Meantone family: Difference between revisions

Revision as of 12:47, 4 June 2021

The 5-limit parent comma of the meantone family is the Didymus or syntonic comma, 81/80. This is the one they all temper out. The period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.

Meantone (12&19, 2.3.5)

Main article: Meantone

Subgroup: 2.3.5

Comma list: 81/80

Mapping: [⟨1 0 -4], ⟨0 1 4]]

Mapping generators: ~2, ~3

Wedgie: ⟨⟨ 1 4 4 ]]

POTE generator: ~3/2 = 696.239

Tuning ranges:

diamond monotone range: [685.714, 720.000] (4\7 to 3\5)
diamond tradeoff range: [694.786, 701.955] (1/3 comma to Pythagorean)
diamond monotone and tradeoff: [694.786, 701.955]

Template:Val list

Badness: 0.007381

Scales: meantone5, meantone7, meantone12

Seven-limit extensions

The 7-limit extensions of meantone are:

Septimal meantone, with normal comma list [[-4 4 -1⟩, [-13 10 0 -1⟩],
Flattone, with normal list [[-4 4 -1⟩, [-17 9 0 1⟩],
Dominant, with normal list [[-4 4 -1⟩, [6 -2 0 -1⟩],
Sharptone, with normal list [[-4 4 -1⟩, [2 -3 0 1⟩],
Injera, with normal list [[-4 4 -1⟩, [-7 8 0 -2⟩],
Mohajira, with normal list [[-4 4 -1⟩, [-23 11 0 2⟩],
Godzilla, with normal list [[-4 4 -1⟩, [-4 -1 0 2⟩],
Mothra, with normal list [[-4 4 -1⟩, [-10 1 0 3⟩],
Squares, with normal list [[-4 4 -1⟩, [-3 9 0 -4⟩], and
Liese, with normal list [[-4 4 -1⟩, [-9 11 0 -3⟩].

Septimal meantone

Deutsch

See also: Meantone

See also: Wikipedia: Septimal meantone temperament

The 7/4 of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, 7/5, C-F#, the tritone, and 21/16, C-E#, the augmented third. Septimal meantone tempers out the common 7-limit commas 126/125 and 225/224 and in fact can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125 and 225/224.

Subgroup: 2.3.5.7

Comma list: 81/80, 126/125

Mapping: [⟨1 0 -4 -13], ⟨0 1 4 10]]

Wedgie: ⟨⟨ 1 4 10 4 13 12 ]]

POTE generator: ~3/2 = 696.495

Minimax tuning:

7- and 9-odd-limit

[[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [-3 0 5/2 0⟩]

Eigenmonzos: 2, 5

Tuning ranges:

Diamond monotone range: [694.737, 700.000] (11\19 to 7\12)
Diamond tradeoff range: [694.786, 701.955] (1/3 comma to Pythagorean)
Diamond monotone and tradeoff: [694.786, 700.000]

Algebraic generator: Cybozem, the real root of 15x³ - 10x² - 18, 503.4257 cents. The recurrence converges quickly.

Template:Val list

Badness: 0.013707

Scales: meantone5, meantone7, meantone12

Unidecimal meantone aka Huygens

See also: Meantone vs meanpop

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 126/125

Mapping: [⟨1 0 -4 -13 -25], ⟨0 1 4 10 18]]

POTE generator: ~3/2 = 696.967

Minimax tuning:

11-odd-limit

[[1 0 0 0 0⟩, [25/16 -1/8 0 0 1/16⟩, [9/4 -1/2 0 0 1/4⟩, [21/8 -5/4 0 0 5/8⟩, [25/8 -9/4 0 0 9/8⟩]

Eigenmonzos: 2, 11/9

Tuning ranges:

Diamond monotone range: [696.774, 700.000] (18\31 to 7\12)
Diamond tradeoff range: [691.202, 701.955]
Diamond monotone and tradeoff: [696.774, 700.000]

Algebraic generator: Traverse, the positive real root of x⁴ + 2x - 13, or 696.9529 cents.

