Porcupine extensions - Revision history

Overthink: /* Interval chain */ Replaced "Porcupine" with "Porcupinefowl"

2025-11-05T22:54:33Z

Interval chain: Replaced "Porcupine" with "Porcupinefowl"

← Older revision		Revision as of 22:54, 5 November 2025
Line 32:		Line 32:
	! colspan="4" \| 13-limit extensions		! colspan="4" \| 13-limit extensions
	\|-		\|-
	! ~~Porcupine~~		! Porcupinefowl
	! Porcupinefish		! Porcupinefish
	! Porkpie		! Porkpie

Overthink: /* Interval chain */ extended bold harmonics to 15, bolded 16/9

2025-11-05T22:53:31Z

Interval chain: extended bold harmonics to 15, bolded 16/9

← Older revision		Revision as of 22:53, 5 November 2025
Line 22:		Line 22:

	== 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 87:		Line 87:
	\| 6		\| 6
	\| 976.9		\| 976.9
	\| '''7/4''', 16/9		\| '''7/4''', '''16/9'''
	\| 26/15		\| 26/15
	\|		\|
Line 103:		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

FloraC: Cleanup

2025-08-06T12:00:45Z

Cleanup

← Older revision		Revision as of 12:00, 6 August 2025
Line 1:		Line 1:
	{{Breadcrumb}}		{{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:		[[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;		* '''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;		* '''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;		* '''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.		* '''Pourcup''' ({{nowrap\| 15f & 22f }}) – tempering out 55/54, 64/63, 100/99, and 196/195.

	Additionally, there are alternative extensions to prime 7:		Additionally, there are alternative extensions to prime 7:
	* '''[[Opossum]]''' ({{nowrap\|8d & 15}}) – tempering out 28/27, 40/39, 55/54, and 66/65.		* '''[[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;		* '''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.		* '''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}}.		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}}.

FloraC: FloraC moved page Porcupine/Extensions to Porcupine extensions over a redirect without leaving a redirect: Reversion for creating broken redirects with no clear gain

2025-08-06T11:59:25Z

FloraC moved page Porcupine/Extensions to Porcupine extensions over a redirect without leaving a redirect: Reversion for creating broken redirects with no clear gain

← Older revision	Revision as of 11:59, 6 August 2025
(No difference)

Lériendil: Text replacement - "Eigenmonzo
(unchanged-interval)" to "Unchanged interval
(eigenmonzo)"

2025-06-21T06:55:59Z

Text replacement - "Eigenmonzo<br>(unchanged-interval)" to "Unchanged interval<br>(eigenmonzo)"

← Older revision		Revision as of 06:55, 21 June 2025
Line 387:		Line 387:
	\|-		\|-
	! Edo<br>generator		! Edo<br>generator
	! ~~Eigenmonzo~~<br>(~~unchanged-interval~~)		! Unchanged interval<br>(eigenmonzo)
	! Generator (¢)		! Generator (¢)
	! Comments		! Comments

ArrowHead294 at 13:19, 7 June 2025

2025-06-07T13:19:33Z

Lériendil: reverted to this version, this follows the precedent of the page for miracle extensions which included an alt 11-limit extension, and additionally the main porcupine page makes the claim that alt 7-limit extensions are to be found here

2025-06-07T04:30:01Z

reverted to this version, this follows the precedent of the page for miracle extensions which included an alt 11-limit extension, and additionally the main porcupine page makes the claim that alt 7-limit extensions are to be found here

← Older revision		Revision as of 04:30, 7 June 2025
Line 1:		Line 1:
	{{Breadcrumb}}		{{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:		[[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;		* '''Porcupinefowl''' (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;		* '''Porcupinefish''' (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;		* '''Porkpie''' (15f & 22) – tempering out 55/54, 64/63, 65/63, and 100/99;
	* '''Pourcup''' (~~{{nowrap\|~~15f & 22f}}) – tempering out 55/54, 64/63, 100/99, and 196/195.		* '''Pourcup''' (15f & 22f) – tempering out 55/54, 64/63, 100/99, and 196/195.

