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'''This page has been deprecated, but it is being kept in place for longevity as a reference material. Please see [[User:BudjarnLambeth/Survey of efficient temperaments by subgroup]] for the new version.''' | |||
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There are at least hundreds, probably thousands, of [[rank-2 temperament]]s described. It can be difficult to know where to start. | There are at least hundreds, probably thousands, of [[rank-2 temperament]]s described. It can be difficult to know where to start. | ||
This page is intended as that starting point. It does not aim to list every temperament. Instead it aims to list only the ones that are of | This page is intended as that starting point. It does not aim to list every temperament. Instead it aims to list only the ones that are of high interest to a sizeable number of composers or theorists. | ||
Composers and theorists disagree amongst themselves about what properties are desirable in a temperament, and you might over time find that you lean more towards one camp or another. This list arranges temperaments by their properties, allowing you the reader to seek out temperaments with whichever properties you value. | |||
== So, which temperaments should I use to make music? == | |||
Ask 5 xenharmonicists, and you'll get 10 different answers. There are many different schools of thought within RTT (regular temperament theory). | |||
Most would agree that a good temperament approximates some subset of [[just intonation]] relatively accurately with a relatively small number of notes. | |||
What they disagree on is ''how'' accurate is "relatively accurate", ''how'' small is "relatively small", and ''which'' JI subsets are interesting enough to be worth approximating. | |||
For example: | |||
'''Xenharmonicist A''' might argue that an error less than 15ish cents on most intervals, and less than 5 cents on the really important ones (like the perfect fifth and the octave), is accurate enough. | |||
And they might argue that 25 notes per [[equave]] is the most that is practical, any more than that is too cumbersome. | |||
They might argue that nobody can hear the harmonic effect of prime harmonics higher than 11. | |||
And they might argue that there's no real reason to use subgroups that are missing primes 2 or 3, because those primes are so important to consonance. | |||
'''Xenharmonicist B''' might argue that the error must be less than 5ish cents on most intervals, anything further out than that sounds out of tune to them. | |||
They might argue that it's perfectly possible to learn up to 50 notes per [[equave]]. | |||
They might argue that they can hear the subtle, delicate effect of prime harmonics up to 23. | |||
And they might argue that subgroups like 3.5.7.11 and 2.5.7.11 are the most fertile ground for new and exciting musical exploration. | |||
Neither xenharmonicist can be objectively shown to be right or wrong. There is an amount of science to this, but there is also a lot of personal subjectivity. Ultimately it's up to you to decide what features you think are important in a temperament. | |||
