Meantone family: Difference between revisions

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.

{{main| Meantone }}

* [[diamond monotone]] range: [685.714, 720.000] (4\7 to 3\5)

* [[diamond tradeoff]] range: [694.786, 701.955] (1/3 comma to Pythagorean)

{{Val list|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b, 212bb, 293bb }}

Scales: [[meantone5]], [[meantone7]], [[meantone12]]

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 }}]]],  

* Liese, with normal list [{{Monzo| -4 4 -1 }}, {{Monzo| -9 11 0 -3 }}].

<div style="float:right">[[:de:septimal-mitteltönig|Deutsch]]</div>

{{see also| Meantone }}

[[Tuning ranges]]:  

[[Algebraic generator]]: Cybozem, the real root of 15''x''³ - 10''x''² - 18, 503.4257 cents. The recurrence converges quickly.

Scales: [[meantone5]], [[meantone7]], [[meantone12]]

{{see also| Meantone vs meanpop }}

Tuning ranges:  

Algebraic generator: Traverse, the positive real root of ''x''⁴ + 2''x'' - 13, or 696.9529 cents.

* [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]

Subgroup: 2.3.5.7.11.13

Badness: 0.018048

Subgroup: 2.3.5.7.11.13

POTE generator: ~3/2 = 697.264

Vals: {{Val list| 12, 19ef, 31, 43, 74 }}

Badness: 0.025899

Subgroup: 2.3.5.7.11.13

Badness: 0.026421

Subgroup: 2.3.5.7.11.13

Badness: 0.031433

{{see also| Meantone vs meanpop }}

Tuning ranges:  

Algebraic generator: Cybozem; or else Radieubiz, the real root of 3''x''³ + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.

* [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]

Subgroup: 2.3.5.7.11.13

Tuning ranges:  

Vals: {{Val list| 12ef, 19, 31, 50, 81, 131bd, 212bbddf }}

Badness: 0.020883

Subgroup: 2.3.5.7.11.13

Badness: 0.027666

Subgroup: 2.3.5.7.11

Badness: 0.021423

Subgroup: 2.3.5.7.11.13

Badness: 0.021182

Subgroup: 2.3.5.7.11.13

Badness: 0.024763

Subgroup: 2.3.5.7.11.13.17

Badness: 0.025535

Subgroup: 2.3.5.7.11.13.17.19

Badness: 0.022302

Subgroup: 2.3.5.7.11.13.17.19.23

Badness: 0.020139

Subgroup: 2.3.5.7.11.13.17.19.23.29

Badness: 0.018168

Subgroup: 2.3.5.7.11.13.17.19.23.29.31

Badness: 0.017069

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37

Badness: 0.016129

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41

Badness: 0.015356

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41.43

Badness: 0.013906

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41.43.47

Badness: 0.013818

Subgroup: 2.3.5.7.11.13

Badness: 0.024243

Subgroup: 2.3.5.7.11

Badness: 0.031539

Subgroup: 2.3.5.7.11.13

Badness: 0.026288

Subgroup: 2.3.5.7.11

Badness: 0.025516

Subgroup: 2.3.5.7.11.13

Badness: 0.028071

Subgroup: 2.3.5.7.11

Badness: 0.038122

Subgroup: 2.3.5.7.11.13

Badness: 0.028817

Subgroup: 2.3.5.7.11.13.17

Badness: 0.022666

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]].

[[Tuning ranges]]:  

Algebraic generator: Squarto, the positive root of 8''x''² - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.

Scales: [[flattone12]]

Subgroup: 2.3.5.7.11

Tuning ranges:  

Vals: {{Val list| 7, 19, 26, 45, 71bc, 116bcde }}

Scales: [[flattone12]]

Subgroup: 2.3.5.7.11.13

Tuning ranges:  

Vals: {{Val list| 7, 19, 26, 45f, 71bcf, 116bcdef }}

Scales: [[flattone12]]

Subgroup: 2.3.5.7.11

Badness: 0.058785

Subgroup: 2.3.5.7.11.13

Badness: 0.034316

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]].

