Hemimean clan: Difference between revisions

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~~<div style="float: right;">~~

[[File:Didacus.png|alt=Didacus.png|thumb|600x560px|A chart of the tuning spectrum of undecimal didacus, showing the offsets of prime harmonics 5, 7, and 11, as a function of the generator; all EDO tunings are shown with vertical lines whose length indicates the EDO's tolerance, i.e. half of its step size in either direction of just, and some small EDOs supporting the temperament are labeled.]]

~~</div>~~

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]].

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* ''[[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) → [[~~Pelogic~~ family #Mohavila|~~Pelogic~~ 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]]

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* ''[[Bischismic]]'' (+32805/32768) → [[Schismatic family #Bischismic|Schismatic family]]

* ''[[Clyde]]'' (+245/243) → [[Kleismic family #Clyde|Kleismic family]]

* [[Parakleismic]] (+4375/4374) → [[~~Ragismic microtemperaments~~ #Parakleismic|~~Ragismic microtemperaments~~]]

* [[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~~]]

* ''[[Decistearn]]'' (+118098/117649) → [[Trisedodge family #Decistearn|Trisedodge family]]

* ''[[Quintagar]]'' (+33554432/33480783) → [[Quindromeda family #Quintagar|Quindromeda family]]

* ''[[Rubidium]]'' (+4194304/4117715) → [[37th-octave temperaments]]

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== Didacus ==

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

[[Subgroup]]: 2.5.7

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

[[Badness]] (~~Dirichlet~~): 0.091

[[Badness]] (Sintel): 0.091

= Strong extensions =

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|-

! rowspan="2" | Extension !! colspan="2" | 5-limit re-restriction !! rowspan="2" | Mapping of 3 !! rowspan="2" | Tuning range*

|-

! Temperament !! 5-limit generator location

|-

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== Hemiwürschmidt ==

''[[#Strong extensions|Return to the map]]''

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Mapping generators: ~2, ~25/14

~~{{Multival|legend=1| 16 2 5 -34 -37 6 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.898

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~~{{todo|verify this & get badness|inline=1}}~~

Badness (Sintel): 0.304

=== 11-limit ===

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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~~, 650ce~~, 811ce }}

Badness: 0.021069

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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

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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

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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

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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

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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

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=== 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

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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

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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

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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

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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 }}

{{Mapping|legend=0| 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

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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 }}

{{Mapping|legend=0| 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

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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 }}

{{Mapping|legend=0| 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

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

~~Mapping:~~ {{~~mapping~~| 2 14 6 9 -10 25 -4 -3 | 0 -16 -2 -5 25 -26 18 17 }}

{{Mapping|legend=0| 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

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== Hemithirds ==

''[[#Strong extensions|Return to the map]]''

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~~{{Multival|legend=1| 15 -2 -5 -38 -50 -6 }}~~

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

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* [[7-odd-limit]]: ~28/25 = {{monzo| 1/10 -1/20 0 1/20 }}

: {{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 }}

: [[Eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.7/3

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3

* [[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|~~eigenmonzo (~~unchanged-interval) basis]]: 2.9/7

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7

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

~~Mapping:~~ {{~~mapping~~| 1 4 2 2 7 | 0 -15 2 5 -22 }}

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

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

~~Mapping:~~ {{~~mapping~~| 1 4 2 2 7 0 | 0 -15 2 5 -22 23 }}

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

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== Spell ==

''[[#Strong extensions|Return to the map]]''

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~~{{Multival|legend=1| 10 2 5 -20 -20 6 }}~~

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

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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

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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

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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

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: mapping generators: ~2, ~75/49

~~{{Multival|legend=1| 17 6 15 -30 -24 18 }}~~

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

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

~~Mapping:~~ {{~~mapping~~| 1 12 6 12 20 | 0 -17 -6 -15 -27 }}

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

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

~~Mapping:~~ {{~~mapping~~| 1 12 6 12 20 -11 | 0 -17 -6 -15 -27 24 }}

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

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

~~Mapping:~~ {{~~mapping~~| 1 12 6 12 20 -11 -10 | 0 -17 -6 -15 -27 24 23 }}

{{Mapping|legend=0| 1 12 6 12 20 -11 -10 | 0 -17 -6 -15 -27 24 23 }}

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

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

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

{{Mapping|legend=0| 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

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

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

{{Mapping|legend=0| 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

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

~~Mapping:~~ {{~~mapping~~| 1 12 6 12 20 8 | 0 -17 -6 -15 -27 -7 }}

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

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: mapping generators: ~2, ~48/35

