Diaschismic extensions: Difference between revisions

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fixed interval tunings
Line 37: Line 37:
! 2.3.5.17 subgroup !! Intervals of 7
! 2.3.5.17 subgroup !! Intervals of 7
|-
|-
| −8 || 220.3 || 100/81 || 21/17, 56/45
| −8 || 370.6 || 100/81 || 21/17, 56/45
|-
|-
| −7 || 440.6 || 125/96 || '''21/16''', 112/85
| −7 || 474.2 || 125/96 || '''21/16''', 112/85
|-
|-
| −6 || 660.9 || 25/18 || 7/5
| −6 || 577.9 || 25/18 || 7/5
|-
|-
| −5 || 881.1 || 25/24 || 21/20
| −5 || 81.6 || 25/24 || 21/20
|-
|-
| −4 || 1101.4 || 10/9, 75/68 || 28/25
| −4 || 185.3 || 10/9, 75/68 || 28/25
|-
|-
| −3 || 1321.7 || 20/17, 32/27 || 119/100
| −3 || 289.0 || 20/17, 32/27 || 119/100
|-
|-
| −2 || 1542.0 || '''5/4''', 34/27, 64/51 || 63/50
| −2 || 392.6 || '''5/4''', 34/27, 64/51 || 63/50
|-
|-
| −1 || 1762.2 || 4/3, 45/34 || 168/125
| −1 || 496.3 || 4/3, 45/34 || 168/125
|-
|-
| 0 || 0.0 || '''1/1''' || 126/125
| 0 || 0.0 || '''1/1''' || 126/125
|-
|-
| 1 || 220.3 || 16/15, '''17/16''', 18/17 ||  
| 1 || 103.7 || 16/15, '''17/16''', 18/17 ||  
|-
|-
| 2 || 440.6 || '''9/8''', 17/15 || 125/112
| 2 || 207.4 || '''9/8''', 17/15 || 125/112
|-
|-
| 3 || 660.9 || 6/5, 81/68 || 25/21
| 3 || 311.0 || 6/5, 81/68 || 25/21
|-
|-
| 4 || 881.1 || 32/25, 51/40, 81/64 || 80/63
| 4 || 414.7 || 32/25, 51/40, 81/64 || 80/63
|-
|-
| 5 || 1101.4 || 27/20, 34/25 || 75/56, 85/63
| 5 || 518.4 || 27/20, 34/25 || 75/56, 85/63
|-
|-
| 6 || 1321.7 || 51/50, 81/80 || 85/84
| 6 || 22.1 || 51/50, 81/80 || 85/84
|-
|-
| 7 || 1542.0 || 27/25 || 15/14, 68/63
| 7 || 125.8 || 27/25 || 15/14, 68/63
|-
|-
| 8 || 1762.2 || 144/125 || 8/7
| 8 || 229.5 || 144/125 || 8/7
|}
|}
<nowiki />* In 3.5.7-targeted [[DKW theory|DKW]] tuning for [[BPS]]
<nowiki />* In 7-limit [[POTE]] tuning
</div>
</div>


Line 82: Line 82:
! 2.3.5.17 subgroup !! Intervals of 7
! 2.3.5.17 subgroup !! Intervals of 7
|-
|-
| &minus;8 || 1762.2 || 125/72 || '''7/4'''
| &minus;8 || 970.6 || 125/72 || '''7/4'''
|-
|-
| &minus;7 || 1542.0 || 50/27 || 28/15, 63/34
| &minus;7 || 1074.2 || 50/27 || 28/15, 63/34
|-
|-
| &minus;6 || 1321.7 || 100/51, 160/81 || 168/85
| &minus;6 || 1177.9 || 100/51, 160/81 || 168/85
|-  
|-  
| &minus;5 || 1101.4 || 40/27, 25/17 || 112/75, 126/85
| &minus;5 || 681.6 || 40/27, 25/17 || 112/75, 126/85
|-
|-
| &minus;4 || 881.1 || '''25/16''', 80/51, 128/81 || 63/40
| &minus;4 || 785.3 || '''25/16''', 80/51, 128/81 || 63/40
|-
|-
| &minus;3 || 660.9 || 5/3, 136/81 || 42/25
| &minus;3 || 889.0 || 5/3, 136/81 || 42/25
|-
|-
| &minus;2 || 440.6 || 16/9, 30/17 || 224/125
| &minus;2 || 992.6 || 16/9, 30/17 || 224/125
|-
|-
| &minus;1 || 220.3 || '''15/8''', 17/9, 32/17 ||
| &minus;1 || 1096.3 || '''15/8''', 17/9, 32/17 ||
|-
|-
| 0 || 0.0 || 17/12, 24/17, 45/32, 64/45 ||
| 0 || 600.0 || 17/12, 24/17, 45/32, 64/45 ||
|-
|-
| 1 || 1762.2 || '''3/2''', 68/45 || 125/84
| 1 || 703.7 || '''3/2''', 68/45 || 125/84
|-
|-
| 2 || 1542.0 || 8/5, 27/17, 51/32 || 100/63
| 2 || 807.4 || 8/5, 27/17, 51/32 || 100/63
|-
|-
| 3 || 1321.7 || 17/10, '''27/16''' || 200/119
| 3 || 911.0 || 17/10, '''27/16''' || 200/119
|-
|-
| 4 || 1101.4 || 9/5, 136/75 || 25/14
| 4 || 1014.7 || 9/5, 136/75 || 25/14
|-
|-
| 5 || 881.1 || 48/25 || 40/21
| 5 || 1118.4 || 48/25 || 40/21
|-
|-
| 6 || 660.9 || 36/25 || 10/7
| 6 || 622.1 || 36/25 || 10/7
|-
|-
| 7 || 440.6 || 192/125 || 32/21, 85/56
| 7 || 725.8 || 192/125 || 32/21, 85/56
|-
|-
| 8 || 220.3 || 81/50 || 34/21, 45/28
| 8 || 829.5 || 81/50 || 34/21, 45/28
|}
|}
<nowiki />* In 3.5.7-targeted [[DKW theory|DKW]] tuning for [[BPS]]
<nowiki />* In 7-limit [[POTE]] tuning
</div>
</div>


