Neutral temperaments: Difference between revisions

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Created page with "'''Dichotic''' is an exotemperament that can be defined to temper out 25/24, the dicot comma, 45/44, and 64/63. As a result it also tempers out 55/54. == Interval chart == This interval chart uses Partch's 11-limit tonality diamond and maps it onto 7edo and 10edo. This also gives an idea of what intervals are possible in dichotic temperament. '''Intervals in bold''' are in the 7-note MOS (symmetric mode) and ''intervals in italics'' are in the 1..."
 
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'''Dichotic''' is an exotemperament that can be defined to temper out [[25/24]], the dicot comma, [[45/44]], and [[64/63]]. As a result it also tempers out [[55/54]].
'''Neutral temperaments''' are any temperaments represented by the edo join 7 & 10, or any reasonable extension of such a temperament, such that the generator is a neutral third of some kind which splits 3/2 into two. They are a subset of and largely cover the ''dicot'' temperament archetype, and impose upon it the condition that the neutral third must be mapped to 2\7 and 3\10. The two most well-known neutral temperaments are the 2.3.11 (Rastmatic) and 2.3.5 (Dicot) versions.


== Interval chart ==
10edo is a contorted 5edo in 2.3.7, hence 7 & 10 in that subgroup represents monocot [[Archy]] temperament.


This interval chart uses Partch's 11-limit tonality diamond and maps it onto [[7edo]] and [[10edo]]. This also gives an idea of what intervals are possible in dichotic temperament. '''Intervals in bold''' are in the 7-note MOS (symmetric mode) and ''intervals in italics'' are in the 10-note MOS (both near-symmetric modes). Note that the interval that can be mapped to 11/8 and 10/7 and the interval mapped to 7/5 and 16/11 cannot coexist in the same 10-note MOS.
== Notation ==
Neutral temperaments may be notated with neutral chain-of-fifths notation.
{| class="wikitable"
|+
!Note
!24edo
!Notation
!2.3...
!5 (Dicot)
!11 (Rastmatic)
!13 (Namo)
!13-limit (no-fives)
!13-limit
|-
|A
|0
|P1
|1/1
|
|
|
|
|
|-
|At
|50
|sA1
|
|81/80
|33/32
|
|
|
|-
|Bb
|100
|m2
|256/243
|
|
|
|
|
|-
|Bd
|150
|n2
|
|10/9, 16/15
|12/11
|
|
|
|-
|B
|200
|M2
|9/8
|
|
|
|8/7
|11/10
|-
|C
|300
|m3
|32/27
|
|
|
|7/6
|
|-
|Ct
|350
|n3
|
|5/4, 6/5
|11/9
|16/13
|
|
|-
|C#
|400
|M3
|81/64
|
|
|
|9/7
|
|-
|Dd
|450
|sd4
|
|
|
|
|14/11
|
|-
|D
|500
|P4
|4/3
|
|
|
|
|
|-
|Dt
|550
|sA4
|
|27/20
|11/8
|18/13
|
|10/7
|-
|Ed
|650
|sd5
|
|40/27
|16/11
|13/9
|
|7/5
|-
|E
|700
|P5
|3/2
|
|
|
|
|
|-
|Et
|750
|sA5
|
|
|
|
|11/7
|
|-
|F
|800
|m6
|128/81
|
|
|
|14/9
|
|-
|Ft
|850
|n6
|
|5/3, 8/5
|18/11
|13/8
|
|
|-
|F#
|900
|M6
|27/16
|
|
|
|12/7
|
|-
|G
|1000
|m7
|16/9
|
|
|
|7/4
|20/11
|-
|Gt
|1050
|n7
|
|15/8, 9/5
|11/6
|
|
|
|-
|G#
|1100
|M7
|243/128
|
|
|
|
|
|-
|Ad
|1150
|sd8
|
|160/81
|64/33
|
|
|
|-
|A
|1200
|P8
|2/1
|
|
|
|
|
|}
These intervals may additionally be arranged on a chart which explains their mappings to 7edo and 10edo:
{| class="wikitable"
! !! 0\7 !! 1\7 !! 2\7 !! 3\7 !! 4\7 !! 5\7 !! 6\7 !! 7\7
|-
! scope="row" | 0\10
| '''P1''' ||  m2||  d3|| || || || ||
|-
! scope="row" | 1\10
|  sA1|| '''n2''' ||  sd3|| || || || ||
|-
! scope="row" | 2\10
|  A1|| M2 || m3 ||  d4|| || || ||
|-
! scope="row" | 3\10
| ||  sA2|| '''n3''' || sd4 || || || ||
|-
! scope="row" | 4\10
| ||  A2|| M3 || '''P4''' ||  d5||  d6|| ||
|-
! scope="row" | 5\10
| || ||  sA3|| sA4 || sd5 ||  sd6|| ||
|-
! scope="row" | 6\10
| || ||  A3||  A4|| '''P5''' || m6 ||  d7||
|-
! scope="row" | 7\10
| || || || || sA5 || '''n6''' ||  sd7||
|-
! scope="row" | 8\10
| || || || ||  A5|| M6 || m7 ||d8
|-
! scope="row" | 9\10
| || || || || ||  sA6|| '''n7''' ||sd8
|-
! scope="row" | 10\10
| || || || || ||  A6||  M7|| '''P8'''
|}
 