[[Tuning ranges]]:  

{{Val list|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }}

[[Badness]]: 0.020690

Subgroup: 2.3.5.7.11

Tuning ranges:  

POTE generator: ~3/2 = 703.254

Badness: 0.024180

Subgroup: 2.3.5.7.11.13

Tuning ranges:  

Vals: {{Val list| 12f, 17c, 29cdef }}

Badness: 0.024108

Subgroup: 2.3.5.7.11.13

Badness: 0.027295

Subgroup: 2.3.5.7.11

Badness: 0.021978

Subgroup: 2.3.5.7.11.13

Badness: 0.027039

Subgroup: 2.3.5.7.11.13.17

Badness: 0.024539

Subgroup: 2.3.5.7.11.13.17.19

Badness: 0.020398

Subgroup: 2.3.5.7.11.13

Badness: 0.018289

Subgroup: 2.3.5.7.11

Badness: 0.036562

Subgroup: 2.3.5.7.11.13

Badness: 0.027435

Subgroup: 2.3.5.7.11

Badness: 0.026141

Subgroup: 2.3.5.7.11.13

Badness: 0.023300

Subgroup: 2.3.5.7.11.13.17

Badness: 0.024535

Subgroup: 2.3.5.7.11.13.17.19

Badness: 0.021098

<span style="display: block; text-align: right;">[[:de:maqamisch|Deutsch]]</span>

{{main| Maqamic }}

Badness: 0.040240

Subgroup: 2.3.5.7.11.13

Badness: 0.027214

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.

[[Badness]]: 0.024848

Subgroup: 2.3.5.7.11

Badness: 0.025167

Subgroup: 2.3.5.7

[[Badness]]: 0.045275

Subgroup: 2.3.5.7.11

Badness: 0.032521

Subgroup: 2.3.5.7

[[Badness]]: 0.134204

Subgroup: 2.3.5.7.11

Badness: 0.063262

Subgroup: 2.3.5.7.11.13

Badness: 0.040324

<span style="display: block; text-align: right;">[[:de:Semiphor,_Semaphor,_Godzilla|Deutsch]]</span>

{{main| Semaphore and Godzilla }}

[[Tuning ranges]]:  

{{Val list|legend=1| 5, 14c, 19 }}

[[Badness]]: 0.026747

Subgroup: 2.3.5.7.11

Tuning ranges:  

Vals: {{Val list| 14c, 19, 33cd, 52cd }}

Badness: 0.028947

Subgroup: 2.3.5.7.11.13

Tuning ranges:  

Vals: {{Val list| 14cf, 19, 33cdff, 52cdff }}

Badness: 0.022503

Subgroup: 2.3.5.7.11

Badness: 0.028510

Subgroup: 2.3.5.7.11

Badness: 0.039647

Subgroup: 2.3.5.7.11.13

Badness: 0.025676

Subgroup: 2.3.5.7.11

Badness: 0.035673

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.

Mapping generators: ~7/5, ~3

[[POTE generator]]: ~3/2 = 694.375

[[Tuning ranges]]:  

{{Val list|legend=1| 12, 26, 38, 102bcd, 140bccd, 178bbccdd }}

* [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3 Two Pairs of Socks] (in [[26edo|26EDO]]) by [[Igliashon Jones]]

Subgroup: 2.3.5.7.11

Tuning ranges:  

Vals: {{Val list| 12, 14c, 26, 90bce, 116bcce }}

Badness: 0.023124

Subgroup: 2.3.5.7.11.13

Tuning ranges:  

Vals: {{Val list| 12f, 14cf, 26, 38e }}

Badness: 0.021565

Subgroup: 2.3.5.7.11.13

Badness: 0.026542

Subgroup: 2.3.5.7.11

Badness: 0.038577

Subgroup: 2.3.5.7.11

Badness: 0.043062

{{see also| Subgroup temperaments #Mohaha }}

Scales: [[mohaha7]], [[mohaha10]]

Subgroup: 2.3.5.11.13

Scales: [[mohaha7]], [[mohaha10]]

