Cassandra triads - Revision history

VectorGraphics at 21:02, 20 April 2025

2025-04-20T21:02:14Z

← Older revision		Revision as of 21:02, 20 April 2025
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	The '''cassandra triads''' are triads in which the two step sizes are (tempered versions of) [[13/11]] and [[14/11]] and the total span is a tempered [[3/2\|perfect fifth]]. There are two of them, ~~one of which is~~ intermediate between major and supermajor, and ~~the other~~ intermediate between minor and subminor.		The '''cassandra triads''' or neogothic triads are triads in which the two step sizes are (tempered versions of) [[13/11]] and [[14/11]] and the total span is a tempered [[3/2\|perfect fifth]]. There are two of them, a major triad intermediate between classical major and supermajor, and likewise a minor triad intermediate between classical minor and subminor.

	This tempers out the [[364/363]], which is one of the [[pentacircle comma]]s. Cassandra triads appear in [[17edo]], and even more closely in [[29edo]]. Cassandra chords are closely related to the 19-limit rootsubminor, rootminor, rootmajor, and roostsupermajor triads; in fact, a Cassandra minor triad could be considered to function as both rootsubminor ''and'' as rootminor, and similarly the cassandra major chord functions as both rootmajor and rootsupermajor.		This tempers out the [[364/363]], which is one of the [[pentacircle comma]]s. Cassandra triads appear in [[17edo]], and even more closely in [[29edo]]. Cassandra chords are closely related to the 19-limit rootsubminor, rootminor, rootmajor, and roostsupermajor triads; in fact, a Cassandra minor triad could be considered to function as both rootsubminor ''and'' as rootminor, and similarly the cassandra major chord functions as both rootmajor and rootsupermajor.
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	== Consonance of cassandra triads ==		== Consonance of cassandra triads ==

			=== JI-based explanation ===
	The 13:11 and 14:11 dyads both reside near local [[harmonic entropy]] maxima due to falling almost halfway between a septimal and a pental consonance. This, as well as the fact that the triadic just approximations are 242:286:364, do not make the Cassandra triads sound appealing as consonances.		The 13:11 and 14:11 dyads both reside near local [[harmonic entropy]] maxima due to falling almost halfway between a septimal and a pental consonance. This, as well as the fact that the triadic just approximations are 242:286:364, do not make the Cassandra triads sound appealing as consonances.

FloraC: Use third-person tone and improve linking

2024-04-11T13:49:11Z

Use third-person tone and improve linking

← Older revision		Revision as of 13:49, 11 April 2024
Line 1:		Line 1:
	The '''cassandra triads''' are triads in which the two step sizes are (tempered versions of) 13:11 and 14:11 and the total span is a tempered perfect fifth. There are two of them, one of which is intermediate between major and supermajor, and the other intermediate between minor and subminor.		The '''cassandra triads''' are triads in which the two step sizes are (tempered versions of) [[13/11]] and [[14/11]] and the total span is a tempered [[3/2\|perfect fifth]]. There are two of them, one of which is intermediate between major and supermajor, and the other intermediate between minor and subminor.

	This tempers out the 364:363, which is one of the [[pentacircle comma]]s. Cassandra triads appear in [[17edo]], and even more closely in [[29edo]]. Cassandra chords are closely related to the 19-limit rootsubminor, rootminor, rootmajor, and roostsupermajor triads; in fact, a Cassandra minor triad could be considered to function as both rootsubminor ''and'' as rootminor, and similarly the ~~Cassandra~~ major chord functions as both rootmajor and rootsupermajor.		This tempers out the [[364/363]], which is one of the [[pentacircle comma]]s. Cassandra triads appear in [[17edo]], and even more closely in [[29edo]]. Cassandra chords are closely related to the 19-limit rootsubminor, rootminor, rootmajor, and roostsupermajor triads; in fact, a Cassandra minor triad could be considered to function as both rootsubminor ''and'' as rootminor, and similarly the cassandra major chord functions as both rootmajor and rootsupermajor.

	The name could have been related to the [[cassandra]] temperament before the discovery that ''cassandra'' and ''andromeda'' were named wrong way around and were later corrected.		The name could have been related to the [[cassandra]] temperament before the discovery that ''cassandra'' and ''andromeda'' were named wrong way around and were later corrected.
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	The 13:11 and 14:11 dyads both reside near local [[harmonic entropy]] maxima due to falling almost halfway between a septimal and a pental consonance. This, as well as the fact that the triadic just approximations are 242:286:364, do not make the Cassandra triads sound appealing as consonances.		The 13:11 and 14:11 dyads both reside near local [[harmonic entropy]] maxima due to falling almost halfway between a septimal and a pental consonance. This, as well as the fact that the triadic just approximations are 242:286:364, do not make the Cassandra triads sound appealing as consonances.

	However, appearances can be deceiving. ~~I (~~[[Mason Green]]) find that ~~Cassandra~~ triads sound ''great'' harmonically, at least as good as 12edo ones if not better, and ~~I have a theory~~ as ~~to why~~.		However, appearances can be deceiving. [[Mason Green]] find that cassandra triads sound ''great'' harmonically, at least as good as 12edo ones if not better, and he theorizes as follows.

