26edo - Revision history

Lériendil: /* JI approximation */

2026-05-31T04:18:00Z

JI approximation

← Older revision		Revision as of 04:18, 31 May 2026
Line 11:		Line 11:
	26edo is characterized by a flat tuning of harmonics 3, 13, 17, and especially 5; and slightly sharp but accurate tunings of 7 and 11. Due to the shared flat tendency, 26edo turns out to be consistent to the 13-odd-limit. It would be consistent to the 17-odd-limit as well, were it not for intervals of 15 = 3*5, which is tuned more than 50% of a step flat. 26edo also inherits many interval approximations from 13edo; notably, 13edo approximates 11/8, 10/9, and the 13:17:21 chord within the 3.5{{c}} [[JND]], and therefore 26edo does so as well.		26edo is characterized by a flat tuning of harmonics 3, 13, 17, and especially 5; and slightly sharp but accurate tunings of 7 and 11. Due to the shared flat tendency, 26edo turns out to be consistent to the 13-odd-limit. It would be consistent to the 17-odd-limit as well, were it not for intervals of 15 = 3*5, which is tuned more than 50% of a step flat. 26edo also inherits many interval approximations from 13edo; notably, 13edo approximates 11/8, 10/9, and the 13:17:21 chord within the 3.5{{c}} [[JND]], and therefore 26edo does so as well.

	With the very flat tuning of 5, the 4:5:6 [[triad]] has more of a submajor quality, and [[25/24]], the distinction between the main [[5-limit]] triads, is reduced to the size of a quartertone. Similarly to [[22edo]], this also serves as the distinction between 6/5 and 7/6 (i.e. minor and subminor), and 7/6 and 8/7 (the primary [[chthonic harmony\|chthonic]] medials), and, as while 7 is accurate, 5 is flat enough in 26edo, [[7/5]] and [[10/7]] are both mapped to the 600¢ half-octave tritone.		With the very flat tuning of 5, the 4:5:6 [[triad]] has more of a submajor quality, and [[25/24]], the distinction between the main [[5-limit]] triads, is reduced to the size of a quartertone. Similarly to [[22edo]], this also serves as the distinction between 6/5 and 7/6 (i.e. minor and subminor), and 7/6 and 8/7 (the primary [[chthonic harmony\|chthonic]] medials); and, as while 7 is accurate, 5 is flat enough in 26edo, [[7/5]] and [[10/7]] are both mapped to the 600¢ half-octave tritone.

	26edo's whole tone, of 4 steps, is close to 10/9, but does triple duty as not only 10/9 and 9/8, but also [[11/10]]; this is characteristic of ''[[Flattone]]'', a form of Meantone that serves a fifth tuned flat of 19edo's. A consequence of this is that [[11/9]] is mapped to the same interval as [[5/4]], and to this we can add the successive mediants, [[16/13]] as well as [[21/17]] and [[26/21]]. The last of these is the most accurate to the 26edo interval, being the product of 13/12 and 8/7, which are both tuned extremely well by 26edo.		26edo's whole tone, of 4 steps, is close to 10/9, but does triple duty as not only 10/9 and 9/8, but also [[11/10]]; this is characteristic of ''[[Flattone]]'', a form of Meantone that serves a fifth tuned flat of 19edo's. A consequence of this is that [[11/9]] is mapped to the same interval as [[5/4]], and to this we can add the successive mediants, [[16/13]] as well as [[21/17]] and [[26/21]]. The last of these is the most accurate to the 26edo interval, being the product of 13/12 and 8/7, which are both tuned extremely well by 26edo.

Lériendil: Undo revision 7291 by Hotcrystal0 (talk)

2026-05-31T04:17:11Z

Undo revision 7291 by Hotcrystal0 (talk)

← Older revision		Revision as of 04:17, 31 May 2026
Line 11:		Line 11:
	26edo is characterized by a flat tuning of harmonics 3, 13, 17, and especially 5; and slightly sharp but accurate tunings of 7 and 11. Due to the shared flat tendency, 26edo turns out to be consistent to the 13-odd-limit. It would be consistent to the 17-odd-limit as well, were it not for intervals of 15 = 3*5, which is tuned more than 50% of a step flat. 26edo also inherits many interval approximations from 13edo; notably, 13edo approximates 11/8, 10/9, and the 13:17:21 chord within the 3.5{{c}} [[JND]], and therefore 26edo does so as well.		26edo is characterized by a flat tuning of harmonics 3, 13, 17, and especially 5; and slightly sharp but accurate tunings of 7 and 11. Due to the shared flat tendency, 26edo turns out to be consistent to the 13-odd-limit. It would be consistent to the 17-odd-limit as well, were it not for intervals of 15 = 3*5, which is tuned more than 50% of a step flat. 26edo also inherits many interval approximations from 13edo; notably, 13edo approximates 11/8, 10/9, and the 13:17:21 chord within the 3.5{{c}} [[JND]], and therefore 26edo does so as well.

