10-form: Difference between revisions

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{| class="wikitable"
{| class="wikitable"
!
!
!1\10
!1
!2\10
!2
!3\10
!3
!4\10
!4
!5\10
!5
!6\10
!6
!7\10
!7
!8\10
!8
!9\10
!9
|-
|-
!sLmLsLmLsL
!sLmLsLmLsL
Line 263: Line 263:


=== Pentawood ===
=== Pentawood ===
Blackwood[10], or pentawood, has the notable feature of every note of the scale having either a major or a minor chord built on it, which not even [[mosdiatonic]] has (as mosdiatonic has a diminished chord). However, this is at the cost of the fifth necessarily being tuned rather sharply. The scale has only two modes, which may be considered major and minor, and as a 1\5-octave scale lacks a single chain of identical intervals capable of describing it. It can be compared to diaschismic[10]; instead of linking the third and antilatus, it makes the antilatus a perfect interval, with no distinctions available within the MOS form of the scale. Pentawood includes the structure of [[archy]] temperament.
Blackwood[10], or pentawood, has the notable feature of every note of the scale having either a major or a minor chord built on it, which not even [[mosdiatonic]] has (as mosdiatonic has a diminished chord). However, this is at the cost of the fifth necessarily being tuned rather sharply. The scale has only two modes, which may be considered major and minor, and as a 1\5-octave scale lacks a single chain of identical intervals capable of describing it. It can be compared to Diaschismic[10]; instead of linking the third and antilatus, it makes the antilatus a perfect interval, with no distinctions available within the MOS form of the scale. Pentawood includes the structure of [[Archy]] temperament.


Additionally, pentawood is a tempering of the aforementioned blackdye.
Additionally, pentawood is a tempering of the aforementioned blackdye.
Interval matrix in 15edo tuning:
{| class="wikitable"
!
!1
!2
!3
!4
!5
!6
!7
!8
!9
|-
!LsLsLsLsLs
|160.0
|240.0
|400.0
|480.0
|640.0
|720.0
|880.0
|960.0
|1120.0
|-
!sLsLsLsLsL
|80.0
|240.0
|320.0
|480.0
|560.0
|720.0
|800.0
|960.0
|1040.0
|}


=== Pajara[10] ===
=== Pajara[10] ===
Called ''jaric'' temperament-agnostically, this scale (with the pattern ssssLssssL) is represented by [[Pajara]] temperament (Diaschismic if the 7-limit interpretations are not accepted). Pajara[10], along with taric, lemon, and lime, gives the 3\10 (representing the simplest 5-limit intervals 5/4 and 6/5) the same distinction as the 8\10 (representing the intervals 12/7 and 7/4), always separating them by a tritone in any given MOS mode. Therefore, the qualities of the two can be linked to form a major/minor dichotomy based upon the harmonic tetrad.
Called ''jaric'' temperament-agnostically, this scale (with the pattern ssssLssssL) is represented by [[Pajara]] temperament (Diaschismic if the 7-limit interpretations are not accepted). Pajara[10], along with taric, lemon, and lime, gives the 3\10 (representing the simplest 5-limit intervals 5/4 and 6/5) the same distinction as the 8\10 (representing the intervals 12/7 and 7/4), always separating them by a tritone in any given MOS mode. Therefore, the qualities of the two can be linked to form a major/minor dichotomy based upon the harmonic tetrad.
Interval matrix in 22edo tuning:
{| class="wikitable"
!
!1
!2
!3
!4
!5
!6
!7
!8
!9
|-
!ssssLssssL
|109.1
|218.2
|327.3
|436.4
|600.0
|709.1
|818.2
|927.3
|1036.4
|-
!sssLssssLs
|109.1
|218.2
|327.3
|490.9
|600.0
|709.1
|818.2
|927.3
|1090.9
|-
!ssLssssLss
|109.1
|218.2
|381.8
|490.9
|600.0
|709.1
|818.2
|981.8
|1090.9
|-
!sLssssLsss
|109.1
|272.7
|381.8
|490.9
|600.0
|709.1
|872.7
|981.8
|1090.9
|-
!LssssLssss
|163.6
|272.7
|381.8
|490.9
|600.0
|763.6
|872.7
|981.8
|1090.9
|}