	There is an ambiguity in how we compute harmonic entropy for triads, depending on whether we use utonal or otonal approximations. The common major triad is otonally a 4:5:6 and utonally a 1 / (10:12:15). Because the otonal form is simpler, it dominates over the utonal form. For the minor triad, the situation is reversed; the minor triad is primarily a utonal 1 / (6:5:4) and secondarily an otonal 10:12:15.		There is an ambiguity in how we compute harmonic entropy for triads, depending on whether we use utonal or otonal approximations. The common major triad is otonally a 4:5:6 and utonally a 1 / (10:12:15). Because the otonal form is simpler, it dominates over the utonal form. For the minor triad, the situation is reversed; the minor triad is primarily a utonal 1 / (6:5:4) and secondarily an otonal 10:12:15.
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	The subminor triad is primarily an otonal 6:7:9 and secondarily an otonal 1 / (14:18:21). The supermajor triad is primarily an otonal 14:18:21 and secondarily a utonal 1 / (6:7:9).		The subminor triad is primarily an otonal 6:7:9 and secondarily an otonal 1 / (14:18:21). The supermajor triad is primarily an otonal 14:18:21 and secondarily a utonal 1 / (6:7:9).

	The consonance of otonal chords comes primarily, ~~I believe~~, from the fact that they share a common fundamental frequency; by the same token, utonal chords share a common overtone. Because these are two different mechanisms, it may be a good idea when calculating harmonic entropy to evaluate the two components ''separately''.		The consonance of otonal chords comes primarily, Mason believes, from the fact that they share a common fundamental frequency; by the same token, utonal chords share a common overtone. Because these are two different mechanisms, it may be a good idea when calculating harmonic entropy to evaluate the two components ''separately''.

	This leaves open the possibility that a chord intermediate between minor and subminor could be heard as both an otonal 6:7:9 ''and'' as a utonal 1 / (6:5:4). As such, as a compromise between these two triads, it has a reasonably strong common overtone ''and'' fundamental.		This leaves open the possibility that a chord intermediate between minor and subminor could be heard as both an otonal 6:7:9 ''and'' as a utonal 1 / (6:5:4). As such, as a compromise between these two triads, it has a reasonably strong common overtone ''and'' fundamental.

	This, in fact, could be how the ~~Cassandra~~ minor chord is perceived and why it is so consonant, even though the individual thirds that make it up are not consonant as dyads. These chords could be called "Borromean" in that sense, and this gives them a certain "indivisible" quality. This makes them sound more cohesive ~~to me~~.		This, in fact, could be how the cassandra minor chord is perceived and why it is so consonant, even though the individual thirds that make it up are not consonant as dyads. These chords could be called "Borromean" in that sense, and this gives them a certain "indivisible" quality. This makes them sound more cohesive.

	In 29edo, the ~~Cassandra~~ triads are actually not even the closest approximations for either minor or subminor; you need to use a non-[[~~Patent_val\|~~patent val]] approximations to use them in this manner, yet despite being non-patent they are reasonably convincing and this makes "val-switching" in compositions a reasonable option. Val-switching adds a great deal of flexibility and new [[comma pump]] options. Also, it means that 12edo melodies and chord progressions can be transferred to 29edo easily if in certain circumstances we use the cassandra chords in place of the ''patent'' minor and major triads.		In 29edo, the cassandra triads are actually not even the closest approximations for either minor or subminor; you need to use a non-[[patent val]] approximations to use them in this manner, yet despite being non-patent they are reasonably convincing and this makes "val-switching" in compositions a reasonable option. Val-switching adds a great deal of flexibility and new [[comma pump]] options. Also, it means that 12edo melodies and chord progressions can be transferred to 29edo easily if in certain circumstances we use the cassandra chords in place of the ''patent'' minor and major triads.

	[[Category:Triad]]		[[Category:Triad]]
	[[Category:Minor minthmic]]		[[Category:Minor minthmic]]
	~~{{Todo\| cleanup }}~~

FloraC: Note the problem in the name

2024-04-11T13:42:40Z

Note the problem in the name

← Older revision		Revision as of 13:42, 11 April 2024
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	The '''cassandra triads''' are triads in which the two step sizes are (tempered versions of) 13:11 and 14:11 and the total span is a tempered perfect fifth. There are two of them, one of which is intermediate between major and supermajor, and the other intermediate between minor and subminor.		The '''cassandra triads''' are triads in which the two step sizes are (tempered versions of) 13:11 and 14:11 and the total span is a tempered perfect fifth. There are two of them, one of which is intermediate between major and supermajor, and the other intermediate between minor and subminor.

	This tempers out the 364:363, which is one of the [[~~pentacircle_comma\|~~pentacircle ~~commas~~]]. Cassandra triads appear in [[~~17edo\|~~17edo]], and even more closely in [[~~29edo\|~~29edo]]. Cassandra chords are closely related to the 19-limit rootsubminor, rootminor, rootmajor, and roostsupermajor triads; in fact, a Cassandra minor triad could be considered to function as both rootsubminor ''and'' as rootminor, and similarly the Cassandra major chord functions as both rootmajor and rootsupermajor.		This tempers out the 364:363, which is one of the [[pentacircle comma]]s. Cassandra triads appear in [[17edo]], and even more closely in [[29edo]]. Cassandra chords are closely related to the 19-limit rootsubminor, rootminor, rootmajor, and roostsupermajor triads; in fact, a Cassandra minor triad could be considered to function as both rootsubminor ''and'' as rootminor, and similarly the Cassandra major chord functions as both rootmajor and rootsupermajor.

	~~==Consonance of Cassandra triads==~~		The name could have been related to the [[cassandra]] temperament before the discovery that ''cassandra'' and ''andromeda'' were named wrong way around and were later corrected.

	The 13:11 and 14:11 dyads both reside near local [[~~Harmonic_Entropy\|~~harmonic entropy]] maxima due to falling almost halfway between a septimal and a pental consonance. This, as well as the fact that the triadic just approximations are 242:286:364, do not make the Cassandra triads sound appealing as consonances.		== Consonance of cassandra triads ==

			The 13:11 and 14:11 dyads both reside near local [[harmonic entropy]] maxima due to falling almost halfway between a septimal and a pental consonance. This, as well as the fact that the triadic just approximations are 242:286:364, do not make the Cassandra triads sound appealing as consonances.