	With the very flat tuning of 5, the 4:5:6 [[triad]] has more of a submajor quality, and [[25/24]], the distinction between the main [[5-limit]] triads, is reduced to the size of a quartertone. Similarly to [[22edo]], this also serves as the distinction between 6/5 and 7/6 (i.e. minor and subminor), and 7/6 and 8/7 (the primary [[chthonic harmony\|chthonic]] medials), and, as while 7 is accurate, 5 is flat enough in 26edo ~~that~~ [[7/5]] and [[10/7]] are both mapped to the 600¢ half-octave tritone.		With the very flat tuning of 5, the 4:5:6 [[triad]] has more of a submajor quality, and [[25/24]], the distinction between the main [[5-limit]] triads, is reduced to the size of a quartertone. Similarly to [[22edo]], this also serves as the distinction between 6/5 and 7/6 (i.e. minor and subminor), and 7/6 and 8/7 (the primary [[chthonic harmony\|chthonic]] medials), and, as while 7 is accurate, 5 is flat enough in 26edo, [[7/5]] and [[10/7]] are both mapped to the 600¢ half-octave tritone.

	26edo's whole tone, of 4 steps, is close to 10/9, but does triple duty as not only 10/9 and 9/8, but also [[11/10]]; this is characteristic of ''[[Flattone]]'', a form of Meantone that serves a fifth tuned flat of 19edo's. A consequence of this is that [[11/9]] is mapped to the same interval as [[5/4]], and to this we can add the successive mediants, [[16/13]] as well as [[21/17]] and [[26/21]]. The last of these is the most accurate to the 26edo interval, being the product of 13/12 and 8/7, which are both tuned extremely well by 26edo.		26edo's whole tone, of 4 steps, is close to 10/9, but does triple duty as not only 10/9 and 9/8, but also [[11/10]]; this is characteristic of ''[[Flattone]]'', a form of Meantone that serves a fifth tuned flat of 19edo's. A consequence of this is that [[11/9]] is mapped to the same interval as [[5/4]], and to this we can add the successive mediants, [[16/13]] as well as [[21/17]] and [[26/21]]. The last of these is the most accurate to the 26edo interval, being the product of 13/12 and 8/7, which are both tuned extremely well by 26edo.

Hotcrystal0: grammar fix

2026-05-22T20:39:18Z

grammar fix

← Older revision		Revision as of 20:39, 22 May 2026
Line 11:		Line 11:
	26edo is characterized by a flat tuning of harmonics 3, 13, 17, and especially 5; and slightly sharp but accurate tunings of 7 and 11. Due to the shared flat tendency, 26edo turns out to be consistent to the 13-odd-limit. It would be consistent to the 17-odd-limit as well, were it not for intervals of 15 = 3*5, which is tuned more than 50% of a step flat. 26edo also inherits many interval approximations from 13edo; notably, 13edo approximates 11/8, 10/9, and the 13:17:21 chord within the 3.5{{c}} [[JND]], and therefore 26edo does so as well.		26edo is characterized by a flat tuning of harmonics 3, 13, 17, and especially 5; and slightly sharp but accurate tunings of 7 and 11. Due to the shared flat tendency, 26edo turns out to be consistent to the 13-odd-limit. It would be consistent to the 17-odd-limit as well, were it not for intervals of 15 = 3*5, which is tuned more than 50% of a step flat. 26edo also inherits many interval approximations from 13edo; notably, 13edo approximates 11/8, 10/9, and the 13:17:21 chord within the 3.5{{c}} [[JND]], and therefore 26edo does so as well.

	With the very flat tuning of 5, the 4:5:6 [[triad]] has more of a submajor quality, and [[25/24]], the distinction between the main [[5-limit]] triads, is reduced to the size of a quartertone. Similarly to [[22edo]], this also serves as the distinction between 6/5 and 7/6 (i.e. minor and subminor), and 7/6 and 8/7 (the primary [[chthonic harmony\|chthonic]] medials), and, as while 7 is accurate, 5 is flat enough in 26edo, [[7/5]] and [[10/7]] are both mapped to the 600¢ half-octave tritone.		With the very flat tuning of 5, the 4:5:6 [[triad]] has more of a submajor quality, and [[25/24]], the distinction between the main [[5-limit]] triads, is reduced to the size of a quartertone. Similarly to [[22edo]], this also serves as the distinction between 6/5 and 7/6 (i.e. minor and subminor), and 7/6 and 8/7 (the primary [[chthonic harmony\|chthonic]] medials), and, as while 7 is accurate, 5 is flat enough in 26edo that [[7/5]] and [[10/7]] are both mapped to the 600¢ half-octave tritone.