@@ Line 7: / Line 7: @@
 The [[5-limit]] parent [[comma]] of the '''meantone family''' is the Didymus or [[Wikipedia: syntonic comma|syntonic comma]], [[81/80]]. This is the one they all temper out. The period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.
-== Meantone (12&amp;19, 2.3.5)  ==
+== Meantone (12&amp;19, 2.3.5) ==
 {{main| Meantone }}
@@ Line 25: / Line 25: @@
 * [[diamond monotone]] range: [685.714, 720.000] (4\7 to 3\5)
 * [[diamond tradeoff]] range: [694.786, 701.955] (1/3 comma to Pythagorean)
-* diamond monotone and tradeoff range: [694.786, 701.955]
+* diamond monotone and tradeoff: [694.786, 701.955]
 {{Val list|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b, 212bb, 293bb }}
@@ Line 33: / Line 33: @@
 Scales: [[meantone5]], [[meantone7]], [[meantone12]]
-=== Seven-limit extensions  ===
+=== Seven-limit extensions ===
 The [[7-limit]] [[extension]]s of meantone are:
 * Septimal meantone, with normal comma list [{{Monzo| -4 4 -1 }}, [[Harrison's comma|{{Monzo| -13 10 0 -1 }}]]],
@@ Line 46: / Line 46: @@
 * Liese, with normal list [{{Monzo| -4 4 -1 }}, {{Monzo| -9 11 0 -3 }}].
-== Septimal meantone  ==
+== Septimal meantone ==
 <div style="float:right">[[:de:septimal-mitteltönig|Deutsch]]</div>
 {{see also| Meantone }}
@@ Line 69: / Line 69: @@
 [[Tuning ranges]]:
-* [[diamond monotone]] range: [694.737, 700.000] (11\19 to 7\12)
+* [[Diamond monotone]] range: [694.737, 700.000] (11\19 to 7\12)
-* [[diamond tradeoff]] range: [694.786, 701.955] (1/3 comma to Pythagorean)
+* [[Diamond tradeoff]] range: [694.786, 701.955] (1/3 comma to Pythagorean)
-* diamond monotone and tradeoff range: [694.786, 700.000]
+* Diamond monotone and tradeoff: [694.786, 700.000]
 [[Algebraic generator]]: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, 503.4257 cents. The recurrence converges quickly.
@@ Line 81: / Line 81: @@
 Scales: [[meantone5]], [[meantone7]], [[meantone12]]
-=== Unidecimal meantone aka Huygens  ===
+=== Unidecimal meantone aka Huygens ===
 {{see also| Meantone vs meanpop }}
@@ Line 98: / Line 98: @@
 Tuning ranges:
-* [[diamond monotone]] range: [696.774, 700.000] (18\31 to 7\12)
+* Diamond monotone range: [696.774, 700.000] (18\31 to 7\12)
-* [[diamond tradeoff]] range: [691.202, 701.955]
+* Diamond tradeoff range: [691.202, 701.955]
-* diamond monotone and tradeoff range: [696.774, 700.000]
+* Diamond monotone and tradeoff: [696.774, 700.000]
 Algebraic generator: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
@@ Line 111: / Line 111: @@
 * [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-74-edo.mp3 Twinkle canon – 74 edo] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]
-==== Tridecimal meantone  ====
+==== Tridecimal meantone ====
 Subgroup: 2.3.5.7.11.13
@@ Line 124: / Line 124: @@
 Badness: 0.018048
-==== Grosstone  ====
+==== Grosstone ====
 Subgroup: 2.3.5.7.11.13
@@ Line 132: / Line 132: @@
 POTE generator: ~3/2 = 697.264
-Tuning ranges:
-* [[diamond monotone]] range: [696.774, 700.000] (18\31 to 7\12)
-* [[diamond tradeoff]] range: [691.202, 701.955]
-* diamond monotone and tradeoff range: [696.774, 700.000]
 Vals: {{Val list| 12, 19ef, 31, 43, 74 }}
@@ Line 142: / Line 137: @@
 Badness: 0.025899
-==== Meridetone  ====
+==== Meridetone ====
 Subgroup: 2.3.5.7.11.13
@@ Line 155: / Line 150: @@
 Badness: 0.026421
-==== Hemimeantone  ====
+==== Hemimeantone ====
 Subgroup: 2.3.5.7.11.13
@@ Line 170: / Line 165: @@
 Badness: 0.031433
-=== Meanpop  ===
+=== Meanpop ===
 {{see also| Meantone vs meanpop }}
@@ Line 189: / Line 184: @@
 Tuning ranges:
-* [[diamond monotone]] range: [694.737, 696.774] (11\19 to 18\31)
+* Diamond monotone range: [694.737, 696.774] (11\19 to 18\31)
