Meantone intervals: Difference between revisions
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{{Wikipedia|List of meantone intervals}} | |||
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. | ||
{| 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 | ||
| | | 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 | ||
| | | 36/35, 50/49, 64/63, 128/125 | ||
|- | |- | ||
| Minor second (m2) | | Minor second (m2) | ||
| 117.53 | | 117.53 | ||
| 16/15 | | 15/14, 16/15 | ||
|- | |- | ||
| Major second (M2) | | Major second (M2) | ||
| 192.99 | | 192.99 | ||
| 10/9 | | 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, 16/9, 25/14 | ||
|- | |- | ||
| Major seventh (M7) | | Major seventh (M7) | ||
| 1082.47 | | 1082.47 | ||
| 15/8 | | 15/8, 28/15 | ||
|- | |- | ||
| Augmented seventh (A7) | | Augmented seventh (A7) | ||
| 1157.94 | | 1157.94 | ||
| 35/18 | | 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 | | 27/14, 40/21, 48/25 | ||
|- | |- | ||
| Perfect octave (P8) | | Perfect octave (P8) | ||