Porcupine/Chords - Revision history

Overthink: complete conflations

2026-02-18T23:50:35Z

complete conflations

← Older revision		Revision as of 23:50, 18 February 2026
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	Below are listed the [[15-odd-limit]] [[dyadic chord]]s of [[11-limit]] [[porcupine\|porcupine temperament]] that do not have generator steps 7 or 13 as [[dyad]]s. Typing the chords requires consideration of the fact that porcupine conflates [[10/9]], [[11/10]] and [[12/11]] and ~~also~~ [[16/9]] ~~and~~ [[7/4]]. If a [[transversal]] can be found which shows the chord to be essentially just, that transversal is listed along with a typing as otonal, utonal, or ambitonal. If a chord is essentially tempered, the chord is analyzed in terms of the transversals 11/10, 6/5, 16/11 and 7/4. Chords that require only [[64/63]] tempering are marked [[archytas chords\|archytas]], by [[100/99]] [[ptolemismic chords\|ptolemismic]], by [[121/120]] [[biyatismic chords\|biyatismic]], by [[176/175]] [[valinorsmic chords\|valinorsmic]], and by [[385/384]] [[keenanismic chords\|keenanismic]]. Chords that require any two of 64/63, 100/99 and 176/175 tempering are marked [[ares chords\|ares]], that require 100/99 and 385/384 tempering are marked [[keemic chords\|keemic]], and that require any two of 121/120, 176/175 and 385/384 are marked [[zeus chords\|zeus]]. Chords that receive tempering by three independent commas above are labeled porcupine.		Below are listed the [[15-odd-limit]] [[dyadic chord]]s of [[11-limit]] [[porcupine\|porcupine temperament]] that do not have generator steps 7 or 13 as [[dyad]]s. Typing the chords requires consideration of the fact that porcupine conflates [[10/9]], [[11/10]] and [[12/11]], [[11/9]] with [[6/5]], [[22/15]] with [[16/11]], and [[16/9]] with [[7/4]]. If a [[transversal]] can be found which shows the chord to be essentially just, that transversal is listed along with a typing as otonal, utonal, or ambitonal. If a chord is essentially tempered, the chord is analyzed in terms of the transversals 11/10, 6/5, 16/11 and 7/4. Chords that require only [[64/63]] tempering are marked [[archytas chords\|archytas]], by [[100/99]] [[ptolemismic chords\|ptolemismic]], by [[121/120]] [[biyatismic chords\|biyatismic]], by [[176/175]] [[valinorsmic chords\|valinorsmic]], and by [[385/384]] [[keenanismic chords\|keenanismic]]. Chords that require any two of 64/63, 100/99 and 176/175 tempering are marked [[ares chords\|ares]], that require 100/99 and 385/384 tempering are marked [[keemic chords\|keemic]], and that require any two of 121/120, 176/175 and 385/384 are marked [[zeus chords\|zeus]]. Chords that receive tempering by three independent commas above are labeled porcupine.

	The transversal is in generator order. This is useful because it tells how common the chords are: For instance, a chord that appears on the sixth generation will appear exactly once in Porcupine[7], twice in Porcupine[8], and nine times in Porcupine[15].		The transversal is in generator order. This is useful because it tells how common the chords are: For instance, a chord that appears on the sixth generation will appear exactly once in Porcupine[7], twice in Porcupine[8], and nine times in Porcupine[15].

