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The '''hemimean clan''' tempers out the no-threes hemimean comma [[3136/3125]]. The head of this clan is the 2.5.7 subgroup temperament didacus. Full 7-limit [[extension]]s of didacus, in general, split the [[syntonic comma]] into two, each for [[126/125]]~[[225/224]], as 3136/3125 = (126/125)/(225/224). These include hemithirds, semisept, emka, decipentic, arch, sengagen, subpental, mowglic, and undetrita, considered below, as well as these considered elsewhere:
{{Technical data page}}
* ''[[Spell]]'', {49/48, 3125/3072} → [[Magic family #Spell|Magic family]]
The '''hemimean clan''' [[Tempering out|tempers out]] the hemimean comma, [[3136/3125]], with [[monzo]] {{monzo| 6 0 -5 2 }}, such that [[7/4]] is split into five steps, of which two make [[5/4]] and three make [[7/5]]; this defines the [[2.5.7 subgroup]] temperament [[didacus]], generated by a tempered hemithird of [[28/25]].
* ''[[Hexe]]'', {50/49, 128/125} → [[Augmented family #Hexe|Augmented family]]
* ''[[Passion]]'', {64/63, 3125/3087} → [[Passion family #Septimal passion|Passion family]]
* [[Meantone]], {81/80, 126/125} → [[Meantone family #Septimal meantone|Meantone family]]
* ''[[Clyde]]'', {245/243, 3136/3125} → [[Kleismic family #Clyde|Kleismic family]]
* ''[[Sycamore]]'', {686/675, 875/864} → [[Sycamore family #Septimal sycamore|Sycamore family]]
* ''[[Bidia]]'', {2048/2025, 3136/3125} → [[Diaschismic family #Bidia|Diaschismic family]]
* ''[[Hemiwürschmidt]]'', {2401/2400, 3136/3125} → [[Würschmidt family #Hemiwürschmidt|Würschmidt family]]
* [[Parakleismic]], {3136/3125, 4375/4374} → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* [[Misty]], {3136/3125, 5120/5103} → [[Misty family #Septimal misty|Misty family]]
* ''[[Bischismic]]'', {3136/3125, 32805/32768} → [[Schismatic family #Bischismic|Schismatic family]]
* ''[[Decistearn]]'', {3136/3125, 118098/117649} → [[Stearnsmic clan #Decistearn|Stearnsmic clan]]
* ''[[Rubidium]]'', {3136/3125, 4194304/4117715} → [[Fractional-octave temperaments #37th-octave temperaments|Fractional-octave temperaments]]
* ''[[Quintagar]]'', {3136/3125, 33554432/33480783} → [[Quindromeda family #Quintagar|Quindromeda family]]


A notable subgroup extension of didacus is [[Chromatic pairs #Roulette|roulette]].
The second comma of the comma list determines which 7-limit family member we are looking at. These [[extension]]s, in general, split the [[syntonic comma]] into two, each for [[126/125]]~[[225/224]], as 3136/3125 = (126/125)/(225/224). Hemiwürschmidt adds [[2401/2400]]; hemithirds adds [[1029/1024]]; spell adds [[49/48]]. These all use the same nominal generator as didacus.  


Septimal passion adds [[64/63]], splitting the hemithird into a further two. Septimal meantone adds [[81/80]] as well as [[126/125]] and [[225/224]], splitting an octave plus the hemithird into two perfect fifths. Sycamore adds [[686/675]], splitting the hemithird into three. Semisept adds [[1728/1715]], splitting an octave plus the hemithird into three. Mohavila adds [[135/128]], whereas cohemimabila adds [[65536/64827]], both splitting two octaves plus the hemithird into three. Emka adds [[84035/82944]], splitting two octaves plus the hemithird into four. Bidia adds [[2048/2025]] with a 1/4-octave period. Misty adds [[5120/5103]] with a 1/3-octave period. Bischismic adds [[32805/32768]] with a semioctave period. Hexe adds [[50/49]] with a 1/6-octave period. Clyde adds [[245/243]] with a generator of ~9/7, five of which make the original. Parakleismic adds [[4375/4374]] with a generator of ~6/5. Arch adds [[5250987/5242880]] with a generator of ~64/63. For these seven generators make the original. Sengagen adds [[420175/419904]] with a generator of ~686/675, splitting the hemithird into eight. Subpental adds [[19683/19600]] with a generator of ~56/45, nine of which make the original.
Didacus has canonical subgroup extensions to primes 11 and 13, at [[#Undecimal didacus|undecimal didacus]]. Other subgroup extensions include rectified hebrew and isra.
Temperaments considered below are hemiwürschmidt, hemithirds, spell, semisept, emka, decipentic, sengagen, subpental, mowglic, and undetrita. Discussed elsewhere are
* ''[[Passion]]'' (+64/63 or 3125/3087) → [[Passion family #Septimal passion|Passion family]]
* [[Meantone]] (+81/80, 126/125, 225/224) → [[Meantone family #Septimal meantone|Meantone family]]
* ''[[Mohavila]]'' (+135/128 or 1323/1250) → [[Mavila family #Mohavila|Mavila family]]
* ''[[Cohemimabila]]'' (+65536/64827) → [[Mabila family #Cohemimabila|Mabila family]]
* ''[[Sycamore]]'' (+686/675 or 875/864) → [[Sycamore family #Septimal sycamore|Sycamore family]]
* ''[[Bidia]]'' (+2048/2025) → [[Diaschismic family #Bidia|Diaschismic family]]
* ''[[Hexe]]'' (+50/49 or 128/125) → [[Augmented family #Hexe|Augmented family]]
* [[Misty]] (+5120/5103) → [[Misty family #Septimal misty|Misty family]]
* ''[[Bischismic]]'' (+32805/32768) → [[Schismatic family #Bischismic|Schismatic family]]
* ''[[Clyde]]'' (+245/243) → [[Kleismic family #Clyde|Kleismic family]]
* [[Parakleismic]] (+4375/4374) → [[Parakleismic family #Parakleismic|Parakleismic family]]
* ''[[Arch]]'' (+5250987/5242880) → [[Escapade family #Arch|Escapade family]]
* ''[[Subpental]]'' (+19683/19600) → [[Sensipent family #Sensipent|Sensipent family]]
* ''[[Doubloon]]'' (+33756345/33554432) → [[Vavoom family #Doubloon|Vavoom family]]
* ''[[Decistearn]]'' (+118098/117649) → [[Stearnsmic clan #Decistearn|Stearnsmic clan]]
* ''[[Quintagar]]'' (+33554432/33480783) → [[Quindromeda family #Quintagar|Quindromeda family]]
* ''[[Rubidium]]'' (+4194304/4117715) → [[37th-octave temperaments]]
= 2.5.7 subgroup =
== Didacus ==
== Didacus ==
Subgroup: 2.5.7
{{main|Didacus}}
 
See also its canonical extension to the 2.5.7.11 subgroup, [[#Undecimal didacus]].
 