It might help to compare these temperaments to [[12edo]], a.k.a. the familiar 12-tone equal temperament which most modern music is tuned to by default. 12edo has, of course, 12 notes per equave, which makes it fairly small by temperament standards (but not abnormally so). | |||
It can be interpreted as a low-to-medium accuracy 5-limit temperament where the most important intervals (the fifth and octave) have an error less than 3 cents, while other notable intervals (like the thirds and sixths) have an error of about 14 cents. | |||
Alternatively, it can be interpreted as a high-accuracy 2.3.17.19 subgroup temperament, where all of the intervals have an error less than 5 cents. | |||
So that should provide a point of comparison to help measure these other temperaments against. | |||
== Guide to tables == | == Guide to tables == | ||
| Line 14: | Line 57: | ||
* Exotemperament: > ~18 [[cents]] of [[error]] on more than one targeted interval | * Exotemperament: > ~18 [[cents]] of [[error]] on more than one targeted interval | ||
* Low accuracy: < ~18c error on most targeted intervals | * Low accuracy: < ~18c error on most targeted intervals | ||
* Medium accuracy: < 12c error on | * Medium accuracy: < 12c error on most targeted intervals | ||
* High accuracy: < 7c error on almost all targeted intervals | * High accuracy: < 7c error on almost all targeted intervals | ||
* Very high accuracy: < 3.5c error on almost all targeted intervals | * Very high accuracy: < 3.5c error on almost all targeted intervals | ||
| Line 43: | Line 86: | ||
! Over 100 notes | ! Over 100 notes | ||
|- | |- | ||
! Exotemperament (18-∞c) | |||
| [[antitonic]], [[bug]], [[dicot]], [[father]] | | [[antitonic]], [[bug]], [[dicot]], [[father]] | ||
| | | | ||
| Line 50: | Line 93: | ||
| | | | ||
|- | |- | ||
! Low accuracy (12-18c) | |||
| [[augmented]], [[blackwood]], [[dimipent]], [[porcupine]], [[whitewood]] | | [[augmented]], [[blackwood]], [[dimipent]], [[porcupine]], [[whitewood]] | ||
| | | [[superpyth]] | ||
| | | | ||
| | | | ||
| | | | ||
|- | |- | ||
! Medium accuracy (7-12c) | |||
| [[meantone]] | | [[meantone]] | ||
| | | [[hanson]], [[magic]] | ||
| | | [[valentine]] | ||
| | | | ||
| | | | ||
|- | |- | ||
! High accuracy (3.5-7c) | |||
| [[diaschismic]] | | [[diaschismic]] | ||
| | | | ||
| Line 71: | Line 114: | ||
| | | | ||
|- | |- | ||
! Very high accuracy (1-3.5c) | |||
| | | | ||
| | | | ||
| Line 78: | Line 121: | ||
| | | | ||
|- | |- | ||
! Microtemperament (0-1c) | |||
| | | | ||
| | | | ||
| | | | ||
| schismic aka [[ | | schismic aka [[Helmholtz temperament|Helmholtz]] | ||
| [[kwazy]] | | [[kwazy]] | ||
|} | |} | ||
| Line 97: | Line 140: | ||
! Over 100 notes | ! Over 100 notes | ||
|- | |- | ||
! Exotemperament (18-∞c) | |||
| [[antitonic]], [[dicot]], [[father]] | | [[antitonic]], [[dicot]], [[father]] | ||
| | | | ||
| Line 104: | Line 147: | ||
| | | | ||
|- | |- | ||
! Low accuracy (12-18c) | |||
| [[blacksmith]] | | [[blacksmith]] | ||
| [[pajara]], [[porcupine]], [[whitewood]] | | [[augene]], [[godzilla]], [[pajara]], [[porcupine]], [[whitewood]] | ||
| | | | ||
| | | | ||
| | | | ||
|- | |- | ||
! Medium accuracy (7-12c) | |||
| | | | ||
| [[meantone]], [[sensi]] | | [[magic]], [[meantone]], [[mothra]], [[sensi]], [[superpyth]] | ||
| | | [[valentine]] | ||
| | | | ||
| | | | ||
|- | |- | ||
! High accuracy (3.5-7c) | |||