== Rastmatic ==
[[File:Mohajira7.png|thumb|Mohajira heptatonic MOS]]
Rastmatic is the neutral temperament in the 2.3.11 subgroup, which equates 11-limit neutral intervals to their exact neutral (2.sqrt(3/2)) counterparts. The generator represents both 11/9 and 27/22; this is one of the most accurate prime subgroups for the temperament. Because the temperament tempers out 243/242, it is a rastmic temperament, hence "rastmic" may be used informally to refer to Rastmatic or its scales. The generator is best tuned around 350 cents, about 3 cents off from the justly tuned 11/9 and 4 cents off from just 27/22. Mohajira extends rastmatic to 2.3.5.11.
 
=== Etymology ===
Rastmatic is named after the rastma, the comma it tempers out, which is in turn named after the maqam ''Rast'' which utilizes a scale with several neutral intervals.
 
== Dicot ==
Dicot<ref group="note>The name ''Interpental'' has been proposed, however it currently is used by 43 & 53, a weak extension of [[Buzzard]].</ref>, not to be confused with the dicot archetype as a whole, is the neutral temperament in the 2.3.5 subgroup. an exotemperament that can be defined to temper out [[25/24]], the Dicot comma. The provided [[edo join]] also tempers out [[45/44]] and [[64/63]] in the 11-limit, representing the extension '''Dichotic''' and also tempering out [[55/54]]. Alternative extensions include 4 & 7 (which conflates 9/7~7/6~6/5~5/4). 7 & 10 and 10 & 17 are both reasonable edo joins, suggesting Dicot as a 3-, 7-, or 10-form temperament.
 
Dicot makes 4:5:6 equidistant, suggesting the simplified structure of [[tertian]] harmony, the same way [[Semaphore]] does for [[chthonic harmony]]. As a result, the temperament archetype [[Neutral third scales|dicot]] is named after it.
 
=== Etymology ===
Dicot originates from the term "dicot" in botany, referring to plants with two embryonic leaves, perhaps by analogy with 3/2 being split into two generators. The name ''Dicot'' would also inspire [[Tetracot]], [[Alphatricot]], and by extension the [[ploidacot]] temperament archetype naming system as a whole.