~~{{Multival|legend=1| 27 8 20 -50 -44 24 }}~~

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

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

~~Mapping:~~ {{~~mapping~~| 1 14 6 12 3 | 0 -27 -8 -20 1 }}

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

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

~~Mapping:~~ {{~~mapping~~| 1 14 6 12 3 6 | 0 -27 -8 -20 1 -5 }}

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

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~~{{Multival|legend=1| 19 -10 -25 -60 -93 -30 }}~~

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

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

~~Mapping:~~ {{~~mapping~~| 1 6 0 -3 3 | 0 -19 10 25 2 }}

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

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

~~Mapping:~~ {{~~mapping~~| 1 6 0 -3 3 3 | 0 -19 10 25 2 3 }}

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

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

~~Mapping:~~ {{~~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

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

~~Mapping:~~ {{~~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

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

~~Mapping:~~ {{~~mapping~~| 1 6 0 -3 3 | 0 -38 20 50 47 }}

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

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

~~Mapping:~~ {{~~mapping~~| 1 6 0 -3 3 8 | 0 -38 20 50 47 -37 }}

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

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~~{{Multival|legend=1| 29 16 40 -42 -18 48 }}~~

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

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

~~Mapping:~~ {{~~mapping~~| 1 1 2 2 3 | 0 29 16 40 23 }}

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

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

~~Mapping:~~ {{~~mapping~~| 1 1 2 2 3 4 | 0 29 16 40 23 -15 }}

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

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

~~Mapping:~~ {{~~mapping~~| 1 1 2 2 3 3 | 0 29 16 40 23 35 }}

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

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== Mowglic ==

The mowglic temperament (19 & 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 & 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

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~~{{Multival|legend=1| 15 22 55 0 45 66 }}~~

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

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

~~Mapping:~~ {{~~mapping~~| 1 0 0 -3 8 | 0 15 22 55 -43 }}

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

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

~~Mapping:~~ {{~~mapping~~| 1 0 0 -3 8 -2 | 0 15 22 55 -43 54 }}

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

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

~~Mapping:~~ {{~~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

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

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

{{Mapping|legend=0| 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

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

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

{{Mapping|legend=0| 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

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

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

{{Mapping|legend=0| 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

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

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

{{Mapping|legend=0| 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

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~~{{Multival|legend=1| 31 24 60 -34 8 72 }}~~

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

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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

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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

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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

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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 }}

{{Mapping|legend=0| 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

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== Undetrita ==

The undetrita temperament (111 & 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]], 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 & 118.

: ''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 [[Subgroup temperaments #No-sevens subgroup|twentcufo temperament]], which is no-sevens version of 111 & 118.

[[Subgroup]]: 2.3.5.7

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~~{{Multival|legend=1| 11 30 75 22 88 90 }}~~

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

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

~~Mapping:~~ {{~~mapping~~| 1 0 -2 -8 0 | 0 11 30 75 24 }}

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

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

~~Mapping:~~ {{~~mapping~~| 1 0 -2 -8 0 5 | 0 11 30 75 24 -9 }}

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

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

~~Mapping:~~ {{~~mapping~~| 1 0 -2 -8 0 -11 | 0 11 30 75 24 102 }}

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

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RMS error: 0.5567 cents

Badness (~~Dirichlet~~): 0.195

Badness (Sintel): 0.195

=== Tridecimal didacus ===

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Optimal ET sequence: {{Optimal ET sequence| 6, 25, 31, 37 }}

Badness (~~Dirichlet~~): 0.324

Badness (Sintel): 0.324

==== Mediantone ====

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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~~

~~Comma list: [[176/175]], [[640/637]], [[221/220]], [[1375/1372]]~~

~~Sval mapping: {{mapping| 1 0 -3 -7 13 -18 | 0 2 5 9 -8 19 }}~~

~~: sval mapping generators: ~2, ~56/25~~

~~Optimal tuning (CWE): ~2 = 1\1, ~28/25 = 194.887~~

~~Optimal ET sequence: {{Optimal ET sequence| 6h, 31gh, 37, 80, 117d }}~~

~~Badness (Dirichlet): 0.612~~

~~===== 2.5.7.11.13.17.19 subgroup =====~~

Subgroup: 2.5.7.11.13.17.19

Line 998:

