Functional harmony in rank-2 temperaments: Difference between revisions

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=== 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 start since the 7-tone mos barely contains anything more a tonic chord.

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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=== Example: orwell ===

[[Orwell]] is generated by ~7/6, so the most basic form of harmony could be 1–7/6–3/2 (0–1–7) and 1–9/7–3/2 (0–6–7). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 7/3. A "dominant" chord is 7/6 over tonic; a "subdominant" chord is 7/6 under tonic. The 9-tone mos contains a tonic and a "dominant" ~~chord~~. The 13-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions.

[[Orwell]] is generated by ~7/6, so the most basic form of harmony could be 1–7/6–3/2 (0–1–7) and 1–9/7–3/2 (0–6–7), or 1–7/4–3 (0–8–7) and 1–12/7–3 (0–(-1)–7), or tetrads such as 1–7/6–3/2–7/4 (0–1–7–8). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 7. A "dominant" chord is either 7/6 or 12/7 over tonic; a "subdominant" chord is either 7/6 or 12/7 under tonic. The 9-tone mos contains a tonic and a "dominant" triad. The 13-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions, triads and tetrads alike.

=== Example: sensi ===

[[Sensi]] is generated by ~9/7, so the most basic form of harmony could be 1–9/7–3/2 (0–1–7) and 1–7/6–3/2 (0–6–7). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 9/7. A "dominant" chord is 9/7 over tonic; a "subdominant" chord is 9/7 under tonic. The 11-tone mos might be a good start since the 8-tone mos barely contains anything more than a tonic chord.

[[Sensi]] is generated by ~9/7, so the most basic form of harmony could be 1–9/7–3/2 (0–1–7) and 1–7/6–3/2 (0–6–7). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 9/7. A "dominant" chord is 9/7 over tonic; a "subdominant" chord is 9/7 under tonic. The 11-tone mos might be a good place to start since the 8-tone mos barely contains anything more than a tonic chord.

== Other role-swapping techniques ==

@@ Line 34: / Line 34: @@
 === 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 start since the 7-tone mos barely contains anything more a tonic chord.
+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 ==
@@ Line 42: / Line 42: @@
 === Example: orwell ===
-[[Orwell]] is generated by ~7/6, so the most basic form of harmony could be 1–7/6–3/2 (0–1–7) and 1–9/7–3/2 (0–6–7). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 7/3. A "dominant" chord is 7/6 over tonic; a "subdominant" chord is 7/6 under tonic. The 9-tone mos contains a tonic and a "dominant" chord. The 13-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions.
+[[Orwell]] is generated by ~7/6, so the most basic form of harmony could be 1–7/6–3/2 (0–1–7) and 1–9/7–3/2 (0–6–7), or 1–7/4–3 (0–8–7) and 1–12/7–3 (0–(-1)–7), or tetrads such as 1–7/6–3/2–7/4 (0–1–7–8). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 7. A "dominant" chord is either 7/6 or 12/7 over tonic; a "subdominant" chord is either 7/6 or 12/7 under tonic. The 9-tone mos contains a tonic and a "dominant" triad. The 13-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions, triads and tetrads alike.
 === Example: sensi ===
-[[Sensi]] is generated by ~9/7, so the most basic form of harmony could be 1–9/7–3/2 (0–1–7) and 1–7/6–3/2 (0–6–7). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 9/7. A "dominant" chord is 9/7 over tonic; a "subdominant" chord is 9/7 under tonic. The 11-tone mos might be a good start since the 8-tone mos barely contains anything more than a tonic chord.
+[[Sensi]] is generated by ~9/7, so the most basic form of harmony could be 1–9/7–3/2 (0–1–7) and 1–7/6–3/2 (0–6–7). The intervals whose roles are being swapped here are 3 and 5 and then 5 and 9/7. A "dominant" chord is 9/7 over tonic; a "subdominant" chord is 9/7 under tonic. The 11-tone mos might be a good place to start since the 8-tone mos barely contains anything more than a tonic chord.
 == Other role-swapping techniques ==