@@ Line 7: / Line 7: @@
 The 5-limit parent comma for the '''schismatic''' (or '''schismic''') '''family''' is the [[schisma]] of 32805/32768, which is the amount by which the [[Pythagorean comma]] exceeds the [[Didymus comma]] (81/80), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its [[monzo]] is {{monzo| -15 8 1 }}, and flipping that yields {{multival| 1 -8 -15 }} for the [[wedgie]]. This tells us the generator is a fifth and [[5/4]] is represented by a diminished fourth. In fact, 10 = (4/3)<sup>8</sup> × 32805/32768.
-== Schismatic aka Helmholtz  ==
+== Schismatic aka Helmholtz ==
-The 5-limit version of the temperament is a [[microtemperament]], sometimes called '''Helmholtz''', '''schismic''' or '''schismatic''', which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. [[53edo]] is a possible tuning for schismatic, but you need [[118edo]] if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering. You could also try 1/9 schisma, with pure minor thirds and a minutely better 5th, or 2/17 schisma, with both thirds flat by 1/17 of a schisma, although the differences would be very hard to distinguish unless using a large gamut.
+The 5-limit version of the temperament is a [[microtemperament]], sometimes called '''Helmholtz''', '''schismic''' or '''schismatic''', which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. [[53edo|53EDO]] is a possible tuning for schismatic, but you need [[118edo|118EDO]] if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering. You could also try 1/9 schisma, with pure minor thirds and a minutely better 5th, or 2/17 schisma, with both thirds flat by 1/17 of a schisma, although the differences would be very hard to distinguish unless using a large gamut.
 Subgroup: 2.3.5
@@ Line 21: / Line 21: @@
 [[Tuning ranges]]:
-* [[diamond monotone]] range: ~3/2 = [700.000, 705.882] (7\12 to 10\17)
+* [[Diamond monotone]] range: ~3/2 = [685.714, 705.882] (4\7 to 10\17)
-* [[diamond tradeoff]] range: ~3/2 = [701.711, 701.955]
+* [[Diamond tradeoff]] range: ~3/2 = [701.711, 701.955]
-* diamond monotone and tradeoff range: ~3/2 = [701.711, 701.955]
+* Diamond monotone and tradeoff: ~3/2 = [701.711, 701.955]
 {{Vals|legend=1| 12, 29, 41, 53, 118, 171, 289, 460, 749, 3456bc, 4205bc, 4954bc, 5703bbc, 6452bbcc }}
@@ Line 29: / Line 29: @@
 [[Badness]]: 0.004259
-=== Seven-limit extensions  ===
+=== Seven-limit extensions ===
 The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at.
 * Garibaldi adds [[garischisma|{{monzo|25 -14 0 -1}}]],
@@ Line 44: / Line 44: @@
 Temperaments not discussed here include [[Sensamagic clan #Salsa|salsa]], [[Gamelismic clan #Guiron|guiron]], [[Porwell temperaments #Hemischis|hemischis]] and [[Turkish maqam music temperaments #Karadeniz temperament|karadeniz]].