=== Tuning considerations ===
A perfect ~351c tuning of the generator, while useful for understanding tertian harmony and suggested by some temperament tuning optimization systems, does not reasonably approximate either 5/4 or 6/5. The optimal tunings of Dicot are roughly bimodal, with ~360c (around 10edo) and ~343c (around 7edo) both being better tunings.
=== History ===
According to Unque, dicot temperament was discussed by Vicentino as a descriptive model of vocal music; the reason it is considered an exotemperament today is, if not members of the xenharmonic community tending to be sensitive to detuning, likely at least partially the fact that the tonality system motivates a distinction between 4:5:6 and 10:12:15. (Note that the tuning error on 5/4 in 10edo is not much greater than the error on 3/2 in the same tuning.)
== Namo ==
[[File:Suhajira7.png|thumb|Suhajira heptatonic MOS]]
''Namo'', ''Intertridecimal'', or ''Harmoneutral'' is the temperament of 512/507, which is 7 & 10 in the 2.3.13 subgroup. It prefers a sharp tuning of the fifth.
It is often framed as a (somewhat inaccurate) extension to Rastmatic. Its generator is best tuned around 355c.
Namo extends to 7 with the temperament Suhajira, which takes the fifth as an [[archy]] fifth.
== Patent vals ==
=== List of patent vals ===
{| class="wikitable"
{| class="wikitable"
! !! 0\7 !! 1\7 !! 2\7 !! 3\7 !! 4\7 !! 5\7 !! 6\7 !! 7\7
|+
!EDO
!Mappings supported
!Generator tuning
!3/2 tuning
|-
|10
|5, 13, 11
|360.0c
|720.0c
|-
|37
|13
|356.8c
|713.5c
|-
|-
! scope="row"| 0\10
|27
| '''1/1''' ||  ||  ||  ||  ||  ||  ||
|13
|355.6c
|711.1c
|-
|-
! scope="row"| 1\10
|71
| || '''12/11, 10/9''' ||  ||  ||  ||  ||  ||
|13
|354.9c
|709.9c
|-
|-
! scope="row"| 2\10
|44
| || ''11/10, 9/8, 8/7'' || 7/6 ||  ||  ||  ||  ||
|13
|354.5c
|709.1c
|-
|-
! scope="row"| 3\10
|61
| ||  || '''6/5, 11/9, 5/4''' || 14/11 ||  ||  ||  ||
|13
|354.1c
|708.2c
|-
|-
! scope="row"| 4\10
|78
| ||  || 9/7 || '''4/3''' ||  ||  ||  ||
|13
|353.8c
|707.7c
|-
|-
! scope="row"| 5\10
|95
| ||  ||  || ''11/8, 10/7'' || ''7/5, 16/11'' ||  ||  ||
|13
|353.7c
|707.4c
|-
|-
! scope="row"| 6\10
|17
| ||  ||  ||  || '''3/2''' || 14/9 ||  ||
|5, 13, 11
|352.9c
|705.9c
|-
|-
! scope="row"| 7\10
|75
| ||  ||  ||  || 11/7 || '''8/5, 18/11, 5/3''' ||  ||
|13
|352.0c
|704.0c
|-
|-
! scope="row"| 8\10
|58
| ||  ||  ||  ||  || 12/7 || ''7/4, 16/9, 20/11'' ||
|13, 11
|351.7c
|703.4c
|-
|-
! scope="row"| 9\10
|41
| ||  ||  ||  ||  ||  || '''9/5, 11/6''' ||
|13, 11
|351.2c
|702.4c
|-
|-
! scope="row"| 10\10
|147
| || || || || || || || '''2/1'''
|11
|351.0c
|702.0c
|-
|106
|11
|350.9c
|701.9c
|-
|171
|11
|350.9c
|701.8c
|-
|65
|13, 11
|350.8c
|701.5c
|-
|219
|11
|350.7c
|701.4c
|-
|154
|11
|350.6c
|701.3c
|-
|243
|11
|350.62c
|701.23c
|-
|332
|11
|350.60c
|701.20c
|-
|89
|11
|350.56c
|701.12c
|-
|380
|11
|350.53c
|701.05c
|-
|291
|11
|350.52c
|701.03c
|-
|202
|11
|350.50c
|700.99c
|-
|517
|11
|350.48c
|700.97c
|-
|315
|11
|350.48c
|700.95c
|-
|428
|11
|350.47c
|700.93c
|-
|541
|11
|350.46c
|700.92c
|-
|113
|11
|350.44c
|700.88c
|-
|476
|11
|350.42c
|700.84c
|-
|363
|11
|350.41c
|700.83c
|-
|250
|11
|350.40c
|700.80c
|-
|387
|11
|350.39c
|700.78c
|-
|137
|11
|350.36c
|700.73c
|-
|435
|11
|350.34c
|700.69c
|-
|298
|11
|350.34c
|700.67c
|-
|459
|11
|350.33c
|700.65c
|-
|161
|11
|350.31c
|700.62c
|-
|346
|11
|350.29c
|700.58c
|-
|185
|11
|350.27c
|700.54c
|-
|394
|11
|350.25c
|700.51c
|-
|209
|11
|350.24c
|700.48c
|-
|233
|11
|350.21c
|700.43c
|-
|257
|11
|350.19c
|700.39c
|-
|281
|11
|350.18c
|700.36c
|-
|305
|11
|350.16c
|700.33c
|-
|329
|11
|350.15c
|700.30c
|-
|353
|11
|350.14c
|700.28c
|-
|24
|13, 11
|350.00c
|700.00c
|-
|247
|11
|349.80c
|699.60c
|-
|223
|11
|349.78c
|699.55c
|-
|199
|11
|349.7c
|699.5c
|-
|175
|11
|349.7c
|699.4c
|-
|151
|11
|349.7c
|699.3c
|-
|127
|11
|349.6c
|699.2c
|-
|103
|11
|349.5c
|699.0c
|-
|79
|11
|349.4c
|698.7c
|-
|55
|13, 11
|349.1c
|698.2c
|-
|31
|13, 11
|348.4c
|696.8c
|-
|38
|13, 11
|347.4c
|694.7c
|-
|45
|13
|346.7c
|693.3c
|-
|7
|5, 13, 11
|342.9c
|685.7c
|}
|}
== Other temperaments with neutral third generators ==
=== Hemififths ===
Hemififths, 41 & 58, is a temperament in the 2.3.5.7 subgroup which tempers out 2401/2400 = S49 (aka the breedsma) equating 49/40 to its 3/2-complement and additionally tempers out 5120/5103 making it an [[aberschismic]] temperament. Note that 7 & 10 in 2.3.5.7 is [[dichotic]].
== Footnotes ==
<references group="note"/>
{{Navbox regtemp}}