	26edo's whole tone, of 4 steps, is close to 10/9, but does triple duty as not only 10/9 and 9/8, but also [[11/10]]; this is characteristic of ''[[Flattone]]'', a form of Meantone that serves a fifth tuned flat of 19edo's. A consequence of this is that [[11/9]] is mapped to the same interval as [[5/4]], and to this we can add the successive mediants, [[16/13]] as well as [[21/17]] and [[26/21]]. The last of these is the most accurate to the 26edo interval, being the product of 13/12 and 8/7, which are both tuned extremely well by 26edo.		26edo's whole tone, of 4 steps, is close to 10/9, but does triple duty as not only 10/9 and 9/8, but also [[11/10]]; this is characteristic of ''[[Flattone]]'', a form of Meantone that serves a fifth tuned flat of 19edo's. A consequence of this is that [[11/9]] is mapped to the same interval as [[5/4]], and to this we can add the successive mediants, [[16/13]] as well as [[21/17]] and [[26/21]]. The last of these is the most accurate to the 26edo interval, being the product of 13/12 and 8/7, which are both tuned extremely well by 26edo.

Lériendil: /* JI approximation */

2026-05-15T13:18:56Z

JI approximation

← Older revision		Revision as of 13:18, 15 May 2026
Line 11:		Line 11:
	26edo is characterized by a flat tuning of harmonics 3, 13, 17, and especially 5; and slightly sharp but accurate tunings of 7 and 11. Due to the shared flat tendency, 26edo turns out to be consistent to the 13-odd-limit. It would be consistent to the 17-odd-limit as well, were it not for intervals of 15 = 3*5, which is tuned more than 50% of a step flat. 26edo also inherits many interval approximations from 13edo; notably, 13edo approximates 11/8, 10/9, and the 13:17:21 chord within the 3.5{{c}} [[JND]], and therefore 26edo does so as well.		26edo is characterized by a flat tuning of harmonics 3, 13, 17, and especially 5; and slightly sharp but accurate tunings of 7 and 11. Due to the shared flat tendency, 26edo turns out to be consistent to the 13-odd-limit. It would be consistent to the 17-odd-limit as well, were it not for intervals of 15 = 3*5, which is tuned more than 50% of a step flat. 26edo also inherits many interval approximations from 13edo; notably, 13edo approximates 11/8, 10/9, and the 13:17:21 chord within the 3.5{{c}} [[JND]], and therefore 26edo does so as well.

	With the very flat tuning of 5, the 4:5:6 [[triad]] has more of a submajor quality, and [[25/24]], the distinction between the main [[5-limit]] triads, is reduced to the size of a quartertone. Similarly to [[22edo]], this also serves as the distinction between 6/5 and 7/6 (i.e. minor and subminor), and 7/6 and 8/7 (the primary [[chthonic harmony\|chthonic]] medials); and ~~notably~~, while 7 is accurate, 5 is flat enough in 26edo ~~that~~ [[7/5]] and [[10/7]] are both mapped to the 600¢ half-octave tritone.		With the very flat tuning of 5, the 4:5:6 [[triad]] has more of a submajor quality, and [[25/24]], the distinction between the main [[5-limit]] triads, is reduced to the size of a quartertone. Similarly to [[22edo]], this also serves as the distinction between 6/5 and 7/6 (i.e. minor and subminor), and 7/6 and 8/7 (the primary [[chthonic harmony\|chthonic]] medials), and, as while 7 is accurate, 5 is flat enough in 26edo, [[7/5]] and [[10/7]] are both mapped to the 600¢ half-octave tritone.

	26edo's whole tone, of 4 steps, is close to 10/9, but does triple duty as not only 10/9 and 9/8, but also [[11/10]]; this is characteristic of ''[[Flattone]]'', a form of Meantone that serves a fifth tuned flat of 19edo's. A consequence of this is that [[11/9]] is mapped to the same interval as [[5/4]], and to this we can add the successive mediants, [[16/13]] as well as [[21/17]] and [[26/21]]. The last of these is the most accurate to the 26edo interval, being the product of 13/12 and 8/7, which are both tuned extremely well by 26edo.		26edo's whole tone, of 4 steps, is close to 10/9, but does triple duty as not only 10/9 and 9/8, but also [[11/10]]; this is characteristic of ''[[Flattone]]'', a form of Meantone that serves a fifth tuned flat of 19edo's. A consequence of this is that [[11/9]] is mapped to the same interval as [[5/4]], and to this we can add the successive mediants, [[16/13]] as well as [[21/17]] and [[26/21]]. The last of these is the most accurate to the 26edo interval, being the product of 13/12 and 8/7, which are both tuned extremely well by 26edo.