Line 972:

Optimal tuning (CWE): ~2 = 1\1, ~19/17 = 194.927

Optimal ET sequence: {{Optimal ET sequence| 6h, 31gh, 37, 80 }}

Optimal ET sequence: {{Optimal ET sequence| 6h, 31gh, 37, 80, 117d* }}

Badness (~~Dirichlet~~): 0.618

<nowiki />* 117d only appears without prime 19

Badness (Sintel): 0.618

==== Roulette ====

Line 1,007:

Line 983:

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~~

~~Comma list: [[170/169]], [[176/175]], [[640/637]], [[1375/1372]]~~

~~Sval mapping: {{mapping| 1 2 2 2 5 7 | 0 2 5 9 -8 -18 }}~~

~~: sval mapping generators: ~2, ~28/25~~

~~Optimal tuning (CWE): ~2 = 1\1, ~28/25 = 194.285~~

~~Optimal ET sequence: {{Optimal ET sequence| 6g, ... 31, 37, 68, 105 }}~~

~~Badness (Dirichlet): 0.685~~

~~===== 2.5.7.11.13.17.19 subgroup =====~~

Subgroup: 2.5.7.11.13.17.19

Line 1,034:

Line 995:

Optimal ET sequence: {{Optimal ET sequence| 6g, ... 31, 37, 68, 105 }}

Badness (~~Dirichlet~~): 0.676

Badness (Sintel): 0.676

== Rectified hebrew ==

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 adds harmonic 13 through tempering out [[4394/4375]] and spliting the generator of didacus in three.

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 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

Line 1,054:

Line 1,015:

== Isra ==

Isra 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 ~~(''iss-RAH'')~~ night journey in the Qur'an, because it is similar to [[luna]] (septimal [[hemithirds]], a didacus extension).

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

Line 1,129:

Line 1,090:

[[Tp tuning #T2 tuning|RMS error]]: 2.003 cents

~~[[Category:Temperament clans]]~~

[[Category:Hemimean clan| ]]

[[Category:Hemimean| ]]

[[Category:~~Rank~~ 2]]

[[Category:Temperament clans]]

[[Category:Catalogs of rank-2 temperaments]]

