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Golden meantone: Difference between revisions

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added a formulation of the golden fifth's size that is more intuitively connected to the diatonic scale
 
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'''Golden Meantone''' is based on making the relation between the whole tone and diatonic semitone intervals be the [http://en.wikipedia.org/wiki/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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<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>
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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.


==Construction==
The process behind constructing golden meantone can be [[Golden sequences and tuning|generalized]] to other MOSes.
Golden Meantone is approximated with increasing accuracy by the infinite sequence of temperaments indicated in the table below. In any meantone temperament the five 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, and because the five intervals sums to an octave, the octave cardinalities in the first column are formed by summing the five numbers to their right. As the cardinality increases the interval sequence better approximates a geometric progression.
 
== 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.  
 
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.


{| class="wikitable"
{| class="wikitable"
! Temperament
! Chroma
! Semitone
! Tone
! Minor third
! Fourth
! Minor sixth
|-
|-
| | <span style="color: #ffffff;"># </span>''Temperament''<span style="color: #ffffff;"># </span>
| [[7edo]]
| | <span style="color: #ffffff;"># </span>''chroma''<span style="color: #ffffff;"># </span>
| 0
| | <span style="color: #ffffff;">#</span>''semitone''<span style="color: #ffffff;"># </span>
| 1
| | <span style="color: #ffffff;">#</span>''tone''<span style="color: #ffffff;"># </span>
| 1
| | <span style="color: #ffffff;">#</span>''minor third''<span style="color: #ffffff;"># </span>
| 2
| | <span style="color: #ffffff;">#</span>''fourth''<span style="color: #ffffff;">#</span>
| 3
| 5
|-
|-
| | <span style="color: #ffffff;"># [[7edo|7edo]]</span>
| [[12edo]]
| | <span style="color: #ffffff;"># </span>0
| 1
| | <span style="color: #ffffff;">#</span>1
| 1
| | <span style="color: #ffffff;">#</span>1
| 2
| | <span style="color: #ffffff;">#</span>2
| 3
| | <span style="color: #ffffff;">#</span>3
| 5
| 8
|-
|-
| | <span style="color: #ffffff;"># [[12edo|12edo]]</span>
| [[19edo]]
| | <span style="color: #ffffff;"># </span>1
| 1
| | <span style="color: #ffffff;">#</span>1
| 2
| | <span style="color: #ffffff;">#</span>2
| 3
| | <span style="color: #ffffff;">#</span>3
| 5
| | <span style="color: #ffffff;">#</span>5
| 8
| 13
|-
|-
| | <span style="color: #ffffff;"># [[19edo|19edo]]</span>
| [[31edo]]
| | <span style="color: #ffffff;"># </span>1
| 2
| | <span style="color: #ffffff;">#</span>2
| 3
| | <span style="color: #ffffff;">#</span>3
| 5
| | <span style="color: #ffffff;">#</span>5
| 8
| | <span style="color: #ffffff;">#</span>8
| 13
| 21
|-
|-
| | <span style="color: #ffffff;"><span style="color: #ffffff;"># </span>[[31edo|31edo]]</span>
| [[50edo]]
| | <span style="color: #ffffff;"># </span>2
| 3
| | <span style="color: #ffffff;">#</span>3
| 5
| | <span style="color: #ffffff;">#</span>5
| 8
| | <span style="color: #ffffff;">#</span>8
| 13
| | <span style="color: #ffffff;">#</span>13
| 21
| 34
|-
|-
| | <span style="color: #ffffff;"># [[50edo|50edo]]</span>
| [[81edo]]
| | <span style="color: #ffffff;"># </span>3
| 5
| | <span style="color: #ffffff;">#</span>5
| 8
| | <span style="color: #ffffff;">#</span>8
| 13
| | <span style="color: #ffffff;">#</span>13
| 21
| | <span style="color: #ffffff;">#</span>21
| 34
| 55
|-
|-
| | <span style="color: #ffffff;"># [[81edo|81edo]]</span>
| [[131edo]]
| | <span style="color: #ffffff;"># </span>5
| 8
| | <span style="color: #ffffff;">#</span>8
| 13
| | <span style="color: #ffffff;">#</span>13
| 21
| | <span style="color: #ffffff;">#</span>21
| 34
| | <span style="color: #ffffff;">#</span>34
| 55
| 89
|-
|-
| | <span style="color: #ffffff;"># [[131edo|131edo]]</span>
| ...
| | <span style="color: #ffffff;"># </span>8
| ...
| | <span style="color: #ffffff;">#</span>13
| ...
| | <span style="color: #ffffff;">#</span>21
| ...
| | <span style="color: #ffffff;">#</span>34
| ...
| | <span style="color: #ffffff;">#</span>55
| ...
|-
| ...
| | <span style="color: #ffffff;"># </span>...
| | <span style="color: #ffffff;"># </span>...
| | ...
| | ...
| | <span style="color: #ffffff;">#</span>...
| | <span style="color: #ffffff;">#</span>...
|}
|}


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 ==
<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]


Note: If reading this above chart, you may be interested in the variety of EDOs that support Meantone in general. There are several not included in the above 'Golden Meantone' sequence. Excluding EDOs such as 62 (31x2), here is a complete list of EDOs 131 and under that support Meantone accurately: (1/4 Comma),112, 81, 131, 31, 50, 119, 69, 88, 107, 126, (1/3 Comma,) 19. They are listed in order of ''overall'' accuracy to 1/4 comma which also equates to overall Just accuracy.
== Music ==


==Evaluation==
=== Modern Renderings ===


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.''
; {{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)


==Listening==
; {{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] - Kornerup himself had no chance to have it - is contained in the [[Warped_Canon|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)


[http://soonlabel.com/xenharmonic/archives/692 Bach's Ricercar a 6] - Tuned into golden meantone by [[Claudi_Meneghin|Claudi Meneghin]]
=== 21st Century ===


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


==Additional reading==
== See also ==
* [[Das Goldene Tonsystem|''Das Goldene Tonsystem'']]
* [[Logarithmic approximants #Golden temperaments]]
* [[Mercury meantone]] – effectively a stretched-octaves version of golden meantone


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


[https://en.xen.wiki/w/Logarithmic_approximants#Golden_temperaments Golden temperaments]
[[Category:Meantone]]
[[Category:Golden meantone| ]]<!-- Main article -->
[[Category:Historical]]
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