	However, appearances can be deceiving. I ([[Mason Green]]) find that Cassandra triads sound ''great'' harmonically, at least as good as 12edo ones if not better, and I have a theory as to why.		However, appearances can be deceiving. I ([[Mason Green]]) find that Cassandra triads sound ''great'' harmonically, at least as good as 12edo ones if not better, and I have a theory as to why.
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	This, in fact, could be how the Cassandra minor chord is perceived and why it is so consonant, even though the individual thirds that make it up are not consonant as dyads. These chords could be called "Borromean" in that sense, and this gives them a certain "indivisible" quality. This makes them sound more cohesive to me.		This, in fact, could be how the Cassandra minor chord is perceived and why it is so consonant, even though the individual thirds that make it up are not consonant as dyads. These chords could be called "Borromean" in that sense, and this gives them a certain "indivisible" quality. This makes them sound more cohesive to me.

	In 29edo, the Cassandra triads are actually not even the closest approximations for either minor or subminor; you need to use a non-[[Patent_val\|patent val]] approximations to use them in this manner, yet despite being non-patent they are reasonably convincing and this makes "val-switching" in compositions a reasonable option. Val-switching adds a great deal of flexibility and new [[~~comma_pump\|~~comma pump]] options. Also, it means that 12edo melodies and chord progressions can be transferred to 29edo easily if in certain circumstances we use the cassandra chords in place of the ''patent'' minor and major triads.		In 29edo, the Cassandra triads are actually not even the closest approximations for either minor or subminor; you need to use a non-[[Patent_val\|patent val]] approximations to use them in this manner, yet despite being non-patent they are reasonably convincing and this makes "val-switching" in compositions a reasonable option. Val-switching adds a great deal of flexibility and new [[comma pump]] options. Also, it means that 12edo melodies and chord progressions can be transferred to 29edo easily if in certain circumstances we use the cassandra chords in place of the ''patent'' minor and major triads.
	[[Category:~~triad~~]]
			[[Category:Triad]]
			[[Category:Minor minthmic]]
			{{Todo\| cleanup }}

BudjarnLambeth: /* Consonance of Cassandra triads */ added link to mason green

2024-01-29T01:43:47Z

Consonance of Cassandra triads: added link to mason green

← Older revision		Revision as of 01:43, 29 January 2024
Line 7:		Line 7:
	The 13:11 and 14:11 dyads both reside near local [[Harmonic_Entropy\|harmonic entropy]] maxima due to falling almost halfway between a septimal and a pental consonance. This, as well as the fact that the triadic just approximations are 242:286:364, do not make the Cassandra triads sound appealing as consonances.		The 13:11 and 14:11 dyads both reside near local [[Harmonic_Entropy\|harmonic entropy]] maxima due to falling almost halfway between a septimal and a pental consonance. This, as well as the fact that the triadic just approximations are 242:286:364, do not make the Cassandra triads sound appealing as consonances.

	However, appearances can be deceiving. I (Mason Green) find that Cassandra triads sound ''great'' harmonically, at least as good as 12edo ones if not better, and I have a theory as to why.		However, appearances can be deceiving. I ([[Mason Green]]) find that Cassandra triads sound ''great'' harmonically, at least as good as 12edo ones if not better, and I have a theory as to why.

	There is an ambiguity in how we compute harmonic entropy for triads, depending on whether we use utonal or otonal approximations. The common major triad is otonally a 4:5:6 and utonally a 1 / (10:12:15). Because the otonal form is simpler, it dominates over the utonal form. For the minor triad, the situation is reversed; the minor triad is primarily a utonal 1 / (6:5:4) and secondarily an otonal 10:12:15.		There is an ambiguity in how we compute harmonic entropy for triads, depending on whether we use utonal or otonal approximations. The common major triad is otonally a 4:5:6 and utonally a 1 / (10:12:15). Because the otonal form is simpler, it dominates over the utonal form. For the minor triad, the situation is reversed; the minor triad is primarily a utonal 1 / (6:5:4) and secondarily an otonal 10:12:15.

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

← Older revision		Revision as of 00:00, 17 July 2018
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	~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~		The '''cassandra triads''' are triads in which the two step sizes are (tempered versions of) 13:11 and 14:11 and the total span is a tempered perfect fifth. There are two of them, one of which is intermediate between major and supermajor, and the other intermediate between minor and subminor.
	~~This is an imported revision from Wikispaces.~~ The ~~revision metadata is included below for reference:<br>~~
	~~: This revision was by author [[User:MasonGreen1\|MasonGreen1]] and made on <tt>2015-12-01 17:48:49 UTC</tt>.<br>~~
	~~: The original revision id was <tt>568515431</tt>.<br>~~
	~~: The revision comment was: <tt></tt><br>~~
	~~The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>~~
	~~<h4>Original Wikitext content:</h4>~~
	<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The cassandra triads are triads in which the two step sizes are (tempered versions of) 13:11 and 14:11 and the total span is a tempered perfect fifth. There are two of them, one of which is intermediate between major and supermajor, and the other intermediate between minor and subminor.

	This tempers out the 364:363, which is one of the [[~~pentacircle comma~~\|pentacircle commas]]. Cassandra triads appear in [[17edo]], and even more closely in [[29edo]]. Cassandra chords are closely related to the 19-limit rootsubminor, rootminor, rootmajor, and roostsupermajor triads; in fact, a Cassandra minor triad could be considered to function as both rootsubminor //and// as rootminor, and similarly the Cassandra major chord functions as both rootmajor and rootsupermajor.		This tempers out the 364:363, which is one of the [[pentacircle_comma\|pentacircle commas]]. Cassandra triads appear in [[17edo\|17edo]], and even more closely in [[29edo\|29edo]]. Cassandra chords are closely related to the 19-limit rootsubminor, rootminor, rootmajor, and roostsupermajor triads; in fact, a Cassandra minor triad could be considered to function as both rootsubminor ''and'' as rootminor, and similarly the Cassandra major chord functions as both rootmajor and rootsupermajor.

