Flattone: Difference between revisions

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'''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.

'''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 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 & 15 rather than the best direct approximations.]]

[[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.]]

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]].

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]].

See [[Meantone family #Flattone]] for technical data.

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

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

! &#~~35;~~

! #

! Cents*

! Approximate ~~Ratios~~

! Approximate ratios

|-

| 0

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| 10/7

|}

<nowiki />* In 13-limit CTE tuning

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

== Scales ==

* [[Flattone12]] ~~–~~ 12-tone chromatic scale in 13-limit POTE tuning

* [[Flattone12]] – 12-tone chromatic scale in 13-limit POTE tuning

== Tunings ==

=== Tuning spectrum ===

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

! Edo ~~Generator~~

! Edo generator

! [[Eigenmonzo|Eigenmonzo (Unchanged-interval)]]*

! [[Eigenmonzo|Eigenmonzo (Unchanged-interval)]]*

! Generator (¢)

! Generator (¢)

! Comments

|-

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| [[Pythagorean tuning]]

|}

<nowiki />* Besides the octave

<nowiki/>* Besides the octave

[[Category:Flattone| ]]

@@ Line 1: / Line 1: @@
-'''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.
+'''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&#x266D;) 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 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 & 15 rather than the best direct approximations.]]
+[[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.]]
-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]].
+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]].
 See [[Meantone family #Flattone]] for technical data.
@@ Line 11: / Line 11: @@
 In the following table, odd harmonics 1–13 are in '''bold'''.
 {| class="wikitable center-1 right-2"
-! &#35;
+! #
 ! Cents*
-! Approximate Ratios
+! Approximate ratios
 |-
 | 0
@@ Line 71: / Line 71: @@
 | 10/7
 |}
-<nowiki />* In 13-limit CTE tuning
+<nowiki/>* In 13-limit CTE tuning, octave reduced
 == Scales ==
-* [[Flattone12]] &ndash; 12-tone chromatic scale in 13-limit POTE tuning
+* [[Flattone12]] – 12-tone chromatic scale in 13-limit POTE tuning
 == Tunings ==
 === Tuning spectrum ===
 {| class="wikitable center-all left-4"
-! Edo<br />Generator
+! Edo<br>generator
-! [[Eigenmonzo|Eigenmonzo<br />(Unchanged-interval)]]*
+! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]*
-! Generator<br />(¢)
+! Generator<br>(¢)
 ! Comments
 |-
@@ Line 269: / Line 269: @@
 | [[Pythagorean tuning]]
 |}
-<nowiki />* Besides the octave
+<nowiki/>* Besides the octave
 [[Category:Flattone| ]] <!-- Main article -->