-* [[diamond tradeoff]] range: [691.202, 701.955]
+* Diamond tradeoff range: [691.202, 701.955]
-* diamond monotone and tradeoff range: [694.737, 696.774]
+* Diamond monotone and tradeoff: [694.737, 696.774]
 Algebraic generator: Cybozem; or else Radieubiz, the real root of 3''x''<sup>3</sup> + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
@@ Line 203: / Line 198: @@
 * [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3 Twinkle canon – 50 edo] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]
-==== Tridecimal Meanpop  ====
+==== Tridecimal Meanpop ====
 Subgroup: 2.3.5.7.11.13
@@ Line 215: / Line 210: @@
 Tuning ranges:
-* [[diamond monotone]] range: [694.737, 696.774] (11\19 to 18\31)
+* Diamond monotone range: [694.737, 696.774] (11\19 to 18\31)
-* [[diamond tradeoff]] range: [691.202, 701.955]
+* Diamond tradeoff range: [691.202, 701.955]
-* diamond monotone and tradeoff range: [694.737, 696.774]
+* Diamond monotone and tradeoff: [694.737, 696.774]
 Vals: {{Val list| 12ef, 19, 31, 50, 81, 131bd, 212bbddf }}
@@ Line 223: / Line 218: @@
 Badness: 0.020883
-==== Meanplop  ====
+==== Meanplop ====
 Subgroup: 2.3.5.7.11.13
@@ Line 236: / Line 231: @@
 Badness: 0.027666
-=== Meanenneadecal  ===
+=== Meanenneadecal ===
 Subgroup: 2.3.5.7.11
@@ Line 249: / Line 244: @@
 Badness: 0.021423
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 262: / Line 257: @@
 Badness: 0.021182
-==== Vincenzo  ====
+==== Vincenzo ====
 Subgroup: 2.3.5.7.11.13
@@ Line 275: / Line 270: @@
 Badness: 0.024763
-===== 17-limit  =====
+===== 17-limit =====
 Subgroup: 2.3.5.7.11.13.17
@@ Line 288: / Line 283: @@
 Badness: 0.025535
-===== 19-limit  =====
+===== 19-limit =====
 Subgroup: 2.3.5.7.11.13.17.19
@@ Line 301: / Line 296: @@
 Badness: 0.022302
-===== 23-limit  =====
+===== 23-limit =====
 Subgroup: 2.3.5.7.11.13.17.19.23
@@ Line 314: / Line 309: @@
 Badness: 0.020139
-===== 29-limit  =====
+===== 29-limit =====
 Subgroup: 2.3.5.7.11.13.17.19.23.29
@@ Line 327: / Line 322: @@
 Badness: 0.018168
-===== 31-limit  =====
+===== 31-limit =====
 Subgroup: 2.3.5.7.11.13.17.19.23.29.31
@@ Line 340: / Line 335: @@
 Badness: 0.017069
-===== 37-limit  =====
+===== 37-limit =====
 Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37
@@ Line 353: / Line 348: @@
 Badness: 0.016129
-===== 41-limit  =====
+===== 41-limit =====
 Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41
@@ Line 366: / Line 361: @@
 Badness: 0.015356
-===== 43-limit  =====
+===== 43-limit =====
 Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41.43
@@ Line 379: / Line 374: @@
 Badness: 0.013906
-===== 47-limit  =====
+===== 47-limit =====
 Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41.43.47
@@ Line 392: / Line 387: @@
 Badness: 0.013818
-==== Meanundec  ====
+==== Meanundec ====
 Subgroup: 2.3.5.7.11.13
@@ Line 405: / Line 400: @@
 Badness: 0.024243
-=== Meanundeci  ===
+=== Meanundeci ===
 Subgroup: 2.3.5.7.11
@@ Line 418: / Line 413: @@
 Badness: 0.031539
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 431: / Line 426: @@
 Badness: 0.026288
-=== Migration  ===
+=== Migration ===
 Subgroup: 2.3.5.7.11
@@ Line 446: / Line 441: @@
 Badness: 0.025516
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 461: / Line 456: @@
 Badness: 0.028071
-=== Bimeantone  ===
+=== Bimeantone ===
-/8 is mapped to half octave minus the [[meantone diesis]].
+/8 is mapped to half octave minus the [[128/125|meantone diesis]].
 Subgroup: 2.3.5.7.11
@@ Line 478: / Line 473: @@
 Badness: 0.038122
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 493: / Line 488: @@
 Badness: 0.028817
-==== 17-limit  ====
+==== 17-limit ====
 Subgroup: 2.3.5.7.11.13.17
@@ Line 508: / Line 503: @@
