Golden meantone - Revision history

Lucius Chiaraviglio: /* Music */ John Bull's ''Fantasia «Ut Re Mi Fa Sol La»'' (late 1500s/early 1600s, from ''Fitzwilliam Virginal Book Vol.1 No.51''): Add rendering in 2026 by Claudi Meneghin in pure golden meantone

2026-05-16T19:15:16Z

Music: John Bull's ''Fantasia «Ut Re Mi Fa Sol La»'' (late 1500s/early 1600s, from ''Fitzwilliam Virginal Book Vol.1 No.51''): Add rendering in 2026 by Claudi Meneghin in pure golden meantone

← Older revision		Revision as of 19:15, 16 May 2026
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	; {{W\|John Bull (composer)\|John Bull}}		; {{W\|John Bull (composer)\|John Bull}}
	* ~~[https://www.youtube.com/watch?v=Ku32F-zEtmU~~ ''Fantasia «Ut Re Mi Fa Sol La»''] (late 1500s/early 1600s, from ''Fitzwilliam Virginal Book Vol.1 No.51'') ~~– rendered by Claudi Meneghin (2020) in~~ a 24 note per octave well-tempered tuning system that uses both golden meantone fifths and [[peppermint]] fifths (tuning specification in video description).		* ''Fantasia «Ut Re Mi Fa Sol La»'' (late 1500s/early 1600s, from ''Fitzwilliam Virginal Book Vol.1 No.51'')
			** [https://www.youtube.com/watch?v=Ku32F-zEtmU In a 24 note per octave well-tempered tuning system] that uses both golden meantone fifths and [[peppermint]] fifths (tuning specification in video description) – rendered by [[Claudi Meneghin]] (2020)
			** [https://www.youtube.com/watch?v=_R-CQp1Bsss In golden meantone (by itself)] – rendered by [[Claudi Meneghin]] (2026)

	; {{W\|Johann Pachelbel}}		; {{W\|Johann Pachelbel}}

Lucius Chiaraviglio: /* Modern renderings */ Add John Bull's ''Fantasia «Ut Re Mi Fa Sol La»'' (late 1500s/early 1600s, from ''Fitzwilliam Virginal Book Vol.1 No.51'') – rendered by Claudi Meneghin (2020) in a 24 note per octave well-tempered system that combines golden meantone and peppermint

2026-05-15T04:31:42Z

Modern renderings: Add John Bull's ''Fantasia «Ut Re Mi Fa Sol La»'' (late 1500s/early 1600s, from ''Fitzwilliam Virginal Book Vol.1 No.51'') – rendered by Claudi Meneghin (2020) in a 24 note per octave well-tempered system that combines golden meantone and peppermint

← Older revision		Revision as of 04:31, 15 May 2026
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	; {{W\|Johann Sebastian Bach}}		; {{W\|Johann Sebastian Bach}}
	* [https://www.youtube.com/watch?v=fwAsBxE7pj0 ''"Ricercar a 6" from The Musical Offering, BWV 1079] (1747) – tuned into golden meantone by [[Claudi Meneghin]] (2021)		* [https://www.youtube.com/watch?v=fwAsBxE7pj0 ''"Ricercar a 6" from The Musical Offering, BWV 1079] (1747) – tuned into golden meantone by [[Claudi Meneghin]] (2021)

			; {{W\|John Bull (composer)\|John Bull}}
			* [https://www.youtube.com/watch?v=Ku32F-zEtmU ''Fantasia «Ut Re Mi Fa Sol La»''] (late 1500s/early 1600s, from ''Fitzwilliam Virginal Book Vol.1 No.51'') – rendered by Claudi Meneghin (2020) in a 24 note per octave well-tempered tuning system that uses both golden meantone fifths and [[peppermint]] fifths (tuning specification in video description).

	; {{W\|Johann Pachelbel}}		; {{W\|Johann Pachelbel}}

Lucius Chiaraviglio: Change spaces to non-breaking spaces in scale IDs to avoid having them possibly split across line wraps

2026-03-04T10:47:19Z

Change spaces to non-breaking spaces in scale IDs to avoid having them possibly split across line wraps

← Older revision		Revision as of 10:47, 4 March 2026
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	cents, approximately 696.214 cents.		cents, approximately 696.214 cents.

	This can be approached by successively taking soft child [[mos scale]]s of pentic ([[2L 3s]], [[5L 2s]], [[7L 5s]], [[12L 7s]], [[19L 12s]], etc). Each time, the generator range "narrows in" on the golden diatonic generator.		This can be approached by successively taking soft child [[mos scale]]s of pentic ([[2L 3s]], [[5L 2s]], [[7L 5s]], [[12L 7s]], [[19L 12s]], etc). Each time, the generator range "narrows in" on the golden diatonic generator.

