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]].  
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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| 10/7
| 10/7
|}
|}
<nowiki>*</nowiki> In 13-limit CTE tuning, octave reduced
<nowiki/>* In 13-limit CTE 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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=== 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)]]*

Revision as of 22:25, 27 November 2025

Lua error in Module:Infobox_regtemp at line 138: attempt to perform arithmetic on local 'generator_size' (a nil value).

Flattone is an alternative extension to 5-limit meantone, the temperament that tempers out the 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.

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 flatter-than-flattone temperament.

See Meantone family #Flattone for technical data.

Interval chain

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

# 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

* 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).

Interval category −2 quartertones −1 quartertone 0 quartertones 1 quartertone 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

Edo
generator 		Eigenmonzo
(unchanged-interval)* 		Generator
(¢) 		Comments
64/63 689.609
13/8 689.868
11/6 689.873
19\33 690.909
13/11 691.079
21/16 691.152
9/5 691.202 1/2 comma
53\92 691.304
21/11 691.467
34\59 691.525
49\85 691.765
11/8 691.886
11/7 692.166 11- and 13-odd-limit minimax
13/12 692.285
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
41\71 692.958
21/20 692.961
13/10 693.223
7/6 693.313
26\45 693.333
7/5 693.653 7-odd-limit minimax
37\64 693.750
9/7 694.099 9-odd-limit minimax
15/13 694.193
15/14 694.246
13/7 694.340
11\19 694.737 Upper bound of 7-, 9-, 11-, 13-odd-limit diamond monotone
5/3 694.786 1/3 comma
25/24 695.810 2/7 comma
5/4 696.578 1/4 comma, 5-odd-limit minimax
15/8 697.654 1/5 comma
7\12 700.000
3/2 701.955 Pythagorean tuning

* Besides the octave

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