[[Subgroup]]: 2.5.7
 
[[Comma list]]: [[3136/3125]]
 
{{Mapping|legend=2| 1 0 -3 | 0 2 5 }}
 
: sval mapping generators: ~2, ~56/25
 
{{Mapping|legend=3| 1 0 0 -3 | 0 0 2 5 }}
 
: [[gencom]]: [2 56/25; 3136/3125]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.772
 
{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 99, 130, 161, 353, 514c, 867c }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
 
[[Badness]] (Sintel): 0.091
 
= Strong extensions =
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Map to strong extensions
|-
! rowspan="2" | Extension !! colspan="2" | 5-limit re-restriction !! rowspan="2" | Mapping of 3 !! rowspan="2" | Tuning range*
|-
! Temperament !! 5-limit generator location
|-
| [[#Hemiwürschmidt|Hemiwürschmidt]] || [[Würschmidt family#Würschmidt|Würschmidt]] || +2 || +16 || ↓ [[31edo|31]]
|-
| [[#Hemithirds|Hemithirds]] || [[Luna family#Luna|Luna]] || +1 || -15 || ↑ 31 <br /> ↓ [[25edo|25]]
|-
| [[#Spell|Spell]] || [[Magic family#Magic|Magic]] || +2 || +10 || ↑ 25
|}
<nowiki />* Defined by intersection with other documented extensions
 
== Hemiwürschmidt ==
''[[#Strong extensions|Return to the map]]''
 
{{See also| Würschmidt family }}
 
'''Hemiwürschmidt''' (sometimes spelled '''hemiwuerschmidt''') is not only one of the more accurate extensions of didacus, but also the most important extension of 5-limit [[würschmidt]], even with the rather large complexity for the fifth. It tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]]. [[68edo]], [[99edo]] and [[130edo]] can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher limit temperament, mapping 11 to 40 generators and 13 to -39.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 3136/3125
 
{{Mapping|legend=1| 1 15 4 7 | 0 -16 -2 -5 }}
 
Mapping generators: ~2, ~25/14
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.898
 
{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328, 557c, 885cc }}
 
[[Badness]]: 0.020307
 
=== 2.3.5.7.23 subgroup ===
As described at the page for [[würschmidt]], there is an extension to prime 23 with essentially no damage, which maps the prime to 28 generators (or 14 generators of würschmidt).
 
Subgroup: 2.3.5.7.23
 
[[Comma list]]: 576/575, 736/735, 1127/1125
 
{{Mapping|legend=1| 1 15 4 7 28 | 0 -16 -2 -5 -28 }}
 
Mapping generators: ~2, ~25/14
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.901
 
{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328 }}
 
Badness (Sintel): 0.304
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 3136/3125
 
Mapping: {{mapping| 1 15 4 7 37 | 0 -16 -2 -5 -40 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.840
 
{{Optimal ET sequence|legend=1| 31, 99e, 130, 811ce }}
 
Badness: 0.021069
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 351/350, 441/440, 3584/3575


[[Comma list]]: 3136/3125
Mapping: {{mapping| 1 15 4 7 37 -29 | 0 -16 -2 -5 -40 39 }}


[[Sval]] [[mapping]]: [{{val| 1 0 -3 }}, {{val| 0 2 5 }}]
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.829


Sval mapping generators: ~2, ~56/25
{{Optimal ET sequence|legend=1| 31, 99e, 130, 291, 421e, 551ce }}


Gencom mapping: [{{val| 1 0 0 -3 }}, {{val| 0 0 2 5 }}]
Badness: 0.023074


[[Gencom]]: [2 56/25; 3136/3125]
==== Hemithir ====
Subgroup: 2.3.5.7.11.13


[[POTE generator]]: ~28/25 = 193.772
Comma list: 121/120, 176/175, 196/195, 275/273


{{Val list|legend=1| 6, 19, 25, 31, 99, 130, 161, 353, 514c, 867c }}
Mapping: {{mapping| 1 15 4 7 37 -3 | 0 -16 -2 -5 -40 8 }}


=== Rectified Hebrew ===
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.918
Rectified Hebrew (37 & 56) is derived from the [https://individual.utoronto.ca/kalendis/hebrew/rect.htm#353 calendar by the same name]. It is leap year pattern takes a stack of 18 Metonic cycle diatonic major scales and truncates the 19th one down to its generator, 11. It tempers out 4394/4375 in the 13-limit.


Subgroup: 2.5.7.13
{{Optimal ET sequence|legend=1| 31, 68e, 99ef }}
 
Badness: 0.031199
 
=== Hemiwur ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 176/175, 1375/1372
 
Mapping: {{mapping| 1 15 4 7 11 | 0 -16 -2 -5 -9 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.884
 
{{Optimal ET sequence|legend=1| 31, 68, 99, 130e, 229e }}
 
Badness: 0.029270
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 176/175, 196/195, 275/273
 
Mapping: {{mapping| 1 15 4 7 11 -3 | 0 -16 -2 -5 -9 8 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 194.004
 
{{Optimal ET sequence|legend=1| 31, 68, 99f, 167ef }}
 
Badness: 0.028432
 
==== Hemiwar ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 66/65, 105/104, 121/120, 1375/1372
 
Mapping: {{mapping| 1 15 4 7 11 23 | 0 -16 -2 -5 -9 -23 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.698
 
{{Optimal ET sequence|legend=1| 6f, 31 }}
 
Badness: 0.044886
 
=== Quadrawürschmidt ===
This has been documented in Graham Breed's temperament finder as ''semihemiwürschmidt'', but ''quadrawürschmidt'' arguably makes more sense.
 
The generator of quadrawürschmidt is essentially a [[septimal meantone]] fifth. However, it is not used to represent [[3/2]], as 3/2 is found at the hemiwürschmidt position, 16 wholetones up. The small comma between the generator and 3/2 is taken to represent [[441/440]].
 
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 3136/3125
 
Mapping: {{mapping| 1 15 4 7 24 | 0 -32 -4 -10 -49 }}
 
: mapping generators: ~2, ~147/110
 
Optimal tuning (POTE): ~2 = 1\1, ~147/110 = 503.0404
 
{{Optimal ET sequence|legend=1| 31, 105be, 136e, 167, 198, 427c }}
 
Badness: 0.034814
 
=== Semihemiwür ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3136/3125, 9801/9800
 
Mapping: {{mapping| 2 14 6 9 -10 | 0 -16 -2 -5 25 }}
 
: mapping generators: ~99/70, ~495/392
 
Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9021
 
{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328 }}
 
Badness: 0.044848
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 1001/1000, 1716/1715, 3136/3125
 
Mapping: {{mapping| 2 14 6 9 -10 25 | 0 -16 -2 -5 25 -26 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9035
 
{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328 }}
 
Badness: 0.023388
 
===== Semihemiwürat =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 289/288, 442/441, 561/560, 676/675, 1632/1625
 
Mapping: {{mapping| 2 14 6 9 -10 25 19 | 0 -16 -2 -5 25 -26 -16 }}
 
Optimal tuning (POTE): ~17/12 = 1\2, ~28/25 = 193.9112
 
{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328g, 526cfgg }}
 
Badness: 0.028987
 
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 289/288, 442/441, 456/455, 476/475, 561/560, 627/625
 
Mapping: {{mapping| 2 14 6 9 -10 25 19 20 | 0 -16 -2 -5 25 -26 -16 -17 }}
 
Optimal tuning (POTE): ~17/12 = 1\2, ~19/17 = 193.9145
 
{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328g, 526cfgg }}
 
Badness: 0.021707
 
===== Semihemiwüram =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 256/255, 676/675, 715/714, 1001/1000, 1225/1224
 
Mapping: {{mapping| 2 14 6 9 -10 25 -4 | 0 -16 -2 -5 25 -26 18 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9112
 
{{Optimal ET sequence|legend=1| 62eg, 68, 130g, 198g }}
 
Badness: 0.029718
 
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 3136/3125, 4394/4375
Comma list: 256/255, 286/285, 400/399, 476/475, 495/494, 1225/1224


Sval mapping: [{{val| 1 2 2 3 }}, {{val| 0 6 15 13 }}]
Mapping: {{mapping| 2 14 6 9 -10 25 -4 -3 | 0 -16 -2 -5 25 -26 18 17 }}


Sval mapping generators: ~2, ~26/25
Optimal tuning (POTE): ~99/70 = 1\2, ~19/17 = 193.9428


POTE generator: ~26/25 = 64.6086
{{Optimal ET sequence|legend=1| 62egh, 68, 130gh, 198gh }}


