User:Pailiaq/Tritone substitution
Tritone Substitution
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.
Why tritone substitution works
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 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.
Both chords therefore resolve to the same tonic with identical voice leading in the critical voices.
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:
- The whole tone = 200¢
- Three whole tones = 600¢
Only 6-EDO multiples that contain a diatonic scale (12, 24, 36, 42, 48, etc.) can satisfy this condition. In these systems, the augmented fourth and diminished fifth are enharmonically equivalent, both measuring exactly 600¢.
In all other EDOs, the diatonic augmented fourth deviates from 600¢.
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.
Examples
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) 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.
In odd-numbered EDOs
Odd-numbered EDOs lack a semioctave entirely. Instead, they contain a small tritone and large tritone that straddle 600¢.
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.
Structural definition of tritone-substitutable dominant chords
A tritone-substitutable dominant chord is any seventh chord where
- two intervals (the third and the seventh OR the root and the fifth) are a tritone apart
- 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 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.
Tritone complements
Two intervals are tritone complements if they sum to 600¢ (or the narrow/wide tritone in odd n EDOs). The intervals from 3rd to 5th, and from 5th to 7th in a tritone-substitutable tetrads are always tritone complements. (e.g. multiplying those two intervals gives you a tritone)
From an RTT lens, for all even-numbered EDOs up to 32, the 600¢ semioctave tempers both septimal tritones 7/5 (~583¢) and 10/7 (~617¢) to the same interval. This means tritone complements, when expressed as frequency ratios, must multiply to equal either 7/5 or 10/7. For example:
- 5/4 × 8/7 = 10/7 | major third + small whole tone
- 6/5 × 7/6 = 7/5 | minor third + subminor third
- 9/7 × 10/9 = 10/7 | supermajor third + small whole tone
This provides a way to identify tritone complement pairs in smaller edos: find two intervals whose product is a septimal tritone. This pair of intervals can be used to create the upper 3 voices (a type of diminished chord) in a dominant chord, which just needs a root a third below the diminished chord to complete.
Relationship to dominant function
Tritone substitution is a specific application of dominant function, which itself emerges from voice-leading forces. A chord exhibits dominant function when it contains:
- A leading tone (~1100¢ above, or ~100¢ the tonic) that resolves upward by a small interval to the tonic
- A fourth (approximately 500¢ above the tonic) that resolves downward by a small interval to the third
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¢):
- The leading tone resolves by a smaller step (~50¢) to the tonic—a tighter, more chromatic pull
- 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
If the leading tone is lowered (e.g., to 1050¢):
- The leading tone resolves by ~150¢—a wider, more whole-tone-like motion
- 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
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
| Tuning type | Tritone sub behavior |
|---|---|
| 12n-EDOs | Diatonic dom7 produces perfect tritone substitution |
| Other even EDOs | Requires nondiatonic chords (e.g., harmonic 7th) for true tritone sub; diatonic approximation possible but imperfect |
| Odd Edos | No exact tritone; "fudged" substitution using small/large tritone pairs |
The principle underlying tritone substitution, that two chords sharing a tritone can substitute for one another, can generalize to any tuning system. What changes is which chords contain a true tritone and whether those chords arise naturally from diatonic structures or must be constructed outside the scale.