18edo - Revision history

Hkm: /* Scales */

2026-03-22T18:43:53Z

Scales

← Older revision		Revision as of 18:43, 22 March 2026
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	~~{{problematic}}~~

	'''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.		'''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.

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	== Scales ==		== Scales ==
			In this section, notations like "2222122221" for taric refer to how many steps of 18edo are between successive degrees of the scale. For example, the first step of taric (between the first two degrees) is two steps of 18edo. The notations in parentheses, like "8L2s", are [[MOS]] or [[MOS substitution]] notation that describes the same thing as the other notation more concisely. All "view" links go to Scale Workshop, a website that lets the user play in a scale through the computer keyboard or touch-screen interface, and download .scl files for use in synths.
	* Straddle-3 diatonic (5L1m1s), 3331332 or 3332331, constructed by an alternating stack of flat and sharp fifths		* Straddle-3 diatonic (5L1m1s), 3331332 or 3332331, constructed by an alternating stack of flat and sharp fifths
	* [[Oneirotonic]] (5L3s), 33133131 (compressed 17edo diatonic)		* [[Oneirotonic]] (5L3s), 33133131 (compressed 17edo diatonic). ([https://scaleworkshop.plainsound.org/scale/RBx7WVRn- view])
	* [[Smitonic]] (4L3s), 3323232 (stretched 19edo diatonic)		* [[Smitonic]] (4L3s), 3323232 (stretched 19edo diatonic). ([https://scaleworkshop.plainsound.org/scale/RBxvWZhiE view])
	* [[Taric]] (8L2s), 2222122221 and the altered MOS pentachordal taric, 2221222221		* [[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		* Hexawood (6L6s), 212121212121 is a "straddle-3 chromatic scale", constructed by an alternating stack of flat and sharp fifths
			* Smitonic MODMOS, 2332233 (inverted step pattern of the scale associated with rast maqam). ([https://scaleworkshop.plainsound.org/scale/RBxV9E9cC view])
			* Antidiatonic, 2242224 (inverted step pattern of diatonic). ([https://scaleworkshop.plainsound.org/scale/RBxzvsm6S view])

	== Edostep interpretations ==		== Edostep interpretations ==

	==== Edostep interpretations ====		==== Edostep interpretations ====
	18edo's edostep has the following interpretations in its patent val:		18edo's edostep has the following interpretations in its [[patent val]]:

	* 25/24 (the difference between 5/4 and 6/5)		* 25/24 (the difference between 5/4 and 6/5)
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	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.		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 ===~~		== Delta-rational theory ==
	~~[https://scaleworkshop.plainsound.org/scale/RBxV9E9cC Smitonic MODMOS] (inverted step pattern of the scale associated with rast maqam)~~

	~~[https://scaleworkshop.plainsound.org/scale/RBxvWZhiE Smitonic] (neutralized antidiatonic)~~

	~~[https://scaleworkshop.plainsound.org/scale/RBxzvsm6S Antidiatonic] (inverted step pattern of diatonic)~~

	~~[https://scaleworkshop.plainsound.org/scale/RBx7WVRn~~- ~~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):		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 ===		=== +1+1 ===

Inthar: /* JI approximation */

2026-03-20T13:37:08Z

JI approximation

← Older revision		Revision as of 13:37, 20 March 2026
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	== JI approximation ==		== 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.		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. Other 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. {{Harmonics in ED\|18\|31\|0}}		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. {{Harmonics in ED\|18\|31\|0}}

Inthar: /* Non-DR theory */

2026-03-20T09:28:58Z

Non-DR theory

← Older revision		Revision as of 09:28, 20 March 2026
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	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. {{Harmonics in ED\|18\|31\|0}}		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. {{Harmonics in ED\|18\|31\|0}}

	== ~~Non-DR theory~~ ==		== Scales and chords ==
	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.		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.