-== Garibaldi  ==
+== Garibaldi ==
 {{main| Garibaldi temperament }}
@@ Line 60: / Line 60: @@
 [[Minimax tuning]]:
-* [[7-odd-limit]]
+* [[7-odd-limit]]: ~3/2 = {{monzo| 2/3 1/15 0 -1/15 }}
 : [{{monzo| 1 0 0 0 }}, {{monzo| 5/3 1/15 0 -1/15 }}, {{monzo| 5/3 -8/15 0 8/15 }}, {{monzo| 5/3 -14/15 0 14/15 }}]
 : [[Eigenmonzo]]s: 2, 7/6
-* [[9-odd-limit]]
+* [[9-odd-limit]]: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }}
 : [{{monzo| 1 0 0 0 }}, {{monzo| 25/16 1/8 0 -1/16 }}, {{monzo| 5/2 -1 0 1/2 }}, {{monzo| 25/8 -7/4 0 7/8 }}]
 : Eigenmonzos: 2, 9/7
 [[Tuning ranges]]:
-* [[diamond monotone]] range: ~3/2 = [700.000, 703.448] (7\12 to 17\29)
+* [[Diamond monotone]] range: ~3/2 = [700.000, 703.448] (7\12 to 17\29)
-* [[diamond tradeoff]] range: ~3/2 = [701.711, 702.915]
+* [[Diamond tradeoff]] range: ~3/2 = [701.711, 702.915]
-* diamond monotone and tradeoff range: ~3/2 = [701.711, 702.915]
+* Diamond monotone and tradeoff: ~3/2 = [701.711, 702.915]
 {{Val list|legend=1| 12, 29, 41, 53, 94, 241c, 335cd, 576ccd }}
@@ Line 88: / Line 88: @@
 Minimax tuning:
-* 11-odd-limit
+* 11-odd-limit: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }}
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 25/16 1/8 0 -1/16 0 }}, {{monzo| 5/2 -1 0 1/2 0 }}, {{monzo| 25/8 -7/4 0 7/8 0 }}, {{monzo| 47/16 23/8 0 -23/16 0 }}]
 : Eigenmonzos: 2, 9/7
+Tuning ranges:
+* Diamond monotone range: ~3/2 = [701.887, 702.439] (31\53 to 24\41)
+* Diamond tradeoff range: ~3/2 = [701.711, 702.915]
+* Diamond monotone and tradeoff: ~3/2 = [701.887, 702.439]
 Vals: {{Val list| 41, 53, 94, 229c, 323c, 417ce }}
@@ Line 96: / Line 101: @@
 Badness: 0.027396
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 106: / Line 111: @@
 POTE generator: ~3/2 = 702.113
+Minimax tuning:
+* 13- and 15-odd-limit: ~3/2 = {{monzo| 19/34 0 0 -1/34 0 1/34 }}
+: Eigenmonzos: 2, 14/13
+Tuning ranges:
+* Diamond monotone range: ~3/2 = [701.887, 702.439] (31\53 to 24\41)
+* 13-odd-limit diamond tradeoff: ~3/2 = [701.711, 703.597]
+* 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 703.597]
+* Diamond monotone and tradeoff: ~3/2 = [701.887, 702.439]
 Vals: {{Val list| 41, 53, 94, 429cdef, 523cdef }}
@@ Line 111: / Line 126: @@
 Badness: 0.020676
-=== Andromeda  ===
+=== Andromeda ===
 Subgroup: 2.3.5.7.11
@@ Line 126: / Line 141: @@
 Badness: 0.023556
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 141: / Line 156: @@
 Badness: 0.020749
-=== Helenus  ===
+=== Helenus ===
 Subgroup: 2.3.5.7.11
@@ Line 156: / Line 171: @@
 Badness: 0.035637
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 171: / Line 186: @@
 Badness: 0.026284
-=== Hemigari  ===
+=== Hemigari ===
 Subgroup: 2.3.5.7.11
@@ Line 186: / Line 201: @@
 Badness: 0.050681
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 201: / Line 216: @@
 Badness: 0.027464
-=== Sanjaab  ===
+=== Sanjaab ===
 Subgroup: 2.3.5.7.11
@@ Line 216: / Line 231: @@
 Badness: 0.058040
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 231: / Line 246: @@
 Badness: 0.033849
-== Schism  ==
+== Schism ==
 {{see also| Archytas clan #Schism }}
