Hemimean clan: Difference between revisions

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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, 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]]
* ''[[Arch]]'', {3136/3125, 5250987/5242880} → [[Escapade family #Arch|Escapade family]]
* ''[[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
 
Mapping: {{mapping| 1 15 4 7 37 -29 | 0 -16 -2 -5 -40 39 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.829
 
{{Optimal ET sequence|legend=1| 31, 99e, 130, 291, 421e, 551ce }}
 
Badness: 0.023074
 
==== Hemithir ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 176/175, 196/195, 275/273
 
Mapping: {{mapping| 1 15 4 7 37 -3 | 0 -16 -2 -5 -40 8 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.918
 
{{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 }}


[[Comma list]]: 3136/3125
Badness: 0.029270


[[Sval]] [[mapping]]: [{{val| 1 0 -3 }}, {{val| 0 2 5 }}]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Sval mapping generators: ~2, ~56/25
Comma list: 121/120, 176/175, 196/195, 275/273


Gencom mapping: [{{val| 1 0 0 -3 }}, {{val| 0 0 2 5 }}]
Mapping: {{mapping| 1 15 4 7 11 -3 | 0 -16 -2 -5 -9 8 }}


[[Gencom]]: [2 56/25; 3136/3125]
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 194.004


[[POTE generator]]: ~28/25 = 193.772
{{Optimal ET sequence|legend=1| 31, 68, 99f, 167ef }}


{{Val list|legend=1| 6, 19, 25, 31, 99, 130, 161, 353, 514c, 867c }}
Badness: 0.028432


=== Rectified Hebrew ===
==== Hemiwar ====
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: 2.3.5.7.11.13


Subgroup: 2.5.7.13
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 79: 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 96: 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 126: 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 139: 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 152: 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 165: 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 178: 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 191: 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 202: 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 223: 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 236: 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 }}


[[POTE generator]]: ~75/64 = 278.800
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/64 = 278.800


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


[[Badness]]: 0.087325
[[Badness]]: 0.087325
Line 266: 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 279: 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 292: 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 305: 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 318: 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 331: 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


== 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 }}


[[POTE generator]]: ~686/675 = 24.217
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~686/675 = 24.217


{{Val list|legend=1| 49, 50, 99, 248, 347, 446 }}
{{Optimal ET sequence|legend=1| 49, 50, 99, 248, 347, 446 }}


[[Badness]]: 0.057978
[[Badness]]: 0.057978
Line 359: 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 372: 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 385: 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 458: 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 471: 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 484: 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 497: 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 510: 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 523: 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 536: 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 566: 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 579: 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 592: 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]]

Latest revision as of 13:34, 9 August 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The hemimean clan tempers out the hemimean comma, 3136/3125, with 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.

The second comma of the comma list determines which 7-limit family member we are looking at. These extensions, 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. 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

2.5.7 subgroup

Didacus

Main article: Didacus

See also its canonical extension to the 2.5.7.11 subgroup, #Undecimal didacus.

Subgroup: 2.5.7

Comma list: 3136/3125

Subgroup-val mapping: [⟨1 0 -3], ⟨0 2 5]]

sval mapping generators: ~2, ~56/25

Gencom mapping: [⟨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: 6, 19, 25, 31, 99, 130, 161, 353, 514c, 867c

RMS error: 0.2138 cents

Badness (Sintel): 0.091

Strong extensions

Map to strong extensions
Extension 5-limit re-restriction Mapping of 3 Tuning range*
Temperament 5-limit generator location
Hemiwürschmidt Würschmidt +2 +16 ↓ 31
Hemithirds Luna +1 -15 ↑ 31
↓ 25
Spell Magic +2 +10 ↑ 25

* Defined by intersection with other documented extensions

Hemiwürschmidt

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: [⟨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: 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: [⟨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: 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: [⟨1 15 4 7 37], ⟨0 -16 -2 -5 -40]]

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.840

Optimal ET sequence: 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

Mapping: [⟨1 15 4 7 37 -29], ⟨0 -16 -2 -5 -40 39]]

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.829

Optimal ET sequence: 31, 99e, 130, 291, 421e, 551ce

Badness: 0.023074

Hemithir

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 196/195, 275/273

Mapping: [⟨1 15 4 7 37 -3], ⟨0 -16 -2 -5 -40 8]]

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.918

Optimal ET sequence: 31, 68e, 99ef

Badness: 0.031199

Hemiwur

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 1375/1372

Mapping: [⟨1 15 4 7 11], ⟨0 -16 -2 -5 -9]]

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.884

Optimal ET sequence: 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: [⟨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: 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: [⟨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: 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: [⟨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: 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: [⟨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: 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: [⟨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: 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: [⟨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: 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: [⟨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: 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: [⟨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: 62eg, 68, 130g, 198g