| | | | ||
| [[ | | [[orwell]] | ||
| [[diaschismic]], [[garibaldi]] | | [[diaschismic]], [[garibaldi]] | ||
| | | | ||
| | | | ||
|- | |- | ||
! Very high accuracy (1-3.5c) | |||
| | | | ||
| | | | ||
| [[miracle | | [[miracle]] | ||
| | | | ||
| | | | ||
|- | |- | ||
! Microtemperament (0-1c) | |||
| | | | ||
| | | | ||
| Line 151: | Line 194: | ||
! Over 100 notes | ! Over 100 notes | ||
|- | |- | ||
! Exotemperament (18-∞c) | |||
| [[antitonic]], [[dicot]], [[father]] | | [[antitonic]], [[dicot]], [[father]] | ||
| | | | ||
| Line 158: | Line 201: | ||
| | | | ||
|- | |- | ||
! Low accuracy (12-18c) | |||
| | | | ||
| [[augene]], [[blacksmith]], [[pajara]], [[porcupine]], [[whitewood]] | |||
| [[godzilla]], [[superpyth]] | |||
| | | | ||
| | | | ||
|- | |- | ||
! Medium accuracy (7-12c) | |||
| | | | ||
| | | | ||
| [[meanpop]], [[meantone]] | | [[magic]], [[meanpop]], [[meantone]], [[mothra]], [[valentine]] | ||
| | | | ||
| | | | ||
|- | |- | ||
! High accuracy (3.5-7c) | |||
| | | | ||
| | | | ||
| [[diaschismic]], [[ | | [[diaschismic]], [[orwell]] | ||
| | | | ||
| | | | ||
|- | |- | ||
! Very high accuracy (1-3.5c) | |||
| | | | ||
| | | | ||
| [[miracle | | [[miracle]] | ||
| | | | ||
| | | | ||
|- | |- | ||
! Microtemperament (0-1c) | |||
| | | | ||
| | | | ||
| Line 205: | Line 248: | ||
! Over 100 notes | ! Over 100 notes | ||
|- | |- | ||
! Exotemperament (18-∞c) | |||
| | | | ||
| | | | ||
| Line 212: | Line 255: | ||
| | | | ||
|- | |- | ||
! Low accuracy (12-18c) | |||
| | | | ||
| [[augene]], [[blacksmith]], [[pajara]], [[porcupine]], [[whitewood]] | |||
| [[superpyth]] | |||
| | | | ||
| | | | ||
|- | |- | ||
! Medium accuracy (7-12c) | |||
| | | | ||
| | | | ||
| [[meantone]] | | [[magic]], [[meantone]], [[mothra]] | ||
| | | | ||
| | | | ||
|- | |- | ||
! High accuracy (3.5-7c) | |||
| | | | ||
| | | | ||
| [[diaschismic]], [[ | | [[diaschismic]], [[orwell]] | ||
| | | | ||
| | | | ||
|- | |- | ||
! Very high accuracy (1-3.5c) | |||
| | |||
| | | | ||
| | | | ||
| | | | ||
| | | | ||
|- | |- | ||
! Microtemperament (0-1c) | |||
| | | | ||
| | | | ||
| Line 259: | Line 302: | ||
! Over 100 notes | ! Over 100 notes | ||
|- | |- | ||
! Exotemperament (18-∞c) | |||
| | | | ||
| | | | ||
| Line 266: | Line 309: | ||
| | | | ||
|- | |- | ||
! Low accuracy (12-18c) | |||
| | | | ||
| | | | ||
| Line 273: | Line 316: | ||
| | | | ||
|- | |- | ||
! Medium accuracy (7-12c) | |||
| | | | ||
| | | | ||
| Line 280: | Line 323: | ||
| | | | ||
|- | |- | ||
! High accuracy (3.5-7c) | |||
| | | | ||
| | | | ||
| Line 287: | Line 330: | ||
| | | | ||
|- | |- | ||
! Very high accuracy (1-3.5c) | |||
| | | | ||
| | | | ||
| Line 294: | Line 337: | ||
| | | | ||
|- | |- | ||
! Microtemperament (0-1c) | |||
| | | | ||
| | | | ||
| Line 313: | Line 356: | ||
! Over 100 notes | ! Over 100 notes | ||
|- | |- | ||
! Exotemperament (18-∞c) | |||
| | | | ||
| | | | ||
| Line 320: | Line 363: | ||
| | | | ||
|- | |- | ||
! Low accuracy (12-18c) | |||
| | | | ||
| | | | ||
| Line 327: | Line 370: | ||
| | | | ||
|- | |- | ||
! Medium accuracy (7-12c) | |||
| | | | ||
| | | | ||
| Line 334: | Line 377: | ||
| | | | ||
|- | |- | ||
! High accuracy (3.5-7c) | |||
| | | | ||
| | | | ||
| Line 341: | Line 384: | ||