@@ Line 250: / Line 265: @@
 [[Badness]]: 0.056648
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 265: / Line 280: @@
 Badness: 0.037482
-== Pontiac  ==
+== Pontiac ==
 Subgroup: 2.3.5.7
@@ Line 279: / Line 294: @@
 [[Minimax tuning]]:
-* [[7-odd-limit]]
+* [[7-odd-limit]]: ~3/2 = {{monzo| 27/47 0 -1/47 1/47 }}
 : [{{monzo| 1 0 0 0 }}, {{monzo| 74/47 0 -1/47 1/47 }}, {{monzo| 113/47 0 8/47 -8/47 }}, {{monzo| 113/47 0 -39/47 39/47 }}]
 : Eigenmonzos: 2, 7/5
-* [[9-odd-limit]]
+* [[9-odd-limit]]: ~3/2 = {{monzo| 1/2 1/5 -1/10 }}
 : [{{monzo| 1 0 0 0 }}, {{monzo| 3/2 1/5 -1/10 0 }}, {{monzo| 3 -8/5 4/5 0 }}, {{monzo| -1/2 39/5 -39/10 0 }}]
 : Eigenmonzos: 2, 10/9
 [[Tuning ranges]]:
-* [[diamond monotone]] range: ~3/2 = [701.538, 701.886] (38\65 to 31\53)
+* [[Diamond monotone]] range: ~3/2 = [701.538, 701.886] (38\65 to 31\53)
-* [[diamond tradeoff]] range: ~3/2 = [701.711, 701.955]
+* [[Diamond tradeoff]] range: ~3/2 = [701.711, 701.955]
-* diamond monotone and tradeoff range: ~3/2 = [701.711, 701.886]
+* Diamond monotone and tradeoff: ~3/2 = [701.711, 701.886]
 {{Val list|legend=1| 53, 118, 171, 1592c, 1763c, 1934c, 2105c, 2276cd, 2447cd, 2618cd, 2789cd, 2960cd, 3131bcd }}
@@ Line 295: / Line 310: @@
 [[Badness]]: 0.014133
-=== Helenoid  ===
+=== Helenoid ===
-The ''helenoid'' temperament (53&amp;118, named by [[User:Xenllium|Xenllium]]) is closely related to the helenus temperament, but with the [[4375/4374|ragisma]] rather than the [[225/224|marvel comma]] tempered out.
+The ''helenoid'' temperament (53&amp;118) is closely related to the helenus temperament, but with the [[4375/4374|ragisma]] rather than the [[225/224|marvel comma]] tempered out.
 Subgroup: 2.3.5.7.11
@@ Line 310: / Line 325: @@
 Badness: 0.038863
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 323: / Line 338: @@
 Badness: 0.033677
-=== Ponta  ===
+=== Ponta ===
-The ''ponta'' temperament (53&amp;224, named by [[User:Xenllium|Xenllium]]) tempers out the [[540/539|swetisma]] and the ragisma.
+The ''ponta'' temperament (53&amp;224) tempers out the [[540/539|swetisma]] and the ragisma.
 Subgroup: 2.3.5.7.11
@@ Line 338: / Line 353: @@
 Badness: 0.048692
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 351: / Line 366: @@
 Badness: 0.023616
-=== Pontic  ===
+=== Pontic ===
-The ''pontic'' temperament (118&amp;171, named by [[User:Xenllium|Xenllium]]) tempers out the [[441/440|werckisma]] and the ragisma.
+The ''pontic'' temperament (118&amp;171) tempers out the [[441/440|werckisma]] and the ragisma.
 Subgroup: 2.3.5.7.11
@@ Line 366: / Line 381: @@
 Badness: 0.049573
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 379: / Line 394: @@
 Badness: 0.045308
-=== Bipont  ===
+=== Bipont ===
-The ''bipont'' temperament (118&amp;224, named by [[User:Xenllium|Xenllium]]) has a period of half octave and tempers out the [[3025/3024|lehmerisma]], 3025/3024 and the [[9801/9800|kalisma]], 9801/9800.
+The ''bipont'' temperament (118&amp;224) has a period of half octave and tempers out the [[3025/3024|lehmerisma]], 3025/3024 and the [[9801/9800|kalisma]], 9801/9800.
