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 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 diminshed fifth (C-G𝄫).  
{{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.  


11- and 13-limit extensions are fairly obvious, using the heavily tempered [[chromatic semitone]] for the undecimal quartertone of [[33/32]] and the tridecimal third tone of [[27/26]]. 11/8 is an augmented fourth, and 13/8 is a minor sixth, conflated with [[8/5]].  
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.]]


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-of-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.  
See [[Meantone family #Flattone]] for technical data.  
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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*
! Approximate Ratios
! Approximate ratios
|-
|-
| 0
| 0
Line 19: Line 35:
|-
|-
| 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
<nowiki/>* In 13-limit CWE 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, 13/11
| 321
| 6/5
| 372
| 5/4, 11/9, 16/13, 26/21
| 423
| 9/7
|-
| Fourth
| 405
| 14/11
| 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
| 11/7
|-
| Sixth
| 777
| 14/9
| 828
| 8/5, 18/11, 13/8, 21/13
| 879
| 5/3
| 930
| 12/7, 22/13
| 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 ==
== Scales ==
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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)]]*
! Generator<br>(¢)
! Generator<br>(¢)
! Comments
! Comments
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|
|
| 690.909
| 690.909
|
| 33c val
|-
|-
|
|
Line 122: Line 278:
|
|
| 691.304
| 691.304
|
| 92bccc val
|-
|-
|
|
Line 132: Line 288:
|
|
| 691.525
| 691.525
|
| 59bc val
|-
|-
| [[85edo|49\85]]
| [[85edo|49\85]]
|
|
| 691.765
| 691.765
|
| 85bccf val
|-
|-
|
|
Line 182: Line 338:
|
|
| 692.958
| 692.958
|
| 71bcf val
|-
|-
|
|
Line 202: Line 358:
|
|
| 693.333
| 693.333
|  
| 45f val
|-
|-
|
|
Line 212: Line 368:
|
|
| 693.750
| 693.750
|
| 64cdef val
|-
|-
|
|
Line 237: Line 393:
|
|
| 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
|-
|-
|
|
Line 262: Line 418:
|
|
| 700.000
| 700.000
|  
| 12d val
|-
|-
|
|
Line 269: Line 425:
| [[Pythagorean tuning]]
| [[Pythagorean tuning]]
|}
|}
<nowiki>*</nowiki> besides the octave
<nowiki/>* Besides the octave


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

Latest revision as of 09:11, 9 February 2026

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);
45/44, 81/80, 385/384 (11-limit);
45/44, 65/64, 78/77, 81/80 (13-limit)
Reduced mapping ⟨1; 1 4 -9 6 -4]
ET join 19 & 26
Generators (CWE) ~3/2 = 693.1 ¢
MOS scales 5L 2s, 7L 5s, 7L 12s
Ploidacot monocot
Pergen (P8, P5)
Minimax error 9-odd-limit: 15.7 ¢;
13-odd-limit: 19.3 ¢
Target scale size 9-odd-limit: 19 notes;
13-odd-limit: 19 notes

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 fokkertone 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
1 693.1 3/2
2 186.1 9/8, 10/9, 11/10
3 879.2 5/3
4 372.2 5/4, 16/13, 26/21
5 1065.3 11/6, 13/7, 15/8, 24/13
6 558.3 11/8, 18/13
7 51.4 25/24, 27/26, 33/32, 36/35, 55/54, 64/63
8 744.4 20/13, 32/21
9 237.5 8/7, 15/13
10 930.5 12/7, 22/13
11 423.6 9/7
12 1116.6 27/14, 40/21
13 609.7 10/7

* In 13-limit CWE 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, 13/11 321 6/5 372 5/4, 11/9, 16/13, 26/21 423 9/7
Fourth 405 14/11 456 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 11/7
Sixth 777 14/9 828 8/5, 18/11, 13/8, 21/13 879 5/3 930 12/7, 22/13 981
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

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 693.5520 ¢ CWE: ~3/2 = 693.7333 ¢ POTE: ~3/2 = 693.7791 ¢
13-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 693.0293 ¢ CWE: ~3/2 = 693.0538 ¢ POTE: ~3/2 = 693.0578 ¢

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 33c val
13/11 691.079
21/16 691.152
9/5 691.202 1/2 comma
53\92 691.304 92bccc val
21/11 691.467
34\59 691.525 59bc val
49\85 691.765 85bccf val
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 71bcf val
21/20 692.961
13/10 693.223
7/6 693.313
26\45 693.333 45f val
7/5 693.653 7-odd-limit minimax
37\64 693.750 64cdef val
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-, and 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 12d val
3/2 701.955 Pythagorean tuning

* Besides the octave

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