| | | | ||
|- | |- | ||
! Very high accuracy (1-3.5c) | |||
| | | | ||
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| Line 348: | Line 391: | ||
| | | | ||
|- | |- | ||
! Microtemperament (0-1c) | |||
| | | | ||
| | | | ||
| Line 356: | Line 399: | ||
|} | |} | ||
== | == 3.5.7 and its extensions == | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 367: | Line 410: | ||
! Over 100 notes | ! Over 100 notes | ||
|- | |- | ||
! Exotemperament (18-∞c) | |||
| | | | ||
| | | | ||
| Line 374: | Line 417: | ||
| | | | ||
|- | |- | ||
! Low accuracy (12-18c) | |||
| | | | ||
| | | | ||
| Line 381: | Line 424: | ||
| | | | ||
|- | |- | ||
! Medium accuracy (7-12c) | |||
| | | | ||
| | | | ||
| Line 388: | Line 431: | ||
| | | | ||
|- | |- | ||
! High accuracy (3.5-7c) | |||
| | | | ||
| | | | ||
| Line 395: | Line 438: | ||
| | | | ||
|- | |- | ||
! Very high accuracy (1-3.5c) | |||
| | | | ||
| | | | ||
| Line 402: | Line 445: | ||
| | | | ||
|- | |- | ||
! Microtemperament (0-1c) | |||
| | | | ||
| | | | ||
| Line 410: | Line 453: | ||
|} | |} | ||
== | == Other no-2s subgroups == | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 421: | Line 464: | ||
! Over 100 notes | ! Over 100 notes | ||
|- | |- | ||
! Exotemperament (18-∞c) | |||
| | | | ||
| | | | ||
| Line 428: | Line 471: | ||
| | | | ||
|- | |- | ||
! Low accuracy (12-18c) | |||
| | | | ||
| | | | ||
| Line 435: | Line 478: | ||
| | | | ||
|- | |- | ||
! Medium accuracy (7-12c) | |||
| | | | ||
| | | | ||
| Line 442: | Line 485: | ||
| | | | ||
|- | |- | ||
! High accuracy (3.5-7c) | |||
| | | | ||
| | | | ||
| Line 449: | Line 492: | ||
| | | | ||
|- | |- | ||
! Very high accuracy (1-3.5c) | |||
| | | | ||
| | | | ||
| Line 456: | Line 499: | ||
| | | | ||
|- | |- | ||
! Microtemperament (0-1c) | |||
| | | | ||
| | | | ||
| Line 464: | Line 507: | ||
|} | |} | ||
== No- | == No-3s subgroups == | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 475: | Line 518: | ||
! Over 100 notes | ! Over 100 notes | ||
|- | |- | ||
! Exotemperament (18-∞c) | |||
| | | | ||
| | | | ||
| Line 482: | Line 525: | ||
| | | | ||
|- | |- | ||
| Low accuracy (12-18c) | ! Low accuracy (12-18c) | ||
| | |||
| | |||
| | |||
| | |||
| | |||
|- | |||
! Medium accuracy (7-12c) | |||
| | |||
| [[didacus]], [[orgone]] | |||
| | |||
| | |||
| | |||
|- | |||
! High accuracy (3.5-7c) | |||
| | |||
| | |||
| | |||
| | |||
| | |||
|- | |||
! Very high accuracy (1-3.5c) | |||
| | |||
| | |||
| | |||
| | |||
| | |||
|- | |||
! Microtemperament (0-1c) | |||
| | |||
| | |||
| | |||
| | |||
| | |||
|} | |||
== 2.3.7 and its extensions == | |||
{| class="wikitable" | |||
|+ | |||
! | |||
! Approx. 10 notes | |||
! Approx. 20 notes | |||
! Approx. 30 notes | |||
! Approx. 70 notes | |||
! Over 100 notes | |||
|- | |||
! Exotemperament (18-∞c) | |||
| | |||
| | |||
| | |||
| | |||
| | |||
|- | |||
! Low accuracy (12-18c) | |||
| [[semaphore]] | | [[semaphore]] | ||
| | | | ||
| Line 489: | Line 586: | ||
| | | | ||
|- | |- | ||
| Medium accuracy (7-12c) | ! Medium accuracy (7-12c) | ||
| | |||
| [[bleu]], [[slendric]] | |||
| | |||
| | |||
| | |||
|- | |||
! High accuracy (3.5-7c) | |||
| | |||
| | |||
| | |||
| | |||
| | |||
|- | |||
! Very high accuracy (1-3.5c) | |||
| | |||
| | |||
| | |||