Optimal GPV sequence: {{val list| 18, 19, 37, 93, 130 }}
Badness: 0.029545


== Hemithirds ==
== Hemithirds ==
{{main| Hemithirds }}
''[[#Strong extensions|Return to the map]]''
{{see also| Luna family #Hemithirds }}


Subgroup: 2.3.5.7
{{Main| Hemithirds }}
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1029/1024, 3136/3125
[[Comma list]]: 1029/1024, 3136/3125


[[Mapping]]: [{{val| 1 4 2 2 }}, {{val| 0 -15 2 5 }}]
{{Mapping|legend=1| 1 4 2 2 | 0 -15 2 5 }}


{{Multival|legend=1| 15 -2 -5 -38 -50 -6 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.244
 
[[POTE generator]]: ~28/25 = 193.244


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[7-odd-limit]]: ~28/25 = {{monzo| 1/10 -1/20 0 1/20 }}
* [[7-odd-limit]]: ~28/25 = {{monzo| 1/10 -1/20 0 1/20 }}
: [[Eigenmonzo]]s (unchanged intervals): 2, 7/6
: {{monzo list| 1 0 0 0 | 5/2 3/4 0 -3/4 | 11/5 -1/10 0 1/10 | 5/2 -1/4 0 1/4 }}
* [[9-odd-limit]]: ~28/25 = {{monzo| 6/25 -2/35 0 1/35 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
: [[Eigenmonzo]]s (unchanged intervals): 2, 9/7
* [[9-odd-limit]]: ~28/25 = {{monzo| 6/35 -2/35 0 1/35 }}
: {{monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 82/35 -4/35 0 2/35 | 20/7 -2/7 0 1/7 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7


{{Val list|legend=1| 25, 31, 87, 118 }}
{{Optimal ET sequence|legend=1| 25, 31, 87, 118 }}


[[Badness]]: 0.044284
[[Badness]]: 0.044284
Line 78: Line 314:
Comma list: 385/384, 441/440, 3136/3125
Comma list: 385/384, 441/440, 3136/3125


Mapping: [{{val| 1 4 2 2 7 }}, {{val| 0 -15 2 5 -22 }}]
Mapping: {{mapping| 1 4 2 2 7 | 0 -15 2 5 -22 }}


POTE generator: ~28/25 = 193.227
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.227


Minimax tuning:
Minimax tuning:
* 11-odd-limit: ~28/25 = {{monzo| 5/27 0 0 1/27 -1/27 }}
* 11-odd-limit: ~28/25 = {{monzo| 5/27 0 0 1/27 -1/27 }}
: Eigenmonzos (unchanged intervals): 2, 11/7
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 11/9 0 0 -5/9 5/9 }}, {{monzo| 64/27 0 0 2/27 -2/27 }}, {{monzo| 79/27 0 0 5/27 -5/27 }}, {{monzo| 79/27 0 0 -22/27 22/27 }}]
: Eigenmonzos (unchanged-intervals): 2, 11/7


Optimal GPV sequence: {{Val list| 25e, 31, 87, 118 }}
{{Optimal ET sequence|legend=1| 25e, 31, 87, 118 }}


Badness: 0.019003
Badness: 0.019003
Line 95: Line 332:
Comma list: 196/195, 352/351, 385/384, 625/624
Comma list: 196/195, 352/351, 385/384, 625/624


Mapping: [{{val| 1 4 2 2 7 0 }}, {{val| 0 -15 2 5 -22 23 }}]
Mapping: {{mapping| 1 4 2 2 7 0 | 0 -15 2 5 -22 23 }}


POTE generator: ~28/25 = 193.166
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.166


Optimal GPV sequence: {{Val list| 31, 56, 87, 118, 205d }}
{{Optimal ET sequence|legend=1| 31, 56, 87, 118, 205d }}


Badness: 0.021738
Badness: 0.021738
== Spell ==
''[[#Strong extensions|Return to the map]]''
{{See also| Magic family }}
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 49/48, 3125/3072
{{Mapping|legend=1| 1 0 2 2 | 0 10 2 5 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 189.927
{{Optimal ET sequence|legend=1| 6, 19, 82dd }}
[[Badness]]: 0.080958
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 49/48, 56/55, 125/121
Mapping: {{mapping| 1 0 2 2 3 | 0 10 2 5 3 }}
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 190.285
{{Optimal ET sequence|legend=1| 6, 19, 44de, 63dee, 82ddee }}
Badness: 0.059791
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 49/48, 56/55, 78/77, 125/121
Mapping: {{mapping| 1 0 2 2 3 4 | 0 10 2 5 3 -2 }}
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 189.928
{{Optimal ET sequence|legend=1| 6, 19, 82ddeeff }}
Badness: 0.045591
==== Cantrip ====
Subgroup: 2.3.5.7.11.13
Comma list: 49/48, 56/55, 91/90, 125/121
Mapping: {{mapping| 1 0 2 2 3 1 | 0 10 2 5 3 17 }}
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 190.360
{{Optimal ET sequence|legend=1| 19, 44de, 63dee, 82ddee }}
Badness: 0.041603
= Weak extensions =


== Semisept ==
== Semisept ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semisept]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semisept]].''


Subgroup: 2.3.5.7
The minimal generator of semisept is half a tempered septimal major sixth (12/7), hence the name. Three such generator steps minus an octave give the hemithird, and six give the classical major third. It can be described as the 31 & 80 temperament, and as one may expect, [[111edo]] makes for a great tuning.
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1728/1715, 3136/3125
[[Comma list]]: 1728/1715, 3136/3125


[[Mapping]]: [{{val| 1 12 6 12 }}, {{val| 0 -17 -6 -15 }}]
{{Mapping|legend=1| 1 12 6 12 | 0 -17 -6 -15 }}


{{Multival|legend=1| 17 6 15 -30 -24 18 }}
: mapping generators: ~2, ~75/49


[[POTE generator]]: ~75/49 = 735.155
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/49 = 735.155


{{Val list|legend=1| 18, 31, 80, 111 }}
{{Optimal ET sequence|legend=1| 18, 31, 80, 111 }}


[[Badness]]: 0.050472
[[Badness]]: 0.050472
Line 125: Line 422:
Comma list: 176/175, 540/539, 1331/1323
Comma list: 176/175, 540/539, 1331/1323


Mapping: [{{val| 1 12 6 12 20 }}, {{val| 0 -17 -6 -15 -27 }}]
Mapping: {{mapping| 1 12 6 12 20 | 0 -17 -6 -15 -27 }}


POTE generator: ~55/36 = 735.125
Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.125


Optimal GPV sequence: {{Val list| 18e, 31, 80, 111, 364cd }}
{{Optimal ET sequence|legend=1| 18e, 31, 80, 111, 364cd }}


Badness: 0.022476
Badness: 0.022476
Line 138: Line 435:
Comma list: 176/175, 351/350, 540/539, 1375/1372
Comma list: 176/175, 351/350, 540/539, 1375/1372


Mapping: [{{val| 1 12 6 12 20 -11 }}, {{val| 0 -17 -6 -15 -27 24 }}]
Mapping: {{mapping| 1 12 6 12 20 -11 | 0 -17 -6 -15 -27 24 }}


POTE generator: ~55/36 = 735.126
Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.126


Optimal GPV sequence: {{Val list| 31, 80, 111 }}
{{Optimal ET sequence|legend=1| 31, 80, 111 }}


Badness: 0.025204
Badness: 0.025204
Line 151: Line 448:
Comma list: 176/175, 256/255, 351/350, 640/637, 715/714
Comma list: 176/175, 256/255, 351/350, 640/637, 715/714


Mapping: [{{val| 1 12 6 12 20 -11 -10 }}, {{val| 0 -17 -6 -15 -27 24 23 }}]
Mapping: {{mapping| 1 12 6 12 20 -11 -10 | 0 -17 -6 -15 -27 24 23 }}