 Badness: 0.022666
-== Flattone  ==
+== Flattone ==
 In flattone, 9 generator steps of 4/3 get to the interval class for 7, meaning that [[7/4]] is a diminished seventh interval (C-Bbb). Other intervals are [[7/6]], a diminished third (C-Ebb), and [[7/5]], a doubly diminshed fifth (C-Gbb). Good tunings for flattone are [[26edo|26EDO]], [[45edo|45EDO]] and [[64edo|64EDO]].
@@ Line 530: / Line 525: @@
 [[Tuning ranges]]:
-* [[diamond monotone]] range: [692.308, 694.737] (15\26 to 11\19)
+* [[Diamond monotone]] range: [692.308, 694.737] (15\26 to 11\19)
-* [[diamond tradeoff]] range: [692.353, 701.955]
+* [[Diamond tradeoff]] range: [692.353, 701.955]
-* diamond monotone and tradeoff range: [692.353, 694.737]
+* Diamond monotone and tradeoff: [692.353, 694.737]
 Algebraic generator: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.
@@ Line 542: / Line 537: @@
 Scales: [[flattone12]]
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 552: / Line 547: @@
 Tuning ranges:
-* [[diamond monotone]] range: [692.308, 694.737] (15\26 to 11\19)
+* Diamond monotone range: [692.308, 694.737] (15\26 to 11\19)
-* [[diamond tradeoff]] range: [682.502, 701.955]
+* Diamond tradeoff range: [682.502, 701.955]
-* diamond monotone and tradeoff range: [692.308, 694.737]
+* Diamond monotone and tradeoff: [692.308, 694.737]
 Vals: {{Val list| 7, 19, 26, 45, 71bc, 116bcde }}
@@ Line 562: / Line 557: @@
 Scales: [[flattone12]]
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 572: / Line 567: @@
 Tuning ranges:
-* [[diamond monotone]] range: [692.308, 694.737] (15\26 to 11\19)
+* Diamond monotone range: [692.308, 694.737] (15\26 to 11\19)
-* [[diamond tradeoff]] range: [682.502, 701.955]
+* Diamond tradeoff range: [682.502, 701.955]
-* diamond monotone and tradeoff range: [692.308, 694.737]
+* Diamond monotone and tradeoff: [692.308, 694.737]
 Vals: {{Val list| 7, 19, 26, 45f, 71bcf, 116bcdef }}
@@ Line 582: / Line 577: @@
 Scales: [[flattone12]]
-=== Ptolemy  ===
+=== Ptolemy ===
 Subgroup: 2.3.5.7.11
@@ Line 595: / Line 590: @@
 Badness: 0.058785
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 608: / Line 603: @@
 Badness: 0.034316
-== Dominant  ==
+== Dominant ==
 The interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh. The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is [[12edo|12EDO]], but it also works well with the Pythagorean tuning of pure [[3/2]] fifths, and with [[29edo|29EDO]], [[41edo|41EDO]], or [[53edo|53EDO]].
@@ Line 622: / Line 617: @@
 [[Tuning ranges]]:
-* [[diamond monotone]] range: [700.000, 720.000] (7\12 to 3\5)
+* [[Diamond monotone]] range: [700.000, 720.000] (7\12 to 3\5)
-* [[diamond tradeoff]] range: [694.786, 715.587]
+* [[Diamond tradeoff]] range: [694.786, 715.587]
-* diamond monotone and tradeoff range: [700.000, 715.587]
+* Diamond monotone and tradeoff: [700.000, 715.587]
 {{Val list|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }}
@@ Line 630: / Line 625: @@
 [[Badness]]: 0.020690
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 638: / Line 633: @@
 Tuning ranges:
-* [[diamond monotone]] range: [700.000, 705.882] (7\12 to 10\17)
+* Diamond monotone range: [700.000, 705.882] (7\12 to 10\17)
-* [[diamond tradeoff]] range: [691.202, 715.587]
+* Diamond tradeoff range: [691.202, 715.587]
-* diamond monotone and tradeoff range: [700.000, 705.882]
+* Diamond monotone and tradeoff: [700.000, 705.882]
 POTE generator: ~3/2 = 703.254
@@ Line 648: / Line 643: @@
 Badness: 0.024180
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 658: / Line 653: @@
 Tuning ranges:
-* [[diamond monotone]] range: 705.882 (10\17)
+* Diamond monotone range: 705.882 (10\17)
-* [[diamond tradeoff]] range: [691.202, 715.587]