@@ Line 1: / Line 1: @@
 {{Technical data page}}
-<div style="float: right;">
-[[File:Didacus.png|alt=Didacus.png|thumb|600x560px|A chart of the tuning spectrum of undecimal didacus, showing the offsets of prime harmonics 5, 7, and 11, as a function of the generator; all EDO tunings are shown with vertical lines whose length indicates the EDO's tolerance, i.e. half of its step size in either direction of just, and some small EDOs supporting the temperament are labeled.]]
-</div>
 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]].
@@ Line 15: / Line 11: @@
 * ''[[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) → [[Pelogic family #Mohavila|Pelogic 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]]
@@ Line 23: / Line 19: @@
 * ''[[Bischismic]]'' (+32805/32768) → [[Schismatic family #Bischismic|Schismatic family]]
 * ''[[Clyde]]'' (+245/243) → [[Kleismic family #Clyde|Kleismic family]]
-* [[Parakleismic]] (+4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
+* [[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]]
+* ''[[Decistearn]]'' (+118098/117649) → [[Trisedodge family #Decistearn|Trisedodge family]]
 * ''[[Quintagar]]'' (+33554432/33480783) → [[Quindromeda family #Quintagar|Quindromeda family]]
 * ''[[Rubidium]]'' (+4194304/4117715) → [[37th-octave temperaments]]
@@ Line 34: / Line 30: @@
 == Didacus ==
 {{main|Didacus}}
+See also its canonical extension to the 2.5.7.11 subgroup, [[#Undecimal didacus]].
 [[Subgroup]]: 2.5.7
@@ Line 53: / Line 51: @@
 [[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
-[[Badness]] (Dirichlet): 0.091
+[[Badness]] (Sintel): 0.091
 = Strong extensions =
@@ Line 60: / Line 58: @@
 |-
 ! rowspan="2" | Extension !! colspan="2" | 5-limit re-restriction !! rowspan="2" | Mapping of 3 !! rowspan="2" | Tuning range*
+|-
 ! Temperament !! 5-limit generator location
 |-
@@ Line 71: / Line 70: @@
 == Hemiwürschmidt ==
+''[[#Strong extensions|Return to the map]]''
 {{See also| Würschmidt family }}
@@ Line 82: / Line 83: @@
 Mapping generators: ~2, ~25/14
-{{Multival|legend=1| 16 2 5 -34 -37 6 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.898
@@ Line 106: / Line 105: @@
 {{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328 }}
-{{todo|verify this & get badness|inline=1}}
+Badness (Sintel): 0.304
 === 11-limit ===
@@ Line 113: / Line 112: @@
 Comma list: 243/242, 441/440, 3136/3125
-Mapping: {{mapping| 1 15 4 7 37 | 0 -16 -2 -5 -40 }}
+{{Mapping|legend=0| 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, 650ce, 811ce }}
+{{Optimal ET sequence|legend=1| 31, 99e, 130, 811ce }}
 Badness: 0.021069
@@ Line 126: / Line 125: @@
 Comma list: 243/242, 351/350, 441/440, 3584/3575
-Mapping: {{mapping| 1 15 4 7 37 -29 | 0 -16 -2 -5 -40 39 }}
+{{Mapping|legend=0| 1 15 4 7 37 -29 | 0 -16 -2 -5 -40 39 }}
 Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.829
@@ Line 139: / Line 138: @@
 Comma list: 121/120, 176/175, 196/195, 275/273
-Mapping: {{mapping| 1 15 4 7 37 -3 | 0 -16 -2 -5 -40 8 }}
+{{Mapping|legend=0| 1 15 4 7 37 -3 | 0 -16 -2 -5 -40 8 }}
 Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.918
@@ Line 152: / Line 151: @@
 Comma list: 121/120, 176/175, 1375/1372
-Mapping: {{mapping| 1 15 4 7 11 | 0 -16 -2 -5 -9 }}
+{{Mapping|legend=0| 1 15 4 7 11 | 0 -16 -2 -5 -9 }}
 Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.884
@@ Line 165: / Line 164: @@
 Comma list: 121/120, 176/175, 196/195, 275/273
-Mapping: {{mapping| 1 15 4 7 11 -3 | 0 -16 -2 -5 -9 8 }}
+{{Mapping|legend=0| 1 15 4 7 11 -3 | 0 -16 -2 -5 -9 8 }}
 Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 194.004
@@ Line 178: / Line 177: @@
 Comma list: 66/65, 105/104, 121/120, 1375/1372
-Mapping: {{mapping| 1 15 4 7 11 23 | 0 -16 -2 -5 -9 -23 }}
