Flattone: Difference between revisions

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| Edo join 1 = 19 | Edo join 2 = 26

| Mapping = 1; 1 4 -9 6 -4

| ~~Generator~~ = 3/2

| Generators = 3/2

| ~~Generator~~ tuning = 693.1

| Generators tuning = 693.1

| Optimization method = CWE

| MOS scales = [[5L 2s]], [[7L 5s]], [[7L 12s]]

| Pergen = (P8, P5)

| Odd limit 1 = 9 ~~| Odd limit 2 = 13~~

| Odd limit 1 = 9 | Mistuning 1 = 15.7 | Complexity 1 = 19

| Mistuning 1 = 15.7 | Mistuning 2 = 19.3

| Odd limit 2 = 13 | Mistuning 2 = 19.3 | Complexity 2 = 19

~~| Complexity 1 = 19~~ | Complexity 2 = 19

}}

'''Flattone''' is an alternative [[extension]] to [[5-limit]] [[meantone]], the [[temperament]] that [[tempering out|tempers out]] the [[81/80|syntonic comma (81/80)]]. It is generated by a fifth that is typically flatter than that of [[septimal meantone]], and nine of those reach the [[pitch class]] of [[8/7]], so that [[7/4]] is a diminished seventh (C–B𝄫), [[7/6]] is a diminished third (C–E𝄫), and [[7/5]] is a doubly diminished fifth (C–G𝄫). Although 7/4 is simpler than in septimal meantone, the full [[9-odd-limit]] [[tonality diamond]] is more complex as the 5 and 7 are reached by going in opposite directions, while also being less accurate.

However, it makes up for that by having simpler 11- and 13-limit interpretations – the whole tone is now flat enough that it can function as [[9/8]], [[10/9]], and [[11/10]], tempering out [[100/99]] and making [[11/8]] an augmented fourth (C–F#). This means the major third functions as both 5/4 and 11/9. Tempering out [[65/64]] means it also represents their [[mediant]] [[16/13]], making [[13/8]] a minor sixth (C–A♭) and a full otonal chord of 8:9:10:11:12:13:14:15:16 accessible with a gamut of 16 notes, compared to 19 for ~~tridecimal meantone~~ or the 29 required by [[meanpop]].

However, it makes up for that by having simpler 11- and 13-limit interpretations – the whole tone is now flat enough that it can function as [[9/8]], [[10/9]], and [[11/10]], tempering out [[100/99]] and making [[11/8]] an augmented fourth (C–F#). This means the major third functions as both 5/4 and 11/9. Tempering out [[65/64]] means it also represents their [[mediant]] [[16/13]], making [[13/8]] a minor sixth (C–A♭) and a full otonal chord of 8:9:10:11:12:13:14:15:16 accessible with a gamut of 16 notes, compared to 19 for [[fokkertone]] or the 29 required by [[meanpop]].

[[File:45EDO_Otonal.mp3|none|thumb|Harmonic scale 8–16 in 45edo, using the flattone mappings for 13 and 15 rather than the best direct approximations.]]

@@ Line 5: / Line 5: @@
 | Edo join 1 = 19 | Edo join 2 = 26
 | Mapping = 1; 1 4 -9 6 -4
-| Generator = 3/2
+| Generators = 3/2
-| Generator tuning = 693.1
+| Generators tuning = 693.1
 | Optimization method = CWE
 | MOS scales = [[5L&nbsp;2s]], [[7L&nbsp;5s]], [[7L&nbsp;12s]]
 | Pergen = (P8, P5)
-| Odd limit 1 = 9 | Odd limit 2 = 13
+| Odd limit 1 = 9 | Mistuning 1 = 15.7 | Complexity 1 = 19
-| Mistuning 1 = 15.7 | Mistuning 2 = 19.3
+| Odd limit 2 = 13 | Mistuning 2 = 19.3 | Complexity 2 = 19
-| Complexity 1 = 19 | Complexity 2 = 19
 }}
 '''Flattone''' is an alternative [[extension]] to [[5-limit]] [[meantone]], the [[temperament]] that [[tempering out|tempers out]] the [[81/80|syntonic comma (81/80)]]. It is generated by a fifth that is typically flatter than that of [[septimal meantone]], and nine of those reach the [[pitch class]] of [[8/7]], so that [[7/4]] is a diminished seventh (C–B𝄫), [[7/6]] is a diminished third (C–E𝄫), and [[7/5]] is a doubly diminished fifth (C–G𝄫). Although 7/4 is simpler than in septimal meantone, the full [[9-odd-limit]] [[tonality diamond]] is more complex as the 5 and 7 are reached by going in opposite directions, while also being less accurate.
-However, it makes up for that by having simpler 11- and 13-limit interpretations – the whole tone is now flat enough that it can function as [[9/8]], [[10/9]], and [[11/10]], tempering out [[100/99]] and making [[11/8]] an augmented fourth (C–F#). This means the major third functions as both 5/4 and 11/9. Tempering out [[65/64]] means it also represents their [[mediant]] [[16/13]], making [[13/8]] a minor sixth (C–A♭) and a full otonal chord of 8:9:10:11:12:13:14:15:16 accessible with a gamut of 16 notes, compared to 19 for tridecimal meantone or the 29 required by [[meanpop]].
+However, it makes up for that by having simpler 11- and 13-limit interpretations – the whole tone is now flat enough that it can function as [[9/8]], [[10/9]], and [[11/10]], tempering out [[100/99]] and making [[11/8]] an augmented fourth (C–F#). This means the major third functions as both 5/4 and 11/9. Tempering out [[65/64]] means it also represents their [[mediant]] [[16/13]], making [[13/8]] a minor sixth (C–A♭) and a full otonal chord of 8:9:10:11:12:13:14:15:16 accessible with a gamut of 16 notes, compared to 19 for [[fokkertone]] or the 29 required by [[meanpop]].
 [[File:45EDO_Otonal.mp3|none|thumb|Harmonic scale 8–16 in 45edo, using the flattone mappings for 13 and 15 rather than the best direct approximations.]]