+* Diamond tradeoff range: [691.202, 715.587]
-* diamond monotone and tradeoff range: 705.882
+* Diamond monotone and tradeoff: 705.882
 Vals: {{Val list| 12f, 17c, 29cdef }}
@@ Line 666: / Line 661: @@
 Badness: 0.024108
-==== Dominion  ====
+==== Dominion ====
 Subgroup: 2.3.5.7.11.13
@@ Line 679: / Line 674: @@
 Badness: 0.027295
-=== Domineering  ===
+=== Domineering ===
 Subgroup: 2.3.5.7.11
@@ Line 692: / Line 687: @@
 Badness: 0.021978
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 705: / Line 700: @@
 Badness: 0.027039
-===== 17-limit  =====
+===== 17-limit =====
 Subgroup: 2.3.5.7.11.13.17
@@ Line 718: / Line 713: @@
 Badness: 0.024539
-===== 19-limit  =====
+===== 19-limit =====
 Subgroup: 2.3.5.7.11.13.17.19
@@ Line 731: / Line 726: @@
 Badness: 0.020398
-==== Dominatrix  ====
+==== Dominatrix ====
 Subgroup: 2.3.5.7.11.13
@@ Line 744: / Line 739: @@
 Badness: 0.018289
-=== Domination  ===
+=== Domination ===
 Subgroup: 2.3.5.7.11
@@ Line 757: / Line 752: @@
 Badness: 0.036562
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 770: / Line 765: @@
 Badness: 0.027435
-=== Arnold  ===
+=== Arnold ===
 Subgroup: 2.3.5.7.11
@@ Line 783: / Line 778: @@
 Badness: 0.026141
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 796: / Line 791: @@
 Badness: 0.023300
-==== 17-limit  ====
+==== 17-limit ====
 Subgroup: 2.3.5.7.11.13.17
@@ Line 809: / Line 804: @@
 Badness: 0.024535
-==== 19-limit  ====
+==== 19-limit ====
 Subgroup: 2.3.5.7.11.13.17.19
@@ Line 822: / Line 817: @@
 Badness: 0.021098
-=== Maqamic  ===
+=== Maqamic ===
 <span style="display: block; text-align: right;">[[:de:maqamisch|Deutsch]]</span>
 {{main| Maqamic }}
@@ Line 842: / Line 837: @@
 Badness: 0.040240
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 857: / Line 852: @@
 Badness: 0.027214
-== Sharptone  ==
+== Sharptone ==
 Sharptone is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, a 7/4 is a major sixth, a 7/6 a whole tone, and a 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. [[12edo|12EDO]] tuning does sharptone about as well as such a thing can be done, of course not in its patent val.
@@ Line 874: / Line 869: @@
 [[Badness]]: 0.024848
-=== Meanertone  ===
+=== Meanertone ===
 Subgroup: 2.3.5.7.11
@@ Line 887: / Line 882: @@
 Badness: 0.025167
-== Plutus  ==
+== Plutus ==
 Subgroup: 2.3.5.7
@@ Line 902: / Line 897: @@
 [[Badness]]: 0.045275
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 915: / Line 910: @@
 Badness: 0.032521
-== Supermean  ==
+== Supermean ==
 Subgroup: 2.3.5.7
@@ Line 928: / Line 923: @@
 [[Badness]]: 0.134204
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 941: / Line 936: @@
 Badness: 0.063262
-=== 13-limit  ===
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
@@ Line 954: / Line 949: @@
 Badness: 0.040324
-== Godzilla  ==
+== Godzilla ==
 <span style="display: block; text-align: right;">[[:de:Semiphor,_Semaphor,_Godzilla|Deutsch]]</span>
 {{main| Semaphore and Godzilla }}
@@ Line 973: / Line 968: @@
 [[Tuning ranges]]:
-* [[diamond monotone]] range: [240.000, 257.143] (1\5 to 3\14)
+* [[Diamond monotone]] range: [240.000, 257.143] (1\5 to 3\14)
-* [[diamond tradeoff]] range: [231.174, 266.871]
+* [[Diamond tradeoff]] range: [231.174, 266.871]
-* diamond monotone and tradeoff range: [240.000, 257.143]
+* Diamond monotone and tradeoff: [240.000, 257.143]
 {{Val list|legend=1| 5, 14c, 19 }}
@@ Line 981: / Line 976: @@
 [[Badness]]: 0.026747
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 993: / Line 988: @@
 Tuning ranges:
-* [[diamond monotone]] range: [252.632, 257.143] (4\19 to 3\14)
+* Diamond monotone range: [252.632, 257.143] (4\19 to 3\14)
-* [[diamond tradeoff]] range: [231.174, 266.871]