{{main|Mohajira}}

Scales: [[mohaha7]], [[mohaha10]]

Subgroup: 2.3.5.7.11

Scales: [[mohaha7]], [[mohaha10]]

Subgroup: 2.3.5.7.11.13

Scales: [[mohaha7]], [[mohaha10]]

Subgroup: 2.3.5.7.11.13.17

Scales: [[mohaha7]], [[mohaha10]]

Subgroup: 2.3.5.7.11.13.17.19

Scales: [[mohaha7]], [[mohaha10]]

Subgroup: 2.3.5.7

Scales: [[mohaha7]], [[mohaha10]]

Subgroup: 2.3.5.7.11

Scales: [[mohaha7]], [[mohaha10]]

Subgroup: 2.3.5.7.11.13

Scales: [[mohaha7]], [[mohaha10]]

Subgroup: 2.3.5.7

[[Badness]]: 0.258825

Subgroup: 2.3.5.7.11

Badness: 0.101499

Subgroup: 2.3.5.7.11.13

Badness: 0.053482

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]].

[[Badness]]: 0.037146

Subgroup: 2.3.5.7.11

Badness: 0.025642

Subgroup: 2.3.5.7.11.13

Badness: 0.023954

Subgroup: 2.3.5.7.11

Badness: 0.055706

Subgroup: 2.3.5.7.11.13

Badness: 0.034124

Subgroup: 2.3.5.7.11

Badness: 0.0313

Subgroup: 2.3.5.7.11.13

Badness: 0.036857

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]].

* [http://clones.soonlabel.com/public/micro/tuning-survey/daily20100603-squares8piano.mp3 Square 8] by [[Chris Vaisvil]]

Subgroup: 2.3.5.7.11

Scales: [[skwares8]], [[skwares11]], [[skwares14]]

Subgroup: 2.3.5.7.11.13

Scales: [[skwares8]], [[skwares11]], [[skwares14]]

Subgroup: 2.3.5.7.11.13

Scales: [[skwares8]], [[skwares11]], [[skwares14]]

Subgroup: 2.3.5.7.11.13.17

Scales: [[skwares8]], [[skwares11]], [[skwares14]]

Subgroup: 2.3.5.7.11.13.17.19

POTE generator: ~9/7 = 426.225

Badness: 0.018839

Scales: [[skwares8]], [[skwares11]], [[skwares14]]

Subgroup: 2.3.5.7.11

Scales: [[skwares8]], [[skwares11]], [[skwares14]]

<span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span>

[[Badness]]: 0.046706

Subgroup: 2.3.5.7.11

Badness: 0.040721

Liesel is a very natural 13-limit tuning, given the generator is so near 13/9.

Badness: 0.027304

Subgroup: 2.3.5.7.11

Badness: 0.041592

Subgroup: 2.3.5.7.11

Badness: 0.026922

Subgroup: 2.3.5.7.11

Badness: 0.054829

Subgroup: 2.3.5.7.11.13

Badness: 0.036144

Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5^1/20, 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.

[[Badness]]: 0.108656

Subgroup: 2.3.5.7.11

Badness: 0.047914

Subgroup: 2.3.5.7.11.13

Badness: 0.029285

Subgroup: 2.3.5.7.11.13.17

Badness: 0.020878

Subgroup: 2.3.5.7.11.13.17.19

Badness: 0.018229

Subgroup: 2.3.5.7

[[Badness]]: 0.102256

Subgroup: 2.3.5.7.11

Badness: 0.070378

Subgroup: 2.3.5.7.11.13

Badness: 0.048829

Subgroup: 2.3.5.7

Mapping generators: ~25/24, ~7

[[POTE generator]]: ~8/7 = 238.396

[[Badness]]: 0.077023

Subgroup: 2.3.5.7.11

Badness: 0.066829

Subgroup: 2.3.5.7.11.13

Badness: 0.045844

Subgroup: 2.3.5.7.11

Badness: 0.036387

Subgroup: 2.3.5.7.11.13