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Quarter-comma meantone: Difference between revisions

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→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'') – performed on Cembalo Universale (harpsichord having 19 notes per octave for extended meantone) by Robert Hill (2015)
 
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{{Wikipedia}}
'''Quarter-comma meantone''' is the tuning of [[meantone]] temperament which makes the perfect fifth ([[3/2]]) the fourth root of 5, or in other words 696.578 [[cent]]s. This means the fifth is flattened by 1/4 of the syntonic comma ([[81/80]] ratio) of 21.506 cents, which is to say by 5.377 cents, hence the name quarter-comma or 1/4-comma meantone. It is the tuning where the major third ([[5/4]]) is tuned pure, and the minor third ([[6/5]]) and the fifth are equally flat by 1/4 syntonic comma. It is also the tuning where the whole tone is the exact [[Geometric mean|logarithmic average]] between the greater tone of [[9/8]] and the lesser tone of [[10/9]], and hence it can be argued it is the only tuning which is strictly "mean tone". It is the minimax tuning for 5-limit meantone, meaning the maximum error on the 5-limit [[tonality diamond]] is minimized. It is also the minimax tuning for septimal meantone in the 7- and 9-limit odd limits, and for meanpop, the version of 11-limit meantone which tunes 11/8 to the doubly diminished fifth, C-Gbb. Moreover, historically it was the predominant tuning of Western common-practice music in the latter part of the Renaissance and the early modern (17th century) era.
'''Quarter-comma meantone''' is the tuning of [[meantone]] temperament which makes the perfect fifth ([[3/2]]) the fourth root of 5, or in other words 696.578 [[cent]]s. This means the fifth is flattened by {{frac|1|4}} of the syntonic comma ([[81/80]] ratio) of 21.506{{cent}}, which is to say by 5.377{{cent}}, hence the name.  


Because of all of these features, it has a certain claim to be considered the canonical meantone tuning and often "meantone" is taken to mean quarter-comma specifically. The traditional interval names, listed below, may be considered to have their older traditional tuning in the quarter-comma meantone tunings listed below. However, other tunings (besides [[12edo]] which is ubiquitous but not an especially good tuning for meantone) may be preferred. Note, for instance, that the doubly augmented second and the doubly diminished fourth are both neutral thirds, about six cents apart. If we want a tuning which will equate the two, we need to use [[31edo]] instead.
Quarter-comma meantone is the tuning where the major third ([[5/4]]) is tuned pure, and the minor third ([[6/5]]) and the fifth are equally flat by {{frac|1|4}} syntonic comma. It is also the tuning where the whole tone is the exact [[geometric mean]] between the greater tone of [[9/8]] and the lesser tone of [[10/9]] (the "semiptolemaic major second" sqrt(5/4)), from which the term "meantone" is derived, and many argue that it is the only tuning which is strictly "mean tone". It is the minimax tuning for 5-limit meantone, meaning the maximum error on the [[5-odd-limit]] [[tonality diamond]] is minimized. It is also the minimax tuning for septimal meantone in the [[7-odd-limit|7-]] and [[9-odd-limit]], and for meanpop (the version of 11-limit meantone which tunes 11/8 to the doubly diminished fifth, C–G𝄫) in the [[11-odd-limit]]. Moreover, historically it was the predominant tuning of Western common-practice music in the latter part of the Renaissance and the early modern (17th century) era.
 
Because of all of these features, it has a certain claim to be considered the canonical meantone tuning and often "meantone" is taken to mean quarter-comma specifically. The traditional interval names, listed below, may be considered to have their older traditional tuning in the quarter-comma meantone tunings listed below. However, other tunings (besides [[12edo]] which is ubiquitous but not an especially good tuning for meantone) may be preferred. Note, for instance, that the doubly augmented second and the doubly diminished fourth are both neutral thirds, about 6{{c}} apart. In [[31edo]], these two intervals are enharmonically equivalent.