+{{Mapping|legend=0| 1 15 4 7 11 23 | 0 -16 -2 -5 -9 -23 }}
 Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.698
@@ Line 188: / Line 187: @@
 === 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
@@ Line 193: / Line 194: @@
 Comma list: 2401/2400, 3025/3024, 3136/3125
-Mapping: {{mapping| 1 15 4 7 24 | 0 -32 -4 -10 -49 }}
+{{Mapping|legend=0| 1 15 4 7 24 | 0 -32 -4 -10 -49 }}
 : mapping generators: ~2, ~147/110
@@ Line 208: / Line 209: @@
 Comma list: 2401/2400, 3136/3125, 9801/9800
-Mapping: {{mapping| 2 14 6 9 -10 | 0 -16 -2 -5 25 }}
+{{Mapping|legend=0| 2 14 6 9 -10 | 0 -16 -2 -5 25 }}
 : mapping generators: ~99/70, ~495/392
@@ Line 223: / Line 224: @@
 Comma list: 676/675, 1001/1000, 1716/1715, 3136/3125
-Mapping: {{mapping| 2 14 6 9 -10 25 | 0 -16 -2 -5 25 -26 }}
+{{Mapping|legend=0| 2 14 6 9 -10 25 | 0 -16 -2 -5 25 -26 }}
 Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9035
@@ Line 236: / Line 237: @@
 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 }}
+{{Mapping|legend=0| 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
@@ Line 249: / Line 250: @@
 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 }}
+{{Mapping|legend=0| 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
@@ Line 262: / Line 263: @@
 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 }}
+{{Mapping|legend=0| 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
@@ Line 275: / Line 276: @@
 Comma list: 256/255, 286/285, 400/399, 476/475, 495/494, 1225/1224
-Mapping: {{mapping| 2 14 6 9 -10 25 -4 -3 | 0 -16 -2 -5 25 -26 18 17 }}
+{{Mapping|legend=0| 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
@@ Line 284: / Line 285: @@
 == Hemithirds ==
+''[[#Strong extensions|Return to the map]]''
 {{Main| Hemithirds }}
@@ Line 291: / Line 294: @@
 {{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
@@ Line 299: / Line 300: @@
 * [[7-odd-limit]]: ~28/25 = {{monzo| 1/10 -1/20 0 1/20 }}
 : {{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 }}
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/3
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
 * [[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|eigenmonzo (unchanged-interval) basis]]: 2.9/7
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7
 {{Optimal ET sequence|legend=1| 25, 31, 87, 118 }}
@@ Line 313: / Line 314: @@
 Comma list: 385/384, 441/440, 3136/3125
-Mapping: {{mapping| 1 4 2 2 7 | 0 -15 2 5 -22 }}
+{{Mapping|legend=0| 1 4 2 2 7 | 0 -15 2 5 -22 }}
 Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.227
@@ Line 331: / Line 332: @@
 Comma list: 196/195, 352/351, 385/384, 625/624
-Mapping: {{mapping| 1 4 2 2 7 0 | 0 -15 2 5 -22 23 }}
+{{Mapping|legend=0| 1 4 2 2 7 0 | 0 -15 2 5 -22 23 }}
 Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.166
@@ Line 340: / Line 341: @@
 == Spell ==
+''[[#Strong extensions|Return to the map]]''
 {{See also| Magic family }}
@@ Line 347: / Line 350: @@
 {{Mapping|legend=1| 1 0 2 2 | 0 10 2 5 }}
-{{Multival|legend=1| 10 2 5 -20 -20 6 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 189.927
@@ Line 361: / Line 362: @@
 Comma list: 49/48, 56/55, 125/121
-Mapping: {{mapping| 1 0 2 2 3 | 0 10 2 5 3 }}
+{{Mapping|legend=0| 1 0 2 2 3 | 0 10 2 5 3 }}
 Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 190.285
@@ Line 374: / Line 375: @@
 Comma list: 49/48, 56/55, 78/77, 125/121
-Mapping: {{mapping| 1 0 2 2 3 4 | 0 10 2 5 3 -2 }}
+{{Mapping|legend=0| 1 0 2 2 3 4 | 0 10 2 5 3 -2 }}
 Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 189.928
@@ Line 387: / Line 388: @@
 Comma list: 49/48, 56/55, 91/90, 125/121
-Mapping: {{mapping| 1 0 2 2 3 1 | 0 10 2 5 3 17 }}
+{{Mapping|legend=0| 1 0 2 2 3 1 | 0 10 2 5 3 17 }}
 Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 190.360
