Meantone intervals: Difference between revisions

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This table shows all the simple intervals of [[POTE]] [[Meantone family #Septimal meantone|septimal meantone]], which includes the entire [[7-limit]] [[tonality diamond]]. Other relevant tables of meantone intervals are the table of [[quarter-comma meantone]] intervals and the table of [[31edo #Intervals|31 edo intervals]].
{{Wikipedia|List of meantone intervals}}


In [[12edo]] the diminished second vanishes, so this cornucopia of intervals collapses to a mere 12. None of the intervals is inherently septimal in 12edo, because they all have simpler 5-limit descriptions.
This table shows all the simple intervals of [[POTE]] [[Meantone family #Septimal meantone|septimal meantone]], which includes the entire [[7-odd-limit]] [[tonality diamond]]. Other relevant tables of meantone intervals are the table of [[quarter-comma meantone]] intervals and the table of [[31edo #Intervals|31edo intervals]].


In [[19edo]], in contrast, the ''double'' diminished second vanishes, so the equivalences are A1~d2, A2~d3, A3~d4, A4~dd5, AA4~d5, A5~d6, A6~d7, and A7~d8. Thus some intervals are undeniably septimal, but ambiguously so because 49/48 vanishes.
In [[12edo]] the diminished second vanishes, so this cornucopia of intervals collapses to a mere 12. None of the intervals is inherently septimal in 12edo, because they all have simpler 5-limit descriptions.
 
In [[19edo]], in contrast, the ''double''-diminished second vanishes, so the equivalences are A1~d2, A2~d3, A3~d4, A4~dd5, AA4~d5, A5~d6, A6~d7, and A7~d8. Thus some intervals are undeniably septimal, but ambiguously so because [[49/48]] vanishes.  


More complex meantone tunings such as [[31edo]] distinguish all intervals listed on this table.
More complex meantone tunings such as [[31edo]] distinguish all intervals listed on this table.
See also [[Wikipedia: List of meantone intervals]]


{| class="wikitable right-2"
{| class="wikitable right-2"
|-
|-
! Name
! Name
! Size <br>(cents)
! Size<br>(cents)
! Ratios
! Ratios
|-
|-
Line 23: Line 23:
| Augmented unison (A1)
| Augmented unison (A1)
| 75.46
| 75.46
| 28/27~25/24~21/20
| 21/20, 25/24, 28/27
|-
|-
| Double augmented unison (AA1)
| Double-augmented unison (AA1)
| 150.93
| 150.93
| Close to 12/11
| Close to 12/11
Line 33: Line 33:
| Diminished second (d2)
| Diminished second (d2)
| 42.06
| 42.06
| 128/125~64/63~50/49~36/35
| 36/35, 50/49, 64/63, 128/125
|-
|-
| Minor second (m2)
| Minor second (m2)
| 117.53
| 117.53
| 16/15~15/14
| 15/14, 16/15
|-
|-
| Major second (M2)
| Major second (M2)
| 192.99
| 192.99
| 10/9~9/8~28/25
| 9/8, 10/9, 28/25
|-
|-
| Augmented second (A2)
| Augmented second (A2)
Line 67: Line 67:
! colspan="3" | Fourths
! colspan="3" | Fourths
|-
|-
| Double diminished fourth (dd4)
| Double-diminished fourth (dd4)
| 352.58
| 352.58
| Close to 11/9
| Close to 11/9
Line 83: Line 83:
| 7/5
| 7/5
|-
|-
| Double augmented fourth (AA4)
| Double-augmented fourth (AA4)
| 654.43
| 654.43
| Close to 16/11
| Close to 16/11
Line 89: Line 89:
! colspan="3" | Fifths
! colspan="3" | Fifths
|-
|-
| Double diminished fifth (dd5)
| Double-diminished fifth (dd5)
| 545.57
| 545.57
| Close to 11/8
| Close to 11/8
Line 105: Line 105:
| 14/9 (a bit 11/7-ish)
| 14/9 (a bit 11/7-ish)
|-
|-
| Double augmented fifth (AA5)
| Double-augmented fifth (AA5)
| 847.42
| 847.42
| Close to 18/11
| Close to 18/11
Line 135: Line 135:
| Minor seventh (m7)
| Minor seventh (m7)
| 1007.01
| 1007.01
| 16/9~9/5~25/14
| 9/5, 16/9, 25/14
|-
|-
| Major seventh (M7)
| Major seventh (M7)
| 1082.47
| 1082.47
| 15/8~28/15
| 15/8, 28/15
|-
|-
| Augmented seventh (A7)
| Augmented seventh (A7)
| 1157.94
| 1157.94
| 35/18~49/25~63/32
| 35/18, 49/25, 63/32
|-
|-
! colspan="3" | Octaves
! colspan="3" | Octaves
|-
|-
| Double diminished octave (dd8)
| Double-diminished octave (dd8)
| 1049.07
| 1049.07
| Close to 11/6
| Close to 11/6
Line 153: Line 153:
| Diminished octave (d8)
| Diminished octave (d8)
| 1124.54
| 1124.54
| 27/14~48/25~40/21
| 27/14, 40/21, 48/25
|-
|-
| Perfect octave (P8)
| Perfect octave (P8)

