User:Pailiaq/Tritone substitution - Revision history

Pailiaq at 22:12, 23 December 2025

2025-12-23T22:12:13Z

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	* The leading tone resolves by a smaller step (~50¢) to the tonic—a tighter, more chromatic pull		* The leading tone resolves by a smaller step (~50¢) to the tonic—a tighter, more chromatic pull
	* The ~~subdominant tendency tone~~ now sits ~550¢ above the tonic, requiring a ~150¢ descent to reach the mediant at 400¢		* The fourth now sits ~550¢ above the tonic, requiring a ~150¢ descent to reach the mediant at 400¢
	* The small leading tone at extremes starts to feel weaker, becoming more like a comma adjustment of the tonic rather than a melodic step		* The small leading tone at extremes starts to feel weaker, becoming more like a comma adjustment of the tonic rather than a melodic step

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	* The leading tone resolves by ~150¢—a wider, more whole-tone-like motion		* The leading tone resolves by ~150¢—a wider, more whole-tone-like motion
	* The ~~subdominant tendency~~ tone at ~450¢ resolves down by ~50¢ to the ~~mediant~~		* The fourth tone at ~450¢ resolves down by ~50¢ to the 400c major third, though may also voicelead more smoothly to the neutral or minor third, letting you target different tonic chords besides major different major
	* The upward resolution feels less urgent		* The upward resolution feels less urgent

	This balancing act means that dominant function in xenharmonic systems can take on different characters depending on where the leading tone sits. ~~Systems~~ with a leading tone ~~close~~ to ~~1100¢ and~~ a tritone ~~close to 600¢ will exhibit~~ the ~~most "12-EDO-like" dominant function; systems with more deviant values will have their own unique dominant flavors~~.		This balancing act means that dominant function in xenharmonic systems can take on different characters depending on where the leading tone sits. If using a wider tritone, like 630c, with the leading tone at 1050c, the fourth tone will then be located at 420c, which will lead very cleanly to a minor or neutral chord. So tritone size as well as its position in the leading tone interval region gives it different properties.

	==Summary==		==Summary==

Pailiaq at 08:55, 23 December 2025

2025-12-23T08:55:43Z

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	In all other EDOs, the diatonic augmented fourth deviates from 600¢.		In all other EDOs, the diatonic augmented fourth deviates from 600¢.

	==~~Tritone substitution in~~ even-numbered EDOs==		==In even-numbered EDOs==

	Even-numbered EDOs always contain an exact semioctave at n/2 steps. As stated, this semioctave typically does not align with the diatonic augmented fourth.		Even-numbered EDOs always contain an exact semioctave at n/2 steps. As stated, this semioctave typically does not align with the diatonic augmented fourth.
	=== Examples ===		=== Examples ===
	In 26-EDO, ~~the semioctave is~~ 13\26 = 600¢, but the diatonic augmented fourth is only 12\26 ≈ 554¢. A diatonic dominant seventh chord therefore lacks a true tritone, breaking the substitution mechanism of the 3rd and 7th swapping.		In 26-EDO, 13\26 = 600¢, but the diatonic augmented fourth is only 12\26 ≈ 554¢. A diatonic dominant seventh chord ((0, 8, 15, 22)\26) therefore lacks a true tritone, breaking the substitution mechanism of the 3rd and 7th swapping.

	The solution is to use the '''harmonic seventh chord''' ((0, 8, 15, 21)\26 ~~approximating 4:5:6:7~~) as your dominant V. In 26-EDO, the harmonic seventh (7/4) lands at 21\26 ≈ 969¢, which is ~0.4¢ from just. With a major third at 8\26, the interval from third to seventh is exactly 13 steps = 600¢. This nondiatonic chord produces a perfect tritone substitution, with the caveat that you must step outside your scale (playing a downminor 7) when playing a V-I progression if you want consistency in sound for when also using tritone substitutions.		The solution is to use the '''harmonic seventh chord''' ((0, 8, 15, 21)\26) as your dominant V. In 26-EDO, the harmonic seventh (7/4) lands at 21\26 ≈ 969¢, which is ~0.4¢ from just. With a major third at 8\26, the interval from third to seventh is exactly 13 steps = 600¢. This nondiatonic chord produces a perfect tritone substitution, with the caveat that you must step outside your scale (playing a downminor 7) when playing a V-I progression if you want consistency in sound for when also using tritone substitutions.

	Howewever, using the diatonic, tritoneless dominant 7th chord also can work; the only difference being that the third and 7ths don't perfectly swap positions, as one of them will drift up or down by an edostep.		Howewever, using the diatonic, tritoneless dominant 7th chord also can work; the only difference being that the third and 7ths don't perfectly swap positions, as one of them will drift up or down by an edostep.