FloraC: A little cleanup

2025-10-12T11:07:18Z

A little cleanup

← Older revision		Revision as of 11:07, 12 October 2025
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	Below are listed the [[15-odd-limit]] [[dyadic chord]]s of [[11-limit]] [[porcupine\|porcupine temperament]] that do not have generator steps 7 or 13 as [[dyad]]s. Typing the chords requires consideration of the fact that porcupine conflates [[10/9]], [[11/10]] and [[12/11]] and also [[16/9]] and [[7/4]]. If a [[transversal]] can be found which shows the chord to be essentially just, that transversal is listed along with a typing as otonal, utonal, or ambitonal. If a chord is essentially tempered, the chord is analyzed in terms of the transversals 11/10, 6/5, 16/11 and 7/4. Chords that require only [[64/63]] tempering are marked [[archytas chords\|archytas]], by [[100/99]] [[ptolemismic chords\|ptolemismic]], by [[121/120]] [[biyatismic chords\|biyatismic]], by [[176/175]] [[valinorsmic chords\|valinorsmic]], and by [[385/384]] [[keenanismic chords\|keenanismic]]. Chords that require 64/63 and 176/175 tempering are marked ares, 100/99 and 385/384 ~~tempered chords~~ are [[keemic chords\|keemic]], and 176/175 and 385/384 ~~tempered chords~~ are marked zeus. Chords that receive tempering by three independent commas above are labeled porcupine.		Below are listed the [[15-odd-limit]] [[dyadic chord]]s of [[11-limit]] [[porcupine\|porcupine temperament]] that do not have generator steps 7 or 13 as [[dyad]]s. Typing the chords requires consideration of the fact that porcupine conflates [[10/9]], [[11/10]] and [[12/11]] and also [[16/9]] and [[7/4]]. If a [[transversal]] can be found which shows the chord to be essentially just, that transversal is listed along with a typing as otonal, utonal, or ambitonal. If a chord is essentially tempered, the chord is analyzed in terms of the transversals 11/10, 6/5, 16/11 and 7/4. Chords that require only [[64/63]] tempering are marked [[archytas chords\|archytas]], by [[100/99]] [[ptolemismic chords\|ptolemismic]], by [[121/120]] [[biyatismic chords\|biyatismic]], by [[176/175]] [[valinorsmic chords\|valinorsmic]], and by [[385/384]] [[keenanismic chords\|keenanismic]]. Chords that require any two of 64/63, 100/99 and 176/175 tempering are marked [[ares chords\|ares]], that require 100/99 and 385/384 tempering are marked [[keemic chords\|keemic]], and that require any two of 121/120, 176/175 and 385/384 are marked [[zeus chords\|zeus]]. Chords that receive tempering by three independent commas above are labeled porcupine.

	The transversal is in generator order. This is useful because it tells how common the chords are: For instance, a chord that appears on the sixth generation will appear exactly once in Porcupine[7], twice in ~~porcupine~~[8], and nine times in Porcupine[15].		The transversal is in generator order. This is useful because it tells how common the chords are: For instance, a chord that appears on the sixth generation will appear exactly once in Porcupine[7], twice in Porcupine[8], and nine times in Porcupine[15].

	The "As generated" column takes the intervals that were generated and places them in size order. The 1st and 2nd inversion (and so on) columns show the inversions of those generated tones. Note that this gives different results than you might be used to: the major chord (1/1–5/4–3/2, or 4:5:6) is the second inversion of the generated 0–2–5 chord.		The "As generated" column takes the intervals that were generated and places them in size order. The 1st and 2nd inversion (and so on) columns show the inversions of those generated tones. Note that this gives different results than you might be used to: the major chord (1–5/4–3/2, or 4:5:6) is the second inversion of the generated 0–2–5 chord.

	Though we are used to thinking of 4:5:6 as the definitive "major chord", with all inversions coming from that, there is nothing definitive about calling these lists below "chord" or "inversion". That is just the way the generators came out.		Though we are used to thinking of 4:5:6 as the definitive "major chord", with all inversions coming from that, there is nothing definitive about calling these lists below "chord" or "inversion". That is just the way the generators came out.