Hkm: Undo revision 5078 by Hkm (talk)

2026-03-19T20:02:27Z

Undo revision 5078 by Hkm (talk)

← Older revision		Revision as of 20:02, 19 March 2026
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	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.		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. {{Harmonics in ED\|18\|11\|0}}		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. {{Harmonics in ED\|18\|31\|0}}

	== Non-DR theory ==		== Non-DR theory ==

Hkm: /* JI approximation */

2026-03-19T20:01:43Z

JI approximation

← Older revision		Revision as of 20:01, 19 March 2026
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	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.		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. {{Harmonics in ED\|18\|31\|0}}		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. {{Harmonics in ED\|18\|11\|0}}

	== Non-DR theory ==		== Non-DR theory ==

Inthar: /* Supersets */

2026-03-18T19:59:30Z

Supersets

← Older revision		Revision as of 19:59, 18 March 2026
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	== Supersets ==		== 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]]. {{Navbox EDO}}		18edo's primes are mostly off by about a twelfth-tone or a sixth-tone. This implies that multiplying it by four 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]]. {{Navbox EDO}}
	{{Cat\|Edos}}		{{Cat\|Edos}}

Inthar: /* +1+?+1 */

2026-03-18T19:59:08Z

+1+?+1

← Older revision		Revision as of 19:59, 18 March 2026
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	* [0 7 10 15]\18 (error 3.3c)		* [0 7 10 15]\18 (error 3.3c)
	* [0 6 12 16]\18 (error 5.2c)		* [0 6 12 16]\18 (error 5.2c)
	* [0 6 11 15]\18 (error 6.8c)		* [0 6 11 15]\18 (error 6.8c) (stretched 4:5:6:7)
	* [0 5 7 11]\18 (error 7.8c)		* [0 5 7 11]\18 (error 7.8c)
	* [0 4 7 10]\18 (error 8.6c)		* [0 4 7 10]\18 (error 8.6c)

Vector at 05:51, 18 March 2026

2026-03-18T05:51:29Z

@@ Line 11: / Line 11: @@
 * [[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
 == DR theory ==
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 === +1+1 ===
 * [0 8 14]\18 (error 6.3c)
-* [0 6 11]\18 (error 7.3c)
+* [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)
@@ Line 33: / Line 122: @@
 * [0 4 7 10]\18 (error 8.6c)
 * [0 4 9 12]\18 (error 8.8c)
 {{Navbox EDO}}
 {{Cat|Edos}}

Inthar: /* DR theory */

2026-03-16T22:22:17Z

DR theory

← Older revision		Revision as of 22:22, 16 March 2026
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	== DR theory ==		== DR theory ==
	18edo has the following approximate [[DR]] chords:		18edo has the following approximate [[DR]] chords (below 10c pairwise logarithmic least-squares error, bounding interval < 1200c, no 1\18 or 17\18):
	* [0 8 14]\18 (~~approx~~. ~~+1+1~~)		=== +1+1 ===
	* [0 6 11 15]\18 (~~approx~~. +1+1+1)		* [0 8 14]\18 (error 6.3c)
	* [0 5 ~~9 16~~]\18 (~~approx~~. +1+1+2)		* [0 6 11]\18 (error 7.3c)
	* [0 ~~4 8~~ 11]\18 (~~approx~~. +1+?+1)		=== +1+2 ===
	* [0 3 11 13]\18 (~~approx~~. ~~+1+?+1~~)		* [0 6 15]\18 (error 1.3c)
	* [0 7 10 15]\18 (~~approx~~. ~~+1+?+1~~)		* [0 3 8]\18 (error 4.8c)
	* [0 ~~3 8~~]\18 (~~approx~~. ~~+1+2~~)		* [0 5 13]\18 (error 8.6c)
	* [0 ~~5 13~~]\18 (~~approx~~. ~~+1+2~~)		=== +2+1 ===
	* [0 5 11]\18 (~~approx~~. ~~+2+3~~)		* [0 7 10]\18 (error 6.8c)
	* [0 ~~6 13~~]\18 (~~very accurately +2+3~~)		* [0 11 15]\18 (error 9.5c)
	~~etc~~.		=== +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)
	{{Navbox EDO}}		{{Navbox EDO}}
	{{Cat\|Edos}}		{{Cat\|Edos}}

Inthar at 16:37, 1 March 2026

2026-03-01T16:37:46Z

← Older revision		Revision as of 16:37, 1 March 2026
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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.		'''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 primes\|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.		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 primes\|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 ==		== Scales ==