Xenharmonic Wiki

Decic: Difference between revisions

← Older edit
VisualWikitext
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
4,516 edits
Removed redirect to Cloudy clan#Decic
Tag: Removed redirect
Fredg999 (talk | contribs)
autopatrolled, Interface administrators, Sysops
6,129 edits
m Update link
 
(13 intermediate revisions by 5 users not shown)
Line 1: Line 1:
'''Decic''' is a temperament for the 7, 11, 13, and 17 [[Harmonic limit|prime limits]]. It is a member of [[marvel temperaments]], [[cloudy clan]], and [[15/14 equal-step tuning|linus temperaments]]. It has a period of 1/10 octave and tempers out 225/224 and 16807/16384. The fifth of decic in size is a [[meantone]] fifth, but four of them are not used to reach the 5th harmonic. Instead, [[14/13]], [[15/14]] and [[16/15]] are equated to 1/10 of an octave, and from this it derives its name. Not only the meantone fifth (flat 3/2) or fourth (sharp 4/3), but also the [[magic]] major third (flat ~5/4) can be used as a generator.
'''Decic''' is a temperament for the 7, 11, 13, and 17 [[Harmonic limit|prime limits]]. It is a member of [[marvel temperaments]], [[cloudy clan]], and [[linus]] temperaments. It has a period of 1/10 octave and tempers out 225/224 and 16807/16384. The fifth of decic in size is a [[meantone]] fifth, but four of them are not used to reach the 5th harmonic. Instead, [[14/13]], [[15/14]] and [[16/15]] are equated to 1/10 of an octave, and from this it derives its name. Not only the meantone fifth (flat 3/2) or fourth (sharp 4/3), but also the [[magic]] major third (flat 5/4) can be used as a generator.
 
There are three mappings for 11, 13 and 17-limit that are comparable in complexity and error: ''decic'' (10&50), ''splendecic'' (10e&50) and ''prodecic'' (10&50e). They share the comma list of the [[No-elevens subgroup temperaments|no-elevens subgroups]], tempering out 105/104, 170/169, 196/195 and 289/288. Decic adds 385/384, splendecic adds 4375/4376, and prodecic adds 441/440 to the 11-limit comma list.


== Temperament data ==
== Temperament data ==
Line 9: Line 11:
[[Comma list]]: 105/104, 144/143, 170/169, 196/195, 221/220
[[Comma list]]: 105/104, 144/143, 170/169, 196/195, 221/220


[[Mapping]]: [{{val| 10 16 23 28 35 37 41 }}, {{val| 0 1 -1 0 2 0 1 }}]
[[Mapping]]: [{{val| 10 16 23 28 35 37 41 }}, {{val| 0 -1 1 0 -2 0 -1 }}]


[[POTE generator]]s:  
[[POTE generator]]s:  
* 7-limit: ~3/2 = 698.69596
* 7-limit: ~15/14 = 120.00000, ~3/2 = 698.69596
* 11-limit: ~3/2 = 696.79119
* 11-limit: ~15/14 = 120.00000, ~3/2 = 696.79119
* 13-limit: ~3/2 = 696.99342
* 13-limit: ~14/13 = 120.00000, ~3/2 = 696.99342
* 17-limit: ~3/2 = 697.08527
* 17-limit: ~14/13 = 120.00000, ~3/2 = 697.08527


[[TOP tuning|TOP generator]]s:  
[[TOP tuning|TOP generator]]s:  
Line 24: Line 26:


[[Diamond monotone]] ranges:  
[[Diamond monotone]] ranges:  
* 7-odd-limit: ~3/2 = [680.00000, 720.00000] (34\60 to 6\10)
* 7-odd-limit: ~3/2 = [680.00000, 720.00000] (17\30 to 6\10)
* 9-odd-limit: ~3/2 = [696.00000, 720.00000] (29\50 to 6\10)
* 9-odd-limit: ~3/2 = [696.00000, 720.00000] (29\50 to 6\10)
* 11, 13, 15, and 17-odd-limit: ~3/2 = [696.00000, 700.00000] (29\50 to 35\60)
* 11, 13, 15, and 17-odd-limit: ~3/2 = [696.00000, 700.00000] (29\50 to 35\60)
Line 38: Line 40:
* 11, 13, 15, and 17-odd-limit: ~3/2 = [696.00000, 700.00000]
* 11, 13, 15, and 17-odd-limit: ~3/2 = [696.00000, 700.00000]