	==Consonance of Cassandra triads==		==Consonance of Cassandra triads==

	The 13:11 and 14:11 dyads both reside near local [[harmonic entropy]] maxima due to falling almost halfway between a septimal and a pental consonance. This, as well as the fact that the triadic just approximations are 242:286:364, do not make the Cassandra triads sound appealing as consonances.		The 13:11 and 14:11 dyads both reside near local [[Harmonic_Entropy\|harmonic entropy]] maxima due to falling almost halfway between a septimal and a pental consonance. This, as well as the fact that the triadic just approximations are 242:286:364, do not make the Cassandra triads sound appealing as consonances.

	However, appearances can be deceiving. I (Mason Green) find that Cassandra triads sound //great// harmonically, at least as good as 12edo ones if not better, and I have a theory as to why.		However, appearances can be deceiving. I (Mason Green) find that Cassandra triads sound ''great'' harmonically, at least as good as 12edo ones if not better, and I have a theory as to why.

	There is an ambiguity in how we compute harmonic entropy for triads, depending on whether we use utonal or otonal approximations. The common major triad is otonally a 4:5:6 and utonally a 1 / (10:12:15). Because the otonal form is simpler, it dominates over the utonal form. For the minor triad, the situation is reversed; the minor triad is primarily a utonal 1 / (6:5:4) and secondarily an otonal 10:12:15.		There is an ambiguity in how we compute harmonic entropy for triads, depending on whether we use utonal or otonal approximations. The common major triad is otonally a 4:5:6 and utonally a 1 / (10:12:15). Because the otonal form is simpler, it dominates over the utonal form. For the minor triad, the situation is reversed; the minor triad is primarily a utonal 1 / (6:5:4) and secondarily an otonal 10:12:15.
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	The subminor triad is primarily an otonal 6:7:9 and secondarily an otonal 1 / (14:18:21). The supermajor triad is primarily an otonal 14:18:21 and secondarily a utonal 1 / (6:7:9).		The subminor triad is primarily an otonal 6:7:9 and secondarily an otonal 1 / (14:18:21). The supermajor triad is primarily an otonal 14:18:21 and secondarily a utonal 1 / (6:7:9).

	The consonance of otonal chords comes primarily, I believe, from the fact that they share a common fundamental frequency; by the same token, utonal chords share a common overtone. Because these are two different mechanisms, it may be a good idea when calculating harmonic entropy to evaluate the two components //separately//.		The consonance of otonal chords comes primarily, I believe, from the fact that they share a common fundamental frequency; by the same token, utonal chords share a common overtone. Because these are two different mechanisms, it may be a good idea when calculating harmonic entropy to evaluate the two components ''separately''.

	This leaves open the possibility that a chord intermediate between minor and subminor could be heard as both an otonal 6:7:9 //and// as a utonal 1 / (6:5:4). As such, as a compromise between these two triads, it has a reasonably strong common overtone //and// fundamental.		This leaves open the possibility that a chord intermediate between minor and subminor could be heard as both an otonal 6:7:9 ''and'' as a utonal 1 / (6:5:4). As such, as a compromise between these two triads, it has a reasonably strong common overtone ''and'' fundamental.

	This, in fact, could be how the Cassandra minor chord is perceived and why it is so consonant, even though the individual thirds that make it up are not consonant as dyads. These chords could be called "Borromean" in that sense, and this gives them a certain "indivisible" quality. This makes them sound more cohesive to me.		This, in fact, could be how the Cassandra minor chord is perceived and why it is so consonant, even though the individual thirds that make it up are not consonant as dyads. These chords could be called "Borromean" in that sense, and this gives them a certain "indivisible" quality. This makes them sound more cohesive to me.

