Functional harmony in rank-2 temperaments: Difference between revisions

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== Functions based on generators ==

In septimal meantone, the most [[consonance and dissonance|consonant]] [[interval]] besides the [[octave]], the [[3/2|perfect fifth (3/2)]], is a [[generator]], ~~meaning it~~ is the most abundant interval in any [[mos scale]]. This is ~~often~~ not true for ~~other~~ temperaments. Here we discuss the technique where "dominant" and "subdominant" are no longer related to tonic by the fifth, but by the generator. ~~This~~ leads to an approach closely adherent to mos scales, and depending on the specific scale pattern, it can sound very xenharmonic.

In septimal meantone, the most [[consonance and dissonance|consonant]] [[interval]] besides the [[octave]], the [[3/2|perfect fifth (3/2)]], is a [[generator]], so that the most consonant interval is also the most abundant interval in any [[mos scale]]. This is not true for any temperament that is not generated by a perfect fifth. In these temperaments, we have to choose what our dominant and subdominant functions are based on.

Here we discuss the technique where "dominant" and "subdominant" are no longer related to tonic by the fifth, but by the generator. Since mos scales are created from stacking generators, this leads to an approach closely adherent to mos scales, and depending on the specific scale pattern, it can sound very xenharmonic. For the other approach see [[#Breaking out of mos]].

=== Example: magic ===

In terms of generator steps of [[magic]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–1–5 and 0–4–5; this is similar, but also in clear contrast to the 0–4–1 and 0–(−3)–1 of meantone. These chords can be used as the basis of harmony, with the roles of 3 and 5 swapped according to their temperamental complexities. Thus a "dominant" chord is 5/4 over tonic; a "subdominant" chord is 5/4 under tonic. The 7-tone mos contains a tonic and a "dominant" chord. The 10-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions. We also have the augmented triad (1–5/4–14/9) as the magic analog of the suspended chord of meantone.

In terms of generator steps of [[magic]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–1–5 and 0–4–5; this is similar, but also in clear contrast to the 0–4–1 and 0–(−3)–1 of meantone. These chords can be used as the basis of harmony, with the roles of 3 and 5 swapped according to their temperamental complexities. Thus a "dominant" chord is 5/4 over tonic; a "subdominant" chord is 5/4 under tonic. The 7-tone mos contains a tonic and a "dominant" chord. The 10-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions.

The suspended chord of meantone is made of two generators stacked. Doing the same in magic, we also have the augmented triad (1–5/4–14/9) as the magic analog of the suspended chord of meantone.

=== Example: hanson ===

In terms of generator steps of [[hanson]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–5–6 and 0–1–6; like in magic, these chords can be used as the basis of harmony, but the intervals whose roles are being swapped here are 3 and 3/5. A "dominant" chord is 6/5 over tonic; a "subdominant" chord is 6/5 under tonic. The 11-tone mos might be a good place to start since the 7-tone mos barely contains anything more a tonic chord.

== Combination of techniques ==

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 == Functions based on generators ==
-In septimal meantone, the most [[consonance and dissonance|consonant]] [[interval]] besides the [[octave]], the [[3/2|perfect fifth (3/2)]], is a [[generator]], meaning it is the most abundant interval in any [[mos scale]]. This is often not true for other temperaments. Here we discuss the technique where "dominant" and "subdominant" are no longer related to tonic by the fifth, but by the generator. This leads to an approach closely adherent to mos scales, and depending on the specific scale pattern, it can sound very xenharmonic.
+In septimal meantone, the most [[consonance and dissonance|consonant]] [[interval]] besides the [[octave]], the [[3/2|perfect fifth (3/2)]], is a [[generator]], so that the most consonant interval is also the most abundant interval in any [[mos scale]]. This is not true for any temperament that is not generated by a perfect fifth. In these temperaments, we have to choose what our dominant and subdominant functions are based on.
+Here we discuss the technique where "dominant" and "subdominant" are no longer related to tonic by the fifth, but by the generator. Since mos scales are created from stacking generators, this leads to an approach closely adherent to mos scales, and depending on the specific scale pattern, it can sound very xenharmonic. For the other approach see [[#Breaking out of mos]].
 === Example: magic ===
-In terms of generator steps of [[magic]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–1–5 and 0–4–5; this is similar, but also in clear contrast to the 0–4–1 and 0–(−3)–1 of meantone. These chords can be used as the basis of harmony, with the roles of 3 and 5 swapped according to their temperamental complexities. Thus a "dominant" chord is 5/4 over tonic; a "subdominant" chord is 5/4 under tonic. The 7-tone mos contains a tonic and a "dominant" chord. The 10-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions. We also have the augmented triad (1–5/4–14/9) as the magic analog of the suspended chord of meantone.
+In terms of generator steps of [[magic]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–1–5 and 0–4–5; this is similar, but also in clear contrast to the 0–4–1 and 0–(−3)–1 of meantone. These chords can be used as the basis of harmony, with the roles of 3 and 5 swapped according to their temperamental complexities. Thus a "dominant" chord is 5/4 over tonic; a "subdominant" chord is 5/4 under tonic. The 7-tone mos contains a tonic and a "dominant" chord. The 10-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions.
+The suspended chord of meantone is made of two generators stacked. Doing the same in magic, we also have the augmented triad (1–5/4–14/9) as the magic analog of the suspended chord of meantone.
 === Example: hanson ===
 In terms of generator steps of [[hanson]], the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2) are 0–5–6 and 0–1–6; like in magic, these chords can be used as the basis of harmony, but the intervals whose roles are being swapped here are 3 and 3/5. A "dominant" chord is 6/5 over tonic; a "subdominant" chord is 6/5 under tonic. The 11-tone mos might be a good place to start since the 7-tone mos barely contains anything more a tonic chord.
 == Combination of techniques ==