 Subgroup: 2.3.5.7.11
@@ Line 394: / Line 409: @@
 Badness: 0.014629
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 407: / Line 422: @@
 Badness: 0.030172
-==== Counterbipont  ====
+==== Counterbipont ====
 Subgroup: 2.3.5.7.11.13
@@ Line 420: / Line 435: @@
 Badness: 0.025547
-==== Quadrapont  ====
+==== Quadrapont ====
 Subgroup: 2.3.5.7.11.13
@@ Line 433: / Line 448: @@
 Badness: 0.021025
-== Grackle  ==
+== Grackle ==
 Subgroup: 2.3.5.7
@@ Line 454: / Line 469: @@
 [[Badness]]: 0.070407
-== Bischismic  ==
+== Bischismic ==
 Subgroup: 2.3.5.7
@@ Line 475: / Line 490: @@
 [[Badness]]: 0.054744
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 490: / Line 505: @@
 Badness: 0.028160
-=== 13-limit  ===
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
@@ Line 505: / Line 520: @@
 Badness: 0.028722
-== Kleischismic  ==
+== Kleischismic ==
 Subgroup: 2.3.5.7
@@ Line 522: / Line 537: @@
 [[Badness]]: 0.110583
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 537: / Line 552: @@
 Badness: 0.036749
-=== 13-limit  ===
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
@@ Line 552: / Line 567: @@
 Badness: 0.037640
-== Pogo  ==
+== Pogo ==
 Subgroup: 2.3.5.7
@@ Line 569: / Line 584: @@
 [[Badness]]: 0.079635
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 584: / Line 599: @@
 Badness: 0.031857
-=== 13-limit  ===
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
@@ Line 599: / Line 614: @@
 Badness: 0.017514
-== Squirrel  ==
+== Squirrel ==
 The squirrel temperament (29&amp;36) has an 11/10 generator, three of which give the fourth (4/3), and thirteen of which give 7/4 with octave reduction.
@@ Line 616: / Line 631: @@
 [[Badness]]: 0.174705
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 629: / Line 644: @@
 Badness: 0.068310
-=== 13-limit  ===
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
@@ Line 642: / Line 657: @@
 Badness: 0.043750
-== Tertiaschis  ==
+== Tertiaschis ==
-The ''tertiaschis'' temperament (94&amp;159, named by [[User:Xenllium|Xenllium]]) has an 11/10 generator and tempers out 1071785/1062882.
+The ''tertiaschis'' temperament (94&amp;159) has an 11/10 generator and tempers out 1071785/1062882.
 Subgroup: 2.3.5.7
@@ Line 659: / Line 674: @@
 [[Badness]]: 0.211859
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 672: / Line 687: @@
 Badness: 0.061336
-=== 13-limit  ===
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
@@ Line 685: / Line 700: @@
 Badness: 0.036700
-=== 17-limit  ===
+=== 17-limit ===
 Subgroup: 2.3.5.7.11.13.17
@@ Line 698: / Line 713: @@
 Badness: 0.026504
-== Term  ==
+== Term ==
 Subgroup: 2.3.5.7
@@ Line 719: / Line 734: @@
 [[Badness]]: 0.019950
-=== Terminal  ===
+=== Terminal ===
-The ''terminal'' temperament (12&amp;159, named by [[User:Xenllium|Xenllium]]) tempers out 441/440 and 4375/4356. In this temperament, 44/35 and 63/50 is represented as one period of 1/3 octave.
+The ''terminal'' temperament (12&amp;159) tempers out 441/440 and 4375/4356. In this temperament, 44/35 and 63/50 is represented as one period of 1/3 octave.
 Subgroup: 2.3.5.7.11
@@ Line 734: / Line 749: @@
 Badness: 0.059502
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 747: / Line 762: @@
 Badness: 0.037082
-==== 17-limit  ====
+==== 17-limit ====
 Subgroup: 2.3.5.7.11.13.17
@@ Line 760: / Line 775: @@
 Badness: 0.027073
-=== Hemiterm  ===
+=== Hemiterm ===
-The ''hemiterm'' temperament (12&amp;342, named by [[User:Xenllium|Xenllium]]) has a period of 1/6 octave and tempers out [[9801/9800]] (kalisma) and 151263/151250 (odiheim comma).
+The ''hemiterm'' temperament (12&amp;342) has a period of 1/6 octave and tempers out [[9801/9800]] (kalisma) and 151263/151250 (odiheim comma).
 Subgroup: 2.3.5.7.11
@@ Line 775: / Line 790: @@
 Badness: 0.029438
-== Sesquiquartififths  ==
+== Sesquiquartififths ==
 Subgroup: 2.3.5.7
@@ Line 796: / Line 811: @@
 [[Badness]]: 0.011244
-=== Sesquart  ===
+=== Sesquart ===
 Subgroup: 2.3.5.7.11
@@ Line 811: / Line 826: @@
 Badness: 0.029306
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 824: / Line 839: @@
 Badness: 0.022396
-=== Bisesqui  ===
+=== Bisesqui ===
 Subgroup: 2.3.5.7.11
@@ Line 837: / Line 852: @@
 Badness: 0.016968
-== Quintilischis  ==
+== Quintilischis ==
-The ''quintilischis'' temperament (12&amp;253, named by [[User:Xenllium|Xenllium]]) slices the fourth (4/3) into five semitones.
