Hemimean clan: Difference between revisions

(23 intermediate revisions by 7 users not shown)

Line 1:

~~<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 ==

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 85:

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 109:

Line 105:

Badness (~~Dirichlet~~): 0.304

Badness (Sintel): 0.304

=== 11-limit ===

Line 116:

Line 112:

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

Line 129:

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 }}

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

Line 142:

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 }}

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

Line 155:

Line 151:

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

Line 168:

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 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 194.004

Line 181:

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 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.698

Line 191:

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 196:

Line 194:

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

Line 211:

Line 209:

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

Line 226:

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 }}

Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9035

Line 239:

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 252:

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 265:

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 278:

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 296:

Line 294:

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

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

Line 304:

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

Line 318:

Line 314:

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

Line 336:

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 }}

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

Line 354:

Line 350:

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

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

Line 368:

Line 362:

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

Line 381:

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 }}

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 189.928

Line 394:

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 }}

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 190.360

Line 416:

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 430:

Line 422:

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

Line 443:

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 }}

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

Line 456:

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 469:

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 482:

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 495:

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 }}

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

Line 515:

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 529:

Line 519:

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

Line 542:

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 }}

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

Line 558:

Line 548:

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

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

Line 572:

Line 560:

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

Line 585:

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 }}

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

Line 598:

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 }}

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

Line 611:

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 }}

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

Line 624:

Line 612:

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

Line 637:

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 }}

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

Line 651:

Line 639:

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

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

Line 665:

Line 651:

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

Line 678:

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 }}

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

Line 691:

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 }}

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

Line 700:

Line 686:

== 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

Line 707:

Line 693:

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

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

Line 721:

Line 705:

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

Line 734:

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 }}

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

Line 747:

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 }}

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

Line 760:

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 773:

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 786:

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 799:

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 816:

Line 800:

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

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

Line 830:

Line 812:

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

Line 843:

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 }}

Optimal tuning (POTE): ~2 = 1\1, ~112/99 = 216.172

Line 856:

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 }}

Optimal tuning (POTE): ~2 = 1\1, ~17/15 = 216.172

Line 869:

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 878:

Line 860:

== 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

Line 885:

Line 869:

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

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

Line 899:

Line 881:

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

Line 912:

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 }}

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

Line 925:

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 }}

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

Line 952:

Line 934:

RMS error: 0.5567 cents

Badness (~~Dirichlet~~): 0.195

Badness (Sintel): 0.195

=== Tridecimal didacus ===

Line 973:

Line 955:

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

Badness (~~Dirichlet~~): 0.324

Badness (Sintel): 0.324

==== Mediantone ====

Line 980:

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 1,005:

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 1,014:

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 1,041:

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 1,061:

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 1,136:

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 85: / 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 109: / Line 105: @@
 {{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328 }}
-Badness (Dirichlet): 0.304
+Badness (Sintel): 0.304
 === 11-limit ===
@@ Line 116: / 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 129: / 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 142: / 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 155: / 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 168: / 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 181: / 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 191: / 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 196: / 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 211: / 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 226: / 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 239: / 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 252: / 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 265: / 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 278: / 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 296: / 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 304: / 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 318: / 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 336: / 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 354: / 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 368: / 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 381: / 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 394: / 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 416: / 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 430: / 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 443: / 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 456: / 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 469: / 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 482: / 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 495: / 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 515: / 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 529: / 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 542: / 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 558: / 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 572: / 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 585: / 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 598: / 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 611: / 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 624: / 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 637: / 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 651: / 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 665: / 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 678: / 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 691: / 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 700: / 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 707: / 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 721: / 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 734: / 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 747: / 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 760: / 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 773: / 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 786: / 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 799: / 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 816: / 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 830: / 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 843: / 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 856: / 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 869: / 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 878: / 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 885: / 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 899: / 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 912: / 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 925: / 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 952: / Line 934: @@
 RMS error: 0.5567 cents
-Badness (Dirichlet): 0.195
+Badness (Sintel): 0.195
 === Tridecimal didacus ===
@@ Line 973: / Line 955: @@
 Optimal ET sequence: {{Optimal ET sequence| 6, 25, 31, 37 }}
-Badness (Dirichlet): 0.324
+Badness (Sintel): 0.324
 ==== Mediantone ====
@@ Line 980: / 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 1,005: / 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 1,014: / 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 1,041: / 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 1,061: / 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 1,136: / 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]]