==== Pentachordal scale ====
==== Pentachordal scale ====
Interval matrix in 22edo tuning:
{| class="wikitable"
!
!1
!2
!3
!4
!5
!6
!7
!8
!9
|-
!sssLsssssL
|109.1
|218.2
|327.3
|490.9
|600.0
|709.1
|818.2
|927.3
|1036.4
|-
!ssLsssssLs
|109.1
|218.2
|381.8
|490.9
|600.0
|709.1
|818.2
|927.3
|1090.9
|-
!sLsssssLss
|109.1
|272.7
|381.8
|490.9
|600.0
|709.1
|818.2
|981.8
|1090.9
|-
!LsssssLsss
|163.6
|272.7
|381.8
|490.9
|600.0
|709.1
|872.7
|981.8
|1090.9
|-
!sssssLsssL
|109.1
|218.2
|327.3
|436.4
|545.5
|709.1
|818.2
|927.3
|1036.4
|-
!ssssLsssLs
|109.1
|218.2
|327.3
|436.4
|600.0
|709.1
|818.2
|927.3
|1090.9
|-
!sssLsssLss
|109.1
|218.2
|327.3
|490.9
|600.0
|709.1
|818.2
|981.8
|1090.9
|-
!ssLsssLsss
|109.1
|218.2
|381.8
|490.9
|600.0
|709.1
|872.7
|981.8
|1090.9
|-
!sLsssLssss
|109.1
|272.7
|381.8
|490.9
|600.0
|763.6
|872.7
|981.8
|1090.9
|-
!LsssLsssss
|163.6
|272.7
|381.8
|490.9
|654.5
|763.6
|872.7
|981.8
|1090.9
|}


=== Taric ===
=== Taric ===
Taric reverses the small and large steps of jaric. Its generator is an oneirotonic fifth rather than a diatonic one.
Interval matrix in 18edo tuning:
{| class="wikitable"
!
!1
!2
!3
!4
!5
!6
!7
!8
!9
|-
!LLLLsLLLLs
|133.3
|266.7
|400.0
|533.3
|600.0
|733.3
|866.7
|1000.0
|1133.3
|-
!LLLsLLLLsL
|133.3
|266.7
|400.0
|466.7
|600.0
|733.3
|866.7
|1000.0
|1066.7
|-
!LLsLLLLsLL
|133.3
|266.7
|333.3
|466.7
|600.0
|733.3
|866.7
|933.3
|1066.7
|-
!LsLLLLsLLL
|133.3
|200.0
|333.3
|466.7
|600.0
|733.3
|800.0
|933.3
|1066.7
|-
!sLLLLsLLLL
|66.7
|200.0
|333.3
|466.7
|600.0
|666.7
|800.0
|933.3
|1066.7
|}
==== Pentachordal taric ====
==== Pentachordal taric ====
Interval matrix in 18edo tuning:
{| class="wikitable"
!
!1
!2
!3
!4
!5
!6
!7
!8
!9
|-
!LLLsLLLLLs
|133.3
|266.7
|400.0
|466.7
|600.0
|733.3
|866.7
|1000.0
|1133.3
|-
!LLsLLLLLsL
|133.3
|266.7
|333.3
|466.7
|600.0
|733.3
|866.7
|1000.0
|1066.7
|-
!LsLLLLLsLL
|133.3
|200.0
|333.3
|466.7
|600.0
|733.3
|866.7
|933.3
|1066.7
|-
!sLLLLLsLLL
|66.7
|200.0
|333.3
|466.7
|600.0
|733.3
|800.0
|933.3
|1066.7
|-
!LLLLLsLLLs
|133.3
|266.7
|400.0
|533.3
|666.7
|733.3
|866.7
|1000.0
|1133.3
|-
!LLLLsLLLsL
|133.3
|266.7
|400.0
|533.3
|600.0
|733.3
|866.7
|1000.0
|1066.7
|-
!LLLsLLLsLL
|133.3
|266.7
|400.0
|466.7
|600.0
|733.3
|866.7
|933.3
|1066.7
|-
!LLsLLLsLLL
|133.3
|266.7
|333.3
|466.7
|600.0
|733.3
|800.0
|933.3
|1066.7
|-
!LsLLLsLLLL
|133.3
|200.0
|333.3
|466.7
|600.0
|666.7
|800.0
|933.3
|1066.7
|-
!sLLLsLLLLL
|66.7
|200.0
|333.3
|466.7
|533.3
|666.7
|800.0
|933.3
|1066.7
|}


== 10-form interval regions ==
== 10-form interval regions ==

Revision as of 08:54, 18 February 2026

The 10-form describes the structure based around a set of 10 pitch classes or high-level interval regions per octave, as opposed to the conventional 7. It is the simplest form that makes the fundamental distinctions necessary to represent the full 7-limit, expanding on the 7-form by adding three new interval classes: the latus, the tritone, and the antilatus. Important lati in this system are 7/6 and 8/7; their complements are 12/7 and 7/4 respectively, which are antilati; 10/7 and 7/5 fall into the tritone category.

