Talk:Cluster temperament - Revision history

Mike Battaglia: 1 revision imported: Moving archived Wikispaces discussion to subpage

2018-10-01T18:02:34Z

1 revision imported: Moving archived Wikispaces discussion to subpage

← Older revision	Revision as of 18:02, 1 October 2018
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WikispacesArchive>Mike Battaglia: Text replacement - "'''All discussion below is archived from the Wikispaces export in its original unaltered form.'''" to "'''All discussion below is archived from the Wikispaces export in its original unaltered form.''' ''...

2018-10-01T18:01:16Z

Text replacement - "'''All discussion below is archived from the Wikispaces export in its original unaltered form.'''" to "'''All discussion below is archived from the Wikispaces export in its original unaltered form.''' <span style="color:#800000">''...

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	~~= ARCHIVED WIKISPACES DISCUSSION BELOW =~~		{{WSArchiveLink}}
	~~'''All discussion below is archived from the Wikispaces export in its original unaltered form.'''~~

	~~<span style="color:#800000">'''PLEASE MAKE ANY NEW COMMENTS <u>ABOVE</u> THIS SECTION.'''</span> Anything below here is for archival purposes only.~~

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	~~== amity ==~~
	~~Keenan, this is an interesting idea, and I'm curious to see you expand on it.~~

	I'm confused about your statement, <tt> For example, the list of amity "thirds" includes ...6/5 (339.5) (363.2) 5/4... where the intervals given in cents are not representable as simple JI intervals (243/200 and 100/81 are about as simple as you can get). </tt>

	But in the 13-limit, 339.5 is 11/9 and 363.2 is 16/13, which are relatively simple. So I wonder if it would be fair to call amity a cluster temperament in the 13-limit, but not in a lower one? I understand it's a fuzzy definition right now, so maybe there are other reasons to not count amity that I don't understand.

	~~- '''Andrew_Heathwaite''' June 20, 2012, 07:37:53 AM UTC-0700~~
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	~~Oh, I guess I missed this back in July. Right you are, Andrew. I guess you'd call that "2.3.5.11.13 hitchcock" temperament. (It doesn't have the same mapping of 11 as 11-limit amity.)~~

	~~- '''keenanpepper''' November 24, 2012, 09:13:41 AM UTC-0800~~
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	Hey Keenan, just saw this page, very interesting! Would you say that the schismatic temperament is a cluster temperament? If so, would 10/9 and 9/8 be a cluster, also 32/27 and 6/5, etc? This seems like an example of "clustering" that's been done subconsciously in the West for centuries.

	Some thoughts on firming things up mathematically: a cluster temperament can be found by taking two commas and tempering out the difference between them. In your slendric example, it's 49/48 and 64/63. In general, it's possible to combine these two commas to get a 3-limit comma via hermite reduction. This gives us 256/243, which implies quasi-5-edo, hence 5 clusters per octave, and pentatonic scales. The 3-limit comma is the sum of 3 7-limit commas, so there are 3 chromas from 9/8 to 32/27.

	I think you could take any comma of the form 2^a·3^b·P^c, with c = ±1, like 81/80 or 64/63 or 33/32, and equate some number of them to any 3-limit comma, and get a cluster temperament. Equating 64/63 and the pythagorean comma implies 12-note scales, and makes 9/8 cluster with 8/7. The amity comma equates the apotome to 5 syntonic commas, hence 7 clusters = heptatonic. Each cluster contains 5 steps between 3-limit ratios, hence 6 ratios: 32/27 - 6/5 - 11/9 - 16/13 - 5/4 - 81/64.

	~~- '''TallKite''' January 24, 2018, 06:34:40 PM UTC-0800~~
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Mike Battaglia: Text replacement - "'''All discussion below is archived from the Wikispaces export in its original unaltered form.'''" to "'''All discussion below is archived from the Wikispaces export in its original unaltered form.''' ''...

2018-09-20T07:00:43Z

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	'''All discussion below is archived from the Wikispaces export in its original unaltered form.'''		'''All discussion below is archived from the Wikispaces export in its original unaltered form.'''

			<span style="color:#800000">'''PLEASE MAKE ANY NEW COMMENTS <u>ABOVE</u> THIS SECTION.'''</span> Anything below here is for archival purposes only.

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Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

New page

= ARCHIVED WIKISPACES DISCUSSION BELOW =
'''All discussion below is archived from the Wikispaces export in its original unaltered form.'''
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== amity ==
Keenan, this is an interesting idea, and I'm curious to see you expand on it.

I'm confused about your statement, <tt> For example, the list of amity "thirds" includes ...6/5 (339.5) (363.2) 5/4... where the intervals given in cents are not representable as simple JI intervals (243/200 and 100/81 are about as simple as you can get). </tt>

But in the 13-limit, 339.5 is 11/9 and 363.2 is 16/13, which are relatively simple. So I wonder if it would be fair to call amity a cluster temperament in the 13-limit, but not in a lower one? I understand it's a fuzzy definition right now, so maybe there are other reasons to not count amity that I don't understand.

- '''Andrew_Heathwaite''' June 20, 2012, 07:37:53 AM UTC-0700
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Oh, I guess I missed this back in July. Right you are, Andrew. I guess you'd call that "2.3.5.11.13 hitchcock" temperament. (It doesn't have the same mapping of 11 as 11-limit amity.)

- '''keenanpepper''' November 24, 2012, 09:13:41 AM UTC-0800
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Hey Keenan, just saw this page, very interesting! Would you say that the schismatic temperament is a cluster temperament? If so, would 10/9 and 9/8 be a cluster, also 32/27 and 6/5, etc? This seems like an example of "clustering" that's been done subconsciously in the West for centuries.

Some thoughts on firming things up mathematically: a cluster temperament can be found by taking two commas and tempering out the difference between them. In your slendric example, it's 49/48 and 64/63. In general, it's possible to combine these two commas to get a 3-limit comma via hermite reduction. This gives us 256/243, which implies quasi-5-edo, hence 5 clusters per octave, and pentatonic scales. The 3-limit comma is the sum of 3 7-limit commas, so there are 3 chromas from 9/8 to 32/27.

I think you could take any comma of the form 2^a·3^b·P^c, with c = ±1, like 81/80 or 64/63 or 33/32, and equate some number of them to any 3-limit comma, and get a cluster temperament. Equating 64/63 and the pythagorean comma implies 12-note scales, and makes 9/8 cluster with 8/7. The amity comma equates the apotome to 5 syntonic commas, hence 7 clusters = heptatonic. Each cluster contains 5 steps between 3-limit ratios, hence 6 ratios: 32/27 - 6/5 - 11/9 - 16/13 - 5/4 - 81/64.

- '''TallKite''' January 24, 2018, 06:34:40 PM UTC-0800
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