	Equivalently, [[edo]] systems in a Fibonacci style recurrence beginning with 7 and 12 are successively better approximations to this ideal.		Equivalently, [[edo]] systems in a Fibonacci style recurrence beginning with 7 and 12 are successively better approximations to this ideal.

FloraC: Wording & style

2026-03-04T10:23:39Z

Wording & style

← Older revision		Revision as of 10:23, 4 March 2026
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	'''Golden meantone''' (or "golden diatonic" temperament-agnostically) is based on making the relation between the whole tone and diatonic semitone intervals be the [[~~Golden~~ ratio~~\|Golden Ratio~~]]		'''Golden meantone''' (or "golden diatonic" temperament-agnostically) is based on making the relation between the [[whole tone]] and [[diatonic semitone]] intervals be the [[golden ratio]]

	<math>\varphi = \frac 1 2 (\sqrt{5}+1) \approx 1.61803\,39887\ldots\,</math>		<math>\varphi = \frac 1 2 (\sqrt{5}+1) \approx 1.61803\,39887\ldots\,</math>

	This makes the ~~Golden~~ fifth exactly		This makes the golden fifth exactly

	<math>(8 - \varphi) / 11</math>		<math>(8 - \varphi) / 11</math>
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	cents, approximately 696.214 cents.		cents, approximately 696.214 cents.

	This can be approached by successively taking soft child ~~MOSes~~ of pentic (2L 3s, 5L 2s, 7L 5s, 12L 7s, 19L 12s, etc). Each time, the generator range "narrows in" on the golden diatonic generator.		This can be approached by successively taking soft child [[mos scale]]s of pentic ([[2L 3s]], [[5L 2s]], [[7L 5s]], [[12L 7s]], [[19L 12s]], etc). Each time, the generator range "narrows in" on the golden diatonic generator.

	Equivalently, ~~EDO~~ systems in a Fibonacci style recurrence beginning with 7 and 12 are successively better approximations to this ideal.		Equivalently, [[edo]] systems in a Fibonacci style recurrence beginning with 7 and 12 are successively better approximations to this ideal.

	The process behind constructing golden meantone can be [[~~Golden~~ sequences and tuning\|generalized]] to other ~~MOSes~~.		The process behind constructing golden meantone can be [[golden sequences and tuning\|generalized]] to other mos scales.

	== Construction ==		== Construction ==
	Golden meantone is approximated with increasing accuracy by the infinite sequence of ~~temperaments~~ indicated in the table below. In any [[meantone]] ~~temperament~~ the intervals in the column headings form part of a Fibonacci sequence (in the sense that each adjacent pair sums to the interval to its immediate right) and in these equal temperaments the sizes of these intervals (expressed in step units) are consecutive numbers from the integer Fibonacci sequence 0, 1, 1, 2, 3, 5... Both the rows and the columns of the table form Fibonacci sequences. As the ~~Octave~~ is the ~~Sum~~ of two ~~Fourth~~ intervals and a ~~Tone~~ this can be rearranged as the ~~Sum~~ of the first five intervals in this table, the sequence of ~~EDOs~~ is a Fibonacci like sequence where terms are the sum of 5 consecutive Fibonacci numbers.		Golden meantone is approximated with increasing accuracy by the infinite sequence of tunings indicated in the table below. In any [[meantone]] tuning the intervals in the column headings form part of a Fibonacci sequence (in the sense that each adjacent pair sums to the interval to its immediate right) and in these equal temperaments the sizes of these intervals (expressed in step units) are consecutive numbers from the integer Fibonacci sequence 0, 1, 1, 2, 3, 5, …. Both the rows and the columns of the table form Fibonacci sequences. As the octave is the sum of two fourth intervals and a tone this can be rearranged as the sum of the first five intervals in this table, the sequence of edos is a Fibonacci like sequence where terms are the sum of 5 consecutive Fibonacci numbers.

	As the cardinality increases the interval sequence better approximates a geometric progression. 81edo marks the point at which the series ceases to display audible differences and approximates all of theses intervals within 1 cent. For 131edo and further, the best ~~5th~~ can no longer be used as the generating interval for a golden meantone tuning.		As the cardinality increases the interval sequence better approximates a geometric progression. [[81edo]] marks the point at which the series ceases to display audible differences and approximates all of theses intervals within 1 cent. For [[131edo]] and further, the best fifth can no longer be used as the generating interval for a golden meantone tuning.

	{\| class="wikitable"		{\| class="wikitable"
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	\|}		\|}

	The success of ~~Golden~~ meantone can be understood in terms of the properties of [[~~Logarithmic~~ approximants\|quadratic approximants]] (q.v.) and the small size of the [[32805/32768\|schisma]].		The success of golden meantone can be understood in terms of the properties of [[logarithmic approximants\|quadratic approximants]] (q.v.) and the small size of the [[32805/32768\|schisma]].

