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'''Diaschismic''', 2.3.5[10 & 12], is a 5-limit temperament equating 45/32 to the half-octave, thereby
'''Diaschismic''', [10 & 12], is the temperament (usually 2.3.5.17) generated by a half-octave representing 45/32~17/12~24/17~64/45 and a semitone representing 135/128~18/17~17/16~16/15, which is tuned to around 705 cents. 3/2 is a period plus a generator; 5/4 is a period minus two generators. (Note that 3/2 can be taken as the generator instead of 17/16.) In other words, diaschismic equates a stack of two 5/4's and two 9/8's to the octave; equivalently, it equates a stack of one 5/4 and two 16/15's to the half-octave.
equating a stack of two 5/4's and two 9/8's to the octave; equivalently, it equates a stack of one 5/4 and two 16/15's to the half-octave.
The generator of Diaschismic can be taken to be ~16/15 or ~3/2.


[[12edo]] and [[22edo]] are notable Diaschismic tunings. Some tunings of Diaschismic that are more accurate in the 5-limit are [[34edo]] and [[46edo]].  
[[12edo]] and [[22edo]] are notable Diaschismic tunings. Some tunings of Diaschismic that are more accurate in the 5-limit are [[34edo]] and [[46edo]].  


== Extensions ==
== Extensions ==
Diaschismic has an [[endoparticular]] extension that adds prime 17, equating the 16/15 generator to 17/16. Adding other primes accurately is harder; [[Pajara]] provides a simple albeit lower-accuracy extension to prime 7 by tempering out 64/63.
 
==== Archytas (10 & 12) (Pajara) ====
''Main article: [[Pajara]]''
 
The edo join [10 & 12] given above represents the extension pajara, which tempers out 64/63 such that a sharpened generator of about 110 cents is also 15/14 and stacks twice to a flatly tuned 8/7. The semioctave is interpreted as 7/5 and 10/7. This is a particularly common practical 7-limit interpretation of diaschismic supported by most of the small diaschismic edos, and especially 22edo.
 
Further extensions of Pajara are discussed on its own page.
 
==== Aberschismic (46 & 58) ====
A more accurate but complex option is to interpret diaschismic as a hemifamity temperament. Here, the comma 81/80 represented by six generators is also 64/63 and half of 50/49, so that 8/7 is a comma above 9/8 (being itself a comma above 10/9)..
 
In the 11-limit, an obvious mapping is to treat the neogothic major third 81/64 as 14/11, as in hemifamity; this sets 11/8 to be offset from the tritone by two commas. As a result, 11/9 is a comma sharp of 6/5. 16/13 is a further comma sharp.
 
Alternatively, another mapping offsets 11 from the fourth by two commas instead, finding 13 at 8 generators up and setting the tritone to 91/64.
 
Only the mappings for the 2.3.5.7.17 group will be given here.
 
== Generator chain ==
{| class="wikitable"
|+
! colspan="5" |Period 1
! colspan="5" |Period 2
|-
! colspan="3" |Up
! colspan="2" |Down
! colspan="3" |Up
! colspan="2" |Down
|-
!#
!Cents
!JI
!Cents
!JI
!#
!Cents
!JI
!Cents
!JI
|-
|0
|0
|1/1
|600
|24/17, 64/45
|0
|600
|17/12, 45/32
|1200
|2/1
|-
|1
|104
|18/17, 17/16, 16/15
|496
|4/3
|1
|704
|3/2
|1096
|15/8, 32/17, 17/9
|-
|2
|208
|9/8
|392
|5/4
|2
|808
|8/5
|992
|16/9
|-
|3
|312
|6/5
|288
|32/27
|3
|912
|27/16
|888
|5/3
|-
|4
|416
|81/64
|184
|10/9
|4
|1016
|9/5
|784
|128/81
|-
|5
|520
|27/20
|80
|21/20, 25/24
|5
|1120
|40/21, 48/25
|680
|40/27
|-
|6
|24
|81/80, 64/63
|576
|7/5
|6
|624
|10/7
|1176
|63/32, 160/81
|-
|7
|128
|27/25, 15/14
|472
|21/16
|7
|728
|32/21
|1072
|50/27, 28/15
|-
|8
|232
|8/7
|368
|21/17
|8
|832
|34/21
|968
|7/4
|-
|9
|336
|17/14
|264
|7/6
|9
|936
|12/7
|864
|28/17
|-
|10
|440
|9/7
|160
|35/32
|10
|1040
|51/28
|760
|14/9
|}


