Phi as a generator - Revision history

FloraC: FloraC moved page Phi as a Generator to Phi as a generator: WP:NCCAPS

2024-03-03T07:45:56Z

FloraC moved page Phi as a Generator to Phi as a generator: WP:NCCAPS

← Older revision	Revision as of 07:45, 3 March 2024
(No difference)

BudjarnLambeth at 03:38, 3 March 2024

2024-03-03T03:38:36Z

← Older revision		Revision as of 03:38, 3 March 2024
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	Generator of phi. Two generators up to 3, two down to 7, other primes are more complex. (One generator up or one down are ambiguous 13.)		Generator of phi. Two generators up to 3, two down to 7, other primes are more complex. (One generator up or one down are ambiguous 13.)

	This temperament is named ~~'''Elderthing'''~~ according to the [[map of rank-2 temperaments]]. It was added to ~~the~~ page by [[Kosmorsky]], so he may be the one who named it.		This temperament is named "elderthing" according to the [[map of rank-2 temperaments]]. It was first added to that page by [[Kosmorsky]], so he may be the one who named it.

	---		---

BudjarnLambeth: Moved information from Map of Rank-2 Temperaments

2024-03-03T03:19:29Z

Moved information from Map of Rank-2 Temperaments

← Older revision		Revision as of 03:19, 3 March 2024
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	Period=[[5edo]]		Period=[[5edo]]

			"Elderthing"

	ET: [[85edo\|85]] (138:163)		ET: [[85edo\|85]] (138:163)

			Generator of phi. Two generators up to 3, two down to 7, other primes are more complex. (One generator up or one down are ambiguous 13.)

	This temperament is named '''Elderthing''' according to the [[map of rank-2 temperaments]]. It was added to the page by [[Kosmorsky]], so he may be the one who named it.		This temperament is named '''Elderthing''' according to the [[map of rank-2 temperaments]]. It was added to the page by [[Kosmorsky]], so he may be the one who named it.

BudjarnLambeth: Added todo cleanup

2024-03-03T03:17:58Z

Added todo cleanup

← Older revision		Revision as of 03:17, 3 March 2024
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	ET: [[97ed5\|97ed5]] (280:347)		ET: [[97ed5\|97ed5]] (280:347)

			{{todo\|cleanup}}
	[[Category:Golden ratio]]		[[Category:Golden ratio]]

BudjarnLambeth: Mentioned the name of one of the temperaments

2024-03-03T02:58:24Z

Mentioned the name of one of the temperaments

← Older revision		Revision as of 02:58, 3 March 2024
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	ET: [[85edo\|85]] (138:163)		ET: [[85edo\|85]] (138:163)

			This temperament is named '''Elderthing''' according to the [[map of rank-2 temperaments]]. It was added to the page by [[Kosmorsky]], so he may be the one who named it.

	---		---

Fredg999 category edits: Removing from Category:Theory using Cat-a-lot

2023-03-05T05:32:46Z

Removing from Category:Theory using Cat-a-lot

← Older revision		Revision as of 05:32, 5 March 2023
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	ET: [[97ed5\|97ed5]] (280:347)		ET: [[97ed5\|97ed5]] (280:347)

	~~[[Category:Theory]]~~
	[[Category:Golden ratio]]		[[Category:Golden ratio]]

Xenllium at 03:58, 24 January 2022

2022-01-24T03:58:32Z

← Older revision		Revision as of 03:58, 24 January 2022
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	Musically, the golden ratio is approximately 833.0903 cents. Just as the ratios of adjacent Fibonacci numbers approximate the golden ratio, the golden ratio in turn may be used to represent those ratios, and, equating whatever subset of these ratios one obtains various commas. Furthermore, the exponents of phi approximate the [http://en.wikipedia.org/wiki/Lucas_number Lucas numbers], closely allied with the Fibonacci numbers, with increasing accuracy, which can be put to good effect in a temperament. Furthermore, the square root of phi, 416.54515, generates the [[Kleismic_family#Sqrtphi\|sqrtphi temperament]], a complex, accurate temperament extending into the higher prime limits, and this contains the phi generated temperament within it.		Musically, the [[golden ratio]] is approximately 833.0903 cents. Just as the ratios of adjacent Fibonacci numbers approximate the golden ratio, the golden ratio in turn may be used to represent those ratios, and, equating whatever subset of these ratios one obtains various commas. Furthermore, the exponents of phi approximate the [http://en.wikipedia.org/wiki/Lucas_number Lucas numbers], closely allied with the Fibonacci numbers, with increasing accuracy, which can be put to good effect in a temperament. Furthermore, the square root of phi, 416.54515, generates the [[Kleismic_family#Sqrtphi\|sqrtphi temperament]], a complex, accurate temperament extending into the higher prime limits, and this contains the phi generated temperament within it.

