User:Pailiaq/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.

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 dominant root, or ~1000¢ above the tonic) resolves down by semitone to the third of the tonic

These two active tones are exactly 600¢ apart—a tritone. 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.

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 "tri-tone" in the literal sense. 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¢.

Tritone substitution 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, 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.

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.

Tritone substitution in odd-numbered EDOs

Odd-numbered EDOs definitionally lack a semioctave entirely. However, tritone substitution can be approximated using 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 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

A tritone-substitutable dominant chord is any seventh chord where

the third and seventh are exactly 600¢ apart (or the system's semioctave)

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.

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.

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 — a major third plus a septimal whole tone
6/5 × 7/6 = 7/5 — a minor third plus a subminor third
9/7 × 10/9 = 10/7 — a supermajor third plus a minor whole tone

This provides a way to identify tritone complement pairs in smaller edos: find two intervals whose product is a septimal tritone.

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 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.

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¢):

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 small leading tone 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 subdominant tendency tone at ~450¢ resolves down by ~50¢ to the mediant
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.

Summary

Caption text
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.