	In 29edo, the Cassandra triads are actually not even the closest approximations for either minor or subminor; you need to use a non-[[patent val]] approximations to use them in this manner, yet despite being non-patent they are reasonably convincing and this makes "val-switching" in compositions a reasonable option. Val-switching adds a great deal of flexibility and new [[comma pump]] options. Also, it means that 12edo melodies and chord progressions can be transferred to 29edo easily if in certain circumstances we use the cassandra chords in place of the //patent// minor and major triads.~~</pre></div>~~		In 29edo, the Cassandra triads are actually not even the closest approximations for either minor or subminor; you need to use a non-[[Patent_val\|patent val]] approximations to use them in this manner, yet despite being non-patent they are reasonably convincing and this makes "val-switching" in compositions a reasonable option. Val-switching adds a great deal of flexibility and new [[comma_pump\|comma pump]] options. Also, it means that 12edo melodies and chord progressions can be transferred to 29edo easily if in certain circumstances we use the cassandra chords in place of the ''patent'' minor and major triads.
	~~<h4>Original HTML content:</h4>~~		[[Category:triad]]
	<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Cassandra triads</title></head><body>The <strong>cassandra triads</strong> are triads in which the two step sizes are (tempered versions of) 13:11 and 14:11 and the total span is a tempered perfect fifth. There are two of them, one of which is intermediate between major and supermajor, and the other intermediate between minor and subminor.<br />
	~~<br />~~
	~~This tempers out the 364~~:363, which is one of the <a class="wiki_link" href="/pentacircle%20comma">pentacircle commas</a>. Cassandra triads appear in <a class="wiki_link" href="/17edo">17edo</a>, and even more closely in <a class="wiki_link" href="/29edo">29edo</a>. Cassandra chords are closely related to the 19-limit rootsubminor, rootminor, rootmajor, and roostsupermajor triads; in fact, a Cassandra minor triad ~~could be considered to function as both rootsubminor <em>and</em> as rootminor, and similarly the Cassandra major chord functions as both rootmajor and rootsupermajor.<br />~~
	~~<br />~~
	<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Consonance of Cassandra triads"></a><!-- ws:end:WikiTextHeadingRule:0 -->Consonance of Cassandra triads</h2>
	~~<br />~~
	The 13:11 and 14:11 dyads both reside near local <a class="wiki_link" href="/harmonic%20entropy">harmonic entropy</a> maxima due to falling almost halfway between a septimal and a pental consonance. This, as well as the fact that the triadic just approximations are 242:286:364, do not make the Cassandra triads sound appealing as consonances.<br />
	~~<br />~~
	However, appearances can be deceiving. I (Mason Green) find that Cassandra triads sound <em>great</em> harmonically, at least as good as 12edo ones if not better, and I have a theory as to why.<br />
	~~<br />~~
	There is an ambiguity in how we compute harmonic entropy for triads, depending on whether we use utonal or otonal approximations. The common major triad is otonally a 4:5:6 and utonally a 1 / (10:12:15). Because the otonal form is simpler, it dominates over the utonal form. For the minor triad, the situation is reversed; the minor triad is primarily a utonal 1 / (6:5:4) and secondarily an otonal 10:12:15.<br />
	~~<br />~~
	~~The subminor triad is primarily an otonal 6:7:9 and secondarily an otonal 1 / (14:18:21). The supermajor triad is primarily an otonal 14:18:21 and secondarily a utonal 1 / (6:7:9).<br />~~
	~~<br />~~
	The consonance of otonal chords comes primarily, I believe, from the fact that they share a common fundamental frequency; by the same token, utonal chords share a common overtone. Because these are two different mechanisms, it may be a good idea when calculating harmonic entropy to evaluate the two components <em>separately</em>.<br />
	~~<br />~~
	This leaves open the possibility that a chord intermediate between minor and subminor could be heard as both an otonal 6:7:9 <em>and</em> as a utonal 1 / (6:5:4). As such, as a compromise between these two triads, it has a reasonably strong common overtone <em>and</em> fundamental.<br />
	~~<br />~~
	This, in fact, could be how the Cassandra minor chord is perceived and why it is so consonant, even though the individual thirds that make it up are not consonant as dyads. These chords could be called &quot;Borromean&quot; in that sense, and this gives them a certain &quot;indivisible&quot; quality. This makes them sound more cohesive to me.<br />
	~~<br />~~
	In 29edo, the Cassandra triads are actually not even the closest approximations for either minor or subminor; you need to use a non-<a class="wiki_link" href="/patent%20val">patent val</a> approximations to use them in this manner, yet despite being non-patent they are reasonably convincing and this makes &quot;val-switching&quot; in compositions a reasonable option. Val-switching adds a great deal of flexibility and new <a class="wiki_link" href="/comma%20pump">comma pump</a> options. Also, it means that 12edo melodies and chord progressions can be transferred to 29edo easily if in certain circumstances we use the cassandra chords in place of the <em>patent</em> minor and major triads.</body></html></pre></div>

Wikispaces>MasonGreen1: Imported revision 568515431 - Original comment:

2015-12-01T17:48:49Z

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2015-12-01T17:47:04Z

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Wikispaces>MasonGreen1: Imported revision 568514967 - Original comment:

2015-12-01T17:46:07Z

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<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The **cassandra triads** are triads in which the two step sizes are (tempered versions of) 13:11 and 14:11 and the total span is a tempered perfect fifth. There are two of them, one of which is intermediate between major and supermajor, and the other intermediate between minor and subminor.

This tempers out the 364:363, which is one of the [[pentacircle comma|pentacircle commas]]. Cassandra triads appear in [[17edo]], and even more closely in [[29edo]]. Cassandra chords are closely related to the 19-limit rootsubminor, rootminor, rootmajor, and roostsupermajor triads; in fact, a Cassandra minor triad could be considered to function as both rootsubminor //and// as rootminor, and similarly the Cassandra major chord functions as both rootmajor and rootsupermajor.

==Consonance of Cassandra triads==

The 13:11 and 14:11 dyads both reside near local [[harmonic entropy]] maxima. This, as well as the fact that the triadic just approximations are 242:286:364, do not make the Cassandra triads sound appealing as consonances.

However, appearances can be deceiving. I (Mason Green) find that Cassandra triads sound //great// harmonically, at least as good as 12edo ones if not better, and I have a theory as to why.

There is an ambiguity in how we compute harmonic entropy for triads, depending on whether we use utonal or otonal approximations. The common major triad is otonally a 4:5:6 and utonally a 1 / (10:12:15). Because the otonal form is simpler, it dominates over the utonal form. For the minor triad, the situation is reversed; the minor triad is primarily a utonal 1 / (6:5:4) and secondarily an otonal 10:12:15.

The subminor triad is primarily an otonal 6:7:9 and secondarily an otonal 1 / (14:18:21). The supermajor triad is primarily an otonal 14:18:21 and a utonal 1 / (6:7:9).

The consonance of otonal chords comes primarily, I believe, from the fact that they share a common fundamental frequency; by the same token, utonal chords share a common overtone. Because these are two different mechanisms, it may be a good idea when calculating harmonic entropy to evaluate the two components //separately//.

This leaves open the possibility that a chord intermediate between minor and subminor could be heard as both an otonal 6:7:9 //and// as a utonal 1 / (6:5:4). As such, as a compromise between these two triads, it has a reasonably strong common overtone //and// fundamental.

