# 18edo

**18 Equal Divisions of the Octave** also known as **The Third-Tone System**.

## Contents

# Basic Properties

18-EDO divides the octave into 18 equal parts of ~66.667 cents each. It does not approximate the 3rd harmonic at all, unless a >30¢-error is considered acceptable, and it approximates the 5th and 7th harmonics equally with 12-TET. It does, however, render a most accurate tuning of 9/8, 7/6, 21/16, 15/11, 12/7, 16/9, and 13/7. It is also the smallest EDO to approximate the harmonic series chord 5:6:7 without tempering out 36/35 (and thus without using the same interval to approximate both 6/5 and 7/6).

In order to access the excellent consonances actually available, one must take a considerably "non-common-practice" approach, meaning to avoid the usual closed-voice "root-3rd-5th" type of chord and instead use chords which are either more compressed or more stretched out. 18-EDO may be treated as a temperament of the 17-limit 4*18 subgroup just intonation subgroup 2.9.75.21.55.39.51. On this subgroup it tempers out exactly the same commas as 72 does on the full 17-limit, and gives precisely the same tunings. The subgroup can be put into a single chord, for example 32:36:39:42:51:55:64:75 (in terms of 18edo, 0-3-5-7-12-14-18-22), and transpositions and inversions of this chord or its subchords provide plenty of harmonic resources.

However, less accurate approximations can be used, and 18edo can be treated as a 7-limit exotemperament with the mapping <18 29 42 51|. This maps 3/2 to 733.33¢ and 7/4 to 1000¢; as a result, 28/27 is tempered out, and weird things happen: 9/8 and 7/6 are both mapped to 266.67¢, while 8/7 gets mapped below both of them to 200¢, making for a rather disordered 7-limit tonality diamond, but hey, whatever floats your boat!

18-EDO contains sub-EDOs 2, 3, 6, and 9, and itself is half of 36-EDO and one-fourth of 72-EDO. It bears some similarities to 13-EDO (with its very flat 4ths and nice subminor 3rds), 11-EDO (with its very sharp minor 3rds, two of which span a very flat 5th), 16-EDO (with its sharp 4ths and flat 5ths), and 17-EDO and 19-EDO (with its narrow semitone, three of which comprise a whole-tone). It is an excellent tuning for those seeking a forceful deviation from the common practice.

## Representations of Just Intervals

Degree | Cents | 5L3s Notation | Nearest Ratio | Error | 17-Limit Ratios* |

0 | 0 | C | 1/1 | 0 | 1/1 |

1 | 66.67 | Db | 27/26 | +1.329 | 78/75, 75/72 |

2 | 133.33 | C# | 27/25 | +0.096 | 51/55, 42/39 |

3 | 200 | D | 9/8 | -3.910 | 9/8 |

4 | 266.67 | Eb | 7/6 | -0.204 | 75/64 |

5 | 333.33 | D# | 17/14 or 40/33 | -2.796 +0.293 | 39/32 |

6 | 400 | E | 5/4 or 44/35 | +13.686 +3.822 | 64/55 |

7 | 466.67 | F | 21/16 | -4.114 | 21/16 |

8 | 533.33 | Gb | 15/11 | -3.617 | 102/75 |

9 | 600 | F# | 17/12 or 24/17 | -3.000 +3.000 | 17/12 |

10 | 666.67 | G | 22/15 | +3.617 | 75/51 |

11 | 733.33 | Hb | 32/21 | +4.114 | 32/21 |

12 | 800 | G# | 8/5 or 35/22 | -13.686 -3.822 | 51/32 |

13 | 866.67 | H | 28/17 or 33/20 | +2.796 -0.293 | 64/39 |

14 | 933.33 | A | 12/7 | +0.204 | 55/32 |

15 | 1000 | Bb | 16/9 | +3.910 | 16/9 |

16 | 1066.67 | A# | 50/27 | -0.096 | 39/21 |

17 | 1133.33 | B | 52/27 | -1.329 | 75/39 |

18 | 1200 | C | 2/1 | 0 | 2/1** |

- based on the above description of 18-EDO as a 2.9.75.21.55.39.51 subgroup temperament

# Notation

18edo can be notated with ups and downs. The notational 5th is the 2nd-best approximation of 3/2, 10\18. This is only 4¢ worse that the best approximation, which becomes the up-fifth. Using this 5th allows conventional notation to be used, including the staff, note names, relative notation, etc. There are two ways to do this.

The first way preserves the __melodic__ meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second way preserves the __harmonic__ meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 18edo "on the fly".