			Additionally, there are alternative extensions to prime 7:
			* '''[[Opossum]]''' (8d & 15) - tempering out 28/27, 40/39, 55/54, and 66/65.
			* '''Porky''' (22 & 29) - tempering out 55/54, 65/64, 91/90, and 100/99;
			* '''Coendou''' (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}}~~.		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 cents, and porkpie flattens it by about 20.

	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.		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. 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 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 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 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.		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 ==

ArrowHead294 at 14:53, 6 June 2025

2025-06-06T14:53:47Z

← Older revision		Revision as of 14:53, 6 June 2025
Line 1:		Line 1:
	{{Breadcrumb~~\|Porcupine~~}}		{{Breadcrumb}}

	[[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:		[[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''' (15 & 22f) – tempering out 40/39, 55/54, 64/63, and 66/65;		* '''Porcupinefowl''' ({{nowrap\|15 & 22f}}) – tempering out 40/39, 55/54, 64/63, and 66/65;
	* '''Porcupinefish''' (15 & 22) – tempering out 55/54, 64/63, 91/90, and 100/99;		* '''Porcupinefish''' ({{nowrap\|15 & 22}}) – tempering out 55/54, 64/63, 91/90, and 100/99;
	* '''Porkpie''' (15f & 22) – tempering out 55/54, 64/63, 65/63, 100/99;		* '''Porkpie''' ({{nowrap\|15f & 22}}) – tempering out 55/54, 64/63, 65/63, 100/99;
	* '''Pourcup''' (15f & 22f) – tempering out 55/54, 64/63, 100/99, and 196/195.		* '''Pourcup''' ({{nowrap\|15f & 22f}}) – tempering out 55/54, 64/63, 100/99, and 196/195.

	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 ~~cents~~, and porkpie flattens it by about 20.		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 ~~cents~~ 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.		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 ~~cents~~.		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 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.		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 ==

FloraC: The reader is encouraged to use the family page for alt 7-limit extensions

2025-06-06T13:46:55Z

The reader is encouraged to use the family page for alt 7-limit extensions

← Older revision		Revision as of 13:46, 6 June 2025
Line 4:		Line 4:
	* '''Porcupinefowl''' (15 & 22f) – tempering out 40/39, 55/54, 64/63, and 66/65;		* '''Porcupinefowl''' (15 & 22f) – tempering out 40/39, 55/54, 64/63, and 66/65;
	* '''Porcupinefish''' (15 & 22) – tempering out 55/54, 64/63, 91/90, and 100/99;		* '''Porcupinefish''' (15 & 22) – tempering out 55/54, 64/63, 91/90, and 100/99;
	* '''Porkpie''' (15f & 22) – tempering out 55/54, 64/63, 65/63, ~~and~~ 100/99;		* '''Porkpie''' (15f & 22) – tempering out 55/54, 64/63, 65/63, 100/99;
	* '''Pourcup''' (15f & 22f) – tempering out 55/54, 64/63, 100/99, and 196/195.		* '''Pourcup''' (15f & 22f) – tempering out 55/54, 64/63, 100/99, and 196/195.

	~~Additionally, there are alternative extensions to prime 7:~~
	* '''[[Opossum]]''' (8d & 15) - tempering out 28/27, 40/39, 55/54, and 66/65.
	* '''Porky''' (22 & 29) - tempering out 55/54, 65/64, 91/90, and 100/99;
	* '''Coendou''' (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 cents, and porkpie flattens it by about 20.		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 cents, and porkpie flattens it by about 20.