+The ''quintilischis'' temperament (12&amp;253) slices the fourth (4/3) into five semitones.
 Subgroup: 2.3.5.7
@@ Line 854: / Line 869: @@
 [[Badness]]: 0.253966
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 867: / Line 882: @@
 Badness: 0.113044
-=== 13-limit  ===
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
@@ Line 880: / Line 895: @@
 Badness: 0.069127
-=== 17-limit  ===
+=== 17-limit ===
 Subgroup: 2.3.5.7.11.13.17
@@ Line 893: / Line 908: @@
 Badness: 0.045992
-=== 19-limit  ===
+=== 19-limit ===
 Subgroup: 2.3.5.7.11.13.17.19
@@ Line 906: / Line 921: @@
 Badness: 0.038155
-== Sextilififths  ==
+== Sextilififths ==
-The sextilififths or ''sextilischis'' (the latter is named by [[User:Xenllium|Xenllium]]) slices the fourth (4/3) into six small semitones, which serves as both 21/20 and 22/21.
+The sextilififths (130&amp;159, also known as ''sextilischis'') slices the fourth (4/3) into six small semitones, which serves as both 21/20 and 22/21.
 Subgroup: 2.3.5.7
@@ Line 925: / Line 940: @@
 [[Badness]]: 0.108794
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 940: / Line 955: @@
 Badness: 0.045457
-=== 13-limit  ===
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
@@ Line 955: / Line 970: @@
 Badness: 0.025276
-== Tsaharuk  ==
+== Tsaharuk ==
 {{See also|Tsaharuk}}
@@ Line 974: / Line 989: @@
 [[Badness]]: 0.030697
-== Quanharuk  ==
+== Quanharuk ==
 Subgroup: 2.3.5.7
@@ Line 991: / Line 1,006: @@
 [[Badness]]: 0.071950
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 1,006: / Line 1,021: @@
 Badness: 0.031549
-=== 13-limit  ===
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
@@ Line 1,021: / Line 1,036: @@
 Badness: 0.021392
-== Quadrant  ==
+== Quadrant ==
-The ''quadrant'' temperament (12&amp;224, named by [[User:Xenllium|Xenllium]]) has a period of quarter octave and tempers out the [[dimcomp comma]], 390625/388962. In this temperament, 25/21 is mapped into quarter octave.
+The ''quadrant'' temperament (12&amp;224) has a period of quarter octave and tempers out the [[dimcomp comma]], 390625/388962. In this temperament, 25/21 is mapped into quarter octave.
 Subgroup: 2.3.5.7
@@ Line 1,040: / Line 1,055: @@
 [[Badness]]: 0.110242
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 1,053: / Line 1,068: @@
 Badness: 0.045738
-=== 13-limit  ===
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
@@ Line 1,066: / Line 1,081: @@
 Badness: 0.027243
-== Septant  ==
+== Septant ==
-The ''septant'' temperament (224&amp;301, named by [[User:Xenllium|Xenllium]]) has a period of 1/7 octave and tempers out the [[akjaysma]], {{monzo|47 -7 -7 -7}}.
+The ''septant'' temperament (224&amp;301) has a period of 1/7 octave and tempers out the [[akjaysma]], {{monzo|47 -7 -7 -7}}.
 Subgroup: 2.3.5.7
@@ Line 1,083: / Line 1,098: @@
 [[Badness]]: 0.111142
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 1,096: / Line 1,111: @@
 Badness: 0.044122
-=== 13-limit  ===
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
@@ Line 1,109: / Line 1,124: @@
 Badness: 0.024706
-== Octant  ==
+== Octant ==
 The octant temperament (224&amp;472) has a period of 1/8 octave. In this temperament, 12/11, 35/27, and 99/70 are mapped into 1\8, 3\8, and 4\8 respectively.
@@ Line 1,128: / Line 1,143: @@
 [[Badness]]: 0.157186
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 1,143: / Line 1,158: @@
 Badness: 0.044778
-=== 13-limit  ===
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13

Schismatic family: Difference between revisions