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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]]. Intervals in limits higher than 7 are in brackets.


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. Except arguably for the tritone, none of the intervals are inherently septimal in 12edo, because they all have simpler 5-limit interpretations.
 
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
! Size<br>(cents)
! Ratios
! Ratios
|-
|-
Line 18: Line 18:
|-
|-
| Perfect unison (P1)
| Perfect unison (P1)
| 0.0
| 0.00
| 1/1
| [[1/1]]
|-
|-
| Augmented unison (A1)
| Augmented unison (A1)
| 75.5
| 75.46
| 28/27~25/24~21/20
| [[21/20]], [[25/24]], [[28/27]]
|-
| Double-augmented unison (AA1)
| 150.93
| [[35/32]], [[12/11|(12/11)]]
|-
|-
! colspan="3" | Seconds
! colspan="3" | Seconds
|-
|-
| Diminished second (d2)
| Diminished second (d2)
| 42.0
| 42.06
| 128/125~64/63~50/49~36/35
| [[36/35]], [[50/49]], [[64/63]], [[128/125]]
|-
|-
| Minor second (m2)
| Minor second (m2)
| 117.5
| 117.53
| 16/15~15/14
| [[15/14]], [[16/15]]
|-
|-
| Major second (M2)
| Major second (M2)
| 193.0
| 192.99
| 10/9~9/8~28/25
| [[9/8]], [[10/9]], [[28/25]]
|-
|-
| Augmented second (A2)
| Augmented second (A2)
| 268.5
| 268.45
| 7/6
| [[7/6]]
|-
|-
! colspan="3" | Thirds
! colspan="3" | Thirds
|-
|-
| Diminished third (d3)
| Diminished third (d3)
| 235.0
| 235.05
| 8/7
| [[8/7]]
|-
|-
| Minor third (m3)
| Minor third (m3)
| 310.5
| 310.52
| 6/5
| [[6/5]]
|-
|-
| Major third (M3)
| Major third (M3)
| 386.0
| 385.98
| 5/4
| [[5/4]]
|-
|-
| Augmented third (A3)
| Augmented third (A3)
| 461.5
| 461.44
| 21/16
| [[21/16]]
|-
|-
! colspan="3" | Fourths
! colspan="3" | Fourths
|-
|-
| Double diminished fourth (dd4)
| Double-diminished fourth (dd4)
| 352.5
| 352.58
| Close to 11/9
| [[49/40]], [[11/9|(11/9)]]
|-
|-
| Diminished fourth (d4)
| Diminished fourth (d4)
| 428.0
| 428.04
| 9/7 (a bit 14/11-ish)
| [[9/7]], [[14/11|(14/11)]]
|-
|-
| Perfect fourth (P4)
| Perfect fourth (P4)
| 503.5
| 503.51
| 4/3
| [[4/3]]
|-
|-
| Augmented fourth (A4)
| Augmented fourth (A4)
| 579.0
| 578.97
| 7/5
| [[7/5]]
|-
|-
| Double augmented fourth (AA4)
| Double-augmented fourth (AA4)
| 654.5
| 654.43
| Close to 16/11
| [[35/24]], [[16/11|(16/11)]]
|-
|-
! colspan="3" | Fifths
! colspan="3" | Fifths
|-
|-
| Double diminished fifth (dd5)
| Double-diminished fifth (dd5)
| 545.5
| 545.57
| Close to 11/8
| [[48/35]], [[11/8|(11/8)]]
|-
|-
| Diminished fifth (d5)
| Diminished fifth (d5)
| 621.0
| 621.03
| 10/7
| [[10/7]]
|-
|-
| Perfect fifth (P5)
| Perfect fifth (P5)
| 696.5
| 696.49
| 3/2
| [[3/2]]
|-
|-
| Augmented fifth (A5)
| Augmented fifth (A5)
| 772.0
| 771.96
| 14/9 (a bit 11/7-ish)
| [[14/9]], [[11/7|(11/7)]]
|-
|-
| Double augmented fifth (AA5)
| Double-augmented fifth (AA5)
| 847.5
| 847.42
| Close to 18/11
| [[80/49]], [[18/11|(18/11)]]
|-
|-
! colspan="3" | Sixths
! colspan="3" | Sixths
|-
|-
| Diminished sixth (d6)
| Diminished sixth (d6)
| 738.5
| 738.56
| 32/21
| [[32/21]]
|-
|-
| Minor sixth (m6)
| Minor sixth (m6)
| 814.0
| 814.02
| 8/5
| [[8/5]]
|-
|-
| Major sixth (M6)
| Major sixth (M6)
| 889.5
| 889.48
| 5/3
| [[5/3]]
|-
|-
| Augmented sixth (A6)
| Augmented sixth (A6)
| 965.0
| 964.95
| 7/4
| [[7/4]]
|-
|-
! colspan="3" | Sevenths
! colspan="3" | Sevenths
|-
|-
| Diminished seventh (d7)
| Diminished seventh (d7)
| 931.5
| 931.55
| 12/7
| [[12/7]]
|-
|-
| Minor seventh (m7)
| Minor seventh (m7)
| 1007.0
| 1007.01
| 16/9~9/5~25/14
| [[9/5]], [[16/9]], [[25/14]]
|-
|-
| Major seventh (M7)
| Major seventh (M7)
| 1082.5
| 1082.47
| 15/8~28/15
| [[15/8]], [[28/15]]
|-
|-
| Augmented seventh (A7)
| Augmented seventh (A7)
| 1158.0
| 1157.94
| 35/18~49/25~63/32
| [[35/18]], [[49/25]], [[63/32]]
|-
|-
! colspan="3" | Octaves
! colspan="3" | Octaves
|-
| Double-diminished octave (dd8)
| 1049.07
| [[64/35]], [[11/6|(11/6)]]
|-
|-
| Diminished octave (d8)
| Diminished octave (d8)
| 1124.5
| 1124.54
| 27/14~48/25~40/21
| [[27/14]], [[40/21]], [[48/25]]
|-
|-
| Perfect octave (P8)
| Perfect octave (P8)
| 1200.0
| 1200.00
| 2/1
| [[2/1]]
|-
| Augmented octave (A8)
| 1275.5
| 25/12~21/10
|}
|}


[[Category:Meantone]]
[[Category:Meantone]]
[[Category:List]]
[[Category:Lists of intervals]]
Retrieved from "https://en.xen.wiki/w/Meantone_intervals"