Badness: 0.029718

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨2 14 6 9 -10 25 -4 -3], ⟨0 -16 -2 -5 25 -26 18 17]]

Optimal tuning (POTE): ~99/70 = 1\2, ~19/17 = 193.9428

Optimal ET sequence: 62egh, 68, 130gh, 198gh

Badness: 0.029545

Hemithirds

Return to the map

Main article: Hemithirds

Subgroup: 2.3.5.7

Comma list: 1029/1024, 3136/3125

Mapping: [⟨1 4 2 2], ⟨0 -15 2 5]]

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.244

Minimax tuning:

[[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⟩]
unchanged-interval (eigenmonzo) basis: 2.7/3
[[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⟩]
unchanged-interval (eigenmonzo) basis: 2.9/7

Optimal ET sequence: 25, 31, 87, 118

Badness: 0.044284

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 3136/3125

Mapping: [⟨1 4 2 2 7], ⟨0 -15 2 5 -22]]

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

Minimax tuning:

  • 11-odd-limit: ~28/25 = [5/27 0 0 1/27 -1/27⟩
[[1 0 0 0 0⟩, [11/9 0 0 -5/9 5/9⟩, [64/27 0 0 2/27 -2/27⟩, [79/27 0 0 5/27 -5/27⟩, [79/27 0 0 -22/27 22/27⟩]
Eigenmonzos (unchanged-intervals): 2, 11/7

Optimal ET sequence: 25e, 31, 87, 118

Badness: 0.019003

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 385/384, 625/624

Mapping: [⟨1 4 2 2 7 0], ⟨0 -15 2 5 -22 23]]

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

Optimal ET sequence: 31, 56, 87, 118, 205d

Badness: 0.021738

Spell

Return to the map

See also: Magic family

Subgroup: 2.3.5.7

Comma list: 49/48, 3125/3072

Mapping: [⟨1 0 2 2], ⟨0 10 2 5]]

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 189.927

Optimal ET sequence: 6, 19, 82dd

Badness: 0.080958

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55, 125/121

Mapping: [⟨1 0 2 2 3], ⟨0 10 2 5 3]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 190.285

Optimal ET sequence: 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: [⟨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: 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: [⟨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: 19, 44de, 63dee, 82ddee

Badness: 0.041603

Weak extensions

Semisept

For the 5-limit version of this temperament, see High badness temperaments #Semisept.

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

Mapping: [⟨1 12 6 12], ⟨0 -17 -6 -15]]

mapping generators: ~2, ~75/49

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

Optimal ET sequence: 18, 31, 80, 111

Badness: 0.050472

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 1331/1323

Mapping: [⟨1 12 6 12 20], ⟨0 -17 -6 -15 -27]]

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

Optimal ET sequence: 18e, 31, 80, 111, 364cd

Badness: 0.022476

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 540/539, 1375/1372

Mapping: [⟨1 12 6 12 20 -11], ⟨0 -17 -6 -15 -27 24]]

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

Optimal ET sequence: 31, 80, 111

Badness: 0.025204

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 176/175, 256/255, 351/350, 640/637, 715/714

Mapping: [⟨1 12 6 12 20 -11 -10], ⟨0 -17 -6 -15 -27 24 23]]

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

Optimal ET sequence: 31, 80, 111

Badness: 0.019919

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 176/175, 286/285, 351/350, 476/475, 540/539, 1331/1323

Mapping: [⟨1 12 6 12 20 -11 -10 -8], ⟨0 -17 -6 -15 -27 24 23 20]]

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

Optimal ET sequence: 31, 80, 111

Badness: 0.016301

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 176/175, 253/252, 286/285, 345/343, 351/350, 391/390, 460/459

Mapping: [⟨1 12 6 12 20 -11 -10 -8 18], ⟨0 -17 -6 -15 -27 24 23 20 -22]]

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

Optimal ET sequence: 31, 80, 111, 191cdh, 302cdgh

Badness: 0.014957

Semishly

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 176/175, 196/195, 275/273

Mapping: [⟨1 12 6 12 20 8], ⟨0 -17 -6 -15 -27 -7]]

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

Optimal ET sequence: 31, 49f, 80f

Badness: 0.028408

Emka

For the 5-limit version of this temperament, see High badness temperaments #Emka.