== Tuning profile ==
== Tuning profile ==
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[[Tuning map]]: {{val| 1200 1896.5784 2786.3137 3365.7843 }}
[[Tuning map]]: {{val| 1200 1896.5784 2786.3137 3365.7843 }}


[[Mistuning map]]: {{val| 0 -5.3766 0 -3.0416 }}
[[Error map]]: {{val| 0 -5.3766 0 -3.0416 }}


== Intervals of quarter-comma meantone ==
== Intervals of quarter-comma meantone ==
Below is a table listing 36 of the notes of quarter-comma meantone, giving the size in cents, the number of fifths up or down on the [[chain of fifths]], the [[Wikipedia:Letter notation|letter notation]] for the interval relative to C as the tonic, the traditional name for the interval, a shorthand form of the name, and the size of the interval in [[fractional monzo]] notation.
Below is a table listing 36 of the notes of quarter-comma meantone, giving the size in cents, the number of fifths up or down on the [[chain of fifths]], the {{w|letter notation}} for the interval relative to C as the tonic, the traditional name for the interval, a shorthand form of the name, and the size of the interval in [[fractional monzo]] notation.


{| class="wikitable right-1 center-2 center-5"
{| class="wikitable right-1 center-2 center-5"
! Size <br> in [[cent]]s
|-
! Fifths <br> dist.
! Size<br />in [[cent]]s
! Note <br> C-based
! Fifths<br />dist.
! Traditional <br> interval name
! Note<br />C-based
! Int. <br> (short)
! Traditional<br />interval name
! [[Fractional monzo|Fract. <br> Monzo]]
! Int.<br />(short)
! [[Fractional monzo|Fract.<br />monzo]]
|-
|-
| 0.00000
| 0.00000
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| Unison
| Unison
| P1
| P1
| [0 0 0&gt;
| {{monzo| 0 0 0 }}
|-
|-
| 41.0590
| 41.0590
| -12
| −12
| Dbb
| Dbb
| Diminished second, diesis
| Diminished second, diesis
| d2
| d2
| [7 0 -3&gt;
| {{monzo| 7 0 -3 }}
|-
|-
| 76.0490
| 76.0490
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| Augmented unison, chromatic semitone
| Augmented unison, chromatic semitone
| A1
| A1
| [-4 0 7/4&gt;
| {{monzo| -4 0 7/4 }}
|-
|-
| 117.108
| 117.108
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| Minor second, diatonic semitone
| Minor second, diatonic semitone
| m2
| m2
| [3 0 -5/4&gt;
| {{monzo| 3 0 -5/4 }}
|-
|-
| 152.098
| 152.098
| 14
| 14
| C##
| Cx
| Augmented chromatic semitone
| Doubly augmented unison, augmented chromatic semitone
| AA1
| AA1
| [-8 0 7/2&gt;
| {{monzo| -8 0 7/2 }}
|-
|-
| 158.167
| 158.167
| -17
| −17
| Ebbb
| Ebbb
| Doubly diminished third
| Doubly diminished third
| dd3
| dd3
| [10 0 -17/4&gt;
| {{monzo| 10 0 -17/4 }}
|-
|-
| 193.157
| 193.157
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| Major second
| Major second
| M2
| M2
| [-1 0 1/2&gt;
| {{monzo| -1 0 1/2 }}
|-
|-
| 234.216
| 234.216
| -10
| −10
| Ebb
| Ebb
| Diminished third
| Diminished third
| d3
| d3
| [6 0 -5/2&gt;
| {{monzo| 6 0 -5/2 }}
|-
|-
| 269.206
| 269.206
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| Augmented second
| Augmented second
| A2
| A2
| [-5 0 9/4&gt;
| {{monzo| -5 0 9/4 }}
|-
|-
| 310.265
| 310.265
| -3
| −3
| Eb
| Eb
| Minor third
| Minor third
| m3
| m3
| [2 0 -3/4&gt;
| {{monzo| 2 0 -3/4 }}
|-
|-
| 345.255
| 345.255
| 16
| 16
| D##
| Dx
| Doubly augmented second
| Doubly augmented second
| AA2
| AA2
| [-9 0 4&gt;
