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12edo: Difference between revisions

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In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.
== Intervals ==
{| class="wikitable"
!Degree
!Cents
! colspan="3" |Interval
!7-limit Ratios*
|-
|0
|0
|unison
|P1
|D
|1/1
|-
|1
|100
|aug 1sn, minor 2nd
|A1, m2
|D#, Eb
|15/14, 16/15, 21/20, 25/24, 28/27
|-
|2
|200
|major 2nd
|M2
|E
|8/7, 9/8, 10/9
|-
|3
|300
|minor 3rd
|m3
|F
|7/6, 6/5
|-
|4
|400
|major 3rd
|M3
|F#
|5/4, 9/7
|-
|5
|500
|perfect 4th
|P4
|G
|4/3
|-
|6
|600
|aug 4th, dim 5th
|A4, d5
|G#, Ab
|7/5, 10/7
|-
|7
|700
|perfect 5th
|P5
|A
|3/2
|-
|8
|800
|minor 6th
|m6
|Bb
|8/5, 14/9
|-
|9
|900
|major 6th
|M6
|B
|5/3, 12/7
|-
|10
|1000
|minor 7th
|m7
|C
|7/4, 9/5, 16/9
|-
|11
|1100
|major 7th
|M7
|C#
|15/8, 28/15, 40/21, 48/25, 27/14
|-
|19
|1200
|perfect 8ve
|P8
|D
|2/1
|}
<nowiki>*</nowiki> based on treating 12-edo as a 2.3.5.7 subgroup temperament; other approaches are possible.


==Rank two temperaments==
==Rank two temperaments==
Line 310: Line 410:
! Error (abs., in [[cent|cents]])
! Error (abs., in [[cent|cents]])
|-
|-
| [[4/3]], [[3/2]]
| '''[[4/3]], [[3/2]]'''
| 1.955
| '''1.955'''
|-
|-
| [[15/13]], [[26/15]]
| [[15/13]], [[26/15]]
Line 325: Line 425:
| 11.731
| 11.731
|-
|-
| [[5/4]], [[8/5]]
| '''[[5/4]], [[8/5]]'''
| 13.686
| '''13.686'''
|-
|-
| [[6/5]], [[5/3]]
| [[6/5]], [[5/3]]
Line 346: Line 446:
| 28.298
| 28.298
|-
|-
| [[8/7]], [[7/4]]
| '''[[8/7]], [[7/4]]'''
| 31.174
| '''31.174'''
|-
|-
| [[7/6]], [[12/7]]
| [[7/6]], [[12/7]]
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| 38.573
| 38.573
|-
|-
| [[16/13]], [[13/8]]
| '''[[16/13]], [[13/8]]'''
| 40.528
| '''40.528'''
|-
|-
| [[13/10]], [[20/13]]
| [[13/10]], [[20/13]]
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| 47.408
| 47.408
|-
|-
| [[11/8]], [[16/11]]
| '''[[11/8]], [[16/11]]'''
| 48.682
| '''48.682'''
|-
|-
| [[12/11]], [[11/6]]
| [[12/11]], [[11/6]]
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