Xenharmonic Wiki

Porcupine extensions: Difference between revisions

← Older edit
VisualWikitext
Review 17-limit extension
Overthink (talk | contribs)
autopatrolled
2,790 edits
→Interval chain: Replaced "Porcupine" with "Porcupinefowl"
 
(17 intermediate revisions by 5 users not shown)
Line 1: Line 1:
{{Breadcrumb|Porcupine}}
{{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.


[[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:
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}}.  


* '''Tridecimal porcupine''' (15 & 22f) – tempering out 40/39, 55/54, 64/63, and 66/65;
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.  
* '''Porkpie''' (15f & 22) – tempering out 55/54, 64/63, 65/63, 100/99;
* '''Porcupinefish''' (15 & 22) – tempering out 55/54, 64/63, 91/90, and 100/99;
* '''Porcup''' (15f & 22f) – tempering out 55/54, 64/63, 100/99, and 196/195.  


Tridecimal porcupine 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, tridecimal porcupine sharpens the interval class of 13 by about 30 cents, and porkpie flattens it by about 20.  
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}}.  


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 cents to tune the 13th harmonic well. Porcup's mapping of 13 is available at +20 generator steps. They unite in [[37edo]], which can be recommended as a tuning for both.  
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 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 cents.  
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 ==
== Interval chain ==
In the following table, odd harmonics and subharmonics 1–13 are in '''bold'''.  
In the following table, odd harmonics and subharmonics 1–15 are in '''bold'''.  


{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
Line 25: Line 32:
! colspan="4" | 13-limit extensions
! colspan="4" | 13-limit extensions
|-
|-
! Porcupine
! Porcupinefowl
! Porcupinefish
! Porcupinefish
! Porkpie
! Porkpie
! Porcup
! Pourcup
|-
|-
| 0
| 0
Line 80: Line 87:
| 6
| 6
| 976.9
| 976.9
| '''7/4''', 16/9
| '''7/4''', '''16/9'''
| 26/15
| 26/15
|  
|  
Line 96: Line 103:
| 8
| 8
| 102.5
| 102.5
| 16/15, 21/20
| '''16/15''', 21/20
| 14/13, 26/25
| 14/13, 26/25
| 27/26
| 27/26
Line 217: Line 224:


== 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 239: 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 252: 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 262: Line 268:
|-
|-
|  
|  
| 18/13
| 13/9
| 159.154
| 159.154
|  
|  
Line 269: 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
|  
|  
| 14/11
| 161.538
| 52bfff val
|-
|
| 11/7
| 161.751
| 161.751
|  
|  
Line 289: Line 300:
|  
|  
| 162.162
| 162.162
|  
| 37ff val
|-
|-
|  
|  
Line 299: Line 310:
|  
|  
| 162.712
| 162.712
|
| 59fff val
|-
|-
|  
|  
Line 319: 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 327: Line 338:
|-
|-
|  
|  
| 16/15
| 15/8
| 163.966
| 163.966
|
|-
| 7\51
|
| 164.706
|  
|  
|-
|-
Line 339: Line 345:
| 11/10
| 11/10
| 165.004
| 165.004
|
|-
| 4\29
|
| 165.517
|  
|  
|-
|-
Line 352: 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 372: 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 383: 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
|
| 150.000
| 8dff val, lower bound of 5-odd-limit diamond monotone
|-
|-
|  
|  
| 12/11
| 11/6
| 150.637
| 150.637
|  
|  
|-
|-
|  
|  
| 6/5
| 5/3
| 157.821
| 157.821
|  
|  
Line 403: 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 416: Line 422:
|-
|-
|  
|  
| 8/7
| 7/4
| 161.471
| 161.471
|  
|  
Line 425: Line 431:
|  
|  
|-
|-
| 7\52
|  
|  
| 14/11
| 161.538
| 52bf val
|-
|
| 11/7
| 161.751
| 161.751
|  
|  
Line 436: Line 447:
|-
|-
|  
|  
| 14/13
| 13/7
| 162.100
| 162.100
|  
|  
Line 448: Line 459:
|  
|  
| 162.162
| 162.162
|  
| Upper bound of 13-odd-limit diamond monotone
|-
|-
|  
|  
Line 456: Line 467:
|-
|-
|  
|  
| 16/13
| 13/8
| 162.322
| 162.322
|  
|  
Line 488: Line 499:
|  
|  
| 163.636
| 163.636
|  
| Upper bound of 7-, 9-, and 11-odd-limit diamond monotone
|-
|-
|  
|  
Line 496: Line 507:
|-
|-
|  
|  
| 16/15
| 15/8
| 163.966
| 163.966
|
|-
| 7\51
|
| 164.706
|  
|  
|-
|-
Line 508: Line 514:
| 11/10
| 11/10
| 165.004
| 165.004
|
|-
| 4\29
|
| 165.517
|  
|  
|-
|-
Line 521: 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 531: Line 537:
|-
|-
|  
|  
| 10/9
| 9/5
| 182.404
| 182.404
|  
|  
Retrieved from "https://en.xen.wiki/w/Porcupine_extensions"