Flattone: Difference between revisions

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The article already starts with "flattone is an alternative extension to 5-limit meantone", so I can't see how this thing helps. Whatever its relation to meantone is, we're documenting this temp as a coherent 7-limit structure
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m Cleanup on infobox
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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 (Cโ€“B๐„ซ), [[7/6]] is a diminished third (Cโ€“E๐„ซ), and [[7/5]] is a doubly diminished fifth (Cโ€“G๐„ซ). 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. ย 
'''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 (Cโ€“B๐„ซ), [[7/6]] is a diminished third (Cโ€“E๐„ซ), and [[7/5]] is a doubly diminished fifth (Cโ€“G๐„ซ). 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. ย 


However, it makes up for that by having simpler 11- and 13-limit interpretations โ€“ 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 (Cโ€“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 (Cโ€“Aโ™ญ) 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 โ€“ 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 (Cโ€“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 (Cโ€“Aโ™ญ) 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 8โ€“16 in 45edo, using the flattone mappings for 13 & 15 rather than the best direct approximations.]]
[[File:45EDO_Otonal.mp3|none|thumb|Harmonic scale 8โ€“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]]. ย 
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]]. ย 
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In the following table, odd harmonics 1โ€“13 are in '''bold'''. ย 
In the following table, odd harmonics 1โ€“13 are in '''bold'''. ย 
{| 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
|}
<nowiki/>* In 13-limit CWE tuning, octave reduced
ย 
=== 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).
ย 
{| class="wikitable right-2 right-4 right-6 right-8 right-10"
|-
! rowspan="2" | Interval category
! colspan="2" | โˆ’2 quartertones
! colspan="2" | โˆ’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
| 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
|}
|}
<nowiki/>* In 13-limit CTE tuning, octave reduced


== 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>(ยข)
! Generator<br>(ยข)
! Comments
! Comments
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|
|
| 690.909
| 690.909
|
| 33c val
|-
|-
|
|
Line 122: 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
|-
|-
|
|
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|
|
| 692.958
| 692.958
|
| 71bcf val
|-
|-
|
|
Line 202: Line 358:
|
|
| 693.333
| 693.333
| ย 
| 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
| ย 
| 12d val
|-
|-
|
|
Line 272: Line 428:


[[Category:Flattone| ]] <!-- Main article -->
[[Category:Flattone| ]] <!-- Main article -->
[[Category:Temperaments]]
[[Category:Rank-2 temperaments]]
[[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 โŸจ1; 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 (Cโ€“B๐„ซ), 7/6 is a diminished third (Cโ€“E๐„ซ), and 7/5 is a doubly diminished fifth (Cโ€“G๐„ซ). 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 โ€“ 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 (Cโ€“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 (Cโ€“Aโ™ญ) 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 8โ€“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 1โ€“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 โˆ’2 quartertones โˆ’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 โ€“ 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
(ยข) 		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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