Latest revision as of 09:48, 4 June 2025

English Wikipedia has an article on:

This table shows all the simple intervals of POTE septimal meantone, which includes the entire 7-odd-limit tonality diamond. Other relevant tables of meantone intervals are the table of quarter-comma meantone intervals and the table of 31edo intervals.

In 12edo the diminished second vanishes, so this cornucopia of intervals collapses to a mere 12. None of the intervals is inherently septimal in 12edo, because they all have simpler 5-limit descriptions.

In 19edo, in contrast, the double-diminished second vanishes, so the equivalences are A1~d2, A2~d3, A3~d4, A4~dd5, AA4~d5, A5~d6, A6~d7, and A7~d8. Thus some intervals are undeniably septimal, but ambiguously so because 49/48 vanishes.

More complex meantone tunings such as 31edo distinguish all intervals listed on this table.

Name Size
(cents) 		Ratios
Unisons
Perfect unison (P1) 0.00 1/1
Augmented unison (A1) 75.46 21/20, 25/24, 28/27
Double-augmented unison (AA1) 150.93 Close to 12/11
Seconds
Diminished second (d2) 42.06 36/35, 50/49, 64/63, 128/125
Minor second (m2) 117.53 15/14, 16/15
Major second (M2) 192.99 9/8, 10/9, 28/25
Augmented second (A2) 268.45 7/6
Thirds
Diminished third (d3) 235.05 8/7
Minor third (m3) 310.52 6/5
Major third (M3) 385.98 5/4
Augmented third (A3) 461.44 21/16
Fourths
Double-diminished fourth (dd4) 352.58 Close to 11/9
Diminished fourth (d4) 428.04 9/7 (a bit 14/11-ish)
Perfect fourth (P4) 503.51 4/3
Augmented fourth (A4) 578.97 7/5
Double-augmented fourth (AA4) 654.43 Close to 16/11
Fifths
Double-diminished fifth (dd5) 545.57 Close to 11/8
Diminished fifth (d5) 621.03 10/7
Perfect fifth (P5) 696.49 3/2
Augmented fifth (A5) 771.96 14/9 (a bit 11/7-ish)
Double-augmented fifth (AA5) 847.42 Close to 18/11
Sixths
Diminished sixth (d6) 738.56 32/21
Minor sixth (m6) 814.02 8/5
Major sixth (M6) 889.48 5/3
Augmented sixth (A6) 964.95 7/4
Sevenths
Diminished seventh (d7) 931.55 12/7
Minor seventh (m7) 1007.01 9/5, 16/9, 25/14
Major seventh (M7) 1082.47 15/8, 28/15
Augmented seventh (A7) 1157.94 35/18, 49/25, 63/32
Octaves
Double-diminished octave (dd8) 1049.07 Close to 11/6
Diminished octave (d8) 1124.54 27/14, 40/21, 48/25
Perfect octave (P8) 1200.00 2/1
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