Revision as of 05:38, 23 October 2024

This page on a regular temperament, temperament collection, or aspect of regular temperament theory is being revised for clarity as part of WikiProject TempClean.

Lua error in Module:Infobox_regtemp at line 138: attempt to perform arithmetic on local 'generator_size' (a nil value).

Srutal, known interchangeably as diaschismic in the 5-limit, is a regular temperament defined by tempering out the comma 2048/2025, 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 therefore 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.

For technical data, see Diaschismic family #Srutal aka diaschismic.

7-limit extensions

The two alternative names for this temperament are assigned to different strong extensions to the 7-limit: srutal (34d&46) and diaschismic (46&58), though there are other mappings that are comparable in complexity and error: pajara (12&22) and keen (22&34).

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 6/5s 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 5/4s less a half octave.

Both of these can be extended straightforwardly to the 11-, 13-, and 17-limit by adding 176/175, 352/351, and 221/220 to the comma list in this order.

For technical data on 7-limit and higher-limit diaschismic: see Diaschismic family #Diaschismic.

For technical data on 7-limit and higher-limit srutal: see Diaschismic family #Srutal.

Interval chains

Diaschismic

First period
# Cents* Approximate Ratios
2.3.5.17 subgroup Intervals of 7
−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

* In 7-limit POTE tuning

Second period
# Cents* Approximate Ratios
2.3.5.17 subgroup Intervals of 7
−8 970.6 125/72 7/4
−7 1074.2 50/27 28/15, 63/34
−6 1177.9 100/51, 160/81 168/85
−5 681.6 40/27, 25/17 112/75, 126/85
−4 785.3 25/16, 80/51, 128/81 63/40
−3 889.0 5/3, 136/81 42/25
−2 992.6 16/9, 30/17 224/125
−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

* In 7-limit POTE tuning

Intervals

Intervals of srutal (34d & 46)
Generator −17 −16 −15 −14 −13 −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 −11 −10 −9 −8 −7 −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 −5 −4 −3 −2 −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,
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

* In 17-limit POTE tuning


Intervals of diaschismic (46 & 58)
Generator −17 −16 −15 −14 −13 −12
Cents* 35.19 139.01 242.82 346.63 450.44 554.25
Ratios 13/12 11/9 22/17, 13/10 11/8
Generator −11 −10 −9 −8 −7 −6
Cents* 58.07 161.88 265.69 369.50 473.32 577.13
Ratios 11/10 7/6 21/17, 26/21 21/16 7/5
Generator −5 −4 −3 −2 −1 − 0
Cents* 80.94 184.75 288.56 392.38 496.19 600.00
Ratios 22/21, 21/20 10/9 20/17, 13/11 5/4 4/3 24/17, 17/12
Generator 0 1 2 3 4 5
Cents* 0.00 103.81 207.62 311.44 415.25 519.06
Ratios 1/1 18/17, 17/16,
16/15 		9/8, 17/15 6/5 14/11
Generator 6 7 8 9 10 11
Cents* 22.87 126.68 230.50 334.31 438.12 541.93
Ratios 15/14, 14/13 8/7 17/14 9/7 15/11
Generator 12 13 14 15 16 17
Cents* 45.75 149.56 253.37 357.18 460.99 564.81
Ratios 12/11 15/13 16/13 17/13 18/13

* In 17-limit POTE tuning

Scales

See also

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