	Let's use the archexample of [[~~46edo\|~~46edo]]. 23edo approximates phi slightly sharp* but nonetheless rather well, and note that it excels on the 2.13.21.34.55.89.144 (etc.) subgroup. Notice also the factorization of these numbers, 21=3x7 34=2x17 55=5x11 144=9x16 89=prime. *Why is this important? It seems that when the generator exceeds phi, the lower fibonacci harmonics are better approximated, and most likely but I'm unsure that if it undershoots phi, the higher harmonics are better approximated to the exclusion of the lower. But I digress. If there were only a way to tease these fibonacci factors apart.		Let's use the archexample of [[46edo]]. 23edo approximates phi slightly sharp* but nonetheless rather well, and note that it excels on the 2.13.21.34.55.89.144 (etc.) subgroup. Notice also the factorization of these numbers, 21=3x7 34=2x17 55=5x11 144=9x16 89=prime. *Why is this important? It seems that when the generator exceeds phi, the lower fibonacci harmonics are better approximated, and most likely but I'm unsure that if it undershoots phi, the higher harmonics are better approximated to the exclusion of the lower. But I digress. If there were only a way to tease these fibonacci factors apart.

	Alas there is! Ratios of 2 are the most common, and these are what the octave reduces, from the altissima 144th harmonic to the solid terrestrial 9/8 major second. So while the octave makes good sense as the equivalence, things do still occur, by and by, which may advantageously be reduced by tritaves, etc., so lets not dismiss the option entirely, and to lesser degrees up the number line.		Alas there is! Ratios of 2 are the most common, and these are what the octave reduces, from the altissima 144th harmonic to the solid terrestrial 9/8 major second. So while the octave makes good sense as the equivalence, things do still occur, by and by, which may advantageously be reduced by tritaves, etc., so lets not dismiss the option entirely, and to lesser degrees up the number line.
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	---		---

	Period=[[~~2edo\|~~2edo]]		Period=[[2edo]]

	ET: [[6edo\|6]] (133:233)		ET: [[6edo\|6]] (133:233)
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	---		---

	Period=[[~~3edo\|~~3edo]]		Period=[[3edo]]

	ET: [[36edo\|36]] (33:36)		ET: [[36edo\|36]] (33:36)
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	---		---

	Period=[[~~4edo\|~~4edo]]		Period=[[4edo]]

	ET: [[36edo\|36]] (32:34)		ET: [[36edo\|36]] (32:34)
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	---		---

	Period=[[~~5edo\|~~5edo]]		Period=[[5edo]]

	ET: [[85edo\|~~85edo~~]] (138:163)		ET: [[85edo\|85]] (138:163)

	---		---

	Period=[[~~7edo\|~~7edo]]		Period=[[7edo]]

	ET: [[49edo\|49]] (24:27)		ET: [[49edo\|49]] (24:27)
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	---		---

	Period=[[~~8edo\|~~8edo]]		Period=[[8edo]]

	ET: [[16edo\|16]] (67:83)		ET: [[16edo\|16]] (67:83)
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	---		---

	Period=[[~~11edo\|~~11edo]]		Period=[[11edo]]

	ET: [[33edo\|33]] (30:40)		ET: [[33edo\|33]] (30:40)
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	---		---

	Period=[[~~edt~~\|3/1]]		Period=[[Edt\|3/1]]

	ET: [[16edt\|16edt]] (126:110)		ET: [[16edt\|16edt]] (126:110)
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	---		---

	Period=[[~~2edt\|~~2edt]]		Period=[[2edt]]

	ET: [[16edt\|16edt]] (118:126)		ET: [[16edt\|16edt]] (118:126)
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	---		---

	Period=[[~~3edt\|~~3edt]]		Period=[[3edt]]