	==~~Tritone substitution in~~ odd-numbered EDOs==		==In odd-numbered EDOs==

	Odd-numbered EDOs ~~definitionally~~ lack a semioctave entirely. ~~However~~, ~~tritone substitution can be approximated using~~ a ~~[[7/6 \|~~ small tritone ]] and ~~[[10/7 \|~~ large tritone]] that straddle 600¢.		Odd-numbered EDOs lack a semioctave entirely. Instead, they contain a small tritone and large tritone that straddle 600¢.

	~~In 31-EDO~~, ~~the small tritone (~~15\31 ≈ 581¢, approximating 7/5) and ~~large tritone (~~16\31 ≈ 619¢, approximating 10/7) ~~differ~~ by ~39¢. ~~A "fudged" tritone substitution uses~~ the small tritone ~~in one~~ chord and the large tritone in the ~~other;~~ the ~~third~~ and ~~seventh ''almost'' swap~~, ~~offset by one step~~. ~~This approximation remains functionally effective~~, ~~though it lacks~~ the ~~perfect symmetry~~ of ~~6n-EDO systems and may not sound as good~~.		This split creates an unavoidable inconsistency when attempting tritone substitution. Composers must make a choice:

			Let's look at 31edo as an example, where 15\31 ≈ 581¢ (approximating 7/5) and 16\31 ≈ 619¢ (approximating 10/7), differing by ~39¢.

			* '''Preserve the common tones'''

			Keep the literal pitches of the third and seventh identical between both chords. If G7 contains B–F (a small tritone), the sub chord Db also uses B and F. Voice leading stays identical—the same pitches resolve to the same destinations. However, B is now a harmonic seventh (~969¢) above Db rather than a minor seventh (~1000¢), so the chord quality shifts from dom7 to harmonic 7th.

			*Preserve the chord quality

			Build a standard dom7 on Db, giving Db–F–Ab–Cb. This chord contains the large tritone (F–Cb) rather than the small tritone (B–F) of the original G7. The root moves by one tritone size, but the chord contains the other—so Cb ≠ B, and the common tones drift by an edostep. Chord quality stays consistent, but voice leading shifts.

			The availability of two tritone sizes also means access to multiple dominant chord types, so long as the intervals fall within the right regions of interval space. This can be generalized to create many different chords with dominant function.

	This also means you have access to multiple dominant 7 chords, so long as the intervals within the chord are situated in the right 'regions' of interval space. This can be generalized to create many different chords with dominant function.
	== Structural definition of tritone-substitutable dominant chords==		== Structural definition of tritone-substitutable dominant chords==

	A '''tritone-substitutable dominant chord''' is any seventh chord where		A '''tritone-substitutable dominant chord''' is any seventh chord where

	* the third and seventh ~~are exactly 600¢ apart (or~~ the ~~system's semioctave)~~		* two intervals (the third and the seventh OR the root and the fifth) are a tritone apart

	* either the third (in the ~~dominant 5 case~~) ~~or seventh (in~~ a tritone ~~sub) functions as a leading tone to your target chord.~~

	~~These are~~ the ~~only structural requirements. The root and fifth can be placed more freely~~.		* either of the two intervals functions as a leading tone to your target chord (as a dominant 5 chord or as a tritone sub).

	Ideally every interval should stay a third apart so as to construct tertian tetrads and avoid sounding like clusters, though following this simple requirement, is surprisingly effective at creating smooth voice leading dominant function even in extreme cases ~~(cluster chords)~~.		Ideally every interval should stay a third apart so as to construct tertian tetrads and avoid sounding like clusters, though following this simple requirement is surprisingly effective at creating smooth voice leading dominant function even in the extreme cases.

	What qualifies as a leading tone also is up for debate - generally intervals between 1050¢ and 1150¢ are used as leading tones.		What qualifies as a leading tone also is up for debate - generally intervals between 1050¢ and 1150¢ are used as leading tones.

	==Tritone complements==		==Tritone complements==

Pailiaq: /* Structural definition of tritone-substitutable dominant chords */

2025-12-23T08:09:00Z

Structural definition of tritone-substitutable dominant chords

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	These are the only structural requirements. The root and fifth can be placed more freely.		These are the only structural requirements. The root and fifth can be placed more freely.

	Ideally every interval should stay a third apart so as to construct tertian tetrads and avoid sounding like clusters, though following this simple requirement, ~~even in pushed to extreme cases~~ is surprisingly effective at creating smooth voice leading dominant function.		Ideally every interval should stay a third apart so as to construct tertian tetrads and avoid sounding like clusters, though following this simple requirement, is surprisingly effective at creating smooth voice leading dominant function even in extreme cases (cluster chords).