This, in fact, could be how the Cassandra minor chord is perceived and why it is so consonant, even though the individual thirds that make it up are not. These chords could be called "Borromean" in that sense, and this gives them a certain "indivisible" quality. This makes them sound more cohesive to me.

In 29edo, the Cassandra triads are actually not even the closest approximations for either minor or subminor; you need to use a non-[[patent val]] approximations to use them in this manner, yet despite being non-patent they are reasonably convincing and this makes "val-switching" in compositions a reasonable option. Val-switching adds a great deal of flexibility and new [[comma pump]] options. Also, it means that 12edo melodies can be transferred to 29edo easily if in certain circumstances we use the cassandra chords in place of the //patent// minor and major triads.</pre></div>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Cassandra triads</title></head><body>The <strong>cassandra triads</strong> are triads in which the two step sizes are (tempered versions of) 13:11 and 14:11 and the total span is a tempered perfect fifth. There are two of them, one of which is intermediate between major and supermajor, and the other intermediate between minor and subminor.<br />
<br />
This tempers out the 364:363, which is one of the <a class="wiki_link" href="/pentacircle%20comma">pentacircle commas</a>. Cassandra triads appear in <a class="wiki_link" href="/17edo">17edo</a>, and even more closely in <a class="wiki_link" href="/29edo">29edo</a>. Cassandra chords are closely related to the 19-limit rootsubminor, rootminor, rootmajor, and roostsupermajor triads; in fact, a Cassandra minor triad could be considered to function as both rootsubminor <em>and</em> as rootminor, and similarly the Cassandra major chord functions as both rootmajor and rootsupermajor.<br />
<br />
<h2 id="toc0"><a name="x-Consonance of Cassandra triads"></a>Consonance of Cassandra triads</h2>
<br />
The 13:11 and 14:11 dyads both reside near local <a class="wiki_link" href="/harmonic%20entropy">harmonic entropy</a> maxima. This, as well as the fact that the triadic just approximations are 242:286:364, do not make the Cassandra triads sound appealing as consonances.<br />
<br />
However, appearances can be deceiving. I (Mason Green) find that Cassandra triads sound <em>great</em> harmonically, at least as good as 12edo ones if not better, and I have a theory as to why.<br />
<br />
There is an ambiguity in how we compute harmonic entropy for triads, depending on whether we use utonal or otonal approximations. The common major triad is otonally a 4:5:6 and utonally a 1 / (10:12:15). Because the otonal form is simpler, it dominates over the utonal form. For the minor triad, the situation is reversed; the minor triad is primarily a utonal 1 / (6:5:4) and secondarily an otonal 10:12:15.<br />
<br />
The subminor triad is primarily an otonal 6:7:9 and secondarily an otonal 1 / (14:18:21). The supermajor triad is primarily an otonal 14:18:21 and a utonal 1 / (6:7:9).<br />
<br />
The consonance of otonal chords comes primarily, I believe, from the fact that they share a common fundamental frequency; by the same token, utonal chords share a common overtone. Because these are two different mechanisms, it may be a good idea when calculating harmonic entropy to evaluate the two components <em>separately</em>.<br />
<br />
This leaves open the possibility that a chord intermediate between minor and subminor could be heard as both an otonal 6:7:9 <em>and</em> as a utonal 1 / (6:5:4). As such, as a compromise between these two triads, it has a reasonably strong common overtone <em>and</em> fundamental.<br />
<br />
This, in fact, could be how the Cassandra minor chord is perceived and why it is so consonant, even though the individual thirds that make it up are not. These chords could be called &quot;Borromean&quot; in that sense, and this gives them a certain &quot;indivisible&quot; quality. This makes them sound more cohesive to me.<br />
<br />
In 29edo, the Cassandra triads are actually not even the closest approximations for either minor or subminor; you need to use a non-<a class="wiki_link" href="/patent%20val">patent val</a> approximations to use them in this manner, yet despite being non-patent they are reasonably convincing and this makes &quot;val-switching&quot; in compositions a reasonable option. Val-switching adds a great deal of flexibility and new <a class="wiki_link" href="/comma%20pump">comma pump</a> options. Also, it means that 12edo melodies can be transferred to 29edo easily if in certain circumstances we use the cassandra chords in place of the <em>patent</em> minor and major triads.</body></html></pre></div>

← Older revision		Revision as of 17:48, 1 December 2015
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	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
	: This revision was by author [[User:MasonGreen1\|MasonGreen1]] and made on <tt>2015-12-01 17:48:06 UTC</tt>.<br>		: This revision was by author [[User:MasonGreen1\|MasonGreen1]] and made on <tt>2015-12-01 17:48:49 UTC</tt>.<br>
	: The original revision id was <tt>~~568515307~~</tt>.<br>		: The original revision id was <tt>568515431</tt>.<br>
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	The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>		The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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	This, in fact, could be how the Cassandra minor chord is perceived and why it is so consonant, even though the individual thirds that make it up are not consonant as dyads. These chords could be called "Borromean" in that sense, and this gives them a certain "indivisible" quality. This makes them sound more cohesive to me.		This, in fact, could be how the Cassandra minor chord is perceived and why it is so consonant, even though the individual thirds that make it up are not consonant as dyads. These chords could be called "Borromean" in that sense, and this gives them a certain "indivisible" quality. This makes them sound more cohesive to me.