@@ Line 409: / Line 410: @@
 : mapping generators: ~2, ~75/49
-{{Multival|legend=1| 17 6 15 -30 -24 18 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/49 = 735.155
@@ Line 423: / Line 422: @@
 Comma list: 176/175, 540/539, 1331/1323
-Mapping: {{mapping| 1 12 6 12 20 | 0 -17 -6 -15 -27 }}
+{{Mapping|legend=0| 1 12 6 12 20 | 0 -17 -6 -15 -27 }}
 Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.125
@@ Line 436: / Line 435: @@
 Comma list: 176/175, 351/350, 540/539, 1375/1372
-Mapping: {{mapping| 1 12 6 12 20 -11 | 0 -17 -6 -15 -27 24 }}
+{{Mapping|legend=0| 1 12 6 12 20 -11 | 0 -17 -6 -15 -27 24 }}
 Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.126
@@ Line 449: / Line 448: @@
 Comma list: 176/175, 256/255, 351/350, 640/637, 715/714
-Mapping: {{mapping| 1 12 6 12 20 -11 -10 | 0 -17 -6 -15 -27 24 23 }}
+{{Mapping|legend=0| 1 12 6 12 20 -11 -10 | 0 -17 -6 -15 -27 24 23 }}
 Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.125
@@ Line 462: / Line 461: @@
 Comma list: 176/175, 286/285, 351/350, 476/475, 540/539, 1331/1323
-Mapping: {{mapping| 1 12 6 12 20 -11 -10 -8 | 0 -17 -6 -15 -27 24 23 20 }}
+{{Mapping|legend=0| 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
@@ Line 475: / Line 474: @@
 Comma list: 176/175, 253/252, 286/285, 345/343, 351/350, 391/390, 460/459
-Mapping: {{mapping| 1 12 6 12 20 -11 -10 -8 18 | 0 -17 -6 -15 -27 24 23 20 -22 }}
+{{Mapping|legend=0| 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
@@ Line 488: / Line 487: @@
 Comma list: 144/143, 176/175, 196/195, 275/273
-Mapping: {{mapping| 1 12 6 12 20 8 | 0 -17 -6 -15 -27 -7 }}
+{{Mapping|legend=0| 1 12 6 12 20 8 | 0 -17 -6 -15 -27 -7 }}
 Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 464.980
@@ Line 508: / Line 507: @@
 : mapping generators: ~2, ~48/35
-{{Multival|legend=1| 27 8 20 -50 -44 24 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 551.782
@@ Line 522: / Line 519: @@
 Comma list: 385/384, 2401/2376, 3136/3125
-Mapping: {{mapping| 1 14 6 12 3 | 0 -27 -8 -20 1 }}
+{{Mapping|legend=0| 1 14 6 12 3 | 0 -27 -8 -20 1 }}
 Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.765
@@ Line 535: / Line 532: @@
 Comma list: 196/195, 364/363, 385/384, 625/624
-Mapping: {{mapping| 1 14 6 12 3 6 | 0 -27 -8 -20 1 -5 }}
+{{Mapping|legend=0| 1 14 6 12 3 6 | 0 -27 -8 -20 1 -5 }}
 Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.758
@@ Line 551: / Line 548: @@
 {{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
@@ Line 565: / Line 560: @@
 Comma list: 441/440, 1344/1331, 3136/3125
-Mapping: {{mapping| 1 6 0 -3 3 | 0 -19 10 25 2 }}
+{{Mapping|legend=0| 1 6 0 -3 3 | 0 -19 10 25 2 }}
 Optimal tuning (POTE): ~2 = 1\1, ~75/64 = 278.799
@@ Line 578: / Line 573: @@
 Comma list: 169/168, 441/440, 832/825, 975/968
-Mapping: {{mapping| 1 6 0 -3 3 3 | 0 -19 10 25 2 3 }}
+{{Mapping|legend=0| 1 6 0 -3 3 3 | 0 -19 10 25 2 3 }}
 Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.802
@@ Line 591: / Line 586: @@
 Comma list: 169/168, 221/220, 256/255, 273/272, 375/374
-Mapping: {{mapping| 1 6 0 -3 3 3 2 | 0 -19 10 25 2 3 9 }}
+{{Mapping|legend=0| 1 6 0 -3 3 3 2 | 0 -19 10 25 2 3 9 }}
 Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.798
@@ Line 604: / Line 599: @@
 Comma list: 169/168, 210/209, 221/220, 256/255, 273/272, 286/285
-Mapping: {{mapping| 1 6 0 -3 3 3 2 1 | 0 -19 10 25 2 3 9 14 }}
+{{Mapping|legend=0| 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
@@ Line 617: / Line 612: @@
 Comma list: 3136/3125, 15488/15435, 16384/16335
-Mapping: {{mapping| 1 6 0 -3 3 | 0 -38 20 50 47 }}
+{{Mapping|legend=0| 1 6 0 -3 3 | 0 -38 20 50 47 }}
 Optimal tuning (POTE): ~2 = 1\1, ~896/825 = 139.403
@@ Line 630: / Line 625: @@
 Comma list: 676/675, 1001/1000, 3136/3125, 15488/15435
