User:Pailiaq/Tritone substitution: Difference between revisions
Created page with "==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..." |
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* 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 dominant root, or ~1000¢ above the tonic) resolves down by semitone to the third of the tonic | ||
These two active tones are exactly 600¢ | 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. | Both chords therefore resolve to the same tonic with identical voice leading in the critical voices. | ||
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The 6n-EDO exception | 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 | 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¢ | * The whole tone = 200¢ | ||
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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: | 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''' | * '''5/4 × 8/7 = 10/7''' | major third + small whole tone | ||
* '''6/5 × 7/6 = 7/5''' | * '''6/5 × 7/6 = 7/5''' | minor third + subminor third | ||
* '''9/7 × 10/9 = 10/7''' | * '''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 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== | == Relationship to dominant function== | ||
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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 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 | * 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¢): ==== | ==== If the leading tone is lowered (e.g., to 1050¢): ==== | ||