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Flattone: Difference between revisions

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{{Infobox regtemp
| Title = Flattone
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
| Comma basis = [[81/80]], [[525/512]] (7-limit);<br>[[45/44]], [[81/80]], [[385/384]] (11-limit);<br>[[45/44]], [[65/64]], [[78/77]], [[81/80]] (13-limit)
| Edo join 1 = 19 | Edo join 2 = 26
| Mapping = 1; 1 4 -9 6 -4
| Generators = 3/2
| 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 | Mistuning 1 = 15.7 | Complexity 1 = 19
| Odd limit 2 = 13 | Mistuning 2 = 19.3 | 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.  
'''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 & 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 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]].  
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'''.  
In the following table, odd harmonics 1–13 are in '''bold'''.  
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
! #
! #
! Cents*
! Cents*
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|-
|-
| 1
| 1
| 693.0
| 693.1
| '''3/2'''
| '''3/2'''
|-
|-
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|-
|-
| 3
| 3
| 879.1
| 879.2
| 5/3
| 5/3
|-
|-
| 4
| 4
| 372.1
| 372.2
| '''5/4''', '''16/13''', 26/21
| '''5/4''', '''16/13''', 26/21
|-
|-
| 5
| 5
| 1065.1
| 1065.3
| 11/6, 13/7, 15/8, 24/13
| 11/6, 13/7, 15/8, 24/13
|-
|-
| 6
| 6
| 558.2
| 558.3
| '''11/8''', 18/13
| '''11/8''', 18/13
|-
|-
| 7
| 7
| 51.2
| 51.4
| 25/24, 27/26, 33/32, 36/35, 55/54, 64/63
| 25/24, 27/26, 33/32, 36/35, 55/54, 64/63
|-
|-
| 8
| 8
| 744.2
| 744.4
| 20/13, 32/21
| 20/13, 32/21
|-
|-
| 9
| 9
| 237.3
| 237.5
| '''8/7''', 15/13
| '''8/7''', 15/13
|-
|-
| 10
| 10
| 930.3
| 930.5
| 12/7, 22/13
| 12/7, 22/13
|-
|-
| 11
| 11
| 423.3
| 423.6
| 9/7
| 9/7
|-
|-
| 12
| 12
| 1116.4
| 1116.6
| 27/14, 40/21
| 27/14, 40/21
|-
|-
| 13
| 13
| 609.4
| 609.7
| 10/7
| 10/7
|}
|}
<nowiki>*</nowiki> In 13-limit CTE tuning, octave reduced
<nowiki/>* In 13-limit CWE tuning, octave reduced


=== As a detemperament of 7et ===
=== As a detemperament of 7et ===
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{| class="wikitable right-2 right-4 right-6 right-8 right-10"
{| class="wikitable right-2 right-4 right-6 right-8 right-10"
|-
! rowspan="2" | Interval category
! 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" | 0 quartertones
! colspan="2" | 1 quartertone
! colspan="2" | 1 quartertone
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|  
|  
| 270
| 270
| 7/6
| 7/6, 13/11
| 321
| 321
| 6/5
| 6/5
| 372
| 372
| 5/4, 16/13, 26/21
| 5/4, 11/9, 16/13, 26/21
| 423
| 423
| 9/7
| 9/7
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| Fourth
| Fourth
| 405
| 405
|  
| 14/11
| 456
| 456
| 13/10, 21/16
| 13/10, 21/16
Line 153: Line 169:
| 20/13, 32/21
| 20/13, 32/21
| 795
| 795
|  
| 11/7
|-
|-
| Sixth
| Sixth
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| 14/9
| 14/9
| 828
| 828
| 8/5, 13/8, 21/13
| 8/5, 18/11, 13/8, 21/13
| 879
| 879
| 5/3
| 5/3
| 930
| 930
| 12/7
| 12/7, 22/13
| 981
| 981
|  
|  
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== Tunings ==
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 693.5520{{c}}
| CWE: ~3/2 = 693.7333{{c}}
| POTE: ~3/2 = 693.7791{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 13-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 693.0293{{c}}
| CWE: ~3/2 = 693.0538{{c}}
| POTE: ~3/2 = 693.0578{{c}}
|}
=== Tuning spectrum ===
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
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|
|
| 690.909
| 690.909
|
| 33c val
|-
|-
|
|
Line 229: Line 278:
|
|
| 691.304
| 691.304
|
| 92bccc val
|-
|-
|
|
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|
|
| 691.525
| 691.525
|
| 59bc val
|-
|-
| [[85edo|49\85]]
| [[85edo|49\85]]
|
|
| 691.765
| 691.765
|
| 85bccf val
|-
|-
|
|
Line 289: Line 338:
|
|
| 692.958
| 692.958
|
| 71bcf val
|-
|-
|
|
Line 309: Line 358:
|
|
| 693.333
| 693.333
|  
| 45f val
|-
|-
|
|
Line 319: Line 368:
|
|
| 693.750
| 693.750
|
| 64cdef val
|-
|-
|
|
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|
|
| 694.737
| 694.737
| Upper bound of 7-, 9-, 11-, 13-odd-limit diamond monotone
| Upper bound of 7-, 9-, 11-, and 13-odd-limit diamond monotone
|-
|-
|
|
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|
|
| 700.000
| 700.000
|  
| 12d val
|-
|-
|
|
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