== List of patent vals ==
== List of patent vals ==
Line 12: Line 169:
{| class="wikitable sortable"
{| class="wikitable sortable"
|-
|-
!Edo!!Generator (3/2)
!Edo
!Extension!!Generator (3/2)
|-
|-
|2||600.000
|2
| ||600.000
|-
|-
|14||685.714
|14
| ||685.714
|-
|-
|12||700.000
|12
|Pajara, Septimal diaschismic||700.000
|-
|-
|70||702.857
|70
|Septimal diaschismic||702.857
|-
|-
|58||703.448
|58
|Septimal diaschismic||703.448
|-
|-
|46||704.348
|46
|Septimal diaschismic (alternate 11 mapping)||704.348
|-
|-
|126||704.762
|126
|(alternate 11 mapping)||704.762
|-
|-
|80||705.000
|80
|(alternate 11 mapping)||705.000
|-
|-
|114||705.263
|114
| ||705.263
|-
|-
|148||705.405
|148
| ||705.405
|-
|-
|34||705.882
|34
|Septimal diaschismic (alternate 11 mapping)||705.882
|-
|-
|124||706.452
|124
| ||706.452
|-
|-
|90||706.667
|90
| ||706.667
|-
|-
|56||707.143
|56
| ||707.143
|-
|-
|78||707.692
|78
| ||707.692
|-
|-
|22||709.091
|22
|Pajara||709.091
|-
|-
|54||711.111
|54
|Pajara||711.111
|-
|-
|32||712.500
|32
|Pajara||712.500
|-
|-
|10||720.000
|10
|Pajara||720.000
|}
|}


{{Navbox regtemp}}
{{Navbox regtemp}}
{{Cat|temperaments}}
{{Cat|temperaments}}

Revision as of 00:25, 29 March 2026

Diaschismic, [10 & 12], is the temperament (usually 2.3.5.17) generated by a half-octave representing 45/32~17/12~24/17~64/45 and a semitone representing 135/128~18/17~17/16~16/15, which is tuned to around 705 cents. 3/2 is a period plus a generator; 5/4 is a period minus two generators. (Note that 3/2 can be taken as the generator instead of 17/16.) In other words, diaschismic equates a stack of two 5/4's and two 9/8's to the octave; equivalently, it equates a stack of one 5/4 and two 16/15's to the half-octave.

12edo and 22edo are notable Diaschismic tunings. Some tunings of Diaschismic that are more accurate in the 5-limit are 34edo and 46edo.

Extensions

Archytas (10 & 12) (Pajara)

Main article: Pajara

The edo join [10 & 12] given above represents the extension pajara, which tempers out 64/63 such that a sharpened generator of about 110 cents is also 15/14 and stacks twice to a flatly tuned 8/7. The semioctave is interpreted as 7/5 and 10/7. This is a particularly common practical 7-limit interpretation of diaschismic supported by most of the small diaschismic edos, and especially 22edo.

Further extensions of Pajara are discussed on its own page.

Aberschismic (46 & 58)

A more accurate but complex option is to interpret diaschismic as a hemifamity temperament. Here, the comma 81/80 represented by six generators is also 64/63 and half of 50/49, so that 8/7 is a comma above 9/8 (being itself a comma above 10/9)..

