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{{Editable user page|If you see any temperaments in the wrong category, please move them to the correct category. <br><br>If you know of a temperament that is much-loved by a sizeable subset of the xen community but is not yet included here, please add it. <br><br>If you see a temperament on here that no one cares about, please remove it. But only remove it if '''nobody''' cares about it. Don't remove it just because you think it's useless. If a few other people do care about it, then don't remove it. <br><br>If you see any ways the wording of the page could be improved, please edit it to make those improvements. <br><br>If you see any typos or grammatical or factual errors, please make an edit to correct those.}}
<div style="border: 1px green solid; background-color: #efe; padding: 0.5em 1em; margin-bottom: 1em">
'''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.'''
</div>




''Stuff I'll add next; the numbers are just shorthand instructions for myself to help me know what to do, don't worry about them:''
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 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:
 


''Hanson M 20 5lim''
'''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.


''Orwell VH 20/30 7,11,13''
And they might argue that 25 notes per [[equave]] is the most that is practical, any more than that is too cumbersome.


''Magic VH 20/30 5,7,11,13''
They might argue that nobody can hear the harmonic effect of prime harmonics higher than 11.


''Valentine VH 20/30 5,7,11''
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.


''Superpyth LW 20/30 5,7,11,13''


''Godzilla LW 20/30 7, 11''
'''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.


''Augene LW 30 7, 11, 13''
They might argue that it's perfectly possible to learn up to 50 notes per [[equave]].


''Didacus MD 20 No3''
They might argue that they can hear the subtle, delicate effect of prime harmonics up to 23.


''Orgone MD 20 No3''
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.


''Slendric MD 20 No5''


''Bleu MD 20 No5''
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).


There are at least hundreds, probably thousands, of [[rank-2 temperament]]s described. It can be difficult to know where to start.  
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.


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.
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.


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 that should provide a point of comparison to help measure these other temperaments against.


== Guide to tables ==
== Guide to tables ==
Line 40: 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 almost all targeted intervals
* 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 78: Line 95:
! Low accuracy (12-18c)
! Low accuracy (12-18c)
| [[augmented]], [[blackwood]], [[dimipent]], [[porcupine]], [[whitewood]]
| [[augmented]], [[blackwood]], [[dimipent]], [[porcupine]], [[whitewood]]
|  
| [[superpyth]]
|  
|  
|  
|  
Line 85: Line 102:
! Medium accuracy (7-12c)
! Medium accuracy (7-12c)
| [[meantone]]
| [[meantone]]
|  
| [[hanson]], [[magic]]
|  
| [[valentine]]
|  
|  
|  
|  
Line 108: Line 125:
|  
|  
|  
|  
| schismic aka [[helmholtz]]
| schismic aka [[Helmholtz temperament|Helmholtz]]
| [[kwazy]]
| [[kwazy]]
|}
|}
Line 132: Line 149:
! Low accuracy (12-18c)
! Low accuracy (12-18c)
| [[blacksmith]]
| [[blacksmith]]
| [[pajara]], [[porcupine]], [[whitewood]]
| [[augene]], [[godzilla]], [[pajara]], [[porcupine]], [[whitewood]]
|  
|  
|  
|  
Line 139: Line 156:
! Medium accuracy (7-12c)
! Medium accuracy (7-12c)
|  
|  
| [[meantone]], [[sensi]]
| [[magic]], [[meantone]], [[mothra]], [[sensi]], [[superpyth]]
|
| [[valentine]]
|  
|  
|  
|  
Line 146: Line 163:
! High accuracy (3.5-7c)
! High accuracy (3.5-7c)
|  
|  
| [[mothra]]
| [[orwell]]
| [[diaschismic]], [[garibaldi]]
| [[diaschismic]], [[garibaldi]]
|  
|  
Line 154: Line 171:
|  
|  
|  
|  
| [[miracle]], [[wuerschmidt]]
| [[miracle]]
|  
|  
|  
|  
Line 186: Line 203:
! Low accuracy (12-18c)
! Low accuracy (12-18c)
|  
|  
| [[blacksmith]], [[pajara]], [[porcupine]], [[whitewood]]
| [[augene]], [[blacksmith]], [[pajara]], [[porcupine]], [[whitewood]]
|  
| [[godzilla]], [[superpyth]]
|  
|  
|  
|  
Line 194: Line 211:
|  
|  
|  
|  
| [[meanpop]], [[meantone]]
| [[magic]], [[meanpop]], [[meantone]], [[mothra]], [[valentine]]
|  
|  
|  
|  
Line 201: Line 218:
|  
|  
|  
|  
| [[diaschismic]], [[mothra]]
| [[diaschismic]], [[orwell]]
|  
|  
|  
|  
Line 208: Line 225:
|  
|  
|  
|  
| [[miracle]], [[wuerschmidt]]
| [[miracle]]
|  
|  
|  
|  
Line 240: Line 257:
! Low accuracy (12-18c)
! Low accuracy (12-18c)
|  
|  
| [[blacksmith]], [[pajara]], [[porcupine]], [[whitewood]]
| [[augene]], [[blacksmith]], [[pajara]], [[porcupine]], [[whitewood]]
|  
| [[superpyth]]
|  
|  
|  
|  
Line 248: Line 265:
|  
|  
|  
|  
| [[meantone]]
| [[magic]], [[meantone]], [[mothra]]
|  
|  
|  
|  
Line 255: Line 272:
|  
|  
|  
|  
| [[diaschismic]], [[mothra]]
| [[diaschismic]], [[orwell]]
|  
|  
|  
|  
Line 262: Line 279:
|  
|  
|  
|  
| [[wuerschmidt]]
|  
|  
|  
|  
|  
Line 382: Line 399:
|}
|}


== No-2s ==
== 3.5.7 and its extensions ==


{| class="wikitable"
{| class="wikitable"
Line 436: Line 453:
|}
|}


== No-3s ==
== Other no-2s subgroups ==


{| class="wikitable"
{| class="wikitable"
Line 490: Line 507:
|}
|}


== No-5s ==
== No-3s subgroups ==
 
{| 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)
|
| [[didacus]], [[orgone]]
|
|
|
|-
! High accuracy (3.5-7c)
|
|
|
|
|
|-
! Very high accuracy (1-3.5c)
|
|
|
|
|
|-
! Microtemperament (0-1c)
|
|
|
|
|
|}
 
== 2.3.7 and its extensions ==


{| class="wikitable"
{| class="wikitable"
Line 510: Line 581:
! Low accuracy (12-18c)
! Low accuracy (12-18c)
| [[semaphore]]
| [[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 ==
{| 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)
|
|  
|  
|  
|  
Line 544: Line 723:
|}
|}


== No-7s ==
== No-7s subgroups ==


{| class="wikitable"
{| class="wikitable"
Line 598: Line 777:
|}
|}


== No-11s ==
== No-11s subgroups ==


{| class="wikitable"
{| class="wikitable"
Line 652: Line 831:
|}
|}


== Higher subgroups ==
== Other subgroups ==


{| class="wikitable"
{| class="wikitable"

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)
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