| | |||
| | |||
|- | |||
! Microtemperament (0-1c) | |||
| | |||
| | |||
| | |||
| | |||
| | |||
|} | |||
== 2.3.11 and its extensions == | |||
{| class="wikitable" | |||
|+ | |||
! | |||
! Approx. 10 notes | |||
! Approx. 20 notes | |||
! Approx. 30 notes | |||
! Approx. 70 notes | |||
! Over 100 notes | |||
|- | |||
! Exotemperament (18-∞c) | |||
| | |||
| | |||
| | |||
| | |||
| | |||
|- | |||
! Low accuracy (12-18c) | |||
| | |||
| | |||
| | |||
| | |||
| | |||
|- | |||
! Medium accuracy (7-12c) | |||
| | |||
| [[neutral]] (no-5 no-7 [[rastmic]]) | |||
| | |||
| | |||
| | |||
|- | |||
! High accuracy (3.5-7c) | |||
| | |||
| no-5 no-7 [[pythrabian]] | |||
| | |||
| | |||
| | |||
|- | |||
! Very high accuracy (1-3.5c) | |||
| | |||
| | |||
| [[tribilo]] (no-5 no-7 [[nexus]]) | |||
| | |||
| | |||
|- | |||
! Microtemperament (0-1c) | |||
| | |||
| | |||
| | |||
| | |||
| no-5 no-7 [[frameshift]] | |||
|} | |||
== 2.3.13, 2.3.17, etc == | |||
{| class="wikitable" | |||
|+ | |||
! | |||
! Approx. 10 notes | |||
! Approx. 20 notes | |||
! Approx. 30 notes | |||
! Approx. 70 notes | |||
! Over 100 notes | |||
|- | |||
! Exotemperament (18-∞c) | |||
| | |||
| | |||
| | |||
| | |||
| | |||
|- | |||
! Low accuracy (12-18c) | |||
| | |||
| | |||
| | |||
| | |||
| | |||
|- | |||
! Medium accuracy (7-12c) | |||
| | | | ||
| | | | ||
| Line 496: | Line 701: | ||
| | | | ||
|- | |- | ||
! High accuracy (3.5-7c) | |||
| | | | ||
| | | | ||
| Line 503: | Line 708: | ||
| | | | ||
|- | |- | ||
! Very high accuracy (1-3.5c) | |||
| | | | ||
| | | | ||
| Line 510: | Line 715: | ||
| | | | ||
|- | |- | ||
! Microtemperament (0-1c) | |||
| | | | ||
| | | | ||
| Line 518: | Line 723: | ||
|} | |} | ||
== No-7s == | == No-7s subgroups == | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 529: | Line 734: | ||
! Over 100 notes | ! Over 100 notes | ||
|- | |- | ||
! Exotemperament (18-∞c) | |||
| | | | ||
| | | | ||
| Line 536: | Line 741: | ||
| | | | ||
|- | |- | ||
! Low accuracy (12-18c) | |||
| | | | ||
| | | | ||
| Line 543: | Line 748: | ||
| | | | ||
|- | |- | ||
! Medium accuracy (7-12c) | |||
| | | | ||
| [[mohaha]] | | [[mohaha]] | ||
| Line 550: | Line 755: | ||
| | | | ||
|- | |- | ||
! High accuracy (3.5-7c) | |||
| | | | ||
| [[sensible]], [[srutal archagall]] | | [[sensible]], [[srutal archagall]] | ||
| Line 557: | Line 762: | ||
| | | | ||
|- | |- | ||
! Very high accuracy (1-3.5c) | |||
| | | | ||
| [[cata]], [[nestoria]], [[sensipent]] | | [[cata]], [[nestoria]], [[sensipent]] | ||
| Line 564: | Line 769: | ||
| | | | ||
|- | |- | ||
! Microtemperament (0-1c) | |||
| | | | ||
| | | | ||
| Line 572: | Line 777: | ||
|} | |} | ||
== No-11s == | == No-11s subgroups == | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 583: | Line 788: | ||
! Over 100 notes | ! Over 100 notes | ||
|- | |- | ||
! Exotemperament (18-∞c) | |||
| | | | ||
| | | | ||
| Line 590: | Line 795: | ||
| | | | ||
|- | |- | ||
! Low accuracy (12-18c) | |||
| | | | ||
| | | | ||
| Line 597: | Line 802: | ||
| | | | ||
|- | |- | ||
! Medium accuracy (7-12c) | |||
| | | | ||
| | | | ||
| Line 604: | Line 809: | ||
| | | | ||
|- | |- | ||
! High accuracy (3.5-7c) | |||
| | | | ||
| | | | ||
| Line 611: | Line 816: | ||
| | | | ||
|- | |- | ||