POTE generator: ~26/17 = 735.125
Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.125


Optimal GPV sequence: {{Val list| 31, 80, 111 }}
{{Optimal ET sequence|legend=1| 31, 80, 111 }}


Badness: 0.019919
Badness: 0.019919
Line 164: Line 461:
Comma list: 176/175, 286/285, 351/350, 476/475, 540/539, 1331/1323
Comma list: 176/175, 286/285, 351/350, 476/475, 540/539, 1331/1323


Mapping: [{{val| 1 12 6 12 20 -11 -10 -8 }}, {{val| 0 -17 -6 -15 -27 24 23 20 }}]
Mapping: {{mapping| 1 12 6 12 20 -11 -10 -8 | 0 -17 -6 -15 -27 24 23 20 }}


POTE generator: ~26/17 = 735.116
Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.116


Optimal GPV sequence: {{Val list| 31, 80, 111 }}
{{Optimal ET sequence|legend=1| 31, 80, 111 }}


Badness: 0.016301
Badness: 0.016301
Line 177: Line 474:
Comma list: 176/175, 253/252, 286/285, 345/343, 351/350, 391/390, 460/459
Comma list: 176/175, 253/252, 286/285, 345/343, 351/350, 391/390, 460/459


Mapping: [{{val| 1 12 6 12 20 -11 -10 -8 18 }}, {{val| 0 -17 -6 -15 -27 24 23 20 -22 }}]
Mapping: {{mapping| 1 12 6 12 20 -11 -10 -8 18 | 0 -17 -6 -15 -27 24 23 20 -22 }}


POTE generator: ~26/17 = 735.106
Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.106


Vals: {{Val list| 31, 80, 111, 191cdh, 302cdgh }}
{{Optimal ET sequence|legend=1| 31, 80, 111, 191cdh, 302cdgh }}


Badness: 0.014957
Badness: 0.014957
Line 190: Line 487:
Comma list: 144/143, 176/175, 196/195, 275/273
Comma list: 144/143, 176/175, 196/195, 275/273


Mapping: [{{val| 1 12 6 12 20 8 }}, {{val| 0 -17 -6 -15 -27 -7 }}]
Mapping: {{mapping| 1 12 6 12 20 8 | 0 -17 -6 -15 -27 -7 }}


POTE generator: ~13/10 = 464.980
Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 464.980


Optimal GPV sequence: {{Val list| 31, 49f, 80f }}
{{Optimal ET sequence|legend=1| 31, 49f, 80f }}


Badness: 0.028408
Badness: 0.028408
Line 201: Line 498:
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Emka]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Emka]].''


[[Emka]] tempers out {{monzo|-50 -8 27}} in the 5-limit. This temperament can be described as 37&amp;50 temperament, which tempers out the hemimean and 84035/82944 (quinzo-ayo). Alternative extension [[Horwell temperaments #Emkay|emkay]] (87&amp;224) tempers out the same 5-limit comma as the emka, but with the horwell (65625/65536) rather than the hemimean tempered out.
Emka tempers out {{monzo| -50 -8 27 }} in the 5-limit. This temperament can be described as 37 &amp; 50 temperament, which tempers out the hemimean and 84035/82944 (quinzo-ayo). Alternative extension [[Horwell temperaments #Emkay|emkay]] (87 &amp; 224) tempers out the same 5-limit comma as the emka, but with the horwell (65625/65536) rather than the hemimean tempered out.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3136/3125, 84035/82944
[[Comma list]]: 3136/3125, 84035/82944


[[Mapping]]: [{{val| 1 14 6 12 }}, {{val| 0 -27 -8 -20 }}]
{{Mapping|legend=1| 1 14 6 12 | 0 -27 -8 -20 }}


{{Multival|legend=1| 27 8 20 -50 -44 24 }}
: mapping generators: ~2, ~48/35


[[POTE generator]]: ~48/35 = 551.782
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 551.782


{{Val list|legend=1| 37, 50, 87, 137d, 224d }}
{{Optimal ET sequence|legend=1| 37, 50, 87, 137d, 224d }}


[[Badness]]: 0.144338
[[Badness]]: 0.144338
Line 222: Line 519:
Comma list: 385/384, 2401/2376, 3136/3125
Comma list: 385/384, 2401/2376, 3136/3125


Mapping: [{{val| 1 14 6 12 3 }}, {{val| 0 -27 -8 -20 1 }}]
Mapping: {{mapping| 1 14 6 12 3 | 0 -27 -8 -20 1 }}


POTE generator: ~11/8 = 551.765
Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.765


Optimal GPV sequence: {{Val list| 37, 50, 87, 224d, 311d }}
{{Optimal ET sequence|legend=1| 37, 50, 87, 224d, 311d }}


Badness: 0.054744
Badness: 0.054744
Line 235: Line 532:
Comma list: 196/195, 364/363, 385/384, 625/624
Comma list: 196/195, 364/363, 385/384, 625/624


Mapping: [{{val| 1 14 6 12 3 6 }}, {{val| 0 -27 -8 -20 1 -5 }}]
Mapping: {{mapping| 1 14 6 12 3 6 | 0 -27 -8 -20 1 -5 }}


POTE generator: ~11/8 = 551.758
Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.758


Optimal GPV sequence: {{Val list| 37, 50, 87, 224d, 311d, 398d }}
{{Optimal ET sequence|legend=1| 37, 50, 87, 224d, 311d, 398d }}


Badness: 0.029741
Badness: 0.029741


== Decipentic ==
== Decipentic ==
The generator for the ''decipentic'' temperament (43&amp;56) is tenth root of fifth harmonic ([[5/1]], pentave), 5<sup>1/10</sup>, tuned between [[75/64]] and [[20/17]] (close to 27/23). Aside from the hemimean comma, this temperament tempers out the [[Bronzismic temperaments|bronzisma]], 2097152/2083725 (satriru-agugu). [[99edo|99EDO]] is a good tuning for decipentic, with generator 23\99, and MOS of 9, 13, 17, 30, 43 or 56 notes are available.
The generator for the decipentic temperament (43 &amp; 56) is the tenth root of the [[5/1|5th harmonic (5/1)]], 5<sup>1/10</sup>, tuned between [[75/64]] and [[20/17]] (close to [[27/23]]). Aside from the hemimean comma, this temperament tempers out the [[bronzisma]], 2097152/2083725. [[99edo]] is a good tuning for decipentic, with generator 23\99, and [[mos scale]]s of 9, 13, 17, 30, 43 or 56 notes are available.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3136/3125, 2097152/2083725
[[Comma list]]: 3136/3125, 2097152/2083725


[[Mapping]]: [{{val| 1 6 0 -3 }}, {{val| 0 -19 10 25 }}]
{{Mapping|legend=1| 1 6 0 -3 | 0 -19 10 25 }}


{{Multival|legend=1| 19 -10 -25 -60 -93 -30 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/64 = 278.800


[[POTE generator]]: ~75/64 = 278.800
{{Optimal ET sequence|legend=1| 13, 43, 56, 99 }}
 
{{Val list|legend=1| 13, 43, 56, 99 }}


[[Badness]]: 0.087325
[[Badness]]: 0.087325
Line 265: Line 560:
Comma list: 441/440, 1344/1331, 3136/3125
Comma list: 441/440, 1344/1331, 3136/3125


Mapping: [{{val| 1 6 0 -3 3 }}, {{val| 0 -19 10 25 2 }}]
Mapping: {{mapping| 1 6 0 -3 3 | 0 -19 10 25 2 }}


POTE generator: ~75/64 = 278.799
Optimal tuning (POTE): ~2 = 1\1, ~75/64 = 278.799


Optimal GPV sequence: {{Val list| 13, 43, 56, 99e }}
{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}