Chords

10-form harmony can be constructed out of:

  • Fundamental triad: 0-3-6\10
  • Fundamental tetrad: 0-3-6-8\10

Notes about distinctions

9/7, while conventionally a third, is generally a kind of imperfect fourth here. Same goes for 7/6 and being a latus, rather than a third.

Important scales

Blackdye

Blackdye constructed from zarlino

Blackdye is constructed as an "indecisive zarlino" of sorts, adding small steps called aberrismas in order to allow for finer control over the intervals used. Alternatively, it may be conceptualized as two Pythagorean pentic scales offset by 10/9.

Interval matrix in JI:

1 2 3 4 5 6 7 8 9
sLmLsLmLsL 81/80 9/8 6/5 4/3 27/20 3/2 8/5 16/9 9/5
LmLsLmLsLs 10/9 32/27 320/243 4/3 40/27 128/81 1280/729 16/9 160/81
mLsLmLsLsL 16/15 32/27 6/5 4/3 64/45 128/81 8/5 16/9 9/5
LsLmLsLsLm 10/9 9/8 5/4 4/3 40/27 3/2 5/3 27/16 15/8
sLmLsLsLmL 81/80 9/8 6/5 4/3 27/20 3/2 243/160 27/16 9/5
LmLsLsLmLs 10/9 32/27 320/243 4/3 40/27 3/2 5/3 16/9 160/81
mLsLsLmLsL 16/15 32/27 6/5 4/3 27/20 3/2 8/5 16/9 9/5
LsLsLmLsLm 10/9 9/8 5/4 81/64 45/32 3/2 5/3 27/16 15/8
sLsLmLsLmL 81/80 9/8 729/640 81/64 27/20 3/2 243/160 27/16 9/5
LsLmLsLmLs 10/9 9/8 5/4 4/3 40/27 3/2 5/3 16/9 160/81

Interval matrix in 34edo tempering:

1 2 3 4 5 6 7 8 9
sLmLsLmLsL 35.3 211.8 317.6 494.1 529.4 705.9 811.8 988.2 1023.5
LmLsLmLsLs 176.5 282.4 458.8 494.1 670.6 776.5 952.9 988.2 1164.7
mLsLmLsLsL 105.9 282.4 317.6 494.1 600.0 776.5 811.8 988.2 1023.5
LsLmLsLsLm 176.5 211.8 388.2 494.1 670.6 705.9 882.4 917.6 1094.1
sLmLsLsLmL 35.3 211.8 317.6 494.1 529.4 705.9 741.2 917.6 1023.5
LmLsLsLmLs 176.5 282.4 458.8 494.1 670.6 705.9 882.4 988.2 1164.7
mLsLsLmLsL 105.9 282.4 317.6 494.1 529.4 705.9 811.8 988.2 1023.5
LsLsLmLsLm 176.5 211.8 388.2 423.5 600.0 705.9 882.4 917.6 1094.1
sLsLmLsLmL 35.3 211.8 247.1 423.5 529.4 705.9 741.2 917.6 1023.5
LsLmLsLmLs 176.5 211.8 388.2 494.1 670.6 705.9 882.4 988.2 1164.7

Note that the 0-3-6-8\10 tetrad includes a wolf interval, e.g. 1/1-6/5-3/2-16/9, on most degrees; only one mode, LsLmLsLmLs, has a dominant tetrad 1/1-5/4-3/2-16/9 on it. Blackdye thus encourages tertian (0-3-6\10-based) harmony.

Pentawood

Blackwood[10], or pentawood, has the notable feature of every note of the scale having either a major or a minor chord built on it, which not even mosdiatonic has (as mosdiatonic has a diminished chord). However, this is at the cost of the fifth necessarily being tuned rather sharply. The scale has only two modes, which may be considered major and minor, and as a 1\5-octave scale lacks a single chain of identical intervals capable of describing it. It can be compared to Diaschismic[10]; instead of linking the third and antilatus, it makes the antilatus a perfect interval, with no distinctions available within the MOS form of the scale. Pentawood includes the structure of Archy temperament.

Additionally, pentawood is a tempering of the aforementioned blackdye.