In the 11-limit, an obvious mapping is to treat the neogothic major third 81/64 as 14/11, as in hemifamity; this sets 11/8 to be offset from the tritone by two commas. As a result, 11/9 is a comma sharp of 6/5. 16/13 is a further comma sharp.

Alternatively, another mapping offsets 11 from the fourth by two commas instead, finding 13 at 8 generators up and setting the tritone to 91/64.

Only the mappings for the 2.3.5.7.17 group will be given here.

Generator chain

Period 1 Period 2
Up Down Up Down
# Cents JI Cents JI # Cents JI Cents JI
0 0 1/1 600 24/17, 64/45 0 600 17/12, 45/32 1200 2/1
1 104 18/17, 17/16, 16/15 496 4/3 1 704 3/2 1096 15/8, 32/17, 17/9
2 208 9/8 392 5/4 2 808 8/5 992 16/9
3 312 6/5 288 32/27 3 912 27/16 888 5/3
4 416 81/64 184 10/9 4 1016 9/5 784 128/81
5 520 27/20 80 21/20, 25/24 5 1120 40/21, 48/25 680 40/27
6 24 81/80, 64/63 576 7/5 6 624 10/7 1176 63/32, 160/81
7 128 27/25, 15/14 472 21/16 7 728 32/21 1072 50/27, 28/15
8 232 8/7 368 21/17 8 832 34/21 968 7/4
9 336 17/14 264 7/6 9 936 12/7 864 28/17
10 440 9/7 160 35/32 10 1040 51/28 760 14/9

List of patent vals

The following patent vals support 2.3.5 Diaschismic, not including vals contorted in 2.3.5.

Edo Extension Generator (3/2)
2 600.000
14 685.714
12 Pajara, Septimal diaschismic 700.000
70 Septimal diaschismic 702.857
58 Septimal diaschismic 703.448
46 Septimal diaschismic (alternate 11 mapping) 704.348
126 (alternate 11 mapping) 704.762
80 (alternate 11 mapping) 705.000
114 705.263
148 705.405
34 Septimal diaschismic (alternate 11 mapping) 705.882
124 706.452
90 706.667
56 707.143
78 707.692
22 Pajara 709.091
54 Pajara 711.111
32 Pajara 712.500
10 Pajara 720.000


View • Talk • EditRegular temperaments
Rank-2
Acot Blackwood (1/5-octave) • Whitewood (1/7-octave) • Compton (1/12-octave)
Monocot Meantone • Schismic • Leapday • Archy
Complexity 2 Diaschismic (diploid monocot) • Pajara (diploid monocot) • Injera (diploid monocot) • Rastmatic (dicot) • Mohajira (dicot) • Intertridecimal (dicot) • Interseptimal (alpha-dicot)
Complexity 3 Augmented (triploid) • Misty (triploid) • Slendric (tricot) • Porcupine (omega-tricot)
Complexity 4 Diminished (tetraploid) • Tetracot (tetracot) • Buzzard (alpha-tetracot) • Squares (beta-tetracot) • Negri (omega-tetracot)
Complexity 5-6 Magic (alpha-pentacot) • Amity (gamma-pentacot) • Kleismic (alpha-hexacot) • Miracle (hexacot)
Higher complexity Orwell (alpha-heptacot) • Sensi (beta-heptacot) • Octacot (octacot) • Wurschmidt (beta-octacot) • Valentine (enneacot) • Ammonite (epsilon-enneacot) • Myna (beta-decacot) • Ennealimmal (enneaploid dicot)
Straddle-3 A-Team (alter-tricot) • Machine (alter-monocot)
No-3 Trismegistus (alpha-triseph) • Orgone (trimech) • Didacus (diseph)
No-octaves Sensamagic (monogem)
Exotemperament Dicot • Mavila • Father
Higher-rank
Rank-3 Hemifamity • Marvel • Parapyth
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