Badness: 0.061413
Badness: 0.061413
Line 278: Line 573:
Comma list: 169/168, 441/440, 832/825, 975/968
Comma list: 169/168, 441/440, 832/825, 975/968


Mapping: [{{val| 1 6 0 -3 3 3 }}, {{val| 0 -19 10 25 2 3 }}]
Mapping: {{mapping| 1 6 0 -3 3 3 | 0 -19 10 25 2 3 }}


POTE generator: ~13/11 = 278.802
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.802


Optimal GPV sequence: {{Val list| 13, 43, 56, 99e }}
{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}


Badness: 0.047611
Badness: 0.047611
Line 291: Line 586:
Comma list: 169/168, 221/220, 256/255, 273/272, 375/374
Comma list: 169/168, 221/220, 256/255, 273/272, 375/374


Mapping: [{{val| 1 6 0 -3 3 3 2 }}, {{val| 0 -19 10 25 2 3 9 }}]
Mapping: {{mapping| 1 6 0 -3 3 3 2 | 0 -19 10 25 2 3 9 }}


POTE generator: ~13/11 = 278.798
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.798


Optimal GPV sequence: {{Val list| 13, 43, 56, 99e }}
{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}


Badness: 0.031191
Badness: 0.031191
Line 304: Line 599:
Comma list: 169/168, 210/209, 221/220, 256/255, 273/272, 286/285
Comma list: 169/168, 210/209, 221/220, 256/255, 273/272, 286/285


Mapping: [{{val| 1 6 0 -3 3 3 2 1 }}, {{val| 0 -19 10 25 2 3 9 14 }}]
Mapping: {{mapping| 1 6 0 -3 3 3 2 1 | 0 -19 10 25 2 3 9 14 }}


POTE generator: ~13/11 = 278.790
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.790


Optimal GPV sequence: {{Val list| 13, 43, 56, 99e }}
{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}


Badness: 0.023899
Badness: 0.023899
Line 317: Line 612:
Comma list: 3136/3125, 15488/15435, 16384/16335
Comma list: 3136/3125, 15488/15435, 16384/16335


Mapping: [{{val| 1 6 0 -3 3 }}, {{val| 0 -38 20 50 47 }}]
Mapping: {{mapping| 1 6 0 -3 3 | 0 -38 20 50 47 }}


POTE generator: ~896/825 = 139.403
Optimal tuning (POTE): ~2 = 1\1, ~896/825 = 139.403


Optimal GPV sequence: {{Val list| 43, 112, 155, 198, 439cd, 637cd }}
{{Optimal ET sequence|legend=1| 43, 112, 155, 198, 439cd, 637cd }}


Badness: 0.092996
Badness: 0.092996
Line 330: Line 625:
Comma list: 676/675, 1001/1000, 3136/3125, 15488/15435
Comma list: 676/675, 1001/1000, 3136/3125, 15488/15435


Mapping: [{{val| 1 6 0 -3 3 8 }}, {{val| 0 -38 20 50 47 -37 }}]
Mapping: {{mapping| 1 6 0 -3 3 8 | 0 -38 20 50 47 -37 }}


POTE generator: ~13/12 = 139.403
Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 139.403


Optimal GPV sequence: {{Val list| 43, 155, 198, 439cdf, 637cdf }}
{{Optimal ET sequence|legend=1| 43, 155, 198, 439cdf, 637cdf }}


Badness: 0.044328
Badness: 0.044328
== Arch ==
{{see also|Escapade family}}
Subgroup: 2.3.5.7
[[Comma list]]: 3136/3125, 5250987/5242880
[[Mapping]]: [{{val| 1 2 2 2 }}, {{val| 0 -18 14 35 }}]
{{Multival|legend=1| 18 -14 -35 -64 -106 -42 }}
[[POTE generator]]: ~64/63 = 27.668
{{Val list|legend=1| 43, 87, 130, 217, 347, 824c, 1171c, 1518cd }}
[[Badness]]: 0.094345
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 441/440, 3136/3125, 4000/3993
Mapping: [{{val| 1 2 2 2 3 }}, {{val| 0 -18 14 35 20 }}]
POTE generator: ~64/63 = 27.663
Optimal GPV sequence: {{Val list| 43, 87, 130, 217, 347e, 911cde }}
Badness: 0.036541
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 676/675, 3136/3125
Mapping: [{{val| 1 2 2 2 3 4 }}, {{val| 0 -18 14 35 20 -13 }}]
POTE generator: ~64/63 = 27.660
Optimal GPV sequence: {{Val list| 43, 87, 130, 217, 347e, 564e }}
Badness: 0.019504


== Sengagen ==
== Sengagen ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3136/3125, 420175/419904
[[Comma list]]: 3136/3125, 420175/419904


[[Mapping]]: [{{val| 1 1 2 2 }}, {{val| 0 29 16 40 }}]
{{Mapping|legend=1| 1 1 2 2 | 0 29 16 40 }}


{{Multival|legend=1| 29 16 40 -42 -18 48 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~686/675 = 24.217


[[POTE generator]]: ~686/675 = 24.217
{{Optimal ET sequence|legend=1| 49, 50, 99, 248, 347, 446 }}
 
{{Val list|legend=1| 49, 50, 99, 248, 347, 446 }}


[[Badness]]: 0.057978
[[Badness]]: 0.057978
Line 401: Line 651:
Comma list: 540/539, 1344/1331, 3136/3125
Comma list: 540/539, 1344/1331, 3136/3125


Mapping: [{{val| 1 1 2 2 3 }}, {{val| 0 29 16 40 23 }}]
Mapping: {{mapping| 1 1 2 2 3 | 0 29 16 40 23 }}


POTE generator: ~99/98 = 24.235
Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.235


Optimal GPV sequence: {{Val list| 49, 50, 99e }}
{{Optimal ET sequence|legend=1| 49, 50, 99e }}


Badness: 0.053828
Badness: 0.053828
Line 414: Line 664:
Comma list: 351/350, 540/539, 975/968, 1344/1331
Comma list: 351/350, 540/539, 975/968, 1344/1331


Mapping: [{{val| 1 1 2 2 3 4 }}, {{val| 0 29 16 40 23 -15 }}]
Mapping: {{mapping| 1 1 2 2 3 4 | 0 29 16 40 23 -15 }}


POTE generator: ~99/98 = 24.181
Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.181


Optimal GPV sequence: {{Val list| 49, 50, 99e, 149e }}
{{Optimal ET sequence|legend=1| 49, 50, 99e, 149e }}


Badness: 0.053531
Badness: 0.053531
Line 427: Line 677:
Comma list: 144/143, 196/195, 364/363, 625/624
Comma list: 144/143, 196/195, 364/363, 625/624


Mapping: [{{val| 1 1 2 2 3 3 }}, {{val| 0 29 16 40 23 35 }}]
Mapping: {{mapping| 1 1 2 2 3 3 | 0 29 16 40 23 35 }}


POTE generator: ~99/98 = 24.234
Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.234


Optimal GPV sequence: {{Val list| 49f, 50, 99ef }}
{{Optimal ET sequence|legend=1| 49f, 50, 99ef }}


Badness: 0.037416
Badness: 0.037416
== Subpental ==
{{see also|Sensipent family}}
Subgroup: 2.3.5.7
[[Comma list]]: 3136/3125, 19683/19600
[[Mapping]]: [{{val| 1 6 8 17 }}, {{val| 0 -14 -18 -45 }}]
{{Multival|legend=1| 14 18 45 -4 32 54 }}
[[POTE generator]]: ~56/45 = 378.467
{{Val list|legend=1| 19, 111, 130, 929c, 1059c, 1189bc, 1319bc }}
[[Badness]]: 0.054303
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 540/539, 3136/3125, 8019/8000
Mapping: [{{val| 1 6 8 17 -6 }}, {{val| 0 -14 -18 -45 30 }}]
POTE generator: ~56/45 = 378.440
Optimal GPV sequence: {{Val list| 19, 111, 130, 241, 371ce, 501cde, 872cde }}
Badness: 0.045352
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 540/539, 676/675, 3136/3125
Mapping: [{{val| 1 6 8 17 -6 16 }}, {{val| 0 -14 -18 -45 30 -39 }}]
POTE generator: ~56/45 = 378.437
Optimal GPV sequence: {{Val list| 19, 111, 130, 241, 371ce }}
Badness: 0.023940