FloraC: Temp rename followup

2025-10-12T11:04:46Z

Temp rename followup

← Older revision		Revision as of 11:04, 12 October 2025
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	Below are listed the [[15-odd-limit]] [[dyadic chord]]s of [[11-limit]] [[porcupine\|porcupine temperament]] that do not have generator steps 7 or 13 as [[dyad]]s. Typing the chords requires consideration of the fact that porcupine conflates [[10/9]], [[11/10]] and [[12/11]] and also [[16/9]] and [[7/4]]. If a [[transversal]] can be found which shows the chord to be essentially just, that transversal is listed along with a typing as otonal, utonal, or ambitonal. If a chord is essentially tempered, the chord is analyzed in terms of the transversals 11/10, 6/5, 16/11 and 7/4. Chords that require only [[64/63]] tempering are marked [[archytas chords\|archytas]], by [[100/99]] [[ptolemismic chords\|ptolemismic]], by [[121/120]] [[biyatismic chords\|biyatismic]], by [[176/175]] [[valinorsmic chords\|valinorsmic]], and by [[385/384]] [[keenanismic chords\|keenanismic]]. Chords that require 64/63 and 176/175 tempering are marked ares, 100/99 and 385/384 tempered chords are [[~~supermagic~~ chords\|~~supermagic~~]], and 176/175 and 385/384 tempered chords are marked zeus. Chords that receive tempering by three independent commas above are labeled porcupine.		Below are listed the [[15-odd-limit]] [[dyadic chord]]s of [[11-limit]] [[porcupine\|porcupine temperament]] that do not have generator steps 7 or 13 as [[dyad]]s. Typing the chords requires consideration of the fact that porcupine conflates [[10/9]], [[11/10]] and [[12/11]] and also [[16/9]] and [[7/4]]. If a [[transversal]] can be found which shows the chord to be essentially just, that transversal is listed along with a typing as otonal, utonal, or ambitonal. If a chord is essentially tempered, the chord is analyzed in terms of the transversals 11/10, 6/5, 16/11 and 7/4. Chords that require only [[64/63]] tempering are marked [[archytas chords\|archytas]], by [[100/99]] [[ptolemismic chords\|ptolemismic]], by [[121/120]] [[biyatismic chords\|biyatismic]], by [[176/175]] [[valinorsmic chords\|valinorsmic]], and by [[385/384]] [[keenanismic chords\|keenanismic]]. Chords that require 64/63 and 176/175 tempering are marked ares, 100/99 and 385/384 tempered chords are [[keemic chords\|keemic]], and 176/175 and 385/384 tempered chords are marked zeus. Chords that receive tempering by three independent commas above are labeled porcupine.

	The transversal is in generator order. This is useful because it tells how common the chords are: For instance, a chord that appears on the sixth generation will appear exactly once in Porcupine[7], twice in porcupine[8], and nine times in Porcupine[15].		The transversal is in generator order. This is useful because it tells how common the chords are: For instance, a chord that appears on the sixth generation will appear exactly once in Porcupine[7], twice in porcupine[8], and nine times in Porcupine[15].
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	\| 0-2-4-6		\| 0-2-4-6
	\| 1-6/5-16/11-7/4		\| 1-6/5-16/11-7/4
	\| ~~supermagic~~		\| keemic
	\| '''1/1-6/5-16/11-7/4'''		\| '''1/1-6/5-16/11-7/4'''
	\| 1/1-6/5-16/11-5/3		\| 1/1-6/5-16/11-5/3
Line 730:		Line 730:
	\| 0-2-3-4-6		\| 0-2-3-4-6
	\| 1-6/5-4/3-16/11-16/9		\| 1-6/5-4/3-16/11-16/9
	\| ~~supermagic~~		\| keemic
	\| C^m,7,11(v5) ''or''<br>C4^7v9 ''or'' C^4v6,9		\| C^m,7,11(v5) ''or''<br>C4^7v9 ''or'' C^4v6,9
	\|-		\|-

Lériendil: Lériendil moved page Chords of porcupine to Porcupine/Chords

2025-06-22T21:51:01Z

Lériendil moved page Chords of porcupine to Porcupine/Chords

← Older revision	Revision as of 21:51, 22 June 2025
(No difference)

TallKite: updated EI to EU

2024-12-29T00:21:50Z

updated EI to EU

← Older revision		Revision as of 00:21, 29 December 2024
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	Though we are used to thinking of 4:5:6 as the definitive "major chord", with all inversions coming from that, there is nothing definitive about calling these lists below "chord" or "inversion". That is just the way the generators came out.		Though we are used to thinking of 4:5:6 as the definitive "major chord", with all inversions coming from that, there is nothing definitive about calling these lists below "chord" or "inversion". That is just the way the generators came out.