[[Optimal GPV sequence]]s:
[[Optimal ET sequence]]s:
* 7-limit: {{Vals| 10, 30b, 40, 50, 60, 110d, 170cdd }}
* 7-limit: {{Optimal ET sequence| 10, 30b, 40, 50, 60, 110d, 170cdd }}
* 11 and 13-limit: {{Vals| 10, 30be, 40, 50 }}
* 11 and 13-limit: {{Optimal ET sequence| 10, 30be, 40, 50 }}
* 17-limit: {{Vals| 10, 30beg, 40, 50 }}
* 17-limit: {{Optimal ET sequence| 10, 30beg, 40, 50 }}


[[Badness]]:  
[[Badness]]:  
Line 50: Line 52:
</div></div>
</div></div>


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">'''Splendecic temperament (10e&amp;50)'''</div>
<div class="mw-collapsible-content">
Subgroup: 2.3.5.7.11.13.17
[[Comma list]]: 105/104, 170/169, 196/195, 289/288, 375/374
[[Mapping]]: [{{val| 10 16 23 28 34 37 41 }}, {{val| 0 -1 1 0 3 0 -1 }}]
[[POTE generator]]s:
* 11-limit: ~15/14 = 120.00000, ~3/2 = 698.51792
* 13-limit: ~14/13 = 120.00000, ~3/2 = 698.36508
* 17-limit: ~14/13 = 120.00000, ~3/2 = 698.37473
[[TOP tuning|TOP generator]]s:
* 11-limit: ~15/14 = 120.18780, ~3/2 = 699.61110
* 13-limit: ~14/13 = 120.15710, ~3/2 = 699.27937
* 17-limit: ~14/13 = 120.15778, ~3/2 = 699.29299
[[Diamond monotone]] ranges:
* 11, 13, 15, and 17-odd-limit: ~3/2 = [696.00000, 700.00000] (29\50 to 35\60)
[[Diamond tradeoff]] ranges:
* 11, 13, and 15-odd-limit: ~3/2 = [693.12909, 702.51219]
* 17-odd-limit: ~3/2 = [693.12909, 704.95541]
Diamond monotone and tradeoff ranges:
* 11, 13, 15, and 17-odd-limit: ~3/2 = [696.00000, 700.00000]
[[Optimal ET sequence]]s:
* 11 and 13-limit: {{Optimal ET sequence| 10e, 30bee, 40e, 50, 60e, 110de }}
* 17-limit: {{Optimal ET sequence| 10e, 30beeg, 40e, 50, 60e, 110deg }}
[[Badness]]:
* 11-limit: 0.059802
* 13-limit: 0.037989
* 17-limit: 0.026050
</div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">'''Prodecic temperament (10&amp;50e)'''</div>
<div class="mw-collapsible-content">
Subgroup: 2.3.5.7.11.13.17
[[Comma list]]: 105/104, 154/153, 170/169, 196/195, 289/288
[[Mapping]]: [{{val| 10 16 23 28 35 37 41 }}, {{val| 0 -1 1 0 -3 0 -1 }}]
[[POTE generator]]s:
* 11-limit: ~15/14 = 120.00000, ~3/2 = 700.20964
* 13-limit: ~14/13 = 120.00000, ~3/2 = 700.50272
* 17-limit: ~14/13 = 120.00000, ~3/2 = 700.47431
[[TOP tuning|TOP generator]]s:
* 11-limit: ~15/14 = 120.20310, ~3/2 = 701.39473
* 13-limit: ~14/13 = 120.16631, ~3/2 = 701.47358
* 17-limit: ~14/13 = 120.15866, ~3/2 = 701.40045
[[Diamond monotone]] ranges:
* 11, 13, 15, and 17-odd-limit: ~3/2 = [700.00000, 702.85714] (35\60 to 41\70)
[[Diamond tradeoff]] ranges:
* 11, 13, 15, and 17-odd-limit: ~3/2 = [693.12909, 707.40794]
Diamond monotone and tradeoff ranges:
