Flattone: Difference between revisions

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{{Infobox regtemp|Comma basis=[[81/80]], [[875/864]]|Edo join 1=19|Edo join 2=26|Generator=3/2|Generator tuning=693.0|Mapping=1; 1 4 -9|MOS scales=[[5L 2s]], [[7L 5s]], [[7L 12s]]|Optimization method=CTE|Pergen=(P8, P5)|Title=Flattone|Subgroups=2.3.5.7}}'''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.

{{Infobox regtemp

| Comma basis=[[81/80]], [[875/864]]

| Edo join 1=19

| Edo join 2=26

| Generator=3/2

| Generator tuning=693.0

| Mapping=1; 1 4 -9

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

| Optimization method=CTE

| Pergen=(P8, P5)

| Title=Flattone

| Subgroups=2.3.5.7

}}

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]].

'''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.

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

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]].

[[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.]]

Reasonable tunings lie between [[19edo]] and [[26edo]]. 19edo is the point where 7/4 and [[12/7]] are conflated. Any tuning whose fifth is sharper than 19edo's has the sizes of 7/4 and 12/7 inverted, so they can be more properly analysed as septimal meantone. Similarly, 26edo is the point where 7/5 and [[10/7]] are conflated. Any tuning whose fifth is flatter than 26edo's has the sizes of 7/5 and 10/7 inverted, so they can be more properly analysed as a [[Meantone family #Flattertone|flatter-than-flattone temperament]].

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In the following table, odd harmonics 1–13 are in '''bold'''.

{| class="wikitable center-1 right-2"

|-

! #

! Cents*

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| 10/7

|}

<nowiki>*~~</nowiki>~~ In 13-limit CTE tuning, octave reduced

<nowiki/>* In 13-limit CTE tuning, octave reduced

=== As a detemperament of 7et ===

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{| class="wikitable right-2 right-4 right-6 right-8 right-10"

|-

! rowspan="2" | Interval category

! colspan="2" | -2 quartertones

! colspan="2" | −2 quartertones

! colspan="2" | -1 quartertone

! colspan="2" | −1 quartertone

! colspan="2" | 0 quartertones

! colspan="2" | 1 quartertone

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=== Tuning spectrum ===

{| class="wikitable center-all left-4"

|-

! Edo<br>generator

! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*

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-{{Infobox regtemp|Comma basis=[[81/80]], [[875/864]]|Edo join 1=19|Edo join 2=26|Generator=3/2|Generator tuning=693.0|Mapping=1; 1 4 -9|MOS scales=[[5L 2s]], [[7L 5s]], [[7L 12s]]|Optimization method=CTE|Pergen=(P8, P5)|Title=Flattone|Subgroups=2.3.5.7}}'''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.
+{{Infobox regtemp
+| Comma basis=[[81/80]], [[875/864]]
+| Edo join 1=19
+| Edo join 2=26
+| Generator=3/2
+| Generator tuning=693.0
+| Mapping=1; 1 4 -9
+| MOS scales=[[5L&nbsp;2s]], [[7L&nbsp;5s]], [[7L&nbsp;12s]]
+| Optimization method=CTE
+| Pergen=(P8, P5)
+| Title=Flattone
+| Subgroups=2.3.5.7
+}}
-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]].
+'''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.
-[[File:45EDO_Otonal.mp3|none|thumb|Harmonic scale 8–16 in 45edo, using the flattone mappings for 13 & 15 rather than the best direct approximations.]]
+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]].
+[[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.]]
 Reasonable tunings lie between [[19edo]] and [[26edo]]. 19edo is the point where 7/4 and [[12/7]] are conflated. Any tuning whose fifth is sharper than 19edo's has the sizes of 7/4 and 12/7 inverted, so they can be more properly analysed as septimal meantone. Similarly, 26edo is the point where 7/5 and [[10/7]] are conflated. Any tuning whose fifth is flatter than 26edo's has the sizes of 7/5 and 10/7 inverted, so they can be more properly analysed as a [[Meantone family #Flattertone|flatter-than-flattone temperament]].
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 In the following table, odd harmonics 1–13 are in '''bold'''.
 {| class="wikitable center-1 right-2"
+|-
 ! #
 ! Cents*
@@ Line 71: / Line 86: @@
 | 10/7
 |}
-<nowiki>*</nowiki> In 13-limit CTE tuning, octave reduced
+<nowiki/>* In 13-limit CTE tuning, octave reduced
 === As a detemperament of 7et ===
@@ Line 77: / Line 92: @@
 {| class="wikitable right-2 right-4 right-6 right-8 right-10"
+|-
 ! rowspan="2" | Interval category
-! colspan="2" | -2 quartertones
+! colspan="2" | −2 quartertones
-! colspan="2" | -1 quartertone
+! colspan="2" | −1 quartertone
 ! colspan="2" | 0 quartertones
 ! colspan="2" | 1 quartertone
@@ Line 186: / Line 202: @@
 === Tuning spectrum ===
 {| class="wikitable center-all left-4"
+|-
 ! Edo<br>generator
 ! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*