	ET: [[9edt\|9edt]] (199:236)		ET: [[9edt\|9edt]] (199:236)
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	---		---

	Period=[[~~5edt\|~~5edt]]		Period=[[5edt]]

	ET: [[25edt\|25edt]] (72:91)		ET: [[25edt\|25edt]] (72:91)
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	---		---

	Period: [[~~6edt\|~~6edt]]		Period: [[6edt]]

	ET: [[48edt\|48edt]] (37:45)		ET: [[48edt\|48edt]] (37:45)
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	---		---

	Period: [[~~9edt\|~~9edt]]		Period: [[9edt]]

	ET: 153edt (122:157)		ET: 153edt (122:157)
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	---		---

	Period: [[~~ed5~~\|5/1]]		Period: [[Ed5\|5/1]]

	[[10ed5\|10ed5]] (259:287)		ET: [[10ed5\|10ed5]] (259:287)

	[[97ed5\|97ed5]] (280:347)		ET: [[97ed5\|97ed5]] (280:347)
	[[Category:~~catalog~~]]
	[[Category:Golden]]		[[Category:Theory]]
			[[Category:Golden ratio]]

Fredg999: Categories, removed "fake header"

2021-08-14T18:52:55Z

Categories, removed "fake header"

← Older revision		Revision as of 18:52, 14 August 2021
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	~~<ul><li>Phi as a Generator</li></ul>~~

	Musically, the golden ratio is approximately 833.0903 cents. Just as the ratios of adjacent Fibonacci numbers approximate the golden ratio, the golden ratio in turn may be used to represent those ratios, and, equating whatever subset of these ratios one obtains various commas. Furthermore, the exponents of phi approximate the [http://en.wikipedia.org/wiki/Lucas_number Lucas numbers], closely allied with the Fibonacci numbers, with increasing accuracy, which can be put to good effect in a temperament. Furthermore, the square root of phi, 416.54515, generates the [[Kleismic_family#Sqrtphi\|sqrtphi temperament]], a complex, accurate temperament extending into the higher prime limits, and this contains the phi generated temperament within it.		Musically, the golden ratio is approximately 833.0903 cents. Just as the ratios of adjacent Fibonacci numbers approximate the golden ratio, the golden ratio in turn may be used to represent those ratios, and, equating whatever subset of these ratios one obtains various commas. Furthermore, the exponents of phi approximate the [http://en.wikipedia.org/wiki/Lucas_number Lucas numbers], closely allied with the Fibonacci numbers, with increasing accuracy, which can be put to good effect in a temperament. Furthermore, the square root of phi, 416.54515, generates the [[Kleismic_family#Sqrtphi\|sqrtphi temperament]], a complex, accurate temperament extending into the higher prime limits, and this contains the phi generated temperament within it.

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	[[97ed5\|97ed5]] (280:347)		[[97ed5\|97ed5]] (280:347)
	[[Category:catalog]]		[[Category:catalog]]
	[[Category:~~phi~~]]		[[Category:Golden]]

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

Show changes

Wikispaces>guest: Imported revision 330710882 - Original comment:

2012-05-06T13:42:51Z

**Imported revision 330710882 - Original comment: **

← Older revision		Revision as of 13:42, 6 May 2012
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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:~~genewardsmith~~\|~~genewardsmith~~]] and made on <tt>~~2011~~-12-31 13:00:44 UTC</tt>.<br>		: This revision was by author [[User:guest\|guest]] and made on <tt>2012-05-06 13:42:51 UTC</tt>.<br>
	: The original revision id was <tt>~~288919029~~</tt>.<br>		: The original revision id was <tt>330710882</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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	Alas there is! Ratios of 2 are the most common, and these are what the octave reduces, from the altissima 144th harmonic to the solid terrestrial 9/8 major second. So while the octave makes good sense as the equivalence, things do still occur, by and by, which may advantageously be reduced by tritaves, etc., so lets not dismiss the option entirely, and to lesser degrees up the number line.		Alas there is! Ratios of 2 are the most common, and these are what the octave reduces, from the altissima 144th harmonic to the solid terrestrial 9/8 major second. So while the octave makes good sense as the equivalence, things do still occur, by and by, which may advantageously be reduced by tritaves, etc., so lets not dismiss the option entirely, and to lesser degrees up the number line.

