18edo

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18edo, or 18 equal divisions of the octave, is the equal tuning featuring steps of (1200/18) ~= 66.7 cents, 18 of which stack to the octave 2/1.

With the sharp fifth 733.3c and the flat fifth 666.7c almost equally detuned from the just fifth, 18edo is often considered the quintessential straddle-3 edo and the straddle-3 version of 12edo. It does not approximate low harmonics well, except 9 and debatably 5; it is also straddle-7, 13, 17, and 19.

Scales

• Straddle-3 diatonic (5L1m1s), 3331332 or 3332331, constructed by an alternating stack of flat and sharp fifths
• Oneirotonic (5L3s), 33133131 (compressed 17edo diatonic)
• Smitonic (4L3s), 3323232 (stretched 19edo diatonic)
• Taric (8L2s), 2222122221 and the altered MOS pentachordal taric, 2221222221
• Hexawood (6L6s) is a "straddle-3 chromatic scale", constructed by an alternating stack of flat and sharp fifths

Edostep interpretations

Edostep interpretations

18edo's edostep has the following interpretations in its patent val:

• 25/24 (the difference between 5/4 and 6/5)
• 56/55 (the difference between 11/8 and 7/5)
• 80/77 (the difference between 7/5 and 16/11)
• 36/35 (the difference between 6/5 and 7/6)

JI approximation

18edo is straddle-3 and -7, but inherits 12edo's approximation of 5 and also approximates 11 to a similar degree of accuracy, making 55/32 tuned accurately. Additionally, 7 and 3 are both sharp about 30 cents, meaning 7/6 is tuned well. Oher intervals tuned well include 28/25 and 25/24. Additionally, because 18edo has both the 5 and 7 from 12edo, 7/5 and 10/7 are tempered together (jubilic temperament) and therefore tuned to 600c (the perfect semioctave). This means that passable tunings of 5:6:7 and its utonal counterpart are available.

Additionally, 18edo's approximate 4:5:6 using the sharp fifth is approximately delta-rational (similar to the case with 15edo), albeit slightly less accurate. It is roughly the isoharmonic JI chord 19:24:29.

Approximation of prime harmonics in 18edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	0.0	+31.4	+13.7	+31.2	-18.0	+26.1	+28.4	-30.8	-28.3	-29.6	-11.7
	Relative (%)	0.0	+47.1	+20.5	+46.8	-27.0	+39.2	+42.6	-46.3	-42.4	-44.4	-17.6
Steps (reduced)		18 (0)	29 (11)	42 (6)	51 (15)	62 (8)	67 (13)	74 (2)	76 (4)	81 (9)	87 (15)	89 (17)

Non-DR theory

18edo's best fifth is 733.3 cents, which generates an oneirotonic scale; it can also be used as the double of 9edo, which has an antidiatonic scale.

In oneirotonic, the "thirds" are the same degree as the "major second" and "perfect fourth" respectively, so that any oneirotonic scale always has two distinct "thirds". This follows from the generator being an odd number of steps. In 13edo, the four possible "thirds" collapse to 3 (as the "diminished fourth" and "major third" are identical) but in 18edo (and in 21edo) there are four distinct qualities, here named with oneirotonic ADIN:

• [0 4 11] - "inframinor" (267c)
• [0 5 11] - "nearminor" (333c)
• [0 6 11] - "nearmajor" (400c)
• [0 7 11] - "ultramajor" (467c)

(listen)

The brightest mode of oneirotonic (other than the very brightest, which doesn't have the 733c fifth) has ultramajor and nearmajor. Most modes have ultramajor and inframinor, corresponding to diatonic sus4 and sus2 chords. The darkest mode has nearminor and inframinor.

Alternately, in antidiatonic, 18edo adds to 9edo a "neutral" third, which is the same as the nearminor oneirotonic third. Therefore, there are 5 different qualities of antidiatonic fifth-bounded triad:

• [0 3 10] - "inframinor" (200c)
• [0 4 10] - "farminor" (267c)
• [0 5 10] - "neutral" (333c)
• [0 6 10] - "farmajor" (400c)
• [0 7 10] - "ultramajor" (467c)

(listen)

These chords also exist in oneirotonic (except for the neutral one), as the antidiatonic fifth is also the oneirotonic major tritone, which appears on 4 different scale degrees.

Additionally, there is the dual-fifth interpretation of 18edo, wherein the basic scale is similar to a 19edo diatonic but with one edostep removed. It is useful to organize the chords based on their appearance on each degree of dual-fifth diatonic.

Degree	LLsLLLm	LLmLLLs
1	[0 6 10] - major antidiatonic	[0 6 11] - nearmajor
2	[0 4 10] - minor antidiatonic	[0 5 11] - nearminor
3	[0 4 10] - minor antidiatonic	[0 5 11] - nearminor
4	[0 6 11] - nearmajor	[0 6 10] - major antidiatonic
5	[0 6 11] - nearmajor	[0 6 10] - major antidiatonic
6	[0 5 11] - nearminor	[0 4 10] - minor antidiatonic
7	[0 5 9] - otonal diminished	[0 4 9] - utonal diminished

The otonal and utonal diminished triads provide tunings of 5:6:7 and 1/(5:6:7) respectively, despite the fact that 18edo tunes neither 5 nor 7 particularly well (in fact, it has the same mappings as 12edo).

To use a scale which includes neutral harmony, an option would be altering some of the antidiatonic degrees to be neutral. This generates "smitonic" (3-3-2-3-2-3-2) or a MODMOS thereof, which can be thought of as the antidiatonic counterpart of mosh. The MOS form of smitonic has five of the seven tertian triads neutral; the remaining two are nearmajor and nearminor triads using the oneirotonic fifth. The MODMOS 2-3-3-2-2-3-3 introduces more variation, adding in a major and a minor antidiatonic triad, leaving neutral triads on three of the degrees.

Scale Workshop links

Smitonic MODMOS (inverted step pattern of the scale associated with rast maqam)

Smitonic (neutralized antidiatonic)

Antidiatonic (inverted step pattern of diatonic)

Oneirotonic

DR theory

18edo has the following approximate DR chords (below 10c pairwise logarithmic least-squares error, bounding interval < 1200c, no 1\18 or 17\18):

+1+1

• [0 8 14]\18 (error 6.3c)
• [0 6 11]\18 (error 7.3c) - this is the "nearmajor" oneirotonic chord, which behaves like a stretched 4:5:6 in a delta-rational context (a similar chord is [0 5 9] of 15edo).

+1+2

• [0 6 15]\18 (error 1.3c)
• [0 3 8]\18 (error 4.8c)
• [0 5 13]\18 (error 8.6c)

+2+1

• [0 7 10]\18 (error 6.8c)
• [0 11 15]\18 (error 9.5c)

+1+?+1

• [0 4 8 11]\18 (error 0.2c)
• [0 3 11 13]\18 (error 0.2c)
• [0 7 10 15]\18 (error 3.3c)
• [0 6 12 16]\18 (error 5.2c)
• [0 6 11 15]\18 (error 6.8c)
• [0 5 7 11]\18 (error 7.8c)
• [0 4 7 10]\18 (error 8.6c)
• [0 4 9 12]\18 (error 8.8c)

Supersets

18edo's primes are mostly off by about a twelfth-tone or a sixth-tone. This implies that multiplying it by six to yield 72edo yields an accurate tuning of just intonation. 36edo contains 72edo's 2.3.7 subgroup and is the next level of structural resolution for said subgroup after 5edo.