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Didacus: Difference between revisions

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=== The hexatonic framework ===
=== The hexatonic framework ===
The 2.5.7 subgroup can be crudely approximated by [[6edo]], which is itself technically a didacus tuning as 5/4 spans 2 steps and 7/5 spans 3. Every other didacus tuning is essentially a dietic inflection of this basic hexatonic structure. Therefore, the intervals of didacus can be organized according to how many steps of [[6edo]], or equivalently the 6-note MOS, they correspond to. They can be labeled "wholetone", "ditone", "tritone", etc., and inflected so that "minor" intervals are those just below a step of 6edo, and "major" intervals are just above.  Below are the intervals of the symmetric mode of Didacus[25] ([[6L 19s]]) in undecimal CEE tuning.
The 2.5.7 subgroup can be crudely approximated by [[6edo]], which is itself technically a didacus tuning as 5/4 spans 2 steps and 7/5 spans 3. Every other didacus tuning is essentially a dietic inflection of this basic hexatonic structure. Therefore, the intervals of didacus can be organized according to how many steps of [[6edo]], or equivalently the 6-note MOS, they correspond to. They can be labeled "wholetone", "ditone", "tritone", etc., and inflected so that "minor" intervals are those just below a step of 6edo, and "major" intervals are just above, whereas the unison, octave, and generators can be labeled "perfect" instead.  Below are the intervals within 10 generators of the unison in undecimal CEE tuning.


{| class="wikitable center-all left-1"
{| class="wikitable center-all left-1"
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|- style="background-color: #DFDFDF;"
|- style="background-color: #DFDFDF;"
! "Augmented" interval
! "Augmented" interval
| 67.79
| 33.89
| 262.14
| 228.24
| 456.49
| 456.49
| 650.84
| 650.84
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|-  
|-  
! JI intervals represented
! JI intervals represented
| 26/25
| 50/49, 56/55, 65/64
| 64/55, 65/56
| 8/7, 25/22
| 13/10, 64/49
| 13/10, 64/49
| 16/11
| 16/11
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|- style="background-color: #DFDFDF;"
|- style="background-color: #DFDFDF;"
! "Major" interval
! "Major" interval
| 33.89
|  
| 228.24
|  
| 422.60
| 422.60
| 616.95
| 616.95
| 811.30
| 811.30
| 1005.65
|  
| ''1200.00''
|  
|-  
|-  
! JI intervals represented
! JI intervals represented
| 50/49, 56/55, 65/64
|  
| 8/7, 25/22
|  
| 14/11, 32/25
| 14/11, 32/25
| 10/7
| 10/7
| 8/5, 35/22
| 8/5, 35/22
|
|
|- style="background-color: #DFDFDF;"
! "Perfect" interval
| ''0.00''
| 194.35
|
|
|
| 1005.65
| ''1200.00''
|-
! JI intervals represented
| ''1/1''
| 28/25
|
|
|
| 25/14
| 25/14
| ''2/1''
| ''2/1''
|- style="background-color: #DFDFDF;"
|- style="background-color: #DFDFDF;"
! "Minor" interval
! "Minor" interval
| ''0.00''
|  
| 194.35
|  
| 388.70
| 388.70
| 583.05
| 583.05
| 777.40
| 777.40
| 971.76
|  
| 1166.11
|  
|-  
|-  
! JI intervals represented
! JI intervals represented
| ''1/1''
|  
| 28/25
|  
| 5/4, 44/35
| 5/4, 44/35
| 7/5
| 7/5
| 11/7, 25/16
| 11/7, 25/16
| 7/4, 44/25
|  
| 49/25, 55/28
|  
|- style="background-color: #DFDFDF;"
|- style="background-color: #DFDFDF;"
! "Diminished" interval
! "Diminished" interval
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| 549.16
| 549.16
| 743.51
| 743.51
| 937.86
| 971.76
| 1132.21
| 1166.11
|-  
|-  
! JI intervals represented
! JI intervals represented
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| 11/8
| 11/8
| 20/13, 49/32
| 20/13, 49/32
| 55/32
| 7/4, 44/25
| 25/13
| 49/25, 55/28
|}
|}


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