View source for Flattone

{{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
}}

'''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]].
[[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]]. 

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

== Interval chain ==
In the following table, odd harmonics 1–13 are in '''bold'''. 
{| class="wikitable center-1 right-2"
|-
! #
! Cents*
! Approximate ratios
|-
| 0
| 0.0
| '''1/1'''
|-
| 1
| 693.0
| '''3/2'''
|-
| 2
| 186.1
| '''9/8''', 10/9, 11/10
|-
| 3
| 879.1
| 5/3
|-
| 4
| 372.1
| '''5/4''', '''16/13''', 26/21
|-
| 5
| 1065.1
| 11/6, 13/7, 15/8, 24/13
|-
| 6
| 558.2
| '''11/8''', 18/13
|-
| 7
| 51.2
| 25/24, 27/26, 33/32, 36/35, 55/54, 64/63
|-
| 8
| 744.2
| 20/13, 32/21
|-
| 9
| 237.3
| '''8/7''', 15/13
|-
| 10
| 930.3
| 12/7, 22/13
|-
| 11
| 423.3
| 9/7
|-
| 12
| 1116.4
| 27/14, 40/21
|-
| 13
| 609.4
| 10/7
|}
<nowiki/>* In 13-limit CTE tuning, octave reduced

=== As a detemperament of 7et ===
Flattone is best analyzed as a 7-form system. It is melodically intuitive compared to standard meantone, in that 7-limit intervals are found as augmented and diminished versions of the category they "should" belong to (that is, the quartertone represents both 25/24 and 64/63).

{| class="wikitable right-2 right-4 right-6 right-8 right-10"
|-
! rowspan="2" | Interval category
! colspan="2" | −2 quartertones
! colspan="2" | −1 quartertone
! colspan="2" | 0 quartertones
! colspan="2" | 1 quartertone
! colspan="2" | 2 quartertones
|-
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
|-
| Unison
| 1098
| 28/15
| 1149
| 48/25, 52/27, 64/33, 35/18, 98/55, 63/32
| 0
| 1/1
| 51
| 25/24, 27/26, 33/32, 36/35, 55/54, 64/63
| 102
| 15/14
|-
| Second
| 84
| 28/27, 21/20
| 135
| 16/15, 12/11, 13/12, 14/13
| 186
| 9/8, 10/9, 11/10
| 237
| 8/7, 15/13
| 288
| 
|-
| Third
| 219
| 
| 270
| 7/6
| 321
| 6/5
| 372
| 5/4, 16/13, 26/21
| 423
| 9/7
|-
| Fourth
| 405
| 
| 456
| 13/10, 21/16
| 507
| 4/3
| 558
| 11/8, 18/13
| 609
| 10/7
|-
| Fifth
| 591
| 7/5
| 642
| 16/11, 13/9
| 693
| 3/2
| 744
| 20/13, 32/21
| 795
| 
|-
| Sixth
| 777
| 14/9
| 828
| 8/5, 13/8, 21/13
| 879
| 5/3
| 930
| 12/7
| 981
| 
|-
| Seventh
| 912
| 
| 963
| 7/4, 26/15
| 1014
| 9/5, 16/9
| 1065
| 15/8, 11/6, 13/7, 24/13
| 1116
| 40/21, 27/14
|}

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

== Tunings ==
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! Generator<br>(¢)
! Comments
|-
|
| [[64/63]]
| 689.609
| 
|-
|
| [[13/8]]
| 689.868
| 
|-
|
| [[11/6]]
| 689.873
| 
|-
| [[33edo|19\33]]
|
| 690.909
|
|-
|
| [[13/11]]
| 691.079
| 
|-
|
| [[21/16]]
| 691.152
| 
|-
|
| [[9/5]]
| 691.202
| [[1/2-comma meantone|1/2 comma]]
|-
| [[92edo|53\92]]
|
| 691.304
|
|-
|
| [[21/11]]
| 691.467
| 
|-
| [[59edo|34\59]]
|
| 691.525
|
|-
| [[85edo|49\85]]
|
| 691.765
|
|-
|
| [[11/8]]
| 691.886
| 
|-
|
| [[11/7]]
| 692.166
| 11- and 13-odd-limit minimax
|-
|
| [[13/12]]
| 692.285
| 
|-
| [[26edo|15\26]]
|
| 692.308
| Lower bound of 7-, 9-, 11-, and 13-odd-limit diamond monotone
|-
|
| [[7/4]]
| 692.353
| 
|-
|
| [[21/13]]
| 692.437
| 
|-
|
| [[36/35]]
| 692.681
| 
|-
|
| [[49/48]]
| 692.858
| 
|-
| [[71edo|41\71]]
|
| 692.958
|
|-
|
| [[21/20]]
| 692.961
| 
|-
|
| [[13/10]]
| 693.223
| 
|-
|
| [[7/6]]
| 693.313
| 
|-
| [[45edo|26\45]]
|
| 693.333
| 
|-
|
| [[7/5]]
| 693.653
| 7-odd-limit minimax
|-
| [[64edo|37\64]]
|
| 693.750
|
|-
|
| [[9/7]]
| 694.099
| 9-odd-limit minimax
|-
|
| [[15/13]]
| 694.193
| 
|-
|
| [[15/14]]
| 694.246
| 
|-
|
| [[13/7]]
| 694.340
| 
|-
| [[19edo|11\19]]
|
| 694.737
| Upper bound of 7-, 9-, 11-, 13-odd-limit diamond monotone
|-
|
| [[5/3]]
| 694.786
| [[1/3-comma meantone|1/3 comma]]
|-
|
| [[25/24]]
| 695.810
| [[2/7-comma meantone|2/7 comma]]
|-
|
| [[5/4]]
| 696.578
| [[Quarter-comma meantone|1/4 comma]], 5-odd-limit minimax
|-
|
| [[15/8]]
| 697.654
| [[1/5-comma meantone|1/5 comma]]
|-
| [[12edo|7\12]]
|
| 700.000
| 
|-
|
| [[3/2]]
| 701.955
| [[Pythagorean tuning]]
|}
<nowiki/>* Besides the octave

[[Category:Flattone| ]] <!-- Main article -->
[[Category:Rank-2 temperaments]]
[[Category:Meantone family]]
[[Category:Avicennmic temperaments]]
[[Category:Keemic temperaments]]