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Bird's eye view of temperaments by accuracy: Difference between revisions

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[[Bird's eye view of temperaments by accuracy#Generator tunings|Generator tunings]]: (3\72, 1\9), (4\99, 1\9), (7\171, 1\9), (11\270, 1\9)
[[Bird's eye view of temperaments by accuracy#Generator tunings|Generator tunings]]: (3\72, 1\9), (4\99, 1\9), (7\171, 1\9), (11\270, 1\9)


Ennealimmal has a 1/9-octave period representing [[27/25]], and two of them represent [[7/6]], tempering out [[4375/4374]]. It is generated by a [[~]][[36/35]] quartertone, with 3/2 being mapped to 6 periods minus 2 generators, 5/4 mapped to 4 periods minus 3 generators, and 7/4 mapped to 8 periods minus 2 generators. It finds a neutral third representing [[49/40]]~[[60/49]] at 3 periods minus 1 generator, tempering out [[2401/2400]]. This temperament therefore tempers out the two smallest superparticular ratios in the [[7-limit]], 2401/2400 and 4375/4374. It is very accurate, with errors of around 0.2 cents in optimized tunings. However, due to the high note count, one may prefer lower accuracy temperaments.
Ennealimmal has a 1/9-octave period representing [[27/25]], and two of them represent [[7/6]], tempering out [[4375/4374]]. It is generated by a [[~]][[36/35]] quartertone, with 3/2 being mapped to 6 periods minus 2 generators, 5/4 mapped to 4 periods minus 3 generators, and 7/4 mapped to 8 periods minus 2 generators. It finds a neutral third representing [[49/40]]~[[60/49]] at 3 periods minus 1 generator, tempering out [[2401/2400]].  
 
This temperament therefore tempers out the two smallest superparticular ratios in the [[7-limit]], 2401/2400 and 4375/4374. It is very accurate, with errors of around 0.2 cents in optimized tunings. However, due to the high note count, one may prefer lower accuracy temperaments.


=== 11-limit focus ===
=== 11-limit focus ===
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Gariwizmic has a 1/2-octave period representing [[99/70]], two of them being ~2/1 and tempering out [[9801/9800]]. It is generated by a slightly sharp fifth with the 2.3.7.11 mappings of [[gary]], tempering out [[19712/19683]] and [[131072/130977]], which essentially maps the pythagorean comma to 64/63, and two of those to 33/32.  
Gariwizmic has a 1/2-octave period representing [[99/70]], two of them being ~2/1 and tempering out [[9801/9800]]. It is generated by a slightly sharp fifth with the 2.3.7.11 mappings of [[gary]], tempering out [[19712/19683]] and [[131072/130977]], which essentially maps the pythagorean comma to 64/63, and two of those to 33/32.  


The kalisma allows the pythagorean comma to be split into two [[2835/2816|fwiwismas]], and this allows reaching a [[352/351]] ~ [[385/384]] minicomma by 47 fifths plus a semioctave, or alternatively put, a ~[[256/243|limma]] minus 3.5 pythcommas, tempering out [[4096/4095]] and [[1716/1715]]. Primes 5 and 13 are thus reached by a diminished fourth (96/77) + minicomma (39 fifths + 1 period), and a triply augmented fourth (44/27) - minicomma (.
The kalisma allows the pythagorean comma to be split into two [[2835/2816|fwiwismas]], and this allows reaching a [[352/351]] ~ [[385/384]] minicomma by 47 fifths plus a semioctave, or alternatively put, a ~[[256/243|limma]] minus 3.5 pythcommas, tempering out [[4096/4095]] and [[1716/1715]]. Primes 5 and 13 are thus reached by a diminished fourth (96/77) + minicomma (39 fifths + 1 period), and a triply augmented fourth (44/27) - minicomma (-27 fifths - 1 period).  


It is best represented in 270edo, which is well known for its astoundingly accurate 13-limit, making it one of the best fifth-based rank-2 temperaments. It can be easily thought as cassandra, but with the minicomma as a "generator" for primes 5 and 13 which is still reachable within the rank-2 structure (It isn't a true generator; were it independent, the temperament would be [[cassaschismic]]).
It is best represented in 270edo, which is well known for its astoundingly accurate 13-limit, making it one of the best fifth-based rank-2 temperaments. It is very complex mapping-wise despite its great accuracy, but it can be easily thought as cassandra, but with the minicomma as a "generator" for primes 5 and 13 (and 19) which is still reachable within the rank-2 structure (It isn't a true generator; were it independent, the temperament would be [[cassaschismic]]).