Emka tempers out [-50 -8 27⟩ in the 5-limit. This temperament can be described as 37 & 50 temperament, which tempers out the hemimean and 84035/82944 (quinzo-ayo). Alternative extension emkay (87 & 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

Comma list: 3136/3125, 84035/82944

Mapping: [⟨1 14 6 12], ⟨0 -27 -8 -20]]

mapping generators: ~2, ~48/35

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

Optimal ET sequence: 37, 50, 87, 137d, 224d

Badness: 0.144338

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2401/2376, 3136/3125

Mapping: [⟨1 14 6 12 3], ⟨0 -27 -8 -20 1]]

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

Optimal ET sequence: 37, 50, 87, 224d, 311d

Badness: 0.054744

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 364/363, 385/384, 625/624

Mapping: [⟨1 14 6 12 3 6], ⟨0 -27 -8 -20 1 -5]]

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

Optimal ET sequence: 37, 50, 87, 224d, 311d, 398d

Badness: 0.029741

Decipentic

The generator for the decipentic temperament (43 & 56) is the tenth root of the 5th harmonic (5/1), 51/10, 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 scales of 9, 13, 17, 30, 43 or 56 notes are available.

Subgroup: 2.3.5.7

Comma list: 3136/3125, 2097152/2083725

Mapping: [⟨1 6 0 -3], ⟨0 -19 10 25]]

Optimal tuning (POTE): ~2 = 1\1, ~75/64 = 278.800

Optimal ET sequence: 13, 43, 56, 99

Badness: 0.087325

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 1344/1331, 3136/3125

Mapping: [⟨1 6 0 -3 3], ⟨0 -19 10 25 2]]

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

Optimal ET sequence: 13, 43, 56, 99e

Badness: 0.061413

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 441/440, 832/825, 975/968

Mapping: [⟨1 6 0 -3 3 3], ⟨0 -19 10 25 2 3]]

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

Optimal ET sequence: 13, 43, 56, 99e

Badness: 0.047611

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 256/255, 273/272, 375/374

Mapping: [⟨1 6 0 -3 3 3 2], ⟨0 -19 10 25 2 3 9]]

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

Optimal ET sequence: 13, 43, 56, 99e

Badness: 0.031191

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 210/209, 221/220, 256/255, 273/272, 286/285

Mapping: [⟨1 6 0 -3 3 3 2 1], ⟨0 -19 10 25 2 3 9 14]]

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

Optimal ET sequence: 13, 43, 56, 99e

Badness: 0.023899

Quasijerome

Subgroup: 2.3.5.7.11

Comma list: 3136/3125, 15488/15435, 16384/16335

Mapping: [⟨1 6 0 -3 3], ⟨0 -38 20 50 47]]

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

Optimal ET sequence: 43, 112, 155, 198, 439cd, 637cd

Badness: 0.092996

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 3136/3125, 15488/15435

Mapping: [⟨1 6 0 -3 3 8], ⟨0 -38 20 50 47 -37]]

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

Optimal ET sequence: 43, 155, 198, 439cdf, 637cdf

Badness: 0.044328

Sengagen

Subgroup: 2.3.5.7

Comma list: 3136/3125, 420175/419904

Mapping: [⟨1 1 2 2], ⟨0 29 16 40]]

Optimal tuning (POTE): ~2 = 1\1, ~686/675 = 24.217

Optimal ET sequence: 49, 50, 99, 248, 347, 446

Badness: 0.057978

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1344/1331, 3136/3125

Mapping: [⟨1 1 2 2 3], ⟨0 29 16 40 23]]

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

Optimal ET sequence: 49, 50, 99e

Badness: 0.053828

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 975/968, 1344/1331

Mapping: [⟨1 1 2 2 3 4], ⟨0 29 16 40 23 -15]]

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

Optimal ET sequence: 49, 50, 99e, 149e

Badness: 0.053531

Sengage

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 364/363, 625/624

Mapping: [⟨1 1 2 2 3 3], ⟨0 29 16 40 23 35]]

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

Optimal ET sequence: 49f, 50, 99ef

Badness: 0.037416

Mowglic

The mowglic temperament (19 & 161) is an extension of the mowgli temperament which tempers out the hemimean comma and the secanticornisma (177147/175000, laruquingu) in the 7-limit.

Subgroup: 2.3.5.7

Comma list: 3136/3125, 177147/175000

Mapping: [⟨1 0 0 -3], ⟨0 15 22 55]]

Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 126.706

Optimal ET sequence: 19, 123d, 142, 161

Badness: 0.129915

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 3136/3125, 72171/71680

Mapping: [⟨1 0 0 -3 8], ⟨0 15 22 55 -43]]

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

Optimal ET sequence: 19, 123de, 142, 161

Badness: 0.094032

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 1701/1690, 3136/3125

Mapping: [⟨1 0 0 -3 8 -2], ⟨0 15 22 55 -43 54]]

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

Optimal ET sequence: 19, 123def, 142f, 161

Badness: 0.051571

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 540/539, 833/832, 1701/1690, 3136/3125

Mapping: [⟨1 0 0 -3 8 -2 10], ⟨0 15 22 55 -43 54 -56]]

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

Optimal ET sequence: 19, 123defg, 142f, 161

Badness: 0.041918

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 0 0 -3 8 -2 10 9], ⟨0 15 22 55 -43 54 -56 -45]]