Latest revision as of 02:10, 2 June 2026

Neutral temperaments are any temperaments represented by the edo join 7 & 10, or any reasonable extension of such a temperament, such that the generator is a neutral third of some kind which splits 3/2 into two. They are a subset of and largely cover the dicot temperament archetype, and impose upon it the condition that the neutral third must be mapped to 2\7 and 3\10. The two most well-known neutral temperaments are the 2.3.11 (Rastmatic) and 2.3.5 (Dicot) versions.

10edo is a contorted 5edo in 2.3.7, hence 7 & 10 in that subgroup represents monocot Archy temperament.

Notation

Neutral temperaments may be notated with neutral chain-of-fifths notation.

Note 24edo Notation 2.3... 5 (Dicot) 11 (Rastmatic) 13 (Namo) 13-limit (no-fives) 13-limit
A 0 P1 1/1
At 50 sA1 81/80 33/32
Bb 100 m2 256/243
Bd 150 n2 10/9, 16/15 12/11
B 200 M2 9/8 8/7 11/10
C 300 m3 32/27 7/6
Ct 350 n3 5/4, 6/5 11/9 16/13
C# 400 M3 81/64 9/7
Dd 450 sd4 14/11
D 500 P4 4/3
Dt 550 sA4 27/20 11/8 18/13 10/7
Ed 650 sd5 40/27 16/11 13/9 7/5
E 700 P5 3/2
Et 750 sA5 11/7
F 800 m6 128/81 14/9
Ft 850 n6 5/3, 8/5 18/11 13/8
F# 900 M6 27/16 12/7
G 1000 m7 16/9 7/4 20/11
Gt 1050 n7 15/8, 9/5 11/6
G# 1100 M7 243/128
Ad 1150 sd8 160/81 64/33
A 1200 P8 2/1