It naturally extends into the 2.3.5.7.11.13.19 subgroup by adding [[1216/1215]] to the comma list, finding the major third to be [[19/15]] and 19/16 to be the minor third + minicomma, thus also working as [[361/360]]. Interestingly, gariwizmic tempers out the smallest superparticular of the 19- and 23-limit: the [[tredekisma]].
It naturally extends into the 2.3.5.7.11.13.19 subgroup by adding [[1216/1215]] to the comma list, finding the major third to be [[19/15]] and 19/16 to be the minor third + minicomma, thus also working as [[361/360]]. Interestingly, gariwizmic tempers out the smallest superparticular of the 19- and 23-limit: the [[tredekisma]].
==== [[Decoid]] ====
Note counts: TBA
Generator tunings: (103\130, 1\10), (111\140, 1\10), (214\270, 1\10)
Decoid has a 1/10-octave period representing [[15/14]], 7 of them being [[13/8]] as in 10edo. The generator can be a [[26/15]] semitritave, tempering out [[676/675]]. Thanks to ''relatively'' good approximation of 10edo of the 2.3.5.7.13, these primes require little change, only 2, -3, -1, and 0 generators respectively. Prime 11 is a bit more complex, at -8 generators. This also tempers out [[2080/2079]], [[4096/4095]] and [[1716/1715]].
The main selling point of decoid is [http://terpstrakeyboard.com/web-app/keys.htm?fundamental=263.09212&right=27&upright=2&size=45&rotation=0&instrument=organ&enum=false&equivSteps=270&spectrum_colors=false&fundamental_color=55FF55&no_labels=false&scale=4.44444%0A8.88889%0A13.33333%0A17.77778%0A22.22222%0A26.66667%0A31.11111%0A35.55556%0A40.%0A44.44444%0A48.88889%0A53.33333%0A57.77778%0A62.22222%0A66.66667%0A71.11111%0A75.55556%0A80.%0A84.44444%0A88.88889%0A93.33333%0A97.77778%0A102.22222%0A106.66667%0A111.11111%0A115.55556%0A120.%0A124.44444%0A128.88889%0A133.33333%0A137.77778%0A142.22222%0A146.66667%0A151.11111%0A155.55556%0A160.%0A164.44444%0A168.88889%0A173.33333%0A177.77778%0A182.22222%0A186.66667%0A191.11111%0A195.55556%0A200.%0A204.44444%0A208.88889%0A213.33333%0A217.77778%0A222.22222%0A226.66667%0A231.11111%0A235.55556%0A240.%0A244.44444%0A248.88889%0A253.33333%0A257.77778%0A262.22222%0A266.66667%0A271.11111%0A275.55556%0A280.%0A284.44444%0A288.88889%0A293.33333%0A297.77778%0A302.22222%0A306.66667%0A311.11111%0A315.55556%0A320.%0A324.44444%0A328.88889%0A333.33333%0A337.77778%0A342.22222%0A346.66667%0A351.11111%0A355.55556%0A360.%0A364.44444%0A368.88889%0A373.33333%0A377.77778%0A382.22222%0A386.66667%0A391.11111%0A395.55556%0A400.%0A404.44444%0A408.88889%0A413.33333%0A417.77778%0A422.22222%0A426.66667%0A431.11111%0A435.55556%0A440.%0A444.44444%0A448.88889%0A453.33333%0A457.77778%0A462.22222%0A466.66667%0A471.11111%0A475.55556%0A480.%0A484.44444%0A488.88889%0A493.33333%0A497.77778%0A502.22222%0A506.66667%0A511.11111%0A515.55556%0A520.%0A524.44444%0A528.88889%0A533.33333%0A537.77778%0A542.22222%0A546.66667%0A551.11111%0A555.55556%0A560.%0A564.44444%0A568.88889%0A573.33333%0A577.77778%0A582.22222%0A586.66667%0A591.11111%0A595.55556%0A600.%0A604.44444%0A608.88889%0A613.33333%0A617.77778%0A622.22222%0A626.66667%0A631.11111%0A635.55556%0A640.%0A644.44444%0A648.88889%0A653.33333%0A657.77778%0A662.22222%0A666.66667%0A671.11111%0A675.55556%0A680.%0A684.44444%0A688.88889%0A693.33333%0A697.77778%0A702.22222%0A706.66667%0A711.11111%0A715.55556%0A720.%0A724.44444%0A728.88889%0A733.33333%0A737.77778%0A742.22222%0A746.66667%0A751.11111%0A755.55556%0A760.%0A764.44444%0A768.88889%0A773.33333%0A777.77778%0A782.22222%0A786.66667%0A791.11111%0A795.55556%0A800.%0A804.44444%0A808.88889%0A813.33333%0A817.77778%0A822.22222%0A826.66667%0A831.11111%0A835.55556%0A840.%0A844.44444%0A848.88889%0A853.33333%0A857.77778%0A862.22222%0A866.66667%0A871.11111%0A875.55556%0A880.%0A884.44444%0A888.88889%0A893.33333%0A897.77778%0A902.22222%0A906.66667%0A911.11111%0A915.55556%0A920.