	What qualifies as a leading tone also is up for debate - generally intervals between 1050¢ and 1150¢ are used as leading tones.		What qualifies as a leading tone also is up for debate - generally intervals between 1050¢ and 1150¢ are used as leading tones.

	==Tritone complements==		==Tritone complements==

Pailiaq: /* Tritone Substitution */

2025-12-23T07:57:36Z

Tritone Substitution

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	==Tritone Substitution==		==Tritone Substitution==
	'''Tritone substitution''' is a chord substitution technique from jazz harmony in which a dominant seventh chord is replaced by another dominant seventh chord whose root is a tritone (600¢) away. The substitution works because both chords share the same tritone interval between their third and seventh, allowing these chord tones to swap roles while maintaining smooth voice leading to the tonic.		A '''Tritone substitution''' is a chord substitution technique from jazz harmony in which a dominant seventh chord is replaced by another dominant seventh chord whose root is a tritone (600¢) away. The substitution works because both chords share the same tritone interval between their third and seventh, allowing these chord tones to swap roles while maintaining smooth voice leading to the tonic.

	In 12n-EDOs, tritone substitution works seamlessly with the diatonic dominant seventh chord. However, in most other tuning systems, the diatonic tritone does not equal the semioctave (half-octave, 600¢), requiring either nondiatonic chord construction or an acceptance of "fudged" approximations.		In 12n-EDOs, tritone substitution works seamlessly with the diatonic dominant seventh chord. However, in most other tuning systems, the diatonic tritone does not equal the semioctave (half-octave, 600¢), requiring either nondiatonic chord construction or an acceptance of "fudged" approximations.

Pailiaq at 07:33, 23 December 2025

2025-12-23T07:33:50Z

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	* A leading tone (~1100¢ above, or ~100¢ the tonic) that resolves upward by a small interval to the tonic		* A leading tone (~1100¢ above, or ~100¢ the tonic) that resolves upward by a small interval to the tonic
	* A ~~subdominant tendency tone~~ (approximately 500¢ above the tonic) that resolves downward by a small interval to the ~~mediant~~		* A fourth (approximately 500¢ above the tonic) that resolves downward by a small interval to the third

	In xenharmonic systems, the ~~interval between~~ the leading tone and the ~~subdominant tendency tone varies~~. We can 'shift' the tritone interval between the 3rd and 7th degree up or down. This creates a trade-off:		In xenharmonic systems, the location of both the leading tone and the fourth can vary. We can 'shift' the tritone interval between the 3rd and 7th degree up or down. This creates a trade-off:

	==== If the leading tone is raised(e.g., to 1150¢ instead of 1100¢): ====		==== If the leading tone is raised(e.g., to 1150¢ instead of 1100¢): ====

Pailiaq at 06:15, 23 December 2025

2025-12-23T06:15:12Z

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	* A subdominant tendency tone (approximately 500¢ above the tonic) that resolves downward by a small interval to the mediant		* A subdominant tendency tone (approximately 500¢ above the tonic) that resolves downward by a small interval to the mediant

	~~In 12-EDO, these two tendency tones are exactly 600¢ apart. The tritone between them is the engine of dominant function, creating instability that demands resolution.~~		In xenharmonic systems, the interval between the leading tone and the subdominant tendency tone varies. We can 'shift' the tritone interval between the 3rd and 7th degree up or down. This creates a trade-off:

	~~==The voice-leading trade-off==~~

	In xenharmonic systems, the interval between the leading tone and the subdominant tendency tone varies. This creates a trade-off:

	==== If the leading tone is raised(e.g., to 1150¢ instead of 1100¢): ====		==== If the leading tone is raised(e.g., to 1150¢ instead of 1100¢): ====

Pailiaq: /* Examples */

2025-12-23T06:08:29Z

Examples

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	In 26-EDO, the semioctave is 13\26 = 600¢, but the diatonic augmented fourth is only 12\26 ≈ 554¢. A diatonic dominant seventh chord therefore lacks a true tritone, breaking the substitution mechanism of the 3rd and 7th swapping.		In 26-EDO, the semioctave is 13\26 = 600¢, but the diatonic augmented fourth is only 12\26 ≈ 554¢. A diatonic dominant seventh chord therefore lacks a true tritone, breaking the substitution mechanism of the 3rd and 7th swapping.