-Mapping: {{mapping| 1 6 0 -3 3 8 | 0 -38 20 50 47 -37 }}
+{{Mapping|legend=0| 1 6 0 -3 3 8 | 0 -38 20 50 47 -37 }}
 Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 139.403
@@ Line 644: / Line 639: @@
 {{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
@@ Line 658: / Line 651: @@
 Comma list: 540/539, 1344/1331, 3136/3125
-Mapping: {{mapping| 1 1 2 2 3 | 0 29 16 40 23 }}
+{{Mapping|legend=0| 1 1 2 2 3 | 0 29 16 40 23 }}
 Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.235
@@ Line 671: / Line 664: @@
 Comma list: 351/350, 540/539, 975/968, 1344/1331
-Mapping: {{mapping| 1 1 2 2 3 4 | 0 29 16 40 23 -15 }}
+{{Mapping|legend=0| 1 1 2 2 3 4 | 0 29 16 40 23 -15 }}
 Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.181
@@ Line 684: / Line 677: @@
 Comma list: 144/143, 196/195, 364/363, 625/624
-Mapping: {{mapping| 1 1 2 2 3 3 | 0 29 16 40 23 35 }}
+{{Mapping|legend=0| 1 1 2 2 3 3 | 0 29 16 40 23 35 }}
 Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.234
@@ Line 693: / Line 686: @@
 == 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
@@ Line 700: / Line 693: @@
 {{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
@@ Line 714: / Line 705: @@
 Comma list: 540/539, 3136/3125, 72171/71680
-Mapping: {{mapping| 1 0 0 -3 8 | 0 15 22 55 -43 }}
+{{Mapping|legend=0| 1 0 0 -3 8 | 0 15 22 55 -43 }}
 Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 126.711
@@ Line 727: / Line 718: @@
 Comma list: 351/350, 540/539, 1701/1690, 3136/3125
-Mapping: {{mapping| 1 0 0 -3 8 -2 | 0 15 22 55 -43 54 }}
+{{Mapping|legend=0| 1 0 0 -3 8 -2 | 0 15 22 55 -43 54 }}
 Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.705
@@ Line 740: / Line 731: @@
 Comma list: 351/350, 540/539, 833/832, 1701/1690, 3136/3125
-Mapping: {{mapping| 1 0 0 -3 8 -2 10 | 0 15 22 55 -43 54 -56 }}
+{{Mapping|legend=0| 1 0 0 -3 8 -2 10 | 0 15 22 55 -43 54 -56 }}
 Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703
@@ Line 753: / Line 744: @@
 Comma list: 351/350, 476/475, 495/494, 513/512, 540/539, 1701/1690
-Mapping: {{mapping| 1 0 0 -3 8 -2 10 9 | 0 15 22 55 -43 54 -56 -45 }}
+{{Mapping|legend=0| 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
@@ Line 766: / Line 757: @@
 Comma list: 276/275, 351/350, 476/475, 495/494, 513/512, 529/528, 540/539
-Mapping: {{mapping| 1 0 0 -3 8 -2 10 9 6 | 0 15 22 55 -43 54 -56 -45 -14 }}
+{{Mapping|legend=0| 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
@@ Line 779: / Line 770: @@
 Comma list: 261/260, 276/275, 351/350, 476/475, 495/494, 513/512, 529/528, 540/539
-Mapping: {{mapping| 1 0 0 -3 8 -2 10 9 6 0 | 0 15 22 55 -43 54 -56 -45 -14 46 }}
+{{Mapping|legend=0| 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
@@ Line 792: / Line 783: @@
 Comma list: 261/260, 276/275, 351/350, 435/434, 476/475, 495/494, 513/512, 529/528, 540/539
-Mapping: {{mapping| 1 0 0 -3 8 -2 10 9 6 0 2 | 0 15 22 55 -43 54 -56 -45 -14 46 28 }}
+{{Mapping|legend=0| 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
@@ Line 809: / Line 800: @@
 {{Mapping|legend=1| 1 -4 -2 -8 | 0 31 24 60 }}
-{{Multival|legend=1| 31 24 60 -34 8 72 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~4375/3888 = 216.173
@@ Line 823: / Line 812: @@
 Comma list: 540/539, 3136/3125, 35937/35840
-Mapping: {{mapping| 1 -4 -2 -8 4 | 0 31 24 60 -3 }}
+{{Mapping|legend=0| 1 -4 -2 -8 4 | 0 31 24 60 -3 }}
 Optimal tuning (POTE): ~2 = 1\1, ~112/99 = 216.168
@@ Line 836: / Line 825: @@
 Comma list: 351/350, 540/539, 847/845, 3136/3125
-Mapping: {{mapping| 1 -4 -2 -8 4 1 | 0 31 24 60 -3 15 }}
+{{Mapping|legend=0| 1 -4 -2 -8 4 1 | 0 31 24 60 -3 15 }}
 Optimal tuning (POTE): ~2 = 1\1, ~112/99 = 216.172
@@ Line 849: / Line 838: @@
 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 }}
+{{Mapping|legend=0| 1 -4 -2 -8 4 1 -6 | 0 31 24 60 -3 15 56 }}