== Mowglic ==
== Mowglic ==
The ''mowglic'' temperament (19&amp;161) is an extension of the [[Syntonic-enneadecal equivalence continuum|mowgli temperament]] which tempers out the hemimean comma and the secanticornisma (177147/175000, laruquingu) in the 7-limit.
The mowglic temperament (19 &amp; 161) is an extension of the [[Syntonic–kleismic equivalence continuum #Mowgli|mowgli temperament]] which tempers out the hemimean comma and the secanticornisma (177147/175000, laruquingu) in the 7-limit.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3136/3125, 177147/175000
[[Comma list]]: 3136/3125, 177147/175000


[[Mapping]]: [{{val| 1 0 0 -3 }}, {{val| 0 15 22 55 }}]
{{Mapping|legend=1| 1 0 0 -3 | 0 15 22 55 }}


{{Multival|legend=1| 15 22 55 0 45 66 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/25 = 126.706


[[POTE generator]]: ~27/25 = 126.706
{{Optimal ET sequence|legend=1| 19, 123d, 142, 161 }}
 
{{Val list|legend=1| 19, 123d, 142, 161 }}


[[Badness]]: 0.129915
[[Badness]]: 0.129915
Line 500: Line 705:
Comma list: 540/539, 3136/3125, 72171/71680
Comma list: 540/539, 3136/3125, 72171/71680


Mapping: [{{val| 1 0 0 -3 8 }}, {{val| 0 15 22 55 -43 }}]
Mapping: {{mapping| 1 0 0 -3 8 | 0 15 22 55 -43 }}


POTE generator: ~27/25 = 126.711
Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 126.711


Optimal GPV sequence: {{Val list| 19, 123de, 142, 161 }}
{{Optimal ET sequence|legend=1| 19, 123de, 142, 161 }}


Badness: 0.094032
Badness: 0.094032
Line 513: Line 718:
Comma list: 351/350, 540/539, 1701/1690, 3136/3125
Comma list: 351/350, 540/539, 1701/1690, 3136/3125


Mapping: [{{val| 1 0 0 -3 8 -2 }}, {{val| 0 15 22 55 -43 54 }}]
Mapping: {{mapping| 1 0 0 -3 8 -2 | 0 15 22 55 -43 54 }}


POTE generator: ~14/13 = 126.705
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.705


Optimal GPV sequence: {{Val list| 19, 123def, 142f, 161 }}
{{Optimal ET sequence|legend=1| 19, 123def, 142f, 161 }}


Badness: 0.051571
Badness: 0.051571
Line 526: Line 731:
Comma list: 351/350, 540/539, 833/832, 1701/1690, 3136/3125
Comma list: 351/350, 540/539, 833/832, 1701/1690, 3136/3125


Mapping: [{{val| 1 0 0 -3 8 -2 10 }}, {{val| 0 15 22 55 -43 54 -56 }}]
Mapping: {{mapping| 1 0 0 -3 8 -2 10 | 0 15 22 55 -43 54 -56 }}


POTE generator: ~14/13 = 126.703
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703


Optimal GPV sequence: {{Val list| 19, 123defg, 142f, 161 }}
{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}


Badness: 0.041918
Badness: 0.041918
Line 539: Line 744:
Comma list: 351/350, 476/475, 495/494, 513/512, 540/539, 1701/1690
Comma list: 351/350, 476/475, 495/494, 513/512, 540/539, 1701/1690


Mapping: [{{val| 1 0 0 -3 8 -2 10 9 }}, {{val| 0 15 22 55 -43 54 -56 -45 }}]
Mapping: {{mapping| 1 0 0 -3 8 -2 10 9 | 0 15 22 55 -43 54 -56 -45 }}


POTE generator: ~14/13 = 126.705
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.705


Optimal GPV sequence: {{Val list| 19, 123defg, 142f, 161 }}
{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}


Badness: 0.032168
Badness: 0.032168
Line 552: Line 757:
Comma list: 276/275, 351/350, 476/475, 495/494, 513/512, 529/528, 540/539
Comma list: 276/275, 351/350, 476/475, 495/494, 513/512, 529/528, 540/539


Mapping: [{{val| 1 0 0 -3 8 -2 10 9 6 }}, {{val| 0 15 22 55 -43 54 -56 -45 -14 }}]
Mapping: {{mapping| 1 0 0 -3 8 -2 10 9 6 | 0 15 22 55 -43 54 -56 -45 -14 }}


POTE generator: ~14/13 = 126.703
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703


Optimal GPV sequence: {{Val list| 19, 123defg, 142f, 161 }}
{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}


Badness: 0.026117
Badness: 0.026117
Line 565: Line 770:
Comma list: 261/260, 276/275, 351/350, 476/475, 495/494, 513/512, 529/528, 540/539
Comma list: 261/260, 276/275, 351/350, 476/475, 495/494, 513/512, 529/528, 540/539


Mapping: [{{val| 1 0 0 -3 8 -2 10 9 6 0 }}, {{val| 0 15 22 55 -43 54 -56 -45 -14 46 }}]
Mapping: {{mapping| 1 0 0 -3 8 -2 10 9 6 0 | 0 15 22 55 -43 54 -56 -45 -14 46 }}


POTE generator: ~14/13 = 126.704
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.704


Optimal GPV sequence: {{Val list| 19, 123defg, 142f, 161 }}
{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}


Badness: 0.021398
Badness: 0.021398
Line 578: Line 783:
Comma list: 261/260, 276/275, 351/350, 435/434, 476/475, 495/494, 513/512, 529/528, 540/539
Comma list: 261/260, 276/275, 351/350, 435/434, 476/475, 495/494, 513/512, 529/528, 540/539


Mapping: [{{val| 1 0 0 -3 8 -2 10 9 6 0 2 }}, {{val| 0 15 22 55 -43 54 -56 -45 -14 46 28 }}]
Mapping: {{mapping| 1 0 0 -3 8 -2 10 9 6 0 2 | 0 15 22 55 -43 54 -56 -45 -14 46 28 }}


POTE generator: ~14/13 = 126.703
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703


Optimal GPV sequence: {{Val list| 19, 123defgk, 142fk, 161 }}
{{Optimal ET sequence|legend=1| 19, 123defgk, 142fk, 161 }}