	== Evaluation ==		== Evaluation ==
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	== Music ==		== Music ==
			=== Modern renderings ===
	=== Modern ~~Renderings~~ ===		; {{W\|Johann Sebastian Bach}}

	; {{w\|Johann Sebastian Bach}}
	* [https://www.youtube.com/watch?v=fwAsBxE7pj0 ''"Ricercar a 6" from The Musical Offering, BWV 1079] (1747) – tuned into golden meantone by [[Claudi Meneghin]] (2021)		* [https://www.youtube.com/watch?v=fwAsBxE7pj0 ''"Ricercar a 6" from The Musical Offering, BWV 1079] (1747) – tuned into golden meantone by [[Claudi Meneghin]] (2021)

	; {{w\|Johann Pachelbel}}		; {{W\|Johann Pachelbel}}
	* [http://www.io.com/~hmiller/midi/canon-golden.mid An acoustic experience]{{dead link}} - Kornerup himself had no chance to have it. In the [[Warped Canon]] collection.		* [http://www.io.com/~hmiller/midi/canon-golden.mid An acoustic experience]{{dead link}} - Kornerup himself had no chance to have it. In the [[Warped Canon]] collection.

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	=== 21st Century ===		=== 21st Century ===

	; [[Alex Ness]]		; [[Alex Ness]]
	* [https://drive.google.com/drive/folders/0BwsXD8q2VCYURkRyZGZJbHhOaUk ''Liber Abaci''] (archived 2017), based on successive equal-tempered approximations of the Golden Meantone temperament		* [https://drive.google.com/drive/folders/0BwsXD8q2VCYURkRyZGZJbHhOaUk ''Liber Abaci''] (archived 2017), based on successive equal-tempered approximations of the Golden Meantone temperament
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	[[Category:Meantone]]		[[Category:Meantone]]
	[[Category:Golden meantone\| ]]<!-- ~~Main~~ article -->		[[Category:Golden meantone\| ]] <!-- main article -->
	[[Category:Historical]]		[[Category:Historical]]

VectorGraphics: added a formulation of the golden fifth's size that is more intuitively connected to the diatonic scale

2025-05-27T01:12:17Z

added a formulation of the golden fifth's size that is more intuitively connected to the diatonic scale

← Older revision		Revision as of 01:12, 27 May 2025
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	<math>(8 - \varphi) / 11</math>		<math>(8 - \varphi) / 11</math>

	octave, or		or

			<math>(3\varphi + 1) / (5\varphi + 2) </math>

			octave, equivalently

	<math>(9600 - 1200 \varphi) / 11</math>		<math>(9600 - 1200 \varphi) / 11</math>

Lucius Chiaraviglio: /* Music */ Attempt to get format to conform with other Music sections; reorganize to have Modern Renderings sub-section; add Mozart's Gigue KV 574 for Harpsichord (Golden Meantone) rendered by Claudi Meneghin

2025-05-26T02:12:32Z

Music: Attempt to get format to conform with other Music sections; reorganize to have Modern Renderings sub-section; add Mozart's Gigue KV 574 for Harpsichord (Golden Meantone) rendered by Claudi Meneghin

← Older revision		Revision as of 02:12, 26 May 2025
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	== Music ==		== Music ==
	* [http://www.io.com/~hmiller/midi/canon-golden.mid An acoustic experience]{{dead link}} - Kornerup himself had no chance to have it ~~– is contained in~~ the [[Warped Canon]] collection.
	* {{w\|~~J. S. Bach~~}}'s [https://www.youtube.com/watch?v=~~fwAsBxE7pj0 "Ricercar a 6" from~~ ''~~The Musical Offering~~''~~, BWV 1079~~] ~~(1747)~~ – tuned into golden meantone by [[Claudi Meneghin]] (~~2021~~)		=== Modern Renderings ===
	* [https://drive.google.com/drive/folders/0BwsXD8q2VCYURkRyZGZJbHhOaUk ''Liber Abaci''] (archived 2017) ~~– composition by [[Alex Ness]]~~, based on successive equal-tempered approximations of the Golden Meantone temperament
			; {{w\|Johann Sebastian Bach}}
			* [https://www.youtube.com/watch?v=fwAsBxE7pj0 ''"Ricercar a 6" from The Musical Offering, BWV 1079] (1747) – tuned into golden meantone by [[Claudi Meneghin]] (2021)

			; {{w\|Johann Pachelbel}}
			* [http://www.io.com/~hmiller/midi/canon-golden.mid An acoustic experience]{{dead link}} - Kornerup himself had no chance to have it. In the [[Warped Canon]] collection.