+* Diamond tradeoff range: [231.174, 266.871]
-* diamond monotone and tradeoff range: [252.632, 257.143]
+* Diamond monotone and tradeoff: [252.632, 257.143]
 Vals: {{Val list| 14c, 19, 33cd, 52cd }}
@@ Line 1,001: / Line 996: @@
 Badness: 0.028947
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 1,013: / Line 1,008: @@
 Tuning ranges:
-* [[diamond monotone]] range: 694.737 (4\19)
+* Diamond monotone range: 694.737 (4\19)
-* [[diamond tradeoff]] range: [621.581, 737.652]
+* Diamond tradeoff range: [621.581, 737.652]
-* diamond monotone and tradeoff range: 694.737
+* Diamond monotone and tradeoff: 694.737
 Vals: {{Val list| 14cf, 19, 33cdff, 52cdff }}
@@ Line 1,021: / Line 1,016: @@
 Badness: 0.022503
-=== Semafour  ===
+=== Semafour ===
 Subgroup: 2.3.5.7.11
@@ Line 1,036: / Line 1,031: @@
 Badness: 0.028510
-=== Varan  ===
+=== Varan ===
 Subgroup: 2.3.5.7.11
@@ Line 1,051: / Line 1,046: @@
 Badness: 0.039647
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 1,066: / Line 1,061: @@
 Badness: 0.025676
-=== Baragon  ===
+=== Baragon ===
 Subgroup: 2.3.5.7.11
@@ Line 1,081: / Line 1,076: @@
 Badness: 0.035673
-== Injera  ==
+== Injera ==
 Injera has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. [[38edo|38EDO]], which is two parallel [[19edo|19EDOs]], is an excellent tuning for injera.
@@ Line 1,093: / Line 1,088: @@
 Mapping generators: ~7/5, ~3
+{{Multival|legend=1| 2 8 8 8 7 -4 }}
 [[POTE generator]]: ~3/2 = 694.375
 [[Tuning ranges]]:
-* [[diamond monotone]] range: [685.714, 700.000] (8\14 to 7\12)
+* [[Diamond monotone]] range: [685.714, 700.000] (8\14 to 7\12)
-* [[diamond tradeoff]] range: [688.957, 701.955]
+* [[Diamond tradeoff]] range: [688.957, 701.955]
-* diamond monotone and tradeoff range: [688.957, 700.000]
+* Diamond monotone and tradeoff: [688.957, 700.000]
-{{Multival|legend=1| 2 8 8 8 7 -4 }}
 {{Val list|legend=1| 12, 26, 38, 102bcd, 140bccd, 178bbccdd }}
@@ Line 1,110: / Line 1,105: @@
 * [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3 Two Pairs of Socks] (in [[26edo|26EDO]]) by [[Igliashon Jones]]
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 1,122: / Line 1,117: @@
 Tuning ranges:
-* [[diamond monotone]] range: [685.714, 700.000] (8\14 to 7\12)
+* Diamond monotone range: [685.714, 700.000] (8\14 to 7\12)
-* [[diamond tradeoff]] range: [682.458, 701.955]
+* Diamond tradeoff range: [682.458, 701.955]
-* diamond monotone and tradeoff range: [685.714, 700.000]
+* Diamond monotone and tradeoff: [685.714, 700.000]
 Vals: {{Val list| 12, 14c, 26, 90bce, 116bcce }}
@@ Line 1,130: / Line 1,125: @@
 Badness: 0.023124
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 1,142: / Line 1,137: @@
 Tuning ranges:
-* [[diamond monotone]] range: 692.308 (15\26)
+* Diamond monotone range: 692.308 (15\26)
-* [[diamond tradeoff]] range: [682.458, 701.955]
+* Diamond tradeoff range: [682.458, 701.955]
-* diamond monotone and tradeoff range: 692.308 (15\26)
+* Diamond monotone and tradeoff: 692.308 (15\26)
 Vals: {{Val list| 12f, 14cf, 26, 38e }}
@@ Line 1,150: / Line 1,145: @@
 Badness: 0.021565
-==== Enjera  ====
+==== Enjera ====
 Subgroup: 2.3.5.7.11.13
@@ Line 1,165: / Line 1,160: @@
 Badness: 0.026542
-=== Injerous  ===
+=== Injerous ===
 Subgroup: 2.3.5.7.11
@@ Line 1,180: / Line 1,175: @@
 Badness: 0.038577
-=== Lahoh  ===
+=== Lahoh ===
 Subgroup: 2.3.5.7.11
@@ Line 1,195: / Line 1,190: @@
 Badness: 0.043062
-== Mohaha  ==
+== Mohaha ==
 {{see also| Subgroup temperaments #Mohaha }}
@@ Line 1,218: / Line 1,213: @@
 Scales: [[mohaha7]], [[mohaha10]]
-=== Mohoho  ===
+=== Mohoho ===
 Subgroup: 2.3.5.11.13
@@ Line 1,237: / Line 1,232: @@
 Scales: [[mohaha7]], [[mohaha10]]
-== Mohajira  ==
+== Mohajira ==
