10edo: Difference between revisions
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'''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]]. | '''10edo''', or 10 equal divisions of the octave (sometimes called '''10-TET''' or '''10-tone equal temperament'''), is the equal tuning featuring steps of (1200/10) = 120 cents, 10 of which stack to the perfect octave [[2/1]]. | ||
As the double of [[5edo]], 10edo has its sharp [[perfect fifth|fifth]] of 720{{c}}, and approximates [[7/4]] well by 960{{c}}. It divides this fifth in two so that it possesses a [[neutral third]] of 360{{c}} that is extremely close to [[16/13]]. Both [[6/5]] and [[5/4]] are mapped to this neutral third, and in this fashion 10edo provides a very primitive basis for the [[7-limit]]. | |||
10edo occupies a position intermediate between the "[[albitonic]]" systems such as [[7edo]], and "chromatic" systems such as [[12edo]]. While scales such as [[3L 4s]] can be used as a structural backbone when working within 10edo, the [[10-form]] itself can serve as a basic form in tuning systems, more analogous to the [[7-form]]. | |||
== Theory == | == Theory == | ||
=== Chords === | |||
10edo is arguably the first edo to have three triads consisting of two [[third]]s to make a fifth. However, aside from the neutral triad, the other two use 5edo intervals, which are equivalent to major seconds and perfect fourths. Treating these intervals as thirds relies on the fact that stacking two sharp fifths minus an octave makes an inframinor third, thus fundamentally using oneirotonic logic. | |||
10edo is arguably the first edo to have three triads consisting of two | |||
Tendo: 0-4-6-(10) | Tendo: 0-4-6-(10) | ||
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See also: [[Oneirotonic#Chords_of_oneirotonic]] | See also: [[Oneirotonic#Chords_of_oneirotonic]] | ||
=== Scales === | |||
Example mosh ( | Example mosh (3L 4s): 0-1-3-4-6-7-9-10 | ||
Mosh is the most characteristic scale in 10edo. | Mosh is the most characteristic scale in 10edo. | ||
Example subaric ( | Example subaric (2L 6s): 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. | 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. | Due to its small size and unique melodic character, it is very easy to detemper 10edo. Example tunings are shown in parenthesis. | ||
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** Soft subaric | ** Soft subaric | ||
== The 10-form == | ==== 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. | 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. | ||
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|2/1 | |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. | 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. | ||
{| class="wikitable" | {| class="wikitable" | ||
Revision as of 13:49, 23 February 2026
10edo, or 10 equal divisions of the octave (sometimes called 10-TET or 10-tone equal temperament), is the equal tuning featuring steps of (1200/10) = 120 cents, 10 of which stack to the perfect octave 2/1.
As the double of 5edo, 10edo has its sharp fifth of 720¢, and approximates 7/4 well by 960¢. It divides this fifth in two so that it possesses a neutral third of 360¢ that is extremely close to 16/13. Both 6/5 and 5/4 are mapped to this neutral third, and in this fashion 10edo provides a very primitive basis for the 7-limit.
10edo occupies a position intermediate between the "albitonic" systems such as 7edo, and "chromatic" systems such as 12edo. While scales such as 3L 4s can be used as a structural backbone when working within 10edo, the 10-form itself can serve as a basic form in tuning systems, more analogous to the 7-form.
Theory
Chords
10edo is arguably the first edo to have three triads consisting of two thirds to make a fifth. However, aside from the neutral triad, the other two use 5edo intervals, which are equivalent to major seconds and perfect fourths. Treating these intervals as thirds relies on the fact that stacking two sharp fifths minus an octave makes an inframinor third, thus fundamentally 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 (3L 4s): 0-1-3-4-6-7-9-10
Mosh is the most characteristic scale in 10edo.
Example subaric (2L 6s): 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 |