%0A924.44444%0A928.88889%0A933.33333%0A937.77778%0A942.22222%0A946.66667%0A951.11111%0A955.55556%0A960.%0A964.44444%0A968.88889%0A973.33333%0A977.77778%0A982.22222%0A986.66667%0A991.11111%0A995.55556%0A1000.%0A1004.44444%0A1008.88889%0A1013.33333%0A1017.77778%0A1022.22222%0A1026.66667%0A1031.11111%0A1035.55556%0A1040.%0A1044.44444%0A1048.88889%0A1053.33333%0A1057.77778%0A1062.22222%0A1066.66667%0A1071.11111%0A1075.55556%0A1080.%0A1084.44444%0A1088.88889%0A1093.33333%0A1097.77778%0A1102.22222%0A1106.66667%0A1111.11111%0A1115.55556%0A1120.%0A1124.44444%0A1128.88889%0A1133.33333%0A1137.77778%0A1142.22222%0A1146.66667%0A1151.11111%0A1155.55556%0A1160.%0A1164.44444%0A1168.88889%0A1173.33333%0A1177.77778%0A1182.22222%0A1186.66667%0A1191.11111%0A1195.55556%0A1200.&names=P1%0Aschismoid%0Akleismoid%0Asemicomma%0Adiacomma%0Asyncomma%0Agpcomma%0A%0A%0A%0A%0A%0A33%2F32%0AsA1%0A28%2F27%0A26%2F25%0A25%2F24%0A%0A22%2F21%0A21%2F20%0Am2%0A19%2F18%0A128%2F121%0A35%2F33%0A33%2F31%0A16%2F15%0AA1%0A%0A29%2F27%0A%0A27%2F25%0A%0A38%2F35%0An2%0A%0A%0A%0A11%2F10%0A32%2F29%0A21%2F19%0A31%2F28%0A10%2F9%0A39%2F35%0A29%2F26%0A28%2F25%0A%0AM2%0A35%2F31%0A%0A%0A%0A%0A8%2F7%0AsA2%0A%0A%0A15%2F13%0A%0A%0A%0A7%2F6%0A%0A%0A%0A%0A13%2F11%0Am3%0A19%2F16%0A%0A%0A%0A6%2F5%0A%0A%0A%0A%0A%0A39%2F32%0A11%2F9%0An3%0A%0A16%2F13%0A%0A%0A%0A%0A%0A5%2F4%0A%0A%0A%0A24%2F19%0AM3%0A80%2F63%0A14%2F11%0A%0A%0A%0A9%2F7%0Asd4%0A%0A%0A13%2F10%0A%0A%0A%0A21%2F16%0A%0A%0A%0A%0A%0A4%2F3%0A%0A%0A%0A%0A%0A%0A%0A%0A15%2F11%0A%0A%0A11%2F8%0AsA4%0A864%2F625%0A18%2F13%0A25%2F18%0A%0A%0A7%2F5%0Ad5%0A45%2F32%0A%0A%0A%0A64%2F45%0AA4%0A10%2F7%0A%0A%0A%0A13%2F9%0A%0Asd5%0A16%2F11%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0AP5%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A20%2F13%0A%0A%0AsA5%0A14%2F9%0A%0A%0A%0A11%2F7%0A%0Am6%0A19%2F12%0A%0A%0A%0A8%2F5%0A%0A%0A%0A21%2F13%0A%0A13%2F8%0A%0An6%0A18%2F11%0A%0A%0A33%2F20%0A%0A%0A%0A5%2F3%0A%0A%0A%0A%0AM6%0A%0A%0A%0A%0A%0A12%2F7%0Asd7%0A%0A19%2F11%0A%0A%0A%0A%0A7%2F4%0A%0A%0A%0A%0A%0Am7%0A%0A%0A%0A%0A9%2F5%0A%0A%0A29%2F16%0A%0A%0A%0A11%2F6%0An7%0A%0A%0A%0A13%2F7%0A%0A%0A%0A15%2F8%0A%0A%0A%0A%0AM7%0A%0A%0A%0A%0A%0A%0Asd8%0A31%2F16%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A2%2F1&note_colors=ffffff%0A000000%0A000000%0A000000%0A000000%0A000000%0A000000%0A000000%0A000000%0A000000%0A000000%0A000000%0A800080%0A000000%0A000000%0A000000%0A000000%0A000000%0A000000%0A000000%0A804000%0A000000%0A000000%0A000000%0A000000%0Affff00%0A000000%0A000000%0A000000%0A000000%0A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its rank-2 layout] which has a a lot of important intervals close together, namely, all the primes except 11 are close to horizontal, and the fact that it is supported too by 270edo, making it an incredible 13-limit temperament. Prime 19 can also be reached by tempering [[1216/1215]], and is thus reached by 7 generators.


=== Higher-limit focus ===
=== Higher-limit focus ===
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=== No-3's focus ===
=== No-3's focus ===
=== No-5's focus ===
=== No-5's focus ===
== '''High accuracy (<4c)''' ==
== '''High accuracy (<4c)''' ==
The bound is the approximate [[JND|melodic JND (Just-Noticeable-Difference)]], though note that this doesn't mean that damage/mistuning is ''imperceptible'' in these temperaments as the harmonic JND can often be significantly smaller, depending largely on context, timbre and who is listening/who you ask.
The bound is the approximate [[JND|melodic JND (Just-Noticeable-Difference)]], though note that this doesn't mean that damage/mistuning is ''imperceptible'' in these temperaments as the harmonic JND can often be significantly smaller, depending largely on context, timbre and who is listening/who you ask.
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