# 10edo

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

Degree | Cents | Approximate Ratios* | Additional Ratios*
of 3, 5 and 9 |
Interval Names | ups and downs notation | ||
---|---|---|---|---|---|---|---|

0 | 0 | 1/1 | 256/243, 50/49, 25/24 | unison | unison, min 2nd | P1, m2 | D, Eb |

1 | 120 | 16/15, 15/14, 13/14 | 10/9, 13/12, 81/80 | small neutral second, large minor second | mid 2nd | ~2 | D^, Ev |

2 | 240 | 8/7, 15/13, 144/125 | 9/8, 7/6 | second/third | maj 2nd, min 3rd | M2,m3 | E, F |

3 | 360 | 16/13 | 5/4 | large neutral third | mid 3rd | ~3 | F^, Gv |

4 | 480 | 64/49, 169/128 | 4/3, 9/7, 13/10 | smaller fourth | maj 3rd, perf 4th | M3,P4 | F#, G |

5 | 600 | 91/64, 128/91, 169/120, 240/169 | 7/5, 10/7, 13/9, 18/13 | tritone | up 4th, down 5th | ^4,v5 | G^, Av |

6 | 720 | 49/32, 256/169 | 3/2, 14/9, 20/13 | bigger fifth | perf 5th, min 6th | P5,m6 | A, Bb |

7 | 840 | 13/8 | 8/5 | neutral sixth | mid 6th | ~6 | A^, Bv |

8 | 960 | 7/4, 26/15, 125/72 | 16/9, 12/7 | sixth/seventh | maj 6th, min 7th | M6,m7 | B, C |

9 | 1080 | 15/8, 28/15, 13/7 | 9/5, 24/13, 160/81 | small major 7th | mid 7th | ~7 | C^, Dv |

10 | 1200 | 2/1 | 243/128, 49/25, 48/25 | octave | maj 7th, octave | M7, P8 | C#, D |

- based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

- 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

Periods
per octave |
Generator | Temperament(s) |
---|---|---|

1 | 1\10 | Messed-up negri (or miracle) |

1 | 3\10 | Dicot/beatles/neutral thirds scale |

2 | 1\10 | Messed-up pajara |

2 | 2\10 | Decimal / messed-up lemba |

5 | 1\10 | Blackwood/blacksmith |

# Commas

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

Rational | Monzo | Size (Cents) | Name 1 | Name 2 | Name 3 |
---|---|---|---|---|---|

256/243 | | 8 -5 > | 90.22 | Limma | Pythagorean Minor 2nd | |

25/24 | | -3 -1 2 > | 70.67 | 5-limit large semitone | 5-limit chromatic semitone | |

16875/16384 | | -14 3 4 > | 51.12 | Negri Comma | Double Augmentation Diesis | |

9931568/9752117 | | -25 7 6 > | 31.57 | Ampersand's Comma | ||

2048/2025 | | 11 -4 -2 > | 19.55 | Diaschisma | ||

525/512 | | -9 1 2 1 > | 43.41 | Avicennma | Avicennma's Enharmonic Diesis | |

49/48 | | -4 -1 0 2 > | 35.70 | Slendro Diesis | ||

50/49 | | 1 0 2 -2 > | 34.98 | Tritonic Diesis | Jubilisma | |

686/675 | | 1 -3 -2 3 > | 27.99 | Senga | ||

64/63 | | 6 -2 0 -1 > | 27.26 | Septimal Comma | Archytas' Comma | Leipziger Komma |

9859966/9733137 | | -10 7 8 -7 > | 22.41 | Blackjackisma | ||

1029/1024 | | -10 1 0 3 > | 8.43 | Gamelisma | ||

225/224 | | -5 2 2 -1 > | 7.71 | Septimal Kleisma | Marvel Comma | |

16875/16807 | | 0 3 4 -5 > | 6.99 | Mirkwai | ||

6772805/6751042 | | 11 -10 -10 10 > | 5.57 | Linus | ||

2401/2400 | | -5 -1 -2 4 > | 0.72 | Breedsma | ||

243/242 | | -1 5 0 0 -2 > | 7.14 | Rastma | ||

385/384 | | -7 -1 1 1 1 > | 4.50 | Keenanisma | ||

441/440 | | -3 2 -1 2 -1 > | 3.93 | Werckisma | ||

540/539 | | 2 3 1 -2 -1 > | 3.21 | Swetisma | ||

3025/3024 | | -4 -3 2 -1 2 > | 0.57 | Lehmerisma | ||

91/90 | | -1 -2 -1 1 0 1 > | 19.13 | Superleap | ||

676/675 | | 2 -3 -2 0 0 2 > | 2.56 | Parizeksma |

# Music

ZIA Space by Elaine Walker "Who Loves You, Me?," "Champagne," and "Avatar"

Ten Fingers *play* by Bill Sethares (synth guitar)

Circle of Thirds *play* by Bill Sethares (synth ens.)

10_fantasy *play* by Aaron Krister Johnson (synth monody)

*Prelude in 10ET* by Aaron Andrew Hunt

Future *play* and Sol *play* by ZIA (synths and voice in 10)

*Prelude* by Rick McGowan (Rhino synthesizer)

*Ideas on the Waterfall of Expression* by Igliashon Jones (synth)

*For two violas and gongs* by Chris Vaisvil (website) more composition information

*Blues 10* by Carlo Serafini (blog entry)

*Waltz 10* by Carlo Serafini (blog entry)

*Smooth 10* by Carlo Serafini (blog entry)

*10preview.ogg* A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)

*decexperiment.ogg* 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)

*10 Earwigs Invasive* by Chris Vaisvil

*Comets Over Flatland 9* by Randy Winchester

*The Dramatic Squirrel Overture* by Chris Vaisvil Details

*Shimmerwing* by Andrew Heathwaite and Chris Vaisvil

*The Csx Freight at 1:20 am* by Zach Curley

*Shall I Refuse My Dinner* by Steve Martin on SoundCloud

**Instruments**

10-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.

A Decaphonic (10-EDO) Classical Guitar |