 Optimal tuning (POTE): ~2 = 1\1, ~17/15 = 216.172
@@ Line 862: / Line 851: @@
 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 }}
+{{Mapping|legend=0| 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
@@ Line 871: / Line 860: @@
 == 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]], 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.
+: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Undetrita (5-limit)]].''
+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
@@ Line 878: / Line 869: @@
 {{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
@@ Line 892: / Line 881: @@
 Comma list: 3025/3024, 3136/3125, 8019/8000
-Mapping: {{mapping| 1 0 -2 -8 0 | 0 11 30 75 24 }}
+{{Mapping|legend=0| 1 0 -2 -8 0 | 0 11 30 75 24 }}
 Optimal tuning (POTE): ~2 = 1\1, ~400/363 = 172.912
@@ Line 905: / Line 894: @@
 Comma list: 352/351, 729/728, 1001/1000, 3025/3024
-Mapping: {{mapping| 1 0 -2 -8 0 5 | 0 11 30 75 24 -9 }}
+{{Mapping|legend=0| 1 0 -2 -8 0 5 | 0 11 30 75 24 -9 }}
 Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 172.930
@@ Line 918: / Line 907: @@
 Comma list: 351/350, 1573/1568, 2080/2079, 3136/3125
-Mapping: {{mapping| 1 0 -2 -8 0 -11 | 0 11 30 75 24 102 }}
+{{Mapping|legend=0| 1 0 -2 -8 0 -11 | 0 11 30 75 24 102 }}
 Optimal tuning (POTE): ~2 = 1\1, ~400/363 = 172.933
@@ Line 945: / Line 934: @@
 RMS error: 0.5567 cents
-Badness (Dirichlet): 0.195
+Badness (Sintel): 0.195
 === Tridecimal didacus ===
@@ Line 966: / Line 955: @@
 Optimal ET sequence: {{Optimal ET sequence| 6, 25, 31, 37 }}
-Badness (Dirichlet): 0.324
+Badness (Sintel): 0.324
 ==== Mediantone ====
@@ Line 973: / Line 962: @@
 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
-Comma list: [[176/175]], [[640/637]], [[221/220]], [[1375/1372]]
-Sval mapping: {{mapping| 1 0 -3 -7 13 -18 | 0 2 5 9 -8 19 }}
-: sval mapping generators: ~2, ~56/25
-Optimal tuning (CWE): ~2 = 1\1, ~28/25 = 194.887
-Optimal ET sequence: {{Optimal ET sequence| 6h, 31gh, 37, 80, 117d }}
-Badness (Dirichlet): 0.612
-===== 2.5.7.11.13.17.19 subgroup =====
 Subgroup: 2.5.7.11.13.17.19
@@ Line 998: / Line 972: @@
 Optimal tuning (CWE): ~2 = 1\1, ~19/17 = 194.927
-Optimal ET sequence: {{Optimal ET sequence| 6h, 31gh, 37, 80 }}
+Optimal ET sequence: {{Optimal ET sequence| 6h, 31gh, 37, 80, 117d* }}
-Badness (Dirichlet): 0.618
+<nowiki />* 117d only appears without prime 19
+Badness (Sintel): 0.618
 ==== Roulette ====
@@ Line 1,007: / Line 983: @@
 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
-Comma list: [[170/169]], [[176/175]], [[640/637]], [[1375/1372]]
-Sval mapping: {{mapping| 1 2 2 2 5 7 | 0 2 5 9 -8 -18 }}
-: sval mapping generators: ~2, ~28/25
-Optimal tuning (CWE): ~2 = 1\1, ~28/25 = 194.285
-Optimal ET sequence: {{Optimal ET sequence| 6g, ... 31, 37, 68, 105 }}
-Badness (Dirichlet): 0.685
-===== 2.5.7.11.13.17.19 subgroup =====
 Subgroup: 2.5.7.11.13.17.19
@@ Line 1,034: / Line 995: @@
 Optimal ET sequence: {{Optimal ET sequence| 6g, ... 31, 37, 68, 105 }}
-Badness (Dirichlet): 0.676
+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.
+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
@@ Line 1,054: / Line 1,015: @@
 == Isra ==
-Isra 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 (''iss-RAH'') night journey in the Qur'an, because it is similar to [[luna]] (septimal [[hemithirds]], a didacus extension).
+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
@@ Line 1,129: / Line 1,090: @@
 [[Tp tuning #T2 tuning|RMS error]]: 2.003 cents
-[[Category:Temperament clans]]
 [[Category:Hemimean clan| ]] <!-- main article -->
 [[Category:Hemimean| ]] <!-- key article -->
-[[Category:Rank 2]]
+[[Category:Temperament clans]]
+[[Category:Catalogs of rank-2 temperaments]]