These intervals may additionally be arranged on a chart which explains their mappings to 7edo and 10edo:

0\7 1\7 2\7 3\7 4\7 5\7 6\7 7\7
0\10 P1 m2 d3
1\10 sA1 n2 sd3
2\10 A1 M2 m3 d4
3\10 sA2 n3 sd4
4\10 A2 M3 P4 d5 d6
5\10 sA3 sA4 sd5 sd6
6\10 A3 A4 P5 m6 d7
7\10 sA5 n6 sd7
8\10 A5 M6 m7 d8
9\10 sA6 n7 sd8
10\10 A6 M7 P8

Rastmatic

Mohajira heptatonic MOS

Rastmatic is the neutral temperament in the 2.3.11 subgroup, which equates 11-limit neutral intervals to their exact neutral (2.sqrt(3/2)) counterparts. The generator represents both 11/9 and 27/22; this is one of the most accurate prime subgroups for the temperament. Because the temperament tempers out 243/242, it is a rastmic temperament, hence "rastmic" may be used informally to refer to Rastmatic or its scales. The generator is best tuned around 350 cents, about 3 cents off from the justly tuned 11/9 and 4 cents off from just 27/22. Mohajira extends rastmatic to 2.3.5.11.

Etymology

Rastmatic is named after the rastma, the comma it tempers out, which is in turn named after the maqam Rast which utilizes a scale with several neutral intervals.

Dicot

Dicot[note 1], not to be confused with the dicot archetype as a whole, is the neutral temperament in the 2.3.5 subgroup. an exotemperament that can be defined to temper out 25/24, the Dicot comma. The provided edo join also tempers out 45/44 and 64/63 in the 11-limit, representing the extension Dichotic and also tempering out 55/54. Alternative extensions include 4 & 7 (which conflates 9/7~7/6~6/5~5/4). 7 & 10 and 10 & 17 are both reasonable edo joins, suggesting Dicot as a 3-, 7-, or 10-form temperament.

Dicot makes 4:5:6 equidistant, suggesting the simplified structure of tertian harmony, the same way Semaphore does for chthonic harmony. As a result, the temperament archetype dicot is named after it.

Etymology

Dicot originates from the term "dicot" in botany, referring to plants with two embryonic leaves, perhaps by analogy with 3/2 being split into two generators. The name Dicot would also inspire Tetracot, Alphatricot, and by extension the ploidacot temperament archetype naming system as a whole.

Tuning considerations

A perfect ~351c tuning of the generator, while useful for understanding tertian harmony and suggested by some temperament tuning optimization systems, does not reasonably approximate either 5/4 or 6/5. The optimal tunings of Dicot are roughly bimodal, with ~360c (around 10edo) and ~343c (around 7edo) both being better tunings.

History

According to Unque, dicot temperament was discussed by Vicentino as a descriptive model of vocal music; the reason it is considered an exotemperament today is, if not members of the xenharmonic community tending to be sensitive to detuning, likely at least partially the fact that the tonality system motivates a distinction between 4:5:6 and 10:12:15. (Note that the tuning error on 5/4 in 10edo is not much greater than the error on 3/2 in the same tuning.)

Namo

Suhajira heptatonic MOS

Namo, Intertridecimal, or Harmoneutral is the temperament of 512/507, which is 7 & 10 in the 2.3.13 subgroup. It prefers a sharp tuning of the fifth.

It is often framed as a (somewhat inaccurate) extension to Rastmatic. Its generator is best tuned around 355c.

Namo extends to 7 with the temperament Suhajira, which takes the fifth as an archy fifth.