	In 29edo, the Cassandra triads are actually not even the closest approximations for either minor or subminor; you need to use a non-[[patent val]] approximations to use them in this manner, yet despite being non-patent they are reasonably convincing and this makes "val-switching" in compositions a reasonable option. Val-switching adds a great deal of flexibility and new [[comma pump]] options. Also, it means that 12edo melodies can be transferred to 29edo easily if in certain circumstances we use the cassandra chords in place of the //patent// minor and major triads.</pre></div>		In 29edo, the Cassandra triads are actually not even the closest approximations for either minor or subminor; you need to use a non-[[patent val]] approximations to use them in this manner, yet despite being non-patent they are reasonably convincing and this makes "val-switching" in compositions a reasonable option. Val-switching adds a great deal of flexibility and new [[comma pump]] options. Also, it means that 12edo melodies and chord progressions can be transferred to 29edo easily if in certain circumstances we use the cassandra chords in place of the //patent// minor and major triads.</pre></div>
	<h4>Original HTML content:</h4>		<h4>Original HTML content:</h4>
	<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Cassandra triads</title></head><body>The <strong>cassandra triads</strong> are triads in which the two step sizes are (tempered versions of) 13:11 and 14:11 and the total span is a tempered perfect fifth. There are two of them, one of which is intermediate between major and supermajor, and the other intermediate between minor and subminor.<br />		<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Cassandra triads</title></head><body>The <strong>cassandra triads</strong> are triads in which the two step sizes are (tempered versions of) 13:11 and 14:11 and the total span is a tempered perfect fifth. There are two of them, one of which is intermediate between major and supermajor, and the other intermediate between minor and subminor.<br />
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	This, in fact, could be how the Cassandra minor chord is perceived and why it is so consonant, even though the individual thirds that make it up are not consonant as dyads. These chords could be called &quot;Borromean&quot; in that sense, and this gives them a certain &quot;indivisible&quot; quality. This makes them sound more cohesive to me.<br />		This, in fact, could be how the Cassandra minor chord is perceived and why it is so consonant, even though the individual thirds that make it up are not consonant as dyads. These chords could be called &quot;Borromean&quot; in that sense, and this gives them a certain &quot;indivisible&quot; quality. This makes them sound more cohesive to me.<br />
	<br />		<br />
	In 29edo, the Cassandra triads are actually not even the closest approximations for either minor or subminor; you need to use a non-<a class="wiki_link" href="/patent%20val">patent val</a> approximations to use them in this manner, yet despite being non-patent they are reasonably convincing and this makes &quot;val-switching&quot; in compositions a reasonable option. Val-switching adds a great deal of flexibility and new <a class="wiki_link" href="/comma%20pump">comma pump</a> options. Also, it means that 12edo melodies can be transferred to 29edo easily if in certain circumstances we use the cassandra chords in place of the <em>patent</em> minor and major triads.</body></html></pre></div>		In 29edo, the Cassandra triads are actually not even the closest approximations for either minor or subminor; you need to use a non-<a class="wiki_link" href="/patent%20val">patent val</a> approximations to use them in this manner, yet despite being non-patent they are reasonably convincing and this makes &quot;val-switching&quot; in compositions a reasonable option. Val-switching adds a great deal of flexibility and new <a class="wiki_link" href="/comma%20pump">comma pump</a> options. Also, it means that 12edo melodies and chord progressions can be transferred to 29edo easily if in certain circumstances we use the cassandra chords in place of the <em>patent</em> minor and major triads.</body></html></pre></div>

← Older revision		Revision as of 17:47, 1 December 2015
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	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
	: This revision was by author [[User:MasonGreen1\|MasonGreen1]] and made on <tt>2015-12-01 17:46:07 UTC</tt>.<br>		: This revision was by author [[User:MasonGreen1\|MasonGreen1]] and made on <tt>2015-12-01 17:47:04 UTC</tt>.<br>
	: The original revision id was <tt>~~568514967~~</tt>.<br>		: The original revision id was <tt>568515141</tt>.<br>
	: The revision comment was: <tt></tt><br>		: The revision comment was: <tt></tt><br>
	The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>		The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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	==Consonance of Cassandra triads==		==Consonance of Cassandra triads==

	The 13:11 and 14:11 dyads both reside near local [[harmonic entropy]] maxima. This, as well as the fact that the triadic just approximations are 242:286:364, do not make the Cassandra triads sound appealing as consonances.		The 13:11 and 14:11 dyads both reside near local [[harmonic entropy]] maxima due to falling almost halfway between a septimal and a pental consonance. This, as well as the fact that the triadic just approximations are 242:286:364, do not make the Cassandra triads sound appealing as consonances.

	However, appearances can be deceiving. I (Mason Green) find that Cassandra triads sound //great// harmonically, at least as good as 12edo ones if not better, and I have a theory as to why.		However, appearances can be deceiving. I (Mason Green) find that Cassandra triads sound //great// harmonically, at least as good as 12edo ones if not better, and I have a theory as to why.
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	<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Consonance of Cassandra triads"></a><!-- ws:end:WikiTextHeadingRule:0 -->Consonance of Cassandra triads</h2>		<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Consonance of Cassandra triads"></a><!-- ws:end:WikiTextHeadingRule:0 -->Consonance of Cassandra triads</h2>
	<br />		<br />
	The 13:11 and 14:11 dyads both reside near local <a class="wiki_link" href="/harmonic%20entropy">harmonic entropy</a> maxima. This, as well as the fact that the triadic just approximations are 242:286:364, do not make the Cassandra triads sound appealing as consonances.<br />		The 13:11 and 14:11 dyads both reside near local <a class="wiki_link" href="/harmonic%20entropy">harmonic entropy</a> maxima due to falling almost halfway between a septimal and a pental consonance. This, as well as the fact that the triadic just approximations are 242:286:364, do not make the Cassandra triads sound appealing as consonances.<br />
	<br />		<br />
	However, appearances can be deceiving. I (Mason Green) find that Cassandra triads sound <em>great</em> harmonically, at least as good as 12edo ones if not better, and I have a theory as to why.<br />		However, appearances can be deceiving. I (Mason Green) find that Cassandra triads sound <em>great</em> harmonically, at least as good as 12edo ones if not better, and I have a theory as to why.<br />

