10edo
10edo, or 10 equal divisions of the octave, is the equal tuning featuring steps of (1200/10) = 120 cents, 10 of which stack to the perfect octave 2/1. It is notable for its good approximation of the 2.7.13 subgroup and for being possibly the smallest edo in the same class as 12edo.
Theory
Chords
10edo is arguably the first edo to have three triads consisting of two thirds to make a fifth. The two found in 5edo rely on the fact that stacking two sharp fifths minus an octave makes an inframinor third, thus using oneirotonic logic.
Tendo: 0-4-6-(10)
Neutral: 0-3-6-(9)
Arto: 0-2-6-(8)
See also: Oneirotonic#Chords_of_oneirotonic
Scales
Example mosh (3L4s): 0-1-3-4-6-7-9-10
Mosh is the most characteristic scale in 10edo.
Example subaric (2L6s): 0-1-2-4-5-6-7-9-10
In subaric the "simic pentachord" 0-1-2-4-6 approximates the diatonic minor pentachord in the only way 10edo is able to. A potential temperament for this interpretation of soft subaric is 10 & 2[-7] 2.3.7.17; in other words, Trienstonian plus a 17/12 half-octave.
Detempers
Due to its small size and unique melodic character, it is very easy to detemper 10edo. Example tunings are shown in parenthesis.
Full octave, neutral third generator
- Sharp (13\43), 3L7s
- Mainly oneirotonic fifth
- Good approximation of 19:22:25:27:29
- Similar to Submajor/Interpental temperament
- Flat (11\37), 7L3s
- Mainly diatonic fifth
- General 13-limit, especially 2.7.13
Half octave, fifth generator
- Flat (3\32)
- Mainly diatonic fifth
- Oceanfront temperament with added 17/14
- Sharp (4\38)
- Mainly oneirotonic fifth
- Soft subaric
The 10-form
Intervals in systems approximating 10edo may be conceptualized using the 10-form. This is arguably a more intuitive way of conceptualizing intervals in the 7-limit than the 7-form is.
The 10-form's key feature is the presence of the tritone as its own interval category separate from fourths and fifths, and the moving of 9/7 and 7/6 away from the category representing thirds. This in effect gives the simplest 7-limit intervals their own pair of categories separate from the simplest 5-limit intervals, much as upgrading from the 5-form to the 7-form gives the simplest 5-limit intervals their own pair of categories.
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 following is a table of 10-note 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 | Rastmic (TODO: find 2.3.5.7 interpretation) | |
| 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 |