* 11, 13, 15, and 17-odd-limit: ~3/2 = [700.00000, 702.85714]
[[Optimal ET sequence]]s:
* 11, 13, and 17-limit: {{Optimal ET sequence| 10, 40ee, 50e, 60e }}
[[Badness]]:
* 11-limit: 0.066600
* 13-limit: 0.041788
* 17-limit: 0.027590
</div></div>
== Interval chains ==
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">'''Intervals of decic (10&amp;50)'''</div>
<div class="mw-collapsible-content">
{| class="wikitable"
|-
! colspan="2" | Generator
! | -3
! | -2
! | -1
! | 0
! | 1
! | 2
! | 3
|-
! rowspan="2" | Period 0
! | Cents*
| | 1131.256
| | 1154.171
| | 1177.085
| | 0.000
| | 22.915
| | 45.829
| | 68.744
|-
! | Ratios
| |
| |
| |
| | 1/1
| |
| | 40/39
| | 28/27, 25/24
|-
! rowspan="2" | Period 1
! | Cents*
| | 51.256
| | 74.171
| | 97.085
| | 120.000
| | 142.915
| | 165.829
| | 188.744
|-
! | Ratios
| | 36/35, 33/32
| | 21/20
| | 18/17, 17/16
| | 16/15, 15/14, 14/13
| | 13/12, 12/11
| |
| | 10/9
|-
! rowspan="2" | Period 2
! | Cents*
| | 171.256
| | 194.171
| | 217.085
| | 240.000
| | 262.915
| | 285.829
| | 308.744
|-
! | Ratios
| | 11/10
| | 9/8
| | 17/15
| | 8/7, 15/13
| | 7/6
| | 20/17, 13/11
| |
|-
! rowspan="2" | Period 3
! | Cents*
| | 291.256
| | 314.171
| | 337.085
| | 360.000
| | 382.915
| | 405.829
| | 428.744
|-
! | Ratios
| |
| | 6/5
| | 17/14
| | 11/9, 16/13, 21/17
| | 26/21, 5/4
| | 14/11
| |
|-
! rowspan="2" | Period 4
! | Cents*
| | 411.256
| | 434.171
| | 457.085
| | 480.000
| | 502.915
| | 525.829
| | 548.744
|-
! | Ratios
| |
| | 9/7
| | 22/17, 13/10, 17/13, 21/16
| |
| | 4/3
| | 15/11
| |
|-
! rowspan="2" | Period 5
! | Cents*
| | 531.256
| | 554.171
| | 577.085
| | 600.000
| | 622.915
| | 645.829
| | 668.744
|-
! | Ratios
| |
| | 11/8, 18/13
| | 7/5
| | 24/17, 17/12
| | 10/7
| | 13/9, 16/11
| |
|}
<nowiki>*</nowiki> in 17-limit POTE tuning
</div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">'''Intervals of splendecic (10e&amp;50)'''</div>
<div class="mw-collapsible-content">
{| class="wikitable"
|-
! colspan="2" | Generator
! | -3
! | -2
! | -1
! | 0
! | 1
! | 2
! | 3
|-
! rowspan="2" | Period 0
! | Cents*
| | 1135.124
| | 1156.749
| | 1178.375
| | 0.000
| | 21.625
| | 43.251
| | 64.876
|-
! | Ratios
| |
| |
| |
| | 1/1
| |
| | 40/39
| | 28/27, 25/24
|-
! rowspan="2" | Period 1
! | Cents*
| | 55.124
| | 76.749
| | 98.375
| | 120.000
| | 141.625
| | 163.251
| | 184.876
|-
! | Ratios
| | 36/35
| | 21/20
| | 18/17, 17/16
| | 16/15, 15/14, 14/13
| | 13/12
| | 11/10
| | 10/9
|-
! rowspan="2" | Period 2
! | Cents*
| | 175.124
| | 196.749
| | 218.375
| | 240.000
| | 261.625
| | 283.251
| | 304.876
|-
! | Ratios
