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Porcupine extensions: Difference between revisions

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[[Porcupine]] has various [[extension]]s to the [[13-limit]].
{{Breadcrumb|Porcupine}}
[[Porcupine]] has various [[extension]]s to the [[13-limit]]. Adding the 13th harmonic to porcupine is not very simple, because 13 tends to fall in between the simple intervals produced by porcupine's generator. The extensions are:
* '''Porcupinefowl''' ({{nowrap| 15 & 22f }}) – tempering out 40/39, 55/54, 64/63, and 66/65;
* '''Porcupinefish''' ({{nowrap| 15 & 22 }}) – tempering out 55/54, 64/63, 91/90, and 100/99;
* '''Porkpie''' ({{nowrap| 15f & 22 }}) – tempering out 55/54, 64/63, 65/63, 100/99;
* '''Pourcup''' ({{nowrap| 15f & 22f }}) – tempering out 55/54, 64/63, 100/99, and 196/195.
 
Additionally, there are alternative extensions to prime 7:
* '''[[Opossum]]''' ({{nowrap| 8d & 15 }}) – tempering out 28/27, 40/39, 55/54, and 66/65.
* '''Porky''' ({{nowrap| 22 & 29 }}) – tempering out 55/54, 65/64, 91/90, and 100/99;
* '''Coendou''' ({{nowrap| 29 & 36ce }}) – tempering out 55/54, 65/64, 100/99, and 105/104.
 
Porcupinefowl maps [[13/8]] to −2 generator steps and conflates it with [[5/3]] and [[18/11]], tempering out [[40/39]]. This is where the generator, representing [[10/9]], [[11/10]], and [[12/11]], goes one step further to stand in for ~[[13/12]]. Porkpie maps 13/8 to +5 generator steps and conflates it with [[8/5]], tempering out [[65/64]]. The generator now represents ~[[14/13]]. Without optimization for the 13-limit, porcupinefowl sharpens the interval class of 13 by about 30{{cent}}, and porkpie flattens it by about 20{{cent}}.
 
The other pair of extensions are of higher complexity, but are well rewarded with better intonation. Porcupinefish's mapping of 13 is available at −17 generator steps. This equates the sharply tuned diatonic major third of porcupine with 13/10 along with 9/7, and requires a much more precise tuning of the porcupine generator to 161.5–163.5{{c}} to tune the 13th harmonic well. Pourcup's mapping of 13 is available at +20 generator steps. They unite in [[37edo]], which can be recommended as a tuning for both.
 
Prime 17 can be found at +8 generator steps, in which case −14 generator steps represent 18/17. This conflates 16/15 with 17/16, tempering out [[256/255]], and 15/14 with 18/17, tempering out [[85/84]]. It can also be found at −14 generator steps, in which case +8 generator steps represent 18/17. This conflates 17/16 with 15/14, tempering out [[120/119]], and 18/17 with 16/15, tempering out [[136/135]]. Both steps tend to be tuned between around 90 and 130{{c}}.
 
Prime 19 can be found at −13 generator steps (25/21, tempering out [[400/399]]), or more crudely at 2 generator steps (6/5, tempering out [[96/95]]).
 
Prime 23 can be found at 4 generator steps (tempering out 256/253) or −11 generator steps (tempering out 161/160). Both of these approximations are rather crude, but may be improved by varying the tuning of the generator. For a more precise (yet more complex) mapping, +26 steps is an option.
 
== Interval chain ==
In the following table, odd harmonics and subharmonics 1–15 are in '''bold'''.
 