Lériendil: added mos loops

2026-05-15T13:15:21Z

added mos loops

← Older revision		Revision as of 13:15, 15 May 2026
Line 233:		Line 233:

	=== Scales ===		=== Scales ===
			==== Organization of MOSes ====
			26edo has six distinct intervals that define octave-periodic generator structures, not counting those of 13edo. These generator structures and consequent [[MOS]] scales organize themselves into two loops of three, each linked by the operation of tripling.
			{\| class="wikitable"
			\|+
			!
			! colspan="3" \|Loop 1
			\|-
			!Temperament
			\|[[Slendric]]
			\|[[Flattone]]
			\|[[Superkleismic]]
			\|-
			!Scale (albitonic)
			\|5-5-5-5-6
			\|4-4-3-4-4-4-3
			\|2-5-2-5-2-5-5
			\|-
			!Generator
			\|5\, 21\
			\|11\, 15\
			\|7\, 19\
			\|-
			!
			! colspan="3" \|Loop 2
			\|-
			!Temperament
			\|[[Quartonic]]
			\|[[Bleu]]
			\|[[Roman]]
			\|-
			!Scale (albitonic)
			\| - (11-note scale has<br>>10 steps interval)
			\|3-3-3-3-3-3-3-3-2
			\|7-1-7-1-1-7-1-1
			\|-
			!Generator
			\|1\, 25\
			\|3\, 23\
			\|9\, 17\
			\|}
	{{WIP}}		{{WIP}}

Lériendil: /* Intervals and notation */

2026-05-15T13:00:19Z

Intervals and notation

← Older revision		Revision as of 13:00, 15 May 2026
Line 101:		Line 101:
	\| 507.7		\| 507.7
	\| F		\| F
	\| 4/3		\| 4/3, ''15/11''
	\| perfect fourth		\| perfect fourth
	\|-		\|-
Line 125:		Line 125:
	\| 692.3		\| 692.3
	\| G		\| G
	\| '''3/2'''		\| '''3/2''', ''22/15''
	\| perfect fifth		\| perfect fifth
	\|-		\|-

Lériendil: /* Intervals and notation */

2026-05-15T12:57:45Z

Intervals and notation

← Older revision		Revision as of 12:57, 15 May 2026
Line 108:		Line 108:
	\| F#		\| F#
	\| '''[11/8]''', 18/13		\| '''[11/8]''', 18/13
	\| ~~subaugmented~~ fourth		\| (sub)augmented fourth
	\|-		\|-
	\| 13		\| 13
Line 120:		Line 120:
	\| Gb		\| Gb
	\| 13/9, [16/11]		\| 13/9, [16/11]
	\| ~~supradiminished~~ fifth		\| (supra)diminished fifth
	\|-		\|-
	\| 15		\| 15

Lériendil: /* Tertian structure */

2026-05-15T07:01:58Z

Tertian structure

← Older revision		Revision as of 07:01, 15 May 2026
Line 223:		Line 223:
	\|9		\|9
	\|}		\|}

			Diatonic thirds are bolded.

	=== Chords ===		=== Chords ===

Lériendil: /* Tertian structure */

2026-05-15T07:01:37Z

Tertian structure

← Older revision		Revision as of 07:01, 15 May 2026
Line 214:		Line 214:
	\|7/6 (+10.0{{c}})		\|7/6 (+10.0{{c}})
	\|'''6/5 (+7.4{{c}})'''		\|'''6/5 (+7.4{{c}})'''
	\|'''11/9 (+22.2{{c}}), 16/13 (+9.8{{c}}), 5/4 (-17.1{{c}})'''		\|'''11/9 (+22.2{{c}}) <br /> 16/13 (+9.8{{c}}) <br /> 5/4 (-17.1{{c}})'''
	\|14/11 (-2.1{{c}}), 9/7 (-19.7{{c}})		\|14/11 (-2.1{{c}}) <br /> 9/7 (-19.7{{c}})
	\|-		\|-
	!Steps		!Steps

Lériendil: added offsets for just thirds

2026-05-15T07:01:18Z

added offsets for just thirds

← Older revision		Revision as of 07:01, 15 May 2026
Line 196:		Line 196:

	== Compositional theory ==		== Compositional theory ==
			=== Tertian structure ===
	{\| class="wikitable"		{\| class="wikitable"
	\|+Thirds in 26edo		\|+Thirds in 26edo
Line 211:		Line 212:
	\|-		\|-
	!Just interpretation		!Just interpretation
	\|7/6		\|7/6 (+10.0{{c}})
	\|'''6/5'''		\|'''6/5 (+7.4{{c}})'''
	\|'''5/4, 16/13'''		\|'''11/9 (+22.2{{c}}), 16/13 (+9.8{{c}}), 5/4 (-17.1{{c}})'''
	\|14/11, 9/7		\|14/11 (-2.1{{c}}), 9/7 (-19.7{{c}})
	\|-		\|-
	!Steps		!Steps