Interval matrix in 15edo tuning:

1 2 3 4 5 6 7 8 9
LsLsLsLsLs 160.0 240.0 400.0 480.0 640.0 720.0 880.0 960.0 1120.0
sLsLsLsLsL 80.0 240.0 320.0 480.0 560.0 720.0 800.0 960.0 1040.0

Pajara[10]

Called jaric temperament-agnostically, this scale (with the pattern ssssLssssL) is represented by Pajara temperament (Diaschismic if the 7-limit interpretations are not accepted). Pajara[10], along with taric, lemon, and lime, gives the 3\10 (representing the simplest 5-limit intervals 5/4 and 6/5) the same distinction as the 8\10 (representing the intervals 12/7 and 7/4), always separating them by a tritone in any given MOS mode. Therefore, the qualities of the two can be linked to form a major/minor dichotomy based upon the harmonic tetrad.

Interval matrix in 22edo tuning:

1 2 3 4 5 6 7 8 9
ssssLssssL 109.1 218.2 327.3 436.4 600.0 709.1 818.2 927.3 1036.4
sssLssssLs 109.1 218.2 327.3 490.9 600.0 709.1 818.2 927.3 1090.9
ssLssssLss 109.1 218.2 381.8 490.9 600.0 709.1 818.2 981.8 1090.9
sLssssLsss 109.1 272.7 381.8 490.9 600.0 709.1 872.7 981.8 1090.9
LssssLssss 163.6 272.7 381.8 490.9 600.0 763.6 872.7 981.8 1090.9

Pentachordal scale

Interval matrix in 22edo tuning:

1 2 3 4 5 6 7 8 9
sssLsssssL 109.1 218.2 327.3 490.9 600.0 709.1 818.2 927.3 1036.4
ssLsssssLs 109.1 218.2 381.8 490.9 600.0 709.1 818.2 927.3 1090.9
sLsssssLss 109.1 272.7 381.8 490.9 600.0 709.1 818.2 981.8 1090.9
LsssssLsss 163.6 272.7 381.8 490.9 600.0 709.1 872.7 981.8 1090.9
sssssLsssL 109.1 218.2 327.3 436.4 545.5 709.1 818.2 927.3 1036.4
ssssLsssLs 109.1 218.2 327.3 436.4 600.0 709.1 818.2 927.3 1090.9
sssLsssLss 109.1 218.2 327.3 490.9 600.0 709.1 818.2 981.8 1090.9
ssLsssLsss 109.1 218.2 381.8 490.9 600.0 709.1 872.7 981.8 1090.9
sLsssLssss 109.1 272.7 381.8 490.9 600.0 763.6 872.7 981.8 1090.9
LsssLsssss 163.6 272.7 381.8 490.9 654.5 763.6 872.7 981.8 1090.9

Taric

Taric reverses the small and large steps of jaric. Its generator is an oneirotonic fifth rather than a diatonic one.

Interval matrix in 18edo tuning:

1 2 3 4 5 6 7 8 9
LLLLsLLLLs 133.3 266.7 400.0 533.3 600.0 733.3 866.7 1000.0 1133.3
LLLsLLLLsL 133.3 266.7 400.0 466.7 600.0 733.3 866.7 1000.0 1066.7
LLsLLLLsLL 133.3 266.7 333.3 466.7 600.0 733.3 866.7 933.3 1066.7
LsLLLLsLLL 133.3 200.0 333.3 466.7 600.0 733.3 800.0 933.3 1066.7
sLLLLsLLLL 66.7 200.0 333.3 466.7 600.0 666.7 800.0 933.3 1066.7

Pentachordal taric

Interval matrix in 18edo tuning:

1 2 3 4 5 6 7 8 9
LLLsLLLLLs 133.3 266.7 400.0 466.7 600.0 733.3 866.7 1000.0 1133.3
LLsLLLLLsL 133.3 266.7 333.3 466.7 600.0 733.3 866.7 1000.0 1066.7
LsLLLLLsLL 133.3 200.0 333.3 466.7 600.0 733.3 866.7 933.3 1066.7
sLLLLLsLLL 66.7 200.0 333.3 466.7 600.0 733.3 800.0 933.3 1066.7
LLLLLsLLLs 133.3 266.7 400.0 533.3 666.7 733.3 866.7 1000.0 1133.3
LLLLsLLLsL 133.3 266.7 400.0 533.3 600.0 733.3 866.7 1000.0 1066.7
LLLsLLLsLL 133.3 266.7 400.0 466.7 600.0 733.3 866.7 933.3 1066.7
LLsLLLsLLL 133.3 266.7 333.3 466.7 600.0 733.3 800.0 933.3 1066.7
LsLLLsLLLL 133.3 200.0 333.3 466.7 600.0 666.7 800.0 933.3 1066.7
sLLLsLLLLL 66.7 200.0 333.3 466.7 533.3 666.7 800.0 933.3 1066.7

10-form interval regions

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