10-form
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.
Interval regions
A table of 10-form interval regions follows; the boundaries are rough and depend heavily on the tuning system and compositional theory in question.
| Step | 10edo | Region | Names (Tellurian) | Names (Diatonic) | Names (Hybrid) | Solfege (Vector) | Notable just intervals |
|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | unison | unison | unison | do | 1/1 |
| 1 | 120 | 50-200 | grade | second | second | re | 16/15, 10/9 |
| 2 | 240 | 200-290 | unilatus | semifourth | unilatus | na | 8/7, 7/6 |
| 3 | 360 | 300-410 | semitres | third | third | mi | 6/5, 5/4 |
| 4 | 480 | 410-540 | bilatus | fourth | fourth | fa | 9/7, 4/3 |
| 5 | 600 | 540-660 | median | tritone | median | zi | 7/5, 10/7, 11/8, 16/11 |
| 6 | 720 | 660-790 | trilatus | fifth | fifth | so | 3/2, 14/9 |
| 7 | 840 | 790-900 | semisept | sixth | sixth | la | 5/3, 8/5 |
| 8 | 960 | 910-1000 | antilatus | semitwelfth | antilatus | be | 7/4, 12/7 |
| 9 | 1080 | 1000-1150 | degrade | seventh | seventh | ti | 15/8, 9/5 |
| 10 | 1200 | 1200 | duplance | octave | octave | do | 2/1 |
The 10-form has clean mappings of 3, 5, 7, and 13, and functions as an organization scheme for the 7-limit. There are also not fixed ranges, the boundaries may vary. The boundaries chosen here are loosely based on 22edo.
| Step | Range (approximate) | JI intervals |
|---|---|---|
| 0 | 0-60c | 1/1 |
| 1 | 60-190c | 10/9, 16/15 |
| 2 | 190-300c | 9/8, 8/7, 7/6 |
| 3 | 300-410c | 5/4, 6/5 |
| 4 | 410-550c | 4/3, 9/7 |
| 5 | 550-650c | 7/5, 10/7 |
| 6 | 650-790c | 3/2, 14/9 |
| 7 | 790-900c | 5/3, 8/5 |
| 8 | 900-1010c | 7/4, 12/7, 16/9 |
| 9 | 1010-1140c | 15/8, 9/5 |
| (10) | 1140-1200c | 2/1 |
Chords
10-form harmony can be constructed out of:
- Fundamental triad: 0-3-6\10, with inversions 0-3-7\10 and 0-4-7\10
- Fundamental tetrad: 0-3-6-8\10, with inversions 0-3-5-7\10, 0-2-4-7\10, and 0-2-5-8\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 is a quasi-diatonic aberrismic scale 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 | Triad quality | |
|---|---|---|---|---|---|---|---|---|---|---|
| sLmLsLmLsL | 81/80 | 9/8 | 6/5 | 4/3 | 27/20 | 3/2 | 8/5 | 16/9 | 9/5 | Minor |
| 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 | Major |
| sLmLsLsLmL | 81/80 | 9/8 | 6/5 | 4/3 | 27/20 | 3/2 | 243/160 | 27/16 | 9/5 | Minor |
| LmLsLsLmLs | 10/9 | 32/27 | 320/243 | 4/3 | 40/27 | 3/2 | 5/3 | 16/9 | 160/81 | Tendo |
| mLsLsLmLsL | 16/15 | 32/27 | 6/5 | 4/3 | 27/20 | 3/2 | 8/5 | 16/9 | 9/5 | Minor |
| LsLsLmLsLm | 10/9 | 9/8 | 5/4 | 81/64 | 45/32 | 3/2 | 5/3 | 27/16 | 15/8 | Major |
| sLsLmLsLmL | 81/80 | 9/8 | 729/640 | 81/64 | 27/20 | 3/2 | 243/160 | 27/16 | 9/5 | Arto |
| LsLmLsLmLs | 10/9 | 9/8 | 5/4 | 4/3 | 40/27 | 3/2 | 5/3 | 16/9 | 160/81 | Major |
Interval matrix in 34edo tempering, which importantly tunes 320/243 and 729/320 as ~13/10 and ~15/13:
| 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 | |
|---|---|---|---|---|---|---|---|---|---|
| Major (LsLsLsLsLs) | 160.0 | 240.0 | 400.0 | 480.0 | 640.0 | 720.0 | 880.0 | 960.0 | 1120.0 |
| Minor (sLsLsLsLsL) | 80.0 | 320.0 | 560.0 | 800.0 | 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 period is 1\2 like jaric, but its generator is an oneirotonic fifth rather than a mosdiatonic one. It corresponds to Octokaidecal, which can be considered what is to Pajara as Mavila is to Meantone.
