Golden meantone: Difference between revisions

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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''' (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>

Line 7:

<math>(8 - \varphi) / 11</math>

octave, or

or

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

octave, equivalently

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

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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 ==

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== Evaluation ==

~~Graham Breed [http://x31eq.com/meantone.htm writes]: ''~~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>

—[http://x31eq.com/meantone.htm#pop Graham Breed]

== Music ==

=== Modern Renderings ===

; {{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

== ~~Listening~~ ==

== See also ==

* [~~http://www.io.com/~hmiller/midi/canon-golden.mid An acoustic experience] - Kornerup himself had no chance to have it - is contained in the~~ [~~[Warped canon~~]] ~~collection.~~

* [[Das Goldene Tonsystem|''Das Goldene Tonsystem'']]

* [~~https://www.youtube.com/watch?v=fwAsBxE7pj0 Bach's Ricercar a 6] - Tuned into golden meantone by~~ [~~[Claudi Meneghin~~]]

* [[Logarithmic approximants #Golden temperaments]]

* [~~https://drive.google.com/drive/folders/0BwsXD8q2VCYURkRyZGZJbHhOaUk Liber Abaci~~] - ~~Composition by Alex Ness, based on successive equal-tempered approximations~~ of ~~the Golden Meantone temperament~~

* [[Mercury meantone]] – effectively a stretched-octaves version of golden meantone

== ~~Additional reading~~ ==

== External links ==

* [http://www.tonalsoft.com/enc/g/golden.aspx ~~Golden meantone -~~ Tonalsoft ~~encyclopedia]~~

* [http://www.tonalsoft.com/enc/g/golden.aspx Tonalsoft Encyclopedia | ''Golden meantone'']

* [https://en.xen.wiki/w/Logarithmic_approximants#Golden_temperaments Golden ~~temperaments]~~

* [[Mercury meantone]~~] - effectively a stretched-octaves version of golden meantone~~

[[Category:Meantone]]

[[Category:Golden meantone| ]]

[[Category:Historical]]

@@ Line 1: / Line 1: @@
-'''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''' (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>
@@ Line 7: / Line 7: @@
 <math>(8 - \varphi) / 11</math>
-octave, or
+or
+<math>(3\varphi + 1) / (5\varphi + 2) </math>
+octave, equivalently
 <math>(9600 - 1200 \varphi) / 11</math>
@@ Line 13: / Line 17: @@
 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 ==
@@ Line 97: / Line 105: @@
 == Evaluation ==
-Graham Breed [http://x31eq.com/meantone.htm writes]: ''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.''
+<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[-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>
+—[http://x31eq.com/meantone.htm#pop Graham Breed]
+== Music ==
+=== Modern Renderings ===
+; {{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
-== Listening ==
+== See also ==
-* [http://www.io.com/~hmiller/midi/canon-golden.mid An acoustic experience] - Kornerup himself had no chance to have it - is contained in the [[Warped canon]] collection.
+* [[Das Goldene Tonsystem|''Das Goldene Tonsystem'']]
-* [https://www.youtube.com/watch?v=fwAsBxE7pj0 Bach's Ricercar a 6] - Tuned into golden meantone by [[Claudi Meneghin]]
+* [[Logarithmic approximants #Golden temperaments]]
-* [https://drive.google.com/drive/folders/0BwsXD8q2VCYURkRyZGZJbHhOaUk Liber Abaci] - Composition by Alex Ness, based on successive equal-tempered approximations of the Golden Meantone temperament
+* [[Mercury meantone]] – effectively a stretched-octaves version of golden meantone
-== Additional reading ==
+== External links ==
-* [http://www.tonalsoft.com/enc/g/golden.aspx Golden meantone - Tonalsoft encyclopedia]
+* [http://www.tonalsoft.com/enc/g/golden.aspx Tonalsoft Encyclopedia | ''Golden meantone'']
-* [https://en.xen.wiki/w/Logarithmic_approximants#Golden_temperaments Golden temperaments]
-* [[Mercury meantone]] - effectively a stretched-octaves version of golden meantone
 [[Category:Meantone]]
 [[Category:Golden meantone| ]]<!-- Main article -->
 [[Category:Historical]]