	While this is but a rank-2 temperament, suppose more generators could be added, apply phi temperament to arbitrarily higher ranks? For the price of a 3/1 you get a 7/1 and so on. That's an interesting idea on its own but it gets even much better, when one considers periods. The period, as I imagine it and maybe I'm way off the mark mathematically, can be seen as an abstract, degenerate rank. It might not be immediately so, 600 cents hardly fills in for 701.955, but eventually it gets there, at very least with phi tunings. For the price of complexity one gets a different kind of simplicity.		While this is but a rank-2 temperament, suppose more generators could be added, apply phi temperament to arbitrarily higher ranks? For the price of a 3/1 you get a 7/1 and so on. That's an interesting idea on its own but it gets even much better, when one considers periods. The period, as I imagine it and maybe I'm way off the mark mathematically, can be seen as an abstract, degenerate rank. It might not be immediately so, 600 cents hardly fills in for 701.955, but eventually it gets there, at very least with phi tunings. For the price of complexity one gets a different kind of simplicity.

	(Heck maybe this is mathematical gibberish for you REAL mathematicians but it works for and makes intuitive sense to me.)		(Heck maybe this is mathematical gibberish for you REAL mathematicians but it works for and makes intuitive sense to me.)
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	ET: [[23edo\|23]] (30:69)		ET: [[23edo\|23]] (30:69)
	ET: [[36edo\|36]] (30:39) best value!		ET: [[36edo\|36]] (30:39) best value!
	ET: [[~~119edo~~\|~~119~~]] (8.749 : 12.675)		ET: [[121edo\|121]] (8.749 : 12.675)

	---		---
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	Alas there is! Ratios of 2 are the most common, and these are what the octave reduces, from the altissima 144th harmonic to the solid terrestrial 9/8 major second. So while the octave makes good sense as the equivalence, things do still occur, by and by, which may advantageously be reduced by tritaves, etc., so lets not dismiss the option entirely, and to lesser degrees up the number line.<br />		Alas there is! Ratios of 2 are the most common, and these are what the octave reduces, from the altissima 144th harmonic to the solid terrestrial 9/8 major second. So while the octave makes good sense as the equivalence, things do still occur, by and by, which may advantageously be reduced by tritaves, etc., so lets not dismiss the option entirely, and to lesser degrees up the number line.<br />
	<br />		<br />
	While this is but a rank-2 temperament, suppose more generators could be added, apply phi temperament to arbitrarily higher ranks? For the price of a 3/1 you get a 7/1 and so on. That's an interesting idea on its own but it gets even much better, when one considers periods. The period, as I imagine it and maybe I'm way off the mark mathematically, can be seen as an abstract, degenerate rank. It might not be immediately so, 600 cents hardly fills in for 701.955, but eventually it gets there, at very least with phi tunings. For the price of complexity one gets a different kind of simplicity. <br />		While this is but a rank-2 temperament, suppose more generators could be added, apply phi temperament to arbitrarily higher ranks? For the price of a 3/1 you get a 7/1 and so on. That's an interesting idea on its own but it gets even much better, when one considers periods. The period, as I imagine it and maybe I'm way off the mark mathematically, can be seen as an abstract, degenerate rank. It might not be immediately so, 600 cents hardly fills in for 701.955, but eventually it gets there, at very least with phi tunings. For the price of complexity one gets a different kind of simplicity.<br />
	<br />		<br />
	(Heck maybe this is mathematical gibberish for you REAL mathematicians but it works for and makes intuitive sense to me.)<br />		(Heck maybe this is mathematical gibberish for you REAL mathematicians but it works for and makes intuitive sense to me.)<br />
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	ET: <a class="wiki_link" href="/23edo">23</a> (30:69)<br />		ET: <a class="wiki_link" href="/23edo">23</a> (30:69)<br />
	ET: <a class="wiki_link" href="/36edo">36</a> (30:39) best value!<br />		ET: <a class="wiki_link" href="/36edo">36</a> (30:39) best value!<br />
	ET: <a class="wiki_link" href="/~~119edo~~">~~119~~</a> (8.749 : 12.675)<br />		ET: <a class="wiki_link" href="/121edo">121</a> (8.749 : 12.675)<br />
	<br />		<br />
	---<br />		---<br />