! Very high accuracy (1-3.5c) | |||
| | | | ||
| | | | ||
| Line 618: | Line 823: | ||
| | | | ||
|- | |- | ||
! Microtemperament (0-1c) | |||
| | | | ||
| | | | ||
| Line 626: | Line 831: | ||
|} | |} | ||
== | == Other subgroups == | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 637: | Line 842: | ||
! Over 100 notes | ! Over 100 notes | ||
|- | |- | ||
! Exotemperament (18-∞c) | |||
| | | | ||
| | | | ||
| Line 644: | Line 849: | ||
| | | | ||
|- | |- | ||
! Low accuracy (12-18c) | |||
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| Line 651: | Line 856: | ||
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|- | |- | ||
! Medium accuracy (7-12c) | |||
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| Line 658: | Line 863: | ||
| | | | ||
|- | |- | ||
! High accuracy (3.5-7c) | |||
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| Line 665: | Line 870: | ||
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|- | |- | ||
! Very high accuracy (1-3.5c) | |||
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| Line 672: | Line 877: | ||
| | | | ||
|- | |- | ||
! Microtemperament (0-1c) | |||
| | | | ||
| | | | ||
Latest revision as of 15:28, 16 April 2025
This page has been deprecated, but it is being kept in place for longevity as a reference material. Please see User:BudjarnLambeth/Survey of efficient temperaments by subgroup for the new version.
There are at least hundreds, probably thousands, of rank-2 temperaments described. It can be difficult to know where to start.
This page is intended as that starting point. It does not aim to list every temperament. Instead it aims to list only the ones that are of high interest to a sizeable number of composers or theorists.
Composers and theorists disagree amongst themselves about what properties are desirable in a temperament, and you might over time find that you lean more towards one camp or another. This list arranges temperaments by their properties, allowing you the reader to seek out temperaments with whichever properties you value.
So, which temperaments should I use to make music?
Ask 5 xenharmonicists, and you'll get 10 different answers. There are many different schools of thought within RTT (regular temperament theory).
Most would agree that a good temperament approximates some subset of just intonation relatively accurately with a relatively small number of notes.
What they disagree on is how accurate is "relatively accurate", how small is "relatively small", and which JI subsets are interesting enough to be worth approximating.
For example:
Xenharmonicist A might argue that an error less than 15ish cents on most intervals, and less than 5 cents on the really important ones (like the perfect fifth and the octave), is accurate enough.
And they might argue that 25 notes per equave is the most that is practical, any more than that is too cumbersome.
They might argue that nobody can hear the harmonic effect of prime harmonics higher than 11.
And they might argue that there's no real reason to use subgroups that are missing primes 2 or 3, because those primes are so important to consonance.
Xenharmonicist B might argue that the error must be less than 5ish cents on most intervals, anything further out than that sounds out of tune to them.