Badness: 0.019331
Badness: 0.019331
== Tremka ==
The name ''tremka'' was initially used for the [[No-sevens subgroup temperaments|no-sevens version]] of 50 &amp; 111 (especially in the 2.3.5.11.13 subgroup), but extending to full 13-limit or higher prime limit does no significant tuning damage, so for that we keep the 2.3.5.11.13 label tremka.
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 3136/3125, 2125764/2100875
{{Mapping|legend=1| 1 -4 -2 -8 | 0 31 24 60 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~4375/3888 = 216.173
{{Optimal ET sequence|legend=1| 50, 111, 161, 272 }}
[[Badness]]: 0.179925
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 540/539, 3136/3125, 35937/35840
Mapping: {{mapping| 1 -4 -2 -8 4 | 0 31 24 60 -3 }}
Optimal tuning (POTE): ~2 = 1\1, ~112/99 = 216.168
{{Optimal ET sequence|legend=1| 50, 111, 161, 272, 433c }}
Badness: 0.068825
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 540/539, 847/845, 3136/3125
Mapping: {{mapping| 1 -4 -2 -8 4 1 | 0 31 24 60 -3 15 }}
Optimal tuning (POTE): ~2 = 1\1, ~112/99 = 216.172
{{Optimal ET sequence|legend=1| 50, 111, 161, 272 }}
Badness: 0.036070
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 351/350, 540/539, 561/560, 847/845, 1089/1088
Mapping: {{mapping| 1 -4 -2 -8 4 1 -6 | 0 31 24 60 -3 15 56 }}
Optimal tuning (POTE): ~2 = 1\1, ~17/15 = 216.172
{{Optimal ET sequence|legend=1| 50, 111, 161, 272 }}
Badness: 0.022528
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 324/323, 351/350, 456/455, 476/455, 495/494, 540/539
Mapping: {{mapping| 1 -4 -2 -8 4 1 -6 -8 | 0 31 24 60 -3 15 56 68 }}
Optimal tuning (POTE): ~2 = 1\1, ~17/15 = 216.170
{{Optimal ET sequence|legend=1| 50, 111, 161, 272h, 433cfh, 705ccdffhh }}
Badness: 0.016900


== Undetrita ==
== Undetrita ==
The ''undetrita'' temperament (111&amp;118) tempers out the hemimean comma (3136/3125) and [[skeetsma]] (14348907/14336000) in the 7-limit; 3025/3024, 3388/3375, and 8019/8000 in the 11-limit. This temperament is related to [[11edt|11EDT]] tuning, and the name ''undetrita'' is a play on the words "undecimus" (Latin for "eleventh") and "[[tritave]]" (third harmonic). It is also related to the [[Subgroup temperaments #No-sevens subgroup|twentcufo temperament]], which is no-sevens version of 111&amp;118.
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Undetrita (5-limit)]].''


Subgroup: 2.3.5.7
The undetrita temperament (111 &amp; 118) tempers out the hemimean comma (3136/3125) and [[scheme comma]] (14348907/14336000) in the 7-limit; 3025/3024, 3388/3375, and 8019/8000 in the 11-limit. This temperament is related to [[11edt]], and the name ''undetrita'' is a play on the words ''undecimus'' (Latin for "eleventh") and ''[[tritave]]'' (3rd harmonic). It is also related to the [[Subgroup temperaments #No-sevens subgroup|twentcufo temperament]], which is no-sevens version of 111 &amp; 118.
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3136/3125, 14348907/14336000
[[Comma list]]: 3136/3125, 14348907/14336000


[[Mapping]]: [{{val|1 0 -2 -8}}, {{val|0 11 30 75}}]
{{Mapping|legend=1| 1 0 -2 -8 | 0 11 30 75 }}


{{Multival|legend=1|11 30 75 22 88 90}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~448/405 = 172.917


[[POTE generator]]: ~448/405 = 172.917
{{Optimal ET sequence|legend=1| 111, 118, 229, 347, 576c }}
 
{{Val list|legend=1| 111, 118, 229, 347, 576c }}


[[Badness]]: 0.114188
[[Badness]]: 0.114188
Line 608: Line 881:
Comma list: 3025/3024, 3136/3125, 8019/8000
Comma list: 3025/3024, 3136/3125, 8019/8000


Mapping: [{{val|1 0 -2 -8 0}}, {{val|0 11 30 75 24}}]
Mapping: {{mapping| 1 0 -2 -8 0 | 0 11 30 75 24 }}


POTE generator: ~400/363 = 172.912
Optimal tuning (POTE): ~2 = 1\1, ~400/363 = 172.912


Optimal GPV sequence: {{Val list| 111, 118, 229, 347 }}
{{Optimal ET sequence|legend=1| 111, 118, 229, 347 }}


Badness: 0.043883
Badness: 0.043883
Line 621: Line 894:
Comma list: 352/351, 729/728, 1001/1000, 3025/3024
Comma list: 352/351, 729/728, 1001/1000, 3025/3024


Mapping: [{{val|1 0 -2 -8 0 5}}, {{val|0 11 30 75 24 -9}}]
Mapping: {{mapping| 1 0 -2 -8 0 5 | 0 11 30 75 24 -9 }}


POTE generator: ~72/65 = 172.930
Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 172.930


Optimal GPV sequence: {{Val list| 111, 229f }}
{{Optimal ET sequence|legend=1| 111, 229f }}


Badness: 0.038771
Badness: 0.038771
Line 634: Line 907:
Comma list: 351/350, 1573/1568, 2080/2079, 3136/3125
Comma list: 351/350, 1573/1568, 2080/2079, 3136/3125


Mapping: [{{val|1 0 -2 -8 0 -11}}, {{val|0 11 30 75 24 102}}]
Mapping: {{mapping| 1 0 -2 -8 0 -11 | 0 11 30 75 24 102 }}


POTE generator: ~400/363 = 172.933
Optimal tuning (POTE): ~2 = 1\1, ~400/363 = 172.933


Optimal GPV sequence: {{Val list| 111, 229 }}
{{Optimal ET sequence|legend=1| 111, 229 }}