	The solution is to use the '''harmonic seventh chord''' ((0, 8, 15, 21)\26 approximating 4:5:6:7) as your dominant. In 26-EDO, the harmonic seventh (7/4) lands at 21\26 ≈ 969¢, which is ~0.4¢ from just. With a major third at 8\26, the interval from third to seventh is exactly 13 steps = 600¢. This nondiatonic chord produces a perfect tritone substitution, with the caveat that you must step outside your scale (playing a downminor 7) when playing a V-I progression if you want consistency in sound for when also using tritone substitutions.		The solution is to use the '''harmonic seventh chord''' ((0, 8, 15, 21)\26 approximating 4:5:6:7) as your dominant V. In 26-EDO, the harmonic seventh (7/4) lands at 21\26 ≈ 969¢, which is ~0.4¢ from just. With a major third at 8\26, the interval from third to seventh is exactly 13 steps = 600¢. This nondiatonic chord produces a perfect tritone substitution, with the caveat that you must step outside your scale (playing a downminor 7) when playing a V-I progression if you want consistency in sound for when also using tritone substitutions.

	Howewever, using the diatonic, tritoneless dominant 7th chord also can work; the only difference being that the third and 7ths don't perfectly swap positions, as one of them will drift up or down by an edostep.		Howewever, using the diatonic, tritoneless dominant 7th chord also can work; the only difference being that the third and 7ths don't perfectly swap positions, as one of them will drift up or down by an edostep.

Pailiaq: /* Examples */

2025-12-23T06:07:16Z

Examples

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	The solution is to use the '''harmonic seventh chord''' ((0, 8, 15, 21)\26 approximating 4:5:6:7) as your dominant. In 26-EDO, the harmonic seventh (7/4) lands at 21\26 ≈ 969¢, which is ~0.4¢ from just. With a major third at 8\26, the interval from third to seventh is exactly 13 steps = 600¢. This nondiatonic chord produces a perfect tritone substitution, with the caveat that you must step outside your scale (playing a downminor 7) when playing a V-I progression if you want consistency in sound for when also using tritone substitutions.		The solution is to use the '''harmonic seventh chord''' ((0, 8, 15, 21)\26 approximating 4:5:6:7) as your dominant. In 26-EDO, the harmonic seventh (7/4) lands at 21\26 ≈ 969¢, which is ~0.4¢ from just. With a major third at 8\26, the interval from third to seventh is exactly 13 steps = 600¢. This nondiatonic chord produces a perfect tritone substitution, with the caveat that you must step outside your scale (playing a downminor 7) when playing a V-I progression if you want consistency in sound for when also using tritone substitutions.

	Howewever, using the diatonic, tritoneless dominant 7th chord also can work; the only difference being that the third and 7ths don't perfectly swap positions, as one of them will drift up or down by ~~a comma~~.		Howewever, using the diatonic, tritoneless dominant 7th chord also can work; the only difference being that the third and 7ths don't perfectly swap positions, as one of them will drift up or down by an edostep.

	==Tritone substitution in odd-numbered EDOs==		==Tritone substitution in odd-numbered EDOs==

Pailiaq: /* Why tritone substitution works */

2025-12-23T05:52:12Z

Why tritone substitution works

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	Both chords therefore resolve to the same tonic with identical voice leading in the critical voices.		Both chords therefore resolve to the same tonic with identical voice leading in the critical voices.

	~~The 6n-EDO exception~~

	For the diatonic dominant seventh chord (That is, a dominant 7th chord that entirely exists within the diatonic scale) to produce a true tritone substitution, the diatonic augmented fourth/diminished fifth must equal exactly half the octave. The augmented fourth is defined as three stacked whole tones, a literal "tri-tone". This equals 600¢ only when:		For the diatonic dominant seventh chord (That is, a dominant 7th chord that entirely exists within the diatonic scale) to produce a true tritone substitution, the diatonic augmented fourth/diminished fifth must equal exactly half the octave. The augmented fourth is defined as three stacked whole tones, a literal "tri-tone". This equals 600¢ only when:

Pailiaq: /* Why tritone substitution works */

2025-12-23T05:49:58Z

Why tritone substitution works

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	The effectiveness of tritone substitution stems from voice leading. In a V7–I resolution in 12-EDO:		The effectiveness of tritone substitution stems from voice leading. In a V7–I resolution in 12-EDO:

	* The '''third''' of the dominant (the leading tone, ~1100¢ above the dominant root) resolves up by semitone (~100¢) to the tonic		* The '''third''' of the dominant (the leading tone, 1100¢ above the dominant root) resolves up by semitone (100¢) to the tonic

	* The '''seventh''' of the dominant (~500¢ above the ~~dominant~~ root, or ~1000¢ above the tonic) resolves down by semitone to the third of the tonic		* The '''seventh''' of the dominant (500¢ above the root, or 1000¢ above the tonic) resolves down by semitone to the third of the tonic

	These two active tones are exactly 600¢, a tritone, apart. When the dominant root is transposed by a tritone, (placed at 100¢) these tones swap positions: what was the third becomes the seventh, and vice versa.		These two active tones are exactly 600¢, a tritone, apart. When the dominant root is transposed by a tritone, (placed at 100¢) these tones swap positions: what was the third becomes the seventh, and vice versa.