They might argue that it's perfectly possible to learn up to 50 notes per equave.
They might argue that they can hear the subtle, delicate effect of prime harmonics up to 23.
And they might argue that subgroups like 3.5.7.11 and 2.5.7.11 are the most fertile ground for new and exciting musical exploration.
Neither xenharmonicist can be objectively shown to be right or wrong. There is an amount of science to this, but there is also a lot of personal subjectivity. Ultimately it's up to you to decide what features you think are important in a temperament.
It might help to compare these temperaments to 12edo, a.k.a. the familiar 12-tone equal temperament which most modern music is tuned to by default. 12edo has, of course, 12 notes per equave, which makes it fairly small by temperament standards (but not abnormally so).
It can be interpreted as a low-to-medium accuracy 5-limit temperament where the most important intervals (the fifth and octave) have an error less than 3 cents, while other notable intervals (like the thirds and sixths) have an error of about 14 cents.
Alternatively, it can be interpreted as a high-accuracy 2.3.17.19 subgroup temperament, where all of the intervals have an error less than 5 cents.
So that should provide a point of comparison to help measure these other temperaments against.
Guide to tables
Rows
The rows categorise temperaments by accuracy. That is, how closely they approximate just intonation intervals. The categories are:
- Exotemperament: > ~18 cents of error on more than one targeted interval
- Low accuracy: < ~18c error on most targeted intervals
- Medium accuracy: < 12c error on most targeted intervals
- High accuracy: < 7c error on almost all targeted intervals
- Very high accuracy: < 3.5c error on almost all targeted intervals
- Microtemperaments: < 1c on all targeted intervals
The definition of "targeted interval" is left deliberately vague, because some temperaments serve a specific purpose and must be assessed differently. In most cases on this page, it refers to the set of intervals that occur in a tonality diamond of the temperament's subgroup.
Columns
The columns categorise temperaments by complexity.
Rank 2 temperaments can generate scales with any number of notes per equave. However, if they have too few notes, they won't be able approximate enough targeted intervals to be useful, and if they have too many notes, they will be filled with extra notes that don't serve much purpose and get in the way. Just how many notes is about right, varies from temperament to temperament. In layman’s terms: More notes needed = more complexity, less notes needed = less complexity. The real definition of complexity is more involved and rigorous than this, but this is good enough for the purposes of a broad overview page.
Subgroup categorisation
If a temperament fits under multiple subgroup headings (e.g. both No-2s and No-5s) it should be placed only under the lowest numbered heading (in this example, No-2s).
5-limit
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | antitonic, bug, dicot, father | ||||
| Low accuracy (12-18c) | augmented, blackwood, dimipent, porcupine, whitewood | superpyth | |||
| Medium accuracy (7-12c) | meantone | hanson, magic | valentine | ||
| High accuracy (3.5-7c) | diaschismic | ||||
| Very high accuracy (1-3.5c) | wuerschmidt | ||||
| Microtemperament (0-1c) | schismic aka Helmholtz | kwazy |
7-limit
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | antitonic, dicot, father | ||||
| Low accuracy (12-18c) | blacksmith | augene, godzilla, pajara, porcupine, whitewood | |||
| Medium accuracy (7-12c) | magic, meantone, mothra, sensi, superpyth | valentine | |||
| High accuracy (3.5-7c) | orwell | diaschismic, garibaldi | |||
| Very high accuracy (1-3.5c) | miracle | ||||
| Microtemperament (0-1c) | ennealimmal | enneadecal, trinity |
11-limit
