 3/2.7.13 | [[Subgroup]]: 3/2.7.13 | ||
[[Comma list]]: 1352/1323 | [[Comma list]]: 1352/1323 ({{monzo| -3 -2 2 }}) | ||
{{Mapping|legend= | {{Mapping|legend=3| 2 0 3 | 0 1 1 }} | ||
: mapping generators: ~26/21, ~7 | |||
: | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~26/21 = 352.6518{{c}}, ~112/81 = 552.6017{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~26/21 = 350.9775{{c}}, ~112/81 = 565.7161{{c}} | |||
[[Optimal | [[Optimal ET sequence]]: [[4edf|*4]], [[10edf|*10]], [[14edf|*14f]], [[46edf|*46dfff]] | ||
: <nowiki/>* Wart for 3/2 | |||
== 5/2-repeating subgroups == | |||
== 5/2- | |||
=== Hyperion === | === Hyperion === | ||
[[Subgroup]]: 5/2.7.11 | [[Subgroup]]: 5/2.7.11 | ||
[[Comma list]]: {{monzo| 11 1 -5 }} | [[Comma list]]: 341796875/329832448 ({{monzo| 11 1 -5 }}) | ||
{{Mapping|legend= | {{Mapping|legend=3| 1 -1 2 | 0 5 1 }} | ||
: mapping generators: ~5/2, ~44/25 | |||
: [[ | [[Optimal tuning]]s: | ||
* [[Tp tuning|Subgroup]] [[WE]]: ~5/2 = 1582.2958{{c}}, ~44/25 = 990.1327{{c}} | |||
[[ | * [[Tp tuning|Subgroup]] [[CWE]]: ~5/2 = 1586.3137{{c}}, ~44/25 = 991.6570{{c}} | ||
[[Optimal ET sequence]]: [[8ed5/2|*8]], [[83ed5/2|*83…]], [[91ed5/2|*91…]], …, [[155ed5/2|*155…]], [[163ed5/2|*163…]] | |||
: <nowiki />* | : <nowiki/>* Wart for 5/2 | ||
= | == See also == | ||
* [[Dual-fifth temperaments]] | * [[Dual-fifth temperaments]] | ||
* [[Irrational-subgroup temperaments]] | |||
* [[Equalizer subgroup]] temperaments | * [[Equalizer subgroup]] temperaments | ||
* [[Substitute harmonic]] temperaments | * [[Substitute harmonic]] temperaments | ||
[[Category: | == References == | ||
[[Category: | <references/> | ||
[[Category:Catalogs of subgroup temperaments]] | |||
[[Category:Catalogs of rank-2 temperaments]] | |||