	The '''bolded''' inversions are named using [[ups and downs]] as described on the [[Pergen]] page. The pergen is (P8, P4/3) third-of-a-4th, #7 in the [http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf notation guide for rank-2 pergens]. One up is -7 generators, octave-reduced, which is a third-sharp. Thus ^<sup>3</sup>C = C# and the enharmonic ~~interval~~ is v<sup>3</sup>A1. The generator is vM2 = 167¢ - ''c''/3, where ''c'' is the amount in cents the tempered fifth exceeds 700¢. ^1 = 33¢ + 2.33''c''. In 22edo, ^1 = 1\22 = 54.5¢.		The '''bolded''' inversions are named using [[ups and downs]] as described on the [[Pergen]] page. The pergen is (P8, P4/3) third-of-a-4th, #7 in the [http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf notation guide for rank-2 pergens]. One up is -7 generators, octave-reduced, which is a third-sharp. Thus ^<sup>3</sup>C = C# and the [[enharmonic unison]] is v<sup>3</sup>A1. The generator is vM2 = 167¢ - ''c''/3, where ''c'' is the amount in cents the tempered fifth exceeds 700¢. ^1 = 33¢ + 2.33''c''. In 22edo, ^1 = 1\22 = 54.5¢.

	In porcupine, 5/4 = vM3, 7/4 = m7 and 11/8 = ^4. Thus ^1 equals ~81/80 and ~33/32. This may not be true for other (P8, P4/3) temperaments. So the ratios in the table below are specific to Porcupine, but the chord names apply to any (P8, P4/3) temperament.		In porcupine, 5/4 = vM3, 7/4 = m7 and 11/8 = ^4. Thus ^1 equals ~81/80 and ~33/32. This may not be true for other (P8, P4/3) temperaments. So the ratios in the table below are specific to Porcupine, but the chord names apply to any (P8, P4/3) temperament.

FloraC: Both tetrads and pentads are incomplete

2024-12-17T14:40:00Z

Both tetrads and pentads are incomplete

← Older revision		Revision as of 14:40, 17 December 2024
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	\| ^Cb		\| ^Cb
	\|}		\|}
	{{Todo\|inline=1\|complete table}}		{{Todo\|inline=1\|complete table\|research\|comment=Both tetrads and pentads are incomplete. Add the missing chords.}}

	== Triads ==		== Triads ==
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	== Pentads ==		== Pentads ==
	~~{{Todo\|inline=1\| research \|comment=investigate what's missing there }}~~

	{\| class="wikitable"		{\| class="wikitable"
	\|-		\|-

FloraC: Correction: 15-odd-limit. Fill in the tables

2024-12-17T14:36:38Z

Correction: 15-odd-limit. Fill in the tables

← Older revision		Revision as of 14:36, 17 December 2024
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	Below are listed the [[11-odd-limit]] [[dyadic chord]]s of [[11-limit]] [[porcupine\|porcupine temperament]] that do not have generator steps 7 or 13 as [[dyad]]s. Typing the chords requires consideration of the fact that porcupine conflates [[10/9]], [[11/10]] and [[12/11]] and also [[16/9]] and [[7/4]]. If a [[transversal]] can be found which shows the chord to be essentially just, that transversal is listed along with a typing as otonal, utonal, or ambitonal. If a chord is essentially tempered, the chord is analyzed in terms of the transversals 11/10, 6/5, 16/11 and 7/4. Chords that require only [[64/63]] tempering are marked [[archytas chords\|archytas]], by [[100/99]] [[ptolemismic chords\|ptolemismic]], by [[121/120]] [[biyatismic chords\|biyatismic]], by [[176/175]] [[valinorsmic chords\|valinorsmic]], and by [[385/384]] [[keenanismic chords\|keenanismic]]. Chords that require 64/63 and 176/175 tempering are marked ares, 100/99 and 385/384 tempered chords are [[supermagic chords\|supermagic]], and 176/175 and 385/384 tempered chords are marked zeus. Chords that receive tempering by three independent commas above are labeled porcupine.		Below are listed the [[15-odd-limit]] [[dyadic chord]]s of [[11-limit]] [[porcupine\|porcupine temperament]] that do not have generator steps 7 or 13 as [[dyad]]s. Typing the chords requires consideration of the fact that porcupine conflates [[10/9]], [[11/10]] and [[12/11]] and also [[16/9]] and [[7/4]]. If a [[transversal]] can be found which shows the chord to be essentially just, that transversal is listed along with a typing as otonal, utonal, or ambitonal. If a chord is essentially tempered, the chord is analyzed in terms of the transversals 11/10, 6/5, 16/11 and 7/4. Chords that require only [[64/63]] tempering are marked [[archytas chords\|archytas]], by [[100/99]] [[ptolemismic chords\|ptolemismic]], by [[121/120]] [[biyatismic chords\|biyatismic]], by [[176/175]] [[valinorsmic chords\|valinorsmic]], and by [[385/384]] [[keenanismic chords\|keenanismic]]. Chords that require 64/63 and 176/175 tempering are marked ares, 100/99 and 385/384 tempered chords are [[supermagic chords\|supermagic]], and 176/175 and 385/384 tempered chords are marked zeus. Chords that receive tempering by three independent commas above are labeled porcupine.