 {{main|Mohajira}}
@@ Line 1,267: / Line 1,262: @@
 Scales: [[mohaha7]], [[mohaha10]]
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 1,289: / Line 1,284: @@
 Scales: [[mohaha7]], [[mohaha10]]
-=== 13-limit  ===
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
@@ Line 1,306: / Line 1,301: @@
 Scales: [[mohaha7]], [[mohaha10]]
-=== 17-limit  ===
+=== 17-limit ===
 Subgroup: 2.3.5.7.11.13.17
@@ Line 1,323: / Line 1,318: @@
 Scales: [[mohaha7]], [[mohaha10]]
-=== 19-limit  ===
+=== 19-limit ===
 Subgroup: 2.3.5.7.11.13.17.19
@@ Line 1,340: / Line 1,335: @@
 Scales: [[mohaha7]], [[mohaha10]]
-== Mohamaq  ==
+== Mohamaq ==
 Subgroup: 2.3.5.7
@@ Line 1,357: / Line 1,352: @@
 Scales: [[mohaha7]], [[mohaha10]]
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 1,374: / Line 1,369: @@
 Scales: [[mohaha7]], [[mohaha10]]
-=== 13-limit  ===
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
@@ Line 1,391: / Line 1,386: @@
 Scales: [[mohaha7]], [[mohaha10]]
-== Orphic  ==
+== Orphic ==
 Subgroup: 2.3.5.7
@@ Line 1,408: / Line 1,403: @@
 [[Badness]]: 0.258825
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 1,423: / Line 1,418: @@
 Badness: 0.101499
-=== 13-limit  ===
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
@@ Line 1,438: / Line 1,433: @@
 Badness: 0.053482
-== Mothra  ==
+== Mothra ==
 Mothra splits the fifth into three 8/7 generators. It uses [[1029/1024]], the gamelisma, to accomplish this deed and also tempers out [[1728/1715]], the orwell comma. Using [[31edo|31EDO]] with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra. In the 2.3.7 subgroup, mothra is identical to [[slendric]].
@@ Line 1,466: / Line 1,461: @@
 [[Badness]]: 0.037146
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 1,481: / Line 1,476: @@
 Badness: 0.025642
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 1,496: / Line 1,491: @@
 Badness: 0.023954
-=== Cynder  ===
+; Music:
+* [http://micro.soonlabel.com/16-ET/mothra/20141028_mothra16br4.mp3 Prelude for solo piano in mothra16, brat 4 tuning] by [http://chrisvaisvil.com/prelude-for-solo-piano-in-mothra16-brat-4-tuning/ Chris Vaisvil]
+=== Cynder ===
 Subgroup: 2.3.5.7.11
@@ Line 1,511: / Line 1,509: @@
 Badness: 0.055706
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 1,526: / Line 1,524: @@
 Badness: 0.034124
-=== Mosura  ===
+=== Mosura ===
 Subgroup: 2.3.5.7.11
@@ Line 1,541: / Line 1,539: @@
 Badness: 0.0313
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 1,556: / Line 1,554: @@
 Badness: 0.036857
-==== Music  ====
+== Squares ==
-* [http://micro.soonlabel.com/16-ET/mothra/20141028_mothra16br4.mp3 Prelude for solo piano in mothra16, brat 4 tuning] by [http://chrisvaisvil.com/prelude-for-solo-piano-in-mothra16-brat-4-tuning/ Chris Vaisvil]
-== Squares  ==
 Squares splits the interval of an eleventh, or 8/3, into four supermajor third ([[9/7]]) intervals, and uses it for a generator. [[31edo|31EDO]], with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out [[2401/2400]], the breedsma, as well as [[2430/2401]].
@@ Line 1,590: / Line 1,585: @@
 * [http://clones.soonlabel.com/public/micro/tuning-survey/daily20100603-squares8piano.mp3 Square 8] by [[Chris Vaisvil]]
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 1,607: / Line 1,602: @@
 Scales: [[skwares8]], [[skwares11]], [[skwares14]]
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 1,624: / Line 1,619: @@
 Scales: [[skwares8]], [[skwares11]], [[skwares14]]
-==== Agora  ====
+==== Agora ====
 Subgroup: 2.3.5.7.11.13
@@ Line 1,641: / Line 1,636: @@
 Scales: [[skwares8]], [[skwares11]], [[skwares14]]
-===== 17-limit  =====
+===== 17-limit =====
 Subgroup: 2.3.5.7.11.13.17
@@ Line 1,658: / Line 1,653: @@