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

Optimal ET sequence: 19, 123defg, 142f, 161

Badness: 0.032168

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 276/275, 351/350, 476/475, 495/494, 513/512, 529/528, 540/539

Mapping: [⟨1 0 0 -3 8 -2 10 9 6], ⟨0 15 22 55 -43 54 -56 -45 -14]]

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

Optimal ET sequence: 19, 123defg, 142f, 161

Badness: 0.026117

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

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

Mapping: [⟨1 0 0 -3 8 -2 10 9 6 0], ⟨0 15 22 55 -43 54 -56 -45 -14 46]]

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

Optimal ET sequence: 19, 123defg, 142f, 161

Badness: 0.021398

31-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31

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

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

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

Optimal ET sequence: 19, 123defgk, 142fk, 161

Badness: 0.019331

Tremka

The name tremka was initially used for the no-sevens version of 50 & 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: [⟨1 -4 -2 -8], ⟨0 31 24 60]]

Optimal tuning (POTE): ~2 = 1\1, ~4375/3888 = 216.173

Optimal ET sequence: 50, 111, 161, 272

Badness: 0.179925

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 3136/3125, 35937/35840

Mapping: [⟨1 -4 -2 -8 4], ⟨0 31 24 60 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~112/99 = 216.168

Optimal ET sequence: 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: [⟨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: 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: [⟨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: 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: [⟨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: 50, 111, 161, 272h, 433cfh, 705ccdffhh

Badness: 0.016900

Undetrita

For the 5-limit version, see Syntonic–chromatic equivalence continuum #Undetrita (5-limit).

The undetrita temperament (111 & 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 twentcufo temperament, which is no-sevens version of 111 & 118.

Subgroup: 2.3.5.7

Comma list: 3136/3125, 14348907/14336000

Mapping: [⟨1 0 -2 -8], ⟨0 11 30 75]]

Optimal tuning (POTE): ~2 = 1\1, ~448/405 = 172.917

Optimal ET sequence: 111, 118, 229, 347, 576c

Badness: 0.114188

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 3136/3125, 8019/8000

Mapping: [⟨1 0 -2 -8 0], ⟨0 11 30 75 24]]

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

Optimal ET sequence: 111, 118, 229, 347

Badness: 0.043883

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 729/728, 1001/1000, 3025/3024

Mapping: [⟨1 0 -2 -8 0 5], ⟨0 11 30 75 24 -9]]

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

Optimal ET sequence: 111, 229f

Badness: 0.038771

Undetritoid

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 1573/1568, 2080/2079, 3136/3125

Mapping: [⟨1 0 -2 -8 0 -11], ⟨0 11 30 75 24 102]]

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

Optimal ET sequence: 111, 229

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)3. In the no-3's 19-limit extension "mediantone", this whole tone generator serves as the two simplest mediants 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: [⟨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: 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 ¢ flat, it makes sense to equate two of it with ~13/10 (tempering out the 8 ¢ 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: [⟨1 0 -3 -7 13], ⟨0 2 5 9 -8]]

sval mapping generators: ~2, ~56/25

Gencom 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: 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)2/2. 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: [⟨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: 6h, 31gh, 37, 80, 117d*

* 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)2 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)3 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: [⟨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: 6g, ... 31, 37, 68, 105

Badness (Sintel): 0.676

Rectified hebrew

Main article: Rectified hebrew

Rectified hebrew (37 & 56) is derived from the 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.

Subgroup: 2.5.7.13

Comma list: 3136/3125, 4394/4375

Sval 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: 18, 19, 37, 93, 130

Isra

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

Comma list: 81/80, 126/125

Subgroup-val mapping: [⟨1 0 -4 -13], ⟨0 1 2 5]]

sval mapping generators: ~2, ~9

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

Optimal ET sequence: 6, 19, 25, 31, 56b, 87b

Tutone

Tutone is every other step of 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

Subgroup-val mapping: [⟨1 0 -4 -13 -25], ⟨0 1 2 5 9]]

Gencom mapping: [⟨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: 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

Subgroup-val mapping: [⟨1 0 -4 -13 -6], ⟨0 1 2 5 3]]

Gencom mapping: [⟨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: 6, 7, 13, 19, 25e, 31e, 56bee, 81beee

RMS error: 3.882 cents

Deutone

See also: Chromatic pairs § Deutone

Deutone is (also) every other step of vincenzo.

Subgroup: 2.9.5.7.13

Comma list: 65/64, 81/80, 91/90

Subgroup-val mapping: [⟨1 0 -4 -13 10], ⟨0 1 2 5 -2]]

Gencom mapping: [⟨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: 6, 7, 13, 19, 25f, 44df

RMS error: 2.003 cents

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