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | antitonic, dicot, father | ||||
| Low accuracy (12-18c) | augene, blacksmith, pajara, porcupine, whitewood | godzilla, superpyth | |||
| Medium accuracy (7-12c) | magic, meanpop, meantone, mothra, valentine | ||||
| High accuracy (3.5-7c) | diaschismic, orwell | ||||
| Very high accuracy (1-3.5c) | miracle | ||||
| Microtemperament (0-1c) | ennealimmal | enneadecal, trinity |
13-limit
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | |||||
| Low accuracy (12-18c) | augene, blacksmith, pajara, porcupine, whitewood | superpyth | |||
| Medium accuracy (7-12c) | magic, meantone, mothra | ||||
| High accuracy (3.5-7c) | diaschismic, orwell | ||||
| Very high accuracy (1-3.5c) | |||||
| Microtemperament (0-1c) | ennealimmal | enneadecal, trinity |
17-limit
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | |||||
| Low accuracy (12-18c) | pajara | ||||
| Medium accuracy (7-12c) | |||||
| High accuracy (3.5-7c) | diaschismic | ||||
| Very high accuracy (1-3.5c) | |||||
| Microtemperament (0-1c) | ennealimmal | trinity |
Higher limits
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | |||||
| Low accuracy (12-18c) | |||||
| Medium accuracy (7-12c) | |||||
| High accuracy (3.5-7c) | |||||
| Very high accuracy (1-3.5c) | |||||
| Microtemperament (0-1c) | ennealimmal | trinity |
3.5.7 and its extensions
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | |||||
| Low accuracy (12-18c) | |||||
| Medium accuracy (7-12c) | |||||
| High accuracy (3.5-7c) | |||||
| Very high accuracy (1-3.5c) | |||||
| Microtemperament (0-1c) |
Other no-2s subgroups
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | |||||
| Low accuracy (12-18c) | |||||
| Medium accuracy (7-12c) | |||||
| High accuracy (3.5-7c) | |||||
| Very high accuracy (1-3.5c) | |||||
| Microtemperament (0-1c) |
No-3s subgroups
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | |||||
| Low accuracy (12-18c) | |||||
| Medium accuracy (7-12c) | didacus, orgone | ||||
| High accuracy (3.5-7c) | |||||
| Very high accuracy (1-3.5c) | |||||
| Microtemperament (0-1c) |
2.3.7 and its extensions
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | |||||
| Low accuracy (12-18c) | semaphore | ||||
| Medium accuracy (7-12c) | bleu, slendric | ||||
| High accuracy (3.5-7c) | |||||
| Very high accuracy (1-3.5c) | |||||
| Microtemperament (0-1c) |
2.3.11 and its extensions
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | |||||
| Low accuracy (12-18c) | |||||
| Medium accuracy (7-12c) | neutral (no-5 no-7 rastmic) | ||||
| High accuracy (3.5-7c) | no-5 no-7 pythrabian | ||||
| Very high accuracy (1-3.5c) | tribilo (no-5 no-7 nexus) | ||||
| Microtemperament (0-1c) | no-5 no-7 frameshift |
2.3.13, 2.3.17, etc
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | |||||
| Low accuracy (12-18c) | |||||
| Medium accuracy (7-12c) | |||||
| High accuracy (3.5-7c) | |||||
| Very high accuracy (1-3.5c) | |||||
| Microtemperament (0-1c) |
No-7s subgroups
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | |||||
| Low accuracy (12-18c) | |||||
| Medium accuracy (7-12c) | mohaha | ||||
| High accuracy (3.5-7c) | sensible, srutal archagall | ||||
| Very high accuracy (1-3.5c) | cata, nestoria, sensipent | ||||
| Microtemperament (0-1c) |
No-11s subgroups
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | |||||
| Low accuracy (12-18c) | |||||
| Medium accuracy (7-12c) | |||||
| High accuracy (3.5-7c) | |||||
| Very high accuracy (1-3.5c) | |||||
| Microtemperament (0-1c) |
Other subgroups
| Approx. 10 notes | Approx. 20 notes | Approx. 30 notes | Approx. 70 notes | Over 100 notes | |
|---|---|---|---|---|---|
| Exotemperament (18-∞c) | |||||
| Low accuracy (12-18c) | |||||
| Medium accuracy (7-12c) | |||||
| High accuracy (3.5-7c) | |||||
| Very high accuracy (1-3.5c) | |||||
| Microtemperament (0-1c) |