Degree | Cents | Up/down notation using the narrow 5th of 10\18,
with major wider than minor |
Up/down notation using the narrow 5th of 10\18,
with major narrower than minor | ||||
---|---|---|---|---|---|---|---|

0 | 0 | perfect unison | P1 | D | perfect unison | P1 | D |

1 | 67 | up unison, downminor 2nd | ^1, vm2 | D^, Ev | up unison, downmajor 2nd | ^1, vM2 | D^, Ev |

2 | 133 | minor 2nd | m2 | E | major 2nd | M2 | E |

3 | 200 | mid 2nd | ~2 | E^ | mid 2nd | ~2 | E^ |

4 | 267 | major 2nd, minor 3rd | M2, m3 | E#, Fb | minor 2nd, major 3rd | m2, M3 | Eb, F# |

5 | 333 | mid 3rd | ~3 | Fv | mid 3rd | ~3 | Fv |

6 | 400 | major 3rd | M3 | F | minor 3rd | m3 | F |

7 | 467 | upmajor 3rd, down 4th | ^M3, v4 | F^, Gv | upminor 3rd, down 4th | ^m3, v4 | F^, Gv |

8 | 533 | perfect 4th | P4 | G | perfect 4th | P4 | G |

9 | 600 | up 4th, down 5th | ^4, v5 | G^, Av | up 4th, down 5th | ^4, v5 | G^, Av |

10 | 667 | perfect 5th | P5 | A | perfect 5th | P5 | A |

11 | 733 | up 5th, downminor 6th | ^5, vm6 | A^, Bv | up fifth, downmajor 6th | ^5, vM6 | A^, Bv |

12 | 800 | minor 6th | m6 | B | major 6th | M6 | B |

13 | 867 | mid 6th | ~6 | B^ | mid 6th | ~6 | B^ |

14 | 933 | major 6th, minor 7th | M6, m7 | B#, Cb | minor 6th, major 7th | m6, M7 | Bb, C# |

15 | 1000 | mid 7th | ~7 | Cv | mid 7th | ~7 | Cv |

16 | 1067 | major 7th | M7 | C | minor 7th | m7 | C |

17 | 1133 | upmajor 7th, down 8ve | ^M7, v8 | C^, Dv | upminor 7th, down 8ve | ^m7, v8 | C^, Dv |

18 | 1200 | perfect 8ve | P8 | D | perfect 8ve | P8 | D |

For alternative notations, see Ups and Downs Notation -"Supersharp" EDOs (pentatonic and nonatonic fifth-generated) and Ups and Downs Notation - Natural Generators (heptatonic third-generated).

## Useful Moment-of-Symmetry Scales

Note: This list excludes scales found in 9-EDO.

### Pentatonic:

3L2s Father Pentatonic: 4 4 3 4 3

### Hexatonic:

4L2s Bicycle: 4 4 1 4 4 1

2L4s Rice Hexatonic: 2 5 2 2 5 2

### Heptatonic:

4L3s Amity/Mish Heptatonic: 3 2 3 2 3 3 2

### Octatonic:

5L3s Father Octatonic: 3 1 3 3 1 3 3 1

2L6s Rice Octatonic: 2 2 3 2 2 2 3 2

### Decatonic:

8L2s Biggie Decatonic: 2 2 1 2 2 2 2 1 2 2

### Dodecatonic:

6L 6s Hexe: 2 1 2 1 2 1 2 1 2 1 2 1

## Application to Guitar

18-EDO is an ideal scale for the first-time refretter, because you can retain all the even-number frets from 12-tET--essentially 1/3 of your work is done for you!

The "Father Octatonic" scale maps very simply to a 6-string guitar tuned in "reverse-standard" tuning (tune using four 466.667¢ intervals, with one 533.333¢ interval between the 2nd and 3rd strings), making for a softer learning-curve than EDOs like 14, 16, or 21 (all of which are most evenly open-tuned using a series of sharpened 4ths and a minor or neutral 3rd, and whose scales thus often require position-shifting and/or larger stretches of the hand).

# Commas

18 EDO tempers out the following commas. (Note: This assumes the val < 18 29 42 51 62 67 |.)

Comma | Monzo | Value (Cents) | Name 1 | Name 2 |
---|---|---|---|---|

128/125 | | 7 0 -3 > | 41.06 | Diesis | Augmented Comma |

| 23 6 -14 > | 3.34 | Vishnuzma | Semisuper | |

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

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

875/864 | | -5 -3 3 1 > | 21.90 | Keema | |

1728/1715 | | 6 3 -1 -3 > | 13.07 | Orwellisma | Orwell Comma |

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

3136/3125 | | 6 0 -5 2 > | 6.08 | Hemimean | |

99/98 | | -1 2 0 -2 1 > | 17.58 | Mothwellsma | |

100/99 | | 2 -2 2 0 -1 > | 17.40 | Ptolemisma | |

65536/65219 | | 16 0 0 -2 -3 > | 8.39 | Orgonisma | |

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

9801/9800 | | -3 4 -2 -2 2 > | 0.18 | Kalisma | Gauss' Comma |

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