10edo

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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. 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 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