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10edo, or 10-tone equal temperament, is a tuning system which divides the octave into 10 equal parts of exactly 120 cents. It can be thought of as two circles of 5edo separated by 120 cents (or 5 circles of 2edo). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of 13/8 and its inversion 16/13; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic moment of symmetry scale of the form 1 2 1 2 1 2 1 (3L 4s - mosh). While not an integral or gap edo, it is a zeta peak edo. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

| Intervals | Images | Linear temperaments | Commas | Music | Instruments

## Intervals

of 3, 5 and 9

*adding the ratios of 3, 5 and 9 introduces greater error while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see Ups and Downs Notation -"Pentatonic" EDOs (pentatonic fifth-based) and Ups and Downs Notation - Natural Generators (heptatonic 3rd-based).

## Images

## Linear temperaments

per octave

## Commas

10 EDO tempers out the following commas. (Note: This assumes the val < 10 16 23 28 35 37 |.)## Music

Ten Fingersplayby Bill Sethares (synth guitar)Circle of Thirds

playby Bill Sethares (synth ens.)10_fantasy

playby Aaron Krister Johnson (synth monody)Prelude in 10ETby Aaron Andrew HuntFuture

playand Solplayby ZIA (synths and voice in 10)Preludeby Rick McGowan (Rhino synthesizer)Ideas on the Waterfall of Expressionby Igliashon Jones (synth)For two violas and gongsby Chris Vaisvil (website) more composition informationBlues 10by Carlo Serafini (blog entry)Waltz 10by Carlo Serafini (blog entry)Smooth 10by Carlo Serafini (blog entry)10 Earwigs Invasiveby Chris VaisvilComets Over Flatland 9by Randy WinchesterThe Dramatic Squirrel Overtureby Chris Vaisvil DetailsShimmerwing by Andrew Heathwaite and Chris Vaisvil

The Csx Freight at 1:20 amby Zach CurleySirmdbidnud2 by martinsj013 on SoundCloud

10tone demo by Clem Fortuna

Instruments10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from 20edo which approximates the 11th harmonic with relative accuracy, among other features.