Badness: 0.042744
Badness: 0.042744
= Subgroup extensions =
== Undecimal didacus ==
In the no-3's [[11-limit]], there is a natural extension with prime 11 by equating [[25/16]] (which is already tuned sharp anyways) with [[11/7]] by tempering out [[176/175]], which is the same route that [[undecimal meantone]] uses, as this is essentially a no-3's restriction of undecimal meantone in the 11-limit, except that undecimal meantone finds ~[[28/25]] at 2 generators (as a flat ~[[9/8]]) while here it is the generator. This is equivalent to finding [[11/4]] as ([[7/5]])<sup>3</sup>. In the no-3's 19-limit extension "mediantone", this whole tone generator serves as the two simplest [[mediant]]s of [[9/8]] and [[10/9]], namely [[19/17]] and [[28/25]], while in undecimal didacus and its extension to the no-3's 13-limit only the latter interpretation is relevant.
Subgroup: 2.5.7.11
Comma list: [[176/175]], [[1375/1372]]
Sval mapping: {{mapping| 1 0 -3 -7 | 0 2 5 9 }}
: sval mapping generators: ~2, ~56/25
Optimal tuning (CWE): 2 = 1\1, ~28/25 = 194.428
Optimal ET sequence: {{Optimal ET sequence| 6, 19e, 25, 31, 37 }}
RMS error: 0.5567 cents
Badness (Sintel): 0.195
=== Tridecimal didacus ===
Tridecimal didacus (formerly ''roulette''; that name has now been reassigned to the no-threes 19-limit extension 37 & 68) is equivalent to [[hemiwur]] or [[grosstone]] with no mapping for prime 3. The mapping of prime 13 is somewhat strange, because it is the only mapping that requires a negative amount of generators (and a large amount of them), but it can be rationalized in a variety of ways, such as that because [[~]][[8/7]] is already tuned almost 3{{cent}} flat, it makes sense to equate two of it with [[~]][[13/10]] (tempering out the 8{{cent}} [[huntma]]). This mapping of 13 increases the [[badness]] of the temperament, but as it does not noticeably affect the optimal generators, it is usually a safe extension to didacus if prime 3 is not included.
Subgroup: 2.5.7.11.13
Comma list: 176/175, 640/637, 1375/1372
Sval mapping: {{mapping| 1 0 -3 -7 13 | 0 2 5 9 -8 }}
: sval mapping generators: ~2, ~56/25
Gencom mapping: {{mapping| 1 0 2 2 2 5 | 0 0 2 5 9 -8 }}
: gencom: [2 28/25; 176/175 1375/1372 640/637]
Optimal tuning (POTE): 2 = 1\1, ~28/25 = 194.594
Optimal ET sequence: {{Optimal ET sequence| 6, 25, 31, 37 }}
Badness (Sintel): 0.324
==== Mediantone ====
Mediantone is named after its whole tone generator serving as the [[mediant]] of [[9/8]] and [[10/9]], namely [[19/17]], in addition to [[28/25]], as well as by the observation that this temperament seems to have been repeatedly rediscovered in parts in a variety of contexts, so that it seems to exist as a "median" of all of these temperaments' logics. It is also an intentional play on "[[meantone]]", as the context one is most likely to first discover this logic is when the tone also represents [[~]][[10/9]][[~]][[9/8]].
In the full no-3's [[19-limit]], this temperament is a structure common to quite a few temperaments. It is a rank-2 version of [[orion]] with a mapping for primes 11 and 13. It is a no-3's version of 19-limit [[grosstone]] which can be seen as an extension of [[undecimal meantone]] according to the "mediant-tone" logic of this temperament, and which as aforementioned effectively doubles the complexity of the temperament as a result of finding the generator of [[~]][[19/17]][[~]][[28/25]] as ([[~]][[3/2]])<sup>2</sup>/[[2/1|2]]. It does not work so well as an extension for [[hemiwur]] to the full 19-limit, but if you want to try anyway (at the cost of primes 17 and 19), a notable patent-val tuning is [[37edo]], which finds prime 3 through the [[würschmidt]] mapping so that [[6/1]] is found at 16 generators.
Subgroup: 2.5.7.11.13.17.19
Comma list: [[176/175]], [[640/637]], [[221/220]], [[476/475]], [[1375/1372]]
Sval mapping: {{mapping| 1 0 -3 -7 13 -18 -19 | 0 2 5 9 -8 19 20 }}
: sval mapping generators: ~2, ~56/25
Optimal tuning (CWE): ~2 = 1\1, ~19/17 = 194.927
Optimal ET sequence: {{Optimal ET sequence| 6h, 31gh, 37, 80, 117d* }}
<nowiki />* 117d only appears without prime 19
Badness (Sintel): 0.618
==== Roulette ====
{{See also | Chromatic pairs #Roulette }}
Roulette is an alternative no-threes 19-limit extension of tridecimal didacus to mediantone (the two mappings converging at [[37edo]]), equating (8/7)<sup>2</sup> to [[17/13]] in addition to 13/10, tempering out [[170/169]] and [[833/832]]; in doing so, it also tempers out the micro-comma [[2000033/2000000]] so that ([[50/49]])<sup>3</sup> is equated to [[17/16]]. The generator is then equated to 19/17 in the same way as in mediantone.
Subgroup: 2.5.7.11.13.17.19
Comma list: [[170/169]], [[176/175]], [[476/475]], [[640/637]], [[1375/1372]]
Sval mapping: {{mapping| 1 2 2 2 5 7 7 | 0 2 5 9 -8 -18 -17 }}
: sval mapping generators: ~2, ~28/25
Optimal tuning (CWE): ~2 = 1\1, ~19/17 = 194.259
Optimal ET sequence: {{Optimal ET sequence| 6g, ... 31, 37, 68, 105 }}
Badness (Sintel): 0.676
== Rectified hebrew ==
{{Main| Rectified hebrew }}
Rectified hebrew (37 &amp; 56) is derived from the [https://individual.utoronto.ca/kalendis/hebrew/rect.htm#353 calendar by the same name]. It is leap year pattern takes a stack of 18 Metonic cycle diatonic major scales and truncates the 19th one down to its generator, 11. It adds harmonic 13 through tempering out [[4394/4375]] and spliting the generator of didacus in three. Notably, it is the no-threes restriction of [[Sycamore family#Septimal sycamore|sycamore]].
Subgroup: 2.5.7.13
Comma list: 3136/3125, 4394/4375
Sval mapping: {{mapping| 1 2 2 3 | 0 6 15 13 }}
: sval mapping generators: ~2, ~26/25
Optimal tuning (POTE): ~2 = 1\1, ~26/25 = 64.6086
{{Optimal ET sequence|legend=1| 18, 19, 37, 93, 130 }}


== Isra ==
== Isra ==
Isra results from taking every other generator of [[septimal meantone]]. It is named after the Isrāʾ (''iss-RAH'') night journey in the Qur'an, because it's similar to [[luna]].  
Isra (''iss-RAH'') results from taking every other generator of [[septimal meantone]], or from [[didacus]] if the generator is interpreted as 9/8. It is named after the Isrāʾ night journey in the Qur'an, because it is similar to [[luna]] (septimal [[hemithirds]], a didacus extension).


Subgroup: 2.9.5.7
[[Subgroup]]: 2.9.5.7


Comma list: 81/80, 126/125
[[Comma list]]: 81/80, 126/125


Mapping: [{{val| 1 3 2 2 }}, {{val| 0 1 2 5 }}]
{{Mapping|legend=2| 1 0 -4 -13 | 0 1 2 5 }}


Mapping generators: ~2/1, ~9/8
: sval mapping generators: ~2, ~9


POTE generator: ~9/8 = 192.9898
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 192.9898


Optimal GPV sequence: {{Val list| 6, 19, 25, 31, 56b, 87b }}
{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 56b, 87b }}


=== Tutone ===
=== Tutone ===
{{See also| Chromatic pairs #Tutone }}
Tutone is every other step of [[Meantone vs meanpop|undecimal meantone]], or undecimal [[didacus]] with the generator interpreted as 9/8.
 
[[Subgroup]]: 2.9.5.7.11
 
[[Comma list]]: 81/80, 99/98, 126/125
 
{{Mapping|legend=2| 1 0 -4 -13 -25 | 0 1 2 5 9 }}
 
{{Mapping|legend=3| 1 3/2 2 2 2 | 0 1/2 2 5 9 }}
 
: [[gencom]]: [2 9/8; 81/80 99/98 126/125]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 193.937
 
{{Optimal ET sequence|legend=1| 6, 19e, 25, 31, 68b, 99b }}
 
[[Badness]]: 0.00536
 
=== Leantone ===
{{See also| Chromatic pairs #Leantone }}
 
Leantone is every other step of [[vincenzo]].
 
[[Subgroup]]: 2.9.5.7.11
 
[[Comma list]]: 45/44, 56/55, 81/80
 
{{Mapping|legend=2| 1 0 -4 -13 -6 | 0 1 2 5 3 }}
 
{{Mapping|legend=3| 1 3/2 2 2 3 | 0 1/2 2 5 3 }}
 
: [[gencom]]: [2 9/8; 45/44 56/55 81/80]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 192.500
 
{{Optimal ET sequence|legend=1| 6, 7, 13, 19, 25e, 31e, 56bee, 81beee }}
 
[[Tp tuning #T2 tuning|RMS error]]: 3.882 cents


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


=== Leantone ===
Deutone is (also) every other step of [[vincenzo]].
{{See also| Chromatic pairs #Leantone }}
 
[[Subgroup]]: 2.9.5.7.13
 
[[Comma list]]: 65/64, 81/80, 91/90
 
{{Mapping|legend=2| 1 0 -4 -13 10 | 0 1 2 5 -2 }}
 
{{Mapping|legend=3| 1 3/2 2 2 0 4 | 0 1/2 2 5 0 -2 }}
 
: [[gencom]]: [2 9/8; 65/64 81/80 91/90]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 191.059
 
{{Optimal ET sequence|legend=1| 6, 7, 13, 19, 25f, 44df }}
 
[[Tp tuning #T2 tuning|RMS error]]: 2.003 cents


[[Category:Regular temperament theory]]
[[Category:Temperament clans]]
[[Category:Temperament clan]]
[[Category:Hemimean clan| ]] <!-- main article -->
[[Category:Hemimean clan| ]] <!-- main article -->
[[Category:Hemimean]]
[[Category:Hemimean| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
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