Lériendil: added mention of 7-limit alternatives

2025-06-05T14:42:18Z

added mention of 7-limit alternatives

← Older revision		Revision as of 14:42, 5 June 2025
Line 4:		Line 4:
	* '''Porcupinefowl''' (15 & 22f) – tempering out 40/39, 55/54, 64/63, and 66/65;		* '''Porcupinefowl''' (15 & 22f) – tempering out 40/39, 55/54, 64/63, and 66/65;
	* '''Porcupinefish''' (15 & 22) – tempering out 55/54, 64/63, 91/90, and 100/99;		* '''Porcupinefish''' (15 & 22) – tempering out 55/54, 64/63, 91/90, and 100/99;
	* '''Porkpie''' (15f & 22) – tempering out 55/54, 64/63, 65/63, 100/99;		* '''Porkpie''' (15f & 22) – tempering out 55/54, 64/63, 65/63, and 100/99;
	* '''Pourcup''' (15f & 22f) – tempering out 55/54, 64/63, 100/99, and 196/195.		* '''Pourcup''' (15f & 22f) – tempering out 55/54, 64/63, 100/99, and 196/195.

			Additionally, there are alternative extensions to prime 7:
			* '''[[Opossum]]''' (8d & 15) - tempering out 28/27, 40/39, 55/54, and 66/65.
			* '''Porky''' (22 & 29) - tempering out 55/54, 65/64, 91/90, and 100/99;
			* '''Coendou''' (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 cents, and porkpie flattens it by about 20.		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 cents, and porkpie flattens it by about 20.

← Older revision		Revision as of 12:00, 6 August 2025
Line 1:		Line 1:
	{{Breadcrumb}}		{{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:		[[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;		* '''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;		* '''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;		* '''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.		* '''Pourcup''' ({{nowrap\| 15f & 22f }}) – tempering out 55/54, 64/63, 100/99, and 196/195.

	Additionally, there are alternative extensions to prime 7:		Additionally, there are alternative extensions to prime 7:
	* '''[[Opossum]]''' ({{nowrap\|8d & 15}}) – tempering out 28/27, 40/39, 55/54, and 66/65.		* '''[[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;		* '''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.		* '''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}}.		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}}.

← Older revision		Revision as of 13:19, 7 June 2025
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	{{Breadcrumb~~\|Porcupine~~}}		{{Breadcrumb}}

	[[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:		[[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''' (15 & 22f) – tempering out 40/39, 55/54, 64/63, and 66/65;		* '''Porcupinefowl''' ({{nowrap\|15 & 22f}}) – tempering out 40/39, 55/54, 64/63, and 66/65;
	* '''Porcupinefish''' (15 & 22) – tempering out 55/54, 64/63, 91/90, and 100/99;		* '''Porcupinefish''' ({{nowrap\|15 & 22}}) – tempering out 55/54, 64/63, 91/90, and 100/99;
	* '''Porkpie''' (15f & 22) – tempering out 55/54, 64/63, 65/63, ~~and~~ 100/99;		* '''Porkpie''' ({{nowrap\|15f & 22}}) – tempering out 55/54, 64/63, 65/63, 100/99;
	* '''Pourcup''' (15f & 22f) – tempering out 55/54, 64/63, 100/99, and 196/195.		* '''Pourcup''' ({{nowrap\|15f & 22f}}) – tempering out 55/54, 64/63, 100/99, and 196/195.

	Additionally, there are alternative extensions to prime 7:		Additionally, there are alternative extensions to prime 7:
	* '''[[Opossum]]''' (8d & 15) - tempering out 28/27, 40/39, 55/54, and 66/65.		* '''[[Opossum]]''' ({{nowrap\|8d & 15}}) – tempering out 28/27, 40/39, 55/54, and 66/65.
	* '''Porky''' (22 & 29) - tempering out 55/54, 65/64, 91/90, and 100/99;		* '''Porky''' ({{nowrap\|22 & 29}}) – tempering out 55/54, 65/64, 91/90, and 100/99;
	* '''Coendou''' (29 & 36ce) - tempering out 55/54, 65/64, 100/99, and 105/104.		* '''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 ~~cents~~, and porkpie flattens it by about 20.		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 ~~cents~~ 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.		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 ~~cents~~.		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 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.		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 ==