Taric is particularly notable since its major tetrad (0-400-733-1000 in 18edo) is close to +1+1+1 DR in certain tunings.
It is also associated with Semibuzzard, a temperament which extends Negripent to the 7-limit by allowing for its ~375c major third plus a tritone to stand for the harmonic seventh.
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 |
Interval matrix in 48edo tuning (supporting Semibuzzard):
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
|---|---|---|---|---|---|---|---|---|---|
| LLLLsLLLLs | 125 | 250 | 375 | 500 | 600 | 725 | 850 | 975 | 1100 |
| LLLsLLLLsL | 125 | 250 | 375 | 475 | 600 | 725 | 850 | 975 | 1075 |
| LLsLLLLsLL | 125 | 250 | 350 | 475 | 600 | 725 | 850 | 950 | 1075 |
| LsLLLLsLLL | 125 | 225 | 350 | 475 | 600 | 725 | 825 | 950 | 1075 |
| sLLLLsLLLL | 100 | 225 | 350 | 475 | 600 | 700 | 825 | 950 | 1075 |
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 |
Interval matrix in 48edo tuning (supporting Semibuzzard):
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
|---|---|---|---|---|---|---|---|---|---|
| LLLsLLLLLs | 125 | 250 | 375 | 475 | 600 | 725 | 850 | 975 | 1100 |
| LLsLLLLLsL | 125 | 250 | 350 | 475 | 600 | 725 | 850 | 975 | 1075 |
| LsLLLLLsLL | 125 | 225 | 350 | 475 | 600 | 725 | 850 | 950 | 1075 |
| sLLLLLsLLL | 100 | 225 | 350 | 475 | 600 | 725 | 825 | 950 | 1075 |
| LLLLLsLLLs | 125 | 250 | 375 | 500 | 625 | 725 | 850 | 975 | 1100 |
| LLLLsLLLsL | 125 | 250 | 375 | 500 | 600 | 725 | 850 | 975 | 1075 |
| LLLsLLLsLL | 125 | 250 | 375 | 475 | 600 | 725 | 850 | 950 | 1075 |
| LLsLLLsLLL | 125 | 250 | 350 | 475 | 600 | 725 | 825 | 950 | 1075 |
| LsLLLsLLLL | 125 | 225 | 350 | 475 | 600 | 700 | 825 | 950 | 1075 |
| sLLLsLLLLL | 100 | 225 | 350 | 475 | 575 | 700 | 825 | 950 | 1075 |
Equidecatonic MOSes
The following is a table of 10-note "equidecatonic" MOSes, the edos they may be found in, and the implied temperaments. All MOSes with an even number of large and small steps were assigned to jubilismic temperaments.
| MOS | EDO | Temperament |
|---|---|---|
| 9L 1s | 29 | Negri |
| 8L 2s | 28 | Semibuzzard |
| 7L 3s | 27 | Beatles (27 & 37) |
| 6L 4s | 26 | Lemba |
| 5L 5s | 25 | Blackwood |
| 4L 6s | 24 | (TODO: find representative temperament) |
| 3L 7s | 23 | Magic |
| 2L 8s | 22 | Pajara |
| 1L 9s | 21 | Miracle |
Solfege
A system of solfege for the 10-form proposed by Vector is as follows:
| Note | Solfege | Solfege (distinct initials) |
|---|---|---|
| 0 | do | do |
| 1 | re | re |
| 2 | na | na |
| 3 | mi | mi |
| 4 | fa | fa |
| 5 | di | zi |
| 6 | sol | so |
| 7 | la | la |
| 8 | bi | be |
| 9 | si | ti |
| (10) | do | do |
As with all of Vector's solfeges, it does not support accidentals and instead applies each syllable to any pitch found at its number of scale steps. In fixed-do systems, accidentals are specified with standard symbols and terminology (#, b, etc), as in standard fixed-do.