	The transversal is in generator order. This is useful because it tells how common the chords are: For instance, a chord that appears on the sixth generation will appear exactly once in ~~porcupine~~[7], twice in porcupine[8], and nine times in ~~porcupine~~[15].		The transversal is in generator order. This is useful because it tells how common the chords are: For instance, a chord that appears on the sixth generation will appear exactly once in Porcupine[7], twice in porcupine[8], and nine times in Porcupine[15].

	The "As generated" column takes the intervals that were generated and places them in size order. The 1st and 2nd inversion (and so on) columns show the inversions of those generated tones. Note that this gives different results than you might be used to: the major chord (1/1–5/4–3/2, or 4:5:6) is the second inversion of the generated 0–2–5 chord.		The "As generated" column takes the intervals that were generated and places them in size order. The 1st and 2nd inversion (and so on) columns show the inversions of those generated tones. Note that this gives different results than you might be used to: the major chord (1/1–5/4–3/2, or 4:5:6) is the second inversion of the generated 0–2–5 chord.
Line 671:		Line 671:
	\| 0-2-4-5		\| 0-2-4-5
	\| 1-6/5-16/11-8/5		\| 1-6/5-16/11-8/5
	\|		\| ptolemismic
	\| 1/1-6/5-16/11-8/5		\| 1/1-6/5-16/11-8/5
	\|		\| 1/1-6/5-4/3-5/3
	\|		\| 1/1-11/10-11/8-5/3
	\| '''1/1-5/4-3/2-9/5'''		\| '''1/1-5/4-3/2-9/5'''
	\| Cv^7		\| Cv^7
Line 716:		Line 716:

	== Pentads ==		== Pentads ==
	{{~~todo~~\|inline=1\| research \|comment=investigate what's missing there }}		{{Todo\|inline=1\| research \|comment=investigate what's missing there }}

	{\| class="wikitable"		{\| class="wikitable"
Line 724:		Line 724:
	! Type		! Type
	! Name		! Name
	\|-
	\|
	\|
	\|
	\|
	\|-		\|-
	\| 0-1-2-3-6		\| 0-1-2-3-6
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	\|-		\|-
	\| 0-2-3-4-6		\| 0-2-3-4-6
	\|		\| 1-6/5-4/3-16/11-16/9
	\|		\| supermagic
	\| C^m,7,11(v5) ''or''<br>C4^7v9 ''or'' C^4v6,9		\| C^m,7,11(v5) ''or''<br>C4^7v9 ''or'' C^4v6,9
	\|-		\|-
	\| 0-3-4-5-6		\| 0-3-4-5-6
	\|		\| 1-4/3-16/11-8/5-16/9
	\|		\| utonal
	\| C^mv9,11		\| C^mv9,11
	\|-		\|-

FloraC: Clarify that the scope of this article is currently the 11-odd-limit. Correct a ratio. Style

2024-12-17T14:04:18Z

Clarify that the scope of this article is currently the 11-odd-limit. Correct a ratio. Style

Show changes

TallKite: added more chord names

2024-12-17T08:38:52Z

added more chord names

Show changes

TallKite: added more chord names, added C as the chord root

2024-12-17T06:08:37Z

added more chord names, added C as the chord root

Show changes