			; {{W\|Wolfgang Amadeus Mozart}}
			* [https://www.youtube.com/watch?v=PSaFMcrivyY&list=PLC6ZSKWKnVz0mOTLQkCUi9ydWGLpBP8gZ&index=3 ''Mozart's Gigue KV 574 for Harpsichord [Golden Meantone''] – tuned into golden meantone by [[Claudi Meneghin]] (2017)

			=== 21st Century ===

			; [[Alex Ness]]
			* [https://drive.google.com/drive/folders/0BwsXD8q2VCYURkRyZGZJbHhOaUk ''Liber Abaci''] (archived 2017), based on successive equal-tempered approximations of the Golden Meantone temperament

	== See also ==		== See also ==

VectorGraphics at 10:01, 7 May 2025

2025-05-07T10:01:33Z

← Older revision		Revision as of 10:01, 7 May 2025
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	cents, approximately 696.214 cents.		cents, approximately 696.214 cents.

	EDO systems in a Fibonacci style recurrence beginning with 7 and 12 are successively better approximations to this ideal.		This can be approached by successively taking soft child MOSes of pentic (2L 3s, 5L 2s, 7L 5s, 12L 7s, 19L 12s, etc). Each time, the generator range "narrows in" on the golden diatonic generator.

			Equivalently, EDO systems in a Fibonacci style recurrence beginning with 7 and 12 are successively better approximations to this ideal.

			The process behind constructing golden meantone can be [[Golden sequences and tuning\|generalized]] to other MOSes.

	== Construction ==		== Construction ==

VectorGraphics: Added the "odd-limit" clarification as bracketed text in the quote instead of a separate line

2025-04-13T21:18:07Z

Added the "odd-limit" clarification as bracketed text in the quote instead of a separate line

← Older revision		Revision as of 21:18, 13 April 2025
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	'''Golden meantone''' is based on making the relation between the whole tone and diatonic semitone intervals be the [[Golden ratio\|Golden Ratio]]		'''Golden meantone''' (or "golden diatonic" temperament-agnostically) is based on making the relation between the whole tone and diatonic semitone intervals be the [[Golden ratio\|Golden Ratio]]

	<math>\varphi = \frac 1 2 (\sqrt{5}+1) \approx 1.61803\,39887\ldots\,</math>		<math>\varphi = \frac 1 2 (\sqrt{5}+1) \approx 1.61803\,39887\ldots\,</math>
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	== Evaluation ==		== Evaluation ==
	<blockquote>		<blockquote>
	I think of this as the standard melodic meantone because the all these ratios are the same. It has the mellow sound of 1/4 comma, but does still have a character of its own. Some algorithms make this almost exactly the optimum 5-limit tuning. It's fairly good as a 7-limit tuning as well. Almost the optimum (according to me) for diminished sevenths. I toyed with this as a guitar tuning, but rejected it because 4:6:9 chords aren't quite good enough. That is, the poor fifth leads to a sludgy major ninth.		I think of this as the standard melodic meantone because the all these ratios are the same. It has the mellow sound of 1/4 comma, but does still have a character of its own. Some algorithms make this almost exactly the optimum 5[-odd]-limit tuning. It's fairly good as a 7[-odd]-limit tuning as well. Almost the optimum (according to me) for diminished sevenths. I toyed with this as a guitar tuning, but rejected it because 4:6:9 chords aren't quite good enough. That is, the poor fifth leads to a sludgy major ninth.
	</blockquote>		</blockquote>
	—[http://x31eq.com/meantone.htm#pop Graham Breed]		—[http://x31eq.com/meantone.htm#pop Graham Breed]

	~~Note that the author uses “''n''-limit” to mean what “''n''-[[odd-limit]]” means on this wiki.~~

	== Music ==		== Music ==

Zhenlige: /* Evaluation */

2025-03-19T05:54:06Z

Evaluation

← Older revision		Revision as of 05:54, 19 March 2025
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	</blockquote>		</blockquote>
	—[http://x31eq.com/meantone.htm#pop Graham Breed]		—[http://x31eq.com/meantone.htm#pop Graham Breed]

			Note that the author uses “''n''-limit” to mean what “''n''-[[odd-limit]]” means on this wiki.

	== Music ==		== Music ==

BudjarnLambeth at 08:09, 7 November 2024

2024-11-07T08:09:26Z

← Older revision		Revision as of 08:09, 7 November 2024
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	'''Golden meantone''' is based on making the relation between the whole tone and diatonic semitone intervals be the [[~~wikipedia:~~Golden ~~Ratio~~\|Golden Ratio]]		'''Golden meantone''' is based on making the relation between the whole tone and diatonic semitone intervals be the [[Golden ratio\|Golden Ratio]]

	<math>\varphi = \frac 1 2 (\sqrt{5}+1) \approx 1.61803\,39887\ldots\,</math>		<math>\varphi = \frac 1 2 (\sqrt{5}+1) \approx 1.61803\,39887\ldots\,</math>