| |
| | 9/8
| | 17/15
| | 8/7, 15/13
| | 7/6
| | 20/17
| |
|-
! rowspan="2" | Period 3
! | Cents*
| | 295.124
| | 316.749
| | 338.375
| | 360.000
| | 381.625
| | 403.251
| | 424.876
|-
! | Ratios
| | 13/11
| | 6/5
| | 17/14
| | 16/13, 21/17
| | 26/21, 5/4
| |
| |
|-
! rowspan="2" | Period 4
! | Cents*
| | 415.124
| | 436.749
| | 458.375
| | 480.000
| | 501.625
| | 523.251
| | 544.876
|-
! | Ratios
| | 14/11
| | 9/7
| | 13/10, 17/13, 21/16
| |
| | 4/3
| |
| | 11/8
|-
! rowspan="2" | Period 5
! | Cents*
| | 535.124
| | 556.749
| | 578.375
| | 600.000
| | 621.625
| | 643.251
| | 664.876
|-
! | Ratios
| | 15/11
| | 18/13
| | 7/5
| | 24/17, 17/12
| | 10/7
| | 13/9
| | 22/15
|}
<nowiki>*</nowiki> in 17-limit POTE tuning
</div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">'''Intervals of prodecic (10&amp;50e)'''</div>
<div class="mw-collapsible-content">
{| class="wikitable"
|-
! colspan="2" | Generator
! | -3
! | -2
! | -1
! | 0
! | 1
! | 2
! | 3
|-
! rowspan="2" | Period 0
! | Cents*
| | 1141.423
| | 1160.949
| | 1180.474
| | 0.000
| | 19.526
| | 39.051
| | 58.577
|-
! | Ratios
| |
| |
| |
| | 1/1
| |
| | 40/39
| | 28/27, 25/24
|-
! rowspan="2" | Period 1
! | Cents*
| | 61.423
| | 80.949
| | 100.474
| | 120.000
| | 139.526
| | 159.051
| | 178.577
|-
! | Ratios
| | 36/35
| | 22/21, 21/20
| | 18/17, 17/16
| | 16/15, 15/14, 14/13
| | 13/12
| | 12/11
| | 10/9
|-
! rowspan="2" | Period 2
! | Cents*
| | 181.423
| | 200.949
| | 220.474
| | 240.000
| | 259.526
| | 279.051
| | 298.577
|-
! | Ratios
| |
| | 9/8
| | 17/15
| | 8/7, 15/13
| | 7/6
| | 20/17
| | 13/11
|-
! rowspan="2" | Period 3
! | Cents*
| | 301.423
| | 320.949
| | 340.474
| | 360.000
| | 379.526
| | 399.051
| | 418.577
|-
! | Ratios
| |
| | 6/5
| | 17/14, 11/9
| | 16/13, 21/17
| | 26/21, 5/4
| |
| | 14/11
|-
! rowspan="2" | Period 4
! | Cents*
| | 421.423
| | 440.949
| | 460.474
| | 480.000
| | 499.526
| | 519.051
| | 538.577
|-
! | Ratios
| |
| | 9/7, 22/17
| | 13/10, 17/13, 21/16
| |
| | 4/3
| |
| | 15/11
|-
! rowspan="2" | Period 5
! | Cents*
| | 541.423
| | 560.949
| | 580.474
| | 600.000
| | 619.526
| | 639.051
| | 658.577
|-
! | Ratios
| | 11/8
| | 18/13
| | 7/5
| | 24/17, 17/12
| | 10/7
| | 13/9
| | 16/11
|}
<nowiki>*</nowiki> in 17-limit POTE tuning
</div></div>
== Scales ==
* [[Decic50]] - [[10L 40s]] scale
[[Category:Decic| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Marvel temperaments]]
[[Category:Cloudy clan]]
[[Category:Cloudy clan]]
[[Category:Marvel temperaments]]
[[Category:Decic| ]]<!-- main article -->
{{IoT}}
Retrieved from "https://en.xen.wiki/w/Decic"