Patent vals

List of patent vals

EDO Mappings supported Generator tuning 3/2 tuning
10 5, 13, 11 360.0c 720.0c
37 13 356.8c 713.5c
27 13 355.6c 711.1c
71 13 354.9c 709.9c
44 13 354.5c 709.1c
61 13 354.1c 708.2c
78 13 353.8c 707.7c
95 13 353.7c 707.4c
17 5, 13, 11 352.9c 705.9c
75 13 352.0c 704.0c
58 13, 11 351.7c 703.4c
41 13, 11 351.2c 702.4c
147 11 351.0c 702.0c
106 11 350.9c 701.9c
171 11 350.9c 701.8c
65 13, 11 350.8c 701.5c
219 11 350.7c 701.4c
154 11 350.6c 701.3c
243 11 350.62c 701.23c
332 11 350.60c 701.20c
89 11 350.56c 701.12c
380 11 350.53c 701.05c
291 11 350.52c 701.03c
202 11 350.50c 700.99c
517 11 350.48c 700.97c
315 11 350.48c 700.95c
428 11 350.47c 700.93c
541 11 350.46c 700.92c
113 11 350.44c 700.88c
476 11 350.42c 700.84c
363 11 350.41c 700.83c
250 11 350.40c 700.80c
387 11 350.39c 700.78c
137 11 350.36c 700.73c
435 11 350.34c 700.69c
298 11 350.34c 700.67c
459 11 350.33c 700.65c
161 11 350.31c 700.62c
346 11 350.29c 700.58c
185 11 350.27c 700.54c
394 11 350.25c 700.51c
209 11 350.24c 700.48c
233 11 350.21c 700.43c
257 11 350.19c 700.39c
281 11 350.18c 700.36c
305 11 350.16c 700.33c
329 11 350.15c 700.30c
353 11 350.14c 700.28c
24 13, 11 350.00c 700.00c
247 11 349.80c 699.60c
223 11 349.78c 699.55c
199 11 349.7c 699.5c
175 11 349.7c 699.4c
151 11 349.7c 699.3c
127 11 349.6c 699.2c
103 11 349.5c 699.0c
79 11 349.4c 698.7c
55 13, 11 349.1c 698.2c
31 13, 11 348.4c 696.8c
38 13, 11 347.4c 694.7c
45 13 346.7c 693.3c
7 5, 13, 11 342.9c 685.7c

Other temperaments with neutral third generators

Hemififths

Hemififths, 41 & 58, is a temperament in the 2.3.5.7 subgroup which tempers out 2401/2400 = S49 (aka the breedsma) equating 49/40 to its 3/2-complement and additionally tempers out 5120/5103 making it an aberschismic temperament. Note that 7 & 10 in 2.3.5.7 is dichotic.

Footnotes

  1. ↑ The name Interpental has been proposed, however it currently is used by 43 & 53, a weak extension of Buzzard.


View • Talk • EditRegular temperaments
Rank-2
Acot Blackwood (1/5-octave) • Whitewood (1/7-octave) • Compton (1/12-octave)
Monocot Meantone • Schismic • Gentle-fifth temperaments • Archy
Complexity 2 Diaschismic (diploid monocot) • Pajara (diploid monocot) • Injera (diploid monocot) • Rastmatic (dicot) • Mohajira (dicot) • Intertridecimal (dicot) • Interseptimal (alpha-dicot)
Complexity 3 Augmented (triploid) • Misty (triploid) • Slendric (tricot) • Porcupine (omega-tricot)
Complexity 4 Diminished (tetraploid) • Tetracot (tetracot) • Buzzard (alpha-tetracot) • Squares (beta-tetracot) • Negri (omega-tetracot)
Complexity 5-6 Magic (alpha-pentacot) • Amity (gamma-pentacot) • Kleismic (alpha-hexacot) • Miracle (hexacot)
Higher complexity Orwell (alpha-heptacot) • Sensi (beta-heptacot) • Octacot (octacot) • Wurschmidt (beta-octacot) • Valentine (enneacot) • Ammonite (delta-enneacot) • Myna (beta-decacot) • Ennealimmal (enneaploid dicot)
Straddle-3 A-Team (alter-tricot) • Machine (alter-monocot)
No-3 Mabilic (alpha-triseph) • Orgone (trimech) • Didacus (diseph) • Sidewalk (epsilon-heptaseph)
No-octaves Sensamagic (monogem)
Exotemperaments Dicot • Mavila • Father
Higher-rank
Rank-3 Aberschismic • Marvel • Parapyth • Pentagoth
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