← Older revision		Revision as of 17:46, 1 December 2015
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	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
	: This revision was by author [[User:MasonGreen1\|MasonGreen1]] and made on <tt>2015-12-01 17:45:22 UTC</tt>.<br>		: This revision was by author [[User:MasonGreen1\|MasonGreen1]] and made on <tt>2015-12-01 17:46:07 UTC</tt>.<br>
	: The original revision id was <tt>~~568514835~~</tt>.<br>		: The original revision id was <tt>568514967</tt>.<br>
	: The revision comment was: <tt></tt><br>		: The revision comment was: <tt></tt><br>
	The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>		The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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	This leaves open the possibility that a chord intermediate between minor and subminor could be heard as both an otonal 6:7:9 //and// as a utonal 1 / (6:5:4). As such, as a compromise between these two triads, it has a reasonably strong common overtone //and// fundamental.		This leaves open the possibility that a chord intermediate between minor and subminor could be heard as both an otonal 6:7:9 //and// as a utonal 1 / (6:5:4). As such, as a compromise between these two triads, it has a reasonably strong common overtone //and// fundamental.

	This, in fact, could be how the Cassandra minor chord is perceived and why it is so consonant, even though the individual thirds that make it up are not. These chords could be called "Borromean" in that sense, and this gives them a certain "indivisible" quality. This makes them sound more cohesive to me.		This, in fact, could be how the Cassandra minor chord is perceived and why it is so consonant, even though the individual thirds that make it up are not consonant as dyads. These chords could be called "Borromean" in that sense, and this gives them a certain "indivisible" quality. This makes them sound more cohesive to me.

	In 29edo, the Cassandra triads are actually not even the closest approximations for either minor or subminor; you need to use a non-[[patent val]] approximations to use them in this manner, yet despite being non-patent they are reasonably convincing and this makes "val-switching" in compositions a reasonable option. Val-switching adds a great deal of flexibility and new [[comma pump]] options. Also, it means that 12edo melodies can be transferred to 29edo easily if in certain circumstances we use the cassandra chords in place of the //patent// minor and major triads.</pre></div>		In 29edo, the Cassandra triads are actually not even the closest approximations for either minor or subminor; you need to use a non-[[patent val]] approximations to use them in this manner, yet despite being non-patent they are reasonably convincing and this makes "val-switching" in compositions a reasonable option. Val-switching adds a great deal of flexibility and new [[comma pump]] options. Also, it means that 12edo melodies can be transferred to 29edo easily if in certain circumstances we use the cassandra chords in place of the //patent// minor and major triads.</pre></div>
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	This leaves open the possibility that a chord intermediate between minor and subminor could be heard as both an otonal 6:7:9 <em>and</em> as a utonal 1 / (6:5:4). As such, as a compromise between these two triads, it has a reasonably strong common overtone <em>and</em> fundamental.<br />		This leaves open the possibility that a chord intermediate between minor and subminor could be heard as both an otonal 6:7:9 <em>and</em> as a utonal 1 / (6:5:4). As such, as a compromise between these two triads, it has a reasonably strong common overtone <em>and</em> fundamental.<br />
	<br />		<br />
	This, in fact, could be how the Cassandra minor chord is perceived and why it is so consonant, even though the individual thirds that make it up are not. These chords could be called &quot;Borromean&quot; in that sense, and this gives them a certain &quot;indivisible&quot; quality. This makes them sound more cohesive to me.<br />		This, in fact, could be how the Cassandra minor chord is perceived and why it is so consonant, even though the individual thirds that make it up are not consonant as dyads. These chords could be called &quot;Borromean&quot; in that sense, and this gives them a certain &quot;indivisible&quot; quality. This makes them sound more cohesive to me.<br />
	<br />		<br />
	In 29edo, the Cassandra triads are actually not even the closest approximations for either minor or subminor; you need to use a non-<a class="wiki_link" href="/patent%20val">patent val</a> approximations to use them in this manner, yet despite being non-patent they are reasonably convincing and this makes &quot;val-switching&quot; in compositions a reasonable option. Val-switching adds a great deal of flexibility and new <a class="wiki_link" href="/comma%20pump">comma pump</a> options. Also, it means that 12edo melodies can be transferred to 29edo easily if in certain circumstances we use the cassandra chords in place of the <em>patent</em> minor and major triads.</body></html></pre></div>		In 29edo, the Cassandra triads are actually not even the closest approximations for either minor or subminor; you need to use a non-<a class="wiki_link" href="/patent%20val">patent val</a> approximations to use them in this manner, yet despite being non-patent they are reasonably convincing and this makes &quot;val-switching&quot; in compositions a reasonable option. Val-switching adds a great deal of flexibility and new <a class="wiki_link" href="/comma%20pump">comma pump</a> options. Also, it means that 12edo melodies can be transferred to 29edo easily if in certain circumstances we use the cassandra chords in place of the <em>patent</em> minor and major triads.</body></html></pre></div>

Cassandra triads - Revision history

VectorGraphics at 21:02, 20 April 2025

FloraC: Use third-person tone and improve linking

FloraC: Note the problem in the name

BudjarnLambeth: /* Consonance of Cassandra triads */ added link to mason green

Wikispaces>FREEZE at 00:00, 17 July 2018

Wikispaces>MasonGreen1: **Imported revision 568515431 - Original comment: **

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Wikispaces>MasonGreen1: **Imported revision 568515141 - Original comment: **

Wikispaces>MasonGreen1: **Imported revision 568514967 - Original comment: **

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Wikispaces>MasonGreen1: Imported revision 568515431 - Original comment:

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