 Scales: [[skwares8]], [[skwares11]], [[skwares14]]
-===== 19-limit  =====
+===== 19-limit =====
 Subgroup: 2.3.5.7.11.13.17.19
@@ Line 1,669: / Line 1,664: @@
 POTE generator: ~9/7 = 426.225
-{{Val list|legend=1| 14cf, 31, 76e }}
+Vals: {{Val list| 14cf, 31, 76e }}
 Badness: 0.018839
@@ Line 1,675: / Line 1,670: @@
 Scales: [[skwares8]], [[skwares11]], [[skwares14]]
-=== Cuboctahedra  ===
+=== Cuboctahedra ===
 Subgroup: 2.3.5.7.11
@@ Line 1,692: / Line 1,687: @@
 Scales: [[skwares8]], [[skwares11]], [[skwares14]]
-== Liese  ==
+== Liese ==
 <span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span>
@@ Line 1,720: / Line 1,715: @@
 [[Badness]]: 0.046706
-=== Liesel  ===
+=== Liesel ===
 Subgroup: 2.3.5.7.11
@@ Line 1,735: / Line 1,730: @@
 Badness: 0.040721
-==== 13-limit  ====
+==== 13-limit ====
 Liesel is a very natural 13-limit tuning, given the generator is so near 13/9.
@@ Line 1,752: / Line 1,747: @@
 Badness: 0.027304
-=== Elisa  ===
+=== Elisa ===
 Subgroup: 2.3.5.7.11
@@ Line 1,767: / Line 1,762: @@
 Badness: 0.041592
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11
@@ Line 1,782: / Line 1,777: @@
 Badness: 0.026922
-=== Lisa  ===
+=== Lisa ===
 Subgroup: 2.3.5.7.11
@@ Line 1,797: / Line 1,792: @@
 Badness: 0.054829
-==== 13-limit  ====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
@@ Line 1,812: / Line 1,807: @@
 Badness: 0.036144
-== Jerome  ==
+== Jerome ==
 Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5<sup>1/20</sup>, or 139.316 cents. While the generator represents both 13/12 and 12/11, the POTE and Hieronymus generators are close to 13/12 in size.
@@ Line 1,831: / Line 1,826: @@
 [[Badness]]: 0.108656
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 1,846: / Line 1,841: @@
 Badness: 0.047914
-=== 13-limit  ===
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
@@ Line 1,861: / Line 1,856: @@
 Badness: 0.029285
-=== 17-limit  ===
+=== 17-limit ===
 Subgroup: 2.3.5.7.11.13.17
@@ Line 1,876: / Line 1,871: @@
 Badness: 0.020878
-=== 19-limit  ===
+=== 19-limit ===
 Subgroup: 2.3.5.7.11.13.17.19
@@ Line 1,891: / Line 1,886: @@
 Badness: 0.018229
-== Cloudtone  ==
+== Cloudtone ==
-The ''cloudtone'' temperament (5&amp;50, named by [[User:Xenllium|Xenllium]]) tempers out the [[cloudy comma]], 16807/16384 and the [[81/80|syntonic comma]], 81/80 in the 7-limit. It can be extended to the 11- and 13-limit by adding 385/384 and 105/104 to the comma list in this order.
+The ''cloudtone'' temperament (5&amp;50) tempers out the [[cloudy comma]], 16807/16384 and the [[81/80|syntonic comma]], 81/80 in the 7-limit. It can be extended to the 11- and 13-limit by adding 385/384 and 105/104 to the comma list in this order.
 Subgroup: 2.3.5.7
@@ Line 1,910: / Line 1,905: @@
 [[Badness]]: 0.102256
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 1,925: / Line 1,920: @@
 Badness: 0.070378
-=== 13-limit  ===
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
@@ Line 1,940: / Line 1,935: @@
 Badness: 0.048829
-== Meanmag  ==
+== Meanmag ==
 Subgroup: 2.3.5.7
@@ Line 1,949: / Line 1,944: @@
 Mapping generators: ~25/24, ~7
-[[Wedgie]]: {{wedgie| 0 0 19 0 30 44 }}
+{{Multival|legend=1| 0 0 19 0 30 44 }}
 [[POTE generator]]: ~8/7 = 238.396
@@ Line 1,957: / Line 1,952: @@
 [[Badness]]: 0.077023
-=== 11-limit  ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 1,972: / Line 1,967: @@
 Badness: 0.066829
-=== 13-limit  ===
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
@@ Line 1,987: / Line 1,982: @@
 Badness: 0.045844
-== Undevigintone  ==
+== Undevigintone ==
 Subgroup: 2.3.5.7.11
@@ Line 2,002: / Line 1,997: @@
 Badness: 0.036387
-=== 13-limit  ===
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13