{| class="wikitable center-1 right-2"
! rowspan="3" | #
! rowspan="3" | Cents*
! colspan="5" | Approximate ratios
|-
! rowspan="2" | 11-limit
! colspan="4" | 13-limit extensions
|-
! Porcupinefowl
! Porcupinefish
! Porkpie
! Pourcup
|-
| 0
| 0.0
| '''1/1'''
|
|
|
|
|-
| 1
| 162.8
| 10/9, 11/10, 12/11
| 13/12
|
| 14/13
|
|-
| 2
| 325.6
| 6/5, 11/9
| 13/11, '''16/13'''
|
| 26/21
|
|-
| 3
| 488.4
| '''4/3'''
| 13/10
|
|
|
|-
| 4
| 651.3
| '''16/11''', 22/15
| 13/9
|
|
|
|-
| 5
| 814.1
| '''8/5'''
| 21/13
|
| '''13/8'''
|
|-
| 6
| 976.9
| '''7/4''', '''16/9'''
| 26/15
|
|
|
|-
| 7
| 1139.7
| 48/25, 64/33, 160/81
| 52/27
| 25/13
| 39/20
|
|-
| 8
| 102.5
| '''16/15''', 21/20
| 14/13, 26/25
| 27/26
| 13/12
|
|-
| 9
| 265.3
| 7/6
|
| 15/13
| 13/11
|
|-
| 10
| 428.2
| 14/11
|
|
| 13/10
|
|-
| 11
| 591.0
| 7/5
|
| 18/13
| 13/9
|
|-
| 12
| 753.8
| 14/9
|
| 20/13
|
|
|-
| 13
| 916.6
| 42/25
|
| 22/13
| 26/15
|
|-
| 14
| 1079.4
| 28/15
|
| 24/13
| 52/27
| 13/7
|-
| 15
| 42.2
| 28/27, 49/48
|
| 40/39
| 26/25
|
|-
| 16
| 205.0
| 28/25
|
|
|
|
|-
| 17
| 367.9
| 49/40, 56/45
|
| '''16/13'''
|
| 26/21
|-
| 18
| 530.7
| 49/36
|
|
|
|
|-
| 19
| 693.5
| 49/33
|
|
|
|
|-
| 20
| 856.3
| 49/30
|
| 21/13
|
| '''13/8'''
|-
| 21
| 1019.1
| 49/27
|
|
|
|
|-
| 22
| 1181.9
| 49/25
|
|
|
| 39/20
|}
<nowiki/>* In 11-limit [[CWE tuning]], octave reduced


== Tuning spectrum ==
== Tuning spectrum ==
=== Tridecimal porcupine ===
=== Porcupinefowl ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|+ style="font-size: 105%;" | Tuning spectrum of 13-limit porcupine
|-
|-
! Edo<br>generator
! Edo<br>generator
Line 24: Line 245:
|  
|  
| 150.000
| 150.000
| Lower bound of 5-odd-limit diamond monotone
| 8d val, lower bound of 5-odd-limit diamond monotone
|-
|-
|  
|  
| 12/11
| 11/6
| 150.637
| 150.637
| Lower bound of 11-odd-limit diamond tradeoff
| Lower bound of 11-odd-limit diamond tradeoff
Line 37: Line 258:
|-
|-
|  
|  
| 6/5
| 5/3
| 157.821
| 157.821
| Lower bound of 5-, 7-, and 9-odd-limit diamond tradeoff
| Lower bound of 5-, 7-, and 9-odd-limit diamond tradeoff
Line 47: Line 268:
|-
|-
|  
|  
| 18/13
| 13/9
| 159.154
| 159.154
|  
|  
Line 54: Line 275:
|  
|  
| 160.000
| 160.000
| Lower bound of 7-, 9-, and 11-odd-limit diamond monotone
| Lower bound of 7-, 9-, 11-, and 13-odd-limit diamond monotone
|-
|-
|  
|  
| 8/7
| 7/4
| 161.471
| 161.471
|  
|  
|-
| 7\52
|
| 161.538
| 52bfff val
|-
|-
|  
|  
| 14/11
| 11/7
| 161.751
| 161.751
|  
|  
Line 74: Line 300:
|  
|  
| 162.162
| 162.162
|  
| 37ff val
|-
|-
|  
|  
Line 84: Line 310:
|  
|  
| 162.712
| 162.712
|
| 59fff val
|-
|-
|  
|  
Line 104: Line 330:
|  
|  
| 163.636
| 163.636
| Upper bound of 7-, 9-, and 11-odd-limit diamond monotone
| 22f val, upper bound of 7-, 9-, 11, and 13-odd-limit diamond monotone
|-
|-
|  
|  
Line 112: Line 338:
|-
|-
|  
|  
| 16/15
| 15/8
| 163.966
| 163.966
|
|-
| 7\51
|
| 164.706
|  
|  
|-
|-
Line 124: Line 345:
| 11/10
| 11/10
| 165.004
| 165.004
|
|-
| 4\29
|
| 165.517
|  
|  
|-
|-
Line 137: Line 353:
|-
|-
|  
|  
| 4/3
| 3/2
| 166.015
| 166.015
| Upper bound of 5- and 7-odd-limit diamond tradeoff
| Upper bound of 5- and 7-odd-limit diamond tradeoff
|-
|-
|  
|  
| 14/13
| 13/7
| 166.037
| 166.037
|  
|  
Line 157: Line 373:
|-
|-
|  
|  
| 16/13
| 13/8
| 179.736
| 179.736
|  
|  
|-
|-
|  
|  
| 10/9
| 9/5
| 182.404
| 182.404
| Upper bound of 9- and 11-odd-limit diamond tradeoff
| Upper bound of 9- and 11-odd-limit diamond tradeoff
Line 168: Line 384:


=== Porcupinefish ===
=== Porcupinefish ===
{| class="wikitable center-all"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
| 1\8
|  
|  
| 12/11
| 150.000
| 8dff val, lower bound of 5-odd-limit diamond monotone
|-
|
| 11/6
| 150.637
| 150.637
|  
|  
|-
|-
|  
|  
| 6/5
| 5/3
| 157.821
| 157.821
|  
|  
Line 188: Line 409:
|  
|  
| 160.000
| 160.000
|  
| Lower bound of 7-, 9-, 11-, and 13-odd-limit diamond monotone
|-
|-
|  
|  
| 18/13
| 13/9
| 160.307
| 160.307
|  
|  
Line 201: Line 422:
|-
|-
|  
|  
| 8/7
| 7/4
| 161.471
| 161.471
|  
|  
Line 210: Line 431:
|  
|  
|-
|-
| 7\52
|  
|  
| 14/11
| 161.538
| 52bf val
|-
|
| 11/7
| 161.751
| 161.751
|  
|  
Line 221: Line 447:
|-
|-
|  
|  
| 14/13
| 13/7
| 162.100
| 162.100
|  
|  
Line 233: Line 459:
|  
|  
| 162.162
| 162.162
|  
| Upper bound of 13-odd-limit diamond monotone
|-
|-
|  
|  
Line 241: Line 467:
|-
|-
|  
|  
| 16/13
| 13/8
| 162.322
| 162.322
|  
|  
Line 273: Line 499:
|  
|  
| 163.636
| 163.636
|  
| Upper bound of 7-, 9-, and 11-odd-limit diamond monotone
|-
|-
|  
|  
Line 281: Line 507:
|-
|-
|  
|  
| 16/15
| 15/8
| 163.966
| 163.966
|
|-
| 7\51
|
| 164.706
|  
|  
|-
|-
Line 293: Line 514:
| 11/10
| 11/10
| 165.004
| 165.004
|
|-
| 4\29
|
| 165.517
|  
|  
|-
|-
Line 306: Line 522:
|-
|-
|  
|  
| 4/3
| 3/2
| 166.015
| 166.015
|  
|  
|-
| 1\7
|
| 171.429
| 7f val, upper bound of 5-odd-limit diamond monotone
|-
|-
|  
|  
Line 316: Line 537:
|-
|-
|  
|  
| 10/9
| 9/5
| 182.404
| 182.404
|  
|  
Retrieved from "https://en.xen.wiki/w/Porcupine_extensions"