Diaschismic extensions: Difference between revisions

{{Breadcrumb|Diaschismic}}

In the [[5-limit]], '''diaschismic''' is a [[regular temperament]] (also known as ''srutal'', though they refer to different extensions in higher limits) defined by [[tempering out]] the comma [[2048/2025]] = [11 -4 -2⟩, the diaschisma. The octave is split into two periods, each representing [[~]][[45/32]]~[[64/45]]; and the [[generator]] can be considered to be a perfect fifth (~[[3/2]]), or a perfect fifth less a period, which is a diatonic semitone of ~[[16/15]]. Tempering out the diaschisma implies that two of these semitones are equated to [[9/8]], and as [[9/8]] = ([[18/17]])([[17/16]]), ~[[16/15]] can very naturally be equated to 17/16 and 18/17 as well, producing a 2.3.5.17 [[subgroup]] extension known as '''srutal archagall''', whose commas are [[136/135]] and [[256/255]]. There are multiple ways to extend diaschismic to primes [[7/1|7]], [[11/1|11]], and [[13/1|13]].

Srutal tempers out [[4375/4374]] in addition to the diaschisma, and therefore [[7/4]] is represented by 15 semitones less a half octave, or five [[6/5]]s less a half octave.  

 

 

Diaschismic sacrifices a slight amount of accuracy by tempering out [[126/125]], but slightly reduces complexity: [[8/7]] is represented by 8 semitones less a half-octave, or we can say 7/4 is equated to four [[5/4]]s less a half octave.

 

 

 

[[Pajara]] combines diaschismic with [[archy]], tempering the fifth to about 709 cents. The interval of two stacked fifths is equated to 16/7, and the harmonic seventh [[7/4]] and the just major third [[5/4]] are separated by a perfect semioctave.

For technical data on 7-limit and higher-limit pajara, see [[Diaschismic family #Pajara]].

=== Keen ===

! rowspan="2" | # !! rowspan="2" | Cents* !! colspan="2" | Approximate Ratios

| −8 || 370.6 || 100/81 || 21/17, 56/45

| −7 || 474.2 || 125/96 || '''21/16''', 112/85

| −6 || 577.9 || 25/18 || 7/5

| −5 || 81.6 || 25/24 || 21/20

| −4 || 185.3 || 10/9, 75/68 || 28/25

| −3 || 289.0 || 20/17, 32/27 || 119/100

| −2 || 392.6 || '''5/4''', 34/27, 64/51 || 63/50

| −1 || 496.3 || 4/3, 45/34 || 168/125

| 0 || 0.0 || '''1/1''' || 126/125

| 1 || 103.7 || 16/15, '''17/16''', 18/17 ||

| 2 || 207.4 || '''9/8''', 17/15 || 125/112

| 3 || 311.0 || 6/5, 81/68 || 25/21

| 4 || 414.7 || 32/25, 51/40, 81/64 || 80/63

| 5 || 518.4 || 27/20, 34/25 || 75/56, 85/63

| 6 || 22.1 || 51/50, 81/80 || 85/84

| 7 || 125.8 || 27/25 || 15/14, 68/63

| 8 || 229.5 || 144/125 || 8/7

<nowiki />* In 7-limit [[POTE]] tuning

! rowspan="2" | &#35; !! rowspan="2" | Cents* !! colspan="2" | Approximate Ratios

! 2.3.5.17 subgroup !! Intervals of 7

| &minus;8 || 970.6 || 125/72 || '''7/4'''

| &minus;7 || 1074.2 || 50/27 || 28/15, 63/34

| &minus;6 || 1177.9 || 100/51, 160/81 || 168/85

| &minus;5 || 681.6 || 40/27, 25/17 || 112/75, 126/85

| &minus;4 || 785.3 || '''25/16''', 80/51, 128/81 || 63/40

| &minus;3 || 889.0 || 5/3, 136/81 || 42/25

| &minus;2 || 992.6 || 16/9, 30/17 || 224/125

| &minus;1 || 1096.3 || '''15/8''', 17/9, 32/17 ||

| 0 || 600.0 || 17/12, 24/17, 45/32, 64/45 ||

| 1 || 703.7 || '''3/2''', 68/45 || 125/84

| 2 || 807.4 || 8/5, 27/17, 51/32 || 100/63

| 3 || 911.0 || 17/10, '''27/16''' || 200/119

| 4 || 1014.7 || 9/5, 136/75 || 25/14

| 5 || 1118.4 || 48/25 || 40/21

| 6 || 622.1 || 36/25 || 10/7

| 7 || 725.8 || 192/125 || 32/21, 85/56

| 8 || 829.5 || 81/50 || 34/21, 45/28

<nowiki />* In 7-limit [[POTE]] tuning

<div class="toccolours mw-collapsible mw-collapsed" style="width: 600px; overflow: auto;">

<div style="line-height: 1.6;">'''Intervals of srutal {{nowrap|(34d &amp; 46)}}'''</div>

! Generator

! &minus;17

! &minus;16

! &minus;15

! &minus;14

! &minus;13

! &minus;12

! Cents*

| 17.73

| 122.57

| 227.40

| 332.24

| 437.08

| 541.92

! Ratios

|  

| 15/14

| 8/7

| 17/14

| 9/7

| 15/11

! Generator

! &minus;11

! &minus;10

! &minus;9

! &minus;8

! &minus;7

! &minus;6

! Cents*

| 46.76

| 151.60

| 256.44

| 361.28

| 466.12

| 570.96

! Ratios

|  

| 12/11

| 15/13

| 16/13

| 17/13

| 18/13

! Generator

! &minus;5

! &minus;4

! &minus;3

! &minus;2

! &minus;1

! 0

! Cents*

| 75.80

| 180.64

| 285.48

| 390.32

| 495.16

| 600.00

! Ratios

| 22/21

| 10/9

| 20/17, 13/11

| 5/4

| 4/3

| 24/17, 17/12

! Generator

! 0

! 1

! 2

! 3

! 4

! 5

! Cents*

| 0.00

| 104.84

| 209.68

| 314.52

| 419.36

| 524.20

! Ratios

| 1/1

| 18/17, 17/16, <br />16/15

| 9/8, 17/15

| 6/5

| 14/11

|  

! Generator

! 6

! 7

! 8

! 9

! 10

! 11

! Cents*

| 29.04

| 133.88

| 238.72

| 343.56

| 448.40

| 553.24

! Ratios

|  

| 14/13, 13/12

|  

| 11/9

| 22/17, 13/10

| 11/8

! Generator

! 12

! 13

! 14

! 15

! 16

! 17

! Cents*

| 58.08

| 162.92

| 267.76

| 372.60

| 477.43

| 582.27

! Ratios

|  

| 11/10

| 7/6

| 21/17

| 21/16

| 7/5

<nowiki />* In 17-limit POTE tuning

* [[Srutal12]] &ndash; proper [[10L 2s]]

* [[Srutal22]] &ndash; improper [[12L 10s]]

* [[Diaschismic12]] &ndash; proper [[10L 2s]]

* [[Diaschismic22]] &ndash; improper [[12L 10s]]

* [[Diaschismic34]] &ndash; improper [[12L 22s]]

[[Category:Temperament extensions]]