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Flattone: Difference between revisions

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| Title = Flattone
| Title = Flattone
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
| Comma basis = [[81/80]], [[525/512]] (2.3.5.7);<br>[[45/44]], [[81/80]], [[385/384]] (2.3.5.7.11);<br>[[45/44]], [[65/64]], [[78/77]], [[81/80]] (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
| Edo join 1 = 19 | Edo join 2 = 26
| Mapping = 1; 1 4 -9 6 -4
| Mapping = 1; 1 4 -9 6 -4
| Generator = 3/2
| Generators = 3/2
| Generator tuning = 693.1
| Generators tuning = 693.1
| Optimization method = CWE
| Optimization method = CWE
| MOS scales = [[5L&nbsp;2s]], [[7L&nbsp;5s]], [[7L&nbsp;12s]]
| MOS scales = [[5L&nbsp;2s]], [[7L&nbsp;5s]], [[7L&nbsp;12s]]
| Pergen = (P8, P5)
| 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.  
'''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.]]
[[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.]]


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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"
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|
|
| 690.909
| 690.909
|
| 33c val
|-
|-
|
|
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|
|
| 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
|-
|-
|
|
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|
|
| 692.958
| 692.958
|
| 71bcf val
|-
|-
|
|
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|
|
| 693.333
| 693.333
|  
| 45f val
|-
|-
|
|
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|
|
| 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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