| {{monzo| -9 0 4 }}
|-
|-
| 351.324
| 351.324
| -15
| −15
| Fbb
| Fbb
| Doubly diminished fourth
| Doubly diminished fourth
| dd4
| dd4
| [9 0 -15/4&gt;
| {{monzo| 9 0 -15/4 }}
|-
|-
| 386.314
| 386.314
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| Major third
| Major third
| M3
| M3
| [-2 0 1&gt;
| {{monzo| -2 0 1 }}
|-
|-
| 427.373
| 427.373
| -8
| −8
| Fb
| Fb
| Diminished fourth
| Diminished fourth
| d4
| d4
| [5 0 -2&gt;
| {{monzo| 5 0 -2 }}
|-
|-
| 462.363
| 462.363
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| Augmented third
| Augmented third
| A3
| A3
| [-6 0 11/4&gt;
| {{monzo| -6 0 11/4 }}
|-
|-
| 503.422
| 503.422
| -1
| −1
| F
| F
| "Perfect" fourth
| Perfect fourth
| P4
| P4
| [1 0 -1/4&gt;
| {{monzo| 1 0 -1/4 }}
|-
|-
| 544.480
| 544.480
| -13
| −13
| Gbb
| Gbb
| Doubly diminished fifth
| Doubly diminished fifth
| dd5
| dd5
| [8 0 -13/4&gt;
| {{monzo| 8 0 -13/4 }}
|-
|-
| 579.471
| 579.471
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| Augmented fourth
| Augmented fourth
| A4
| A4
| [-3 0 3/2&gt;
| {{monzo| -3 0 3/2 }}
|-
|-
| 620.529
| 620.529
| -6
| −6
| Gb
| Gb
| Diminished fifth
| Diminished fifth
| d5
| d5
| [4 0 -3/2&gt;
| {{monzo| 4 0 -3/2 }}
|-
|-
| 655.520
| 655.520
| 13
| 13
| F##
| Fx
| Doubly augmented fourth
| Doubly augmented fourth
| AA4
| AA4
| [-7 0 13/4&gt;
| {{monzo| -7 0 13/4 }}
|-
|-
| 696.578
| 696.578
| 1
| 1
| G
| G
| "Perfect" fifth
| Perfect fifth
| P5
| P5
| [0 0 1/4&gt;
| {{monzo| 0 0 1/4 }}
|-
|-
| 737.637
| 737.637
| -11
| −11
| Abb
| Abb
| Diminshed sixth
| Diminished sixth
| d6
| d6
| [7 0 -11/4&gt;
| {{monzo| 7 0 -11/4 }}
|-
|-
| 772.627
| 772.627
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| Augmented fifth
| Augmented fifth
| A5
| A5
| [-4 0 2&gt;
| {{monzo| -4 0 2 }}
|-
|-
| 813.686
| 813.686
| -4
| −4
| Ab
| Ab
| Minor sixth
| Minor sixth
| m6
| m6
| [3 0 -1&gt;
| {{monzo| 3 0 -1 }}
|-
|-
| 848.676
| 848.676
| 15
| 15
| G##
| Gx
| Doubly augmented fifth
| Doubly augmented fifth
| AA5
| AA5
| [-8 0 15/4&gt;
| {{monzo| -8 0 15/4 }}
|-
|-
| 854.745
| 854.745
| -16
| −16
| Bbbb
| Bbbb
| Doubly diminished seventh
| Doubly diminished seventh
| dd7
| dd7
| [10 0 -4&gt;
| {{monzo| 10 0 -4 }}
|-
|-
| 889.735
| 889.735
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| Major sixth
| Major sixth
| M6
| M6
| [-1 0 3/4&gt;
| {{monzo| -1 0 3/4 }}
|-
|-
| 930.794
| 930.794
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| Diminished seventh
| Diminished seventh
| d7
| d7
| [6 0 -9/4&gt;
| {{monzo| 6 0 -9/4 }}
|-
|-
| 965.784
| 965.784
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| Augmented sixth
| Augmented sixth
| A6
| A6
| [-5 0 5/2&gt;
| {{monzo| -5 0 5/2 }}
|-
|-
| 1006.84
| 1006.84
| -2
| −2
| Bb
| Bb
| Minor seventh
| Minor seventh
| m7
| m7
| [2 0 -1/2&gt;
| {{monzo| 2 0 -1/2 }}
|-
|-
| 1041.83
| 1041.83
| 17
| 17
| A##
| Ax
| Doubly augmented sixth
| Doubly augmented sixth
| AA6
| AA6
| [-9 0 17/4&gt;
| {{monzo| -9 0 17/4 }}
|-
|-
| 1047.90
| 1047.90
| -14
| −14
| C'bb
| C'bb
| Doubly diminished octave
| Doubly diminished octave
| dd8
| dd8
| [9 0 -7/2&gt;
| {{monzo| 9 0 -7/2 }}
|-
|-
| 1082.89
| 1082.89
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| Major seventh
| Major seventh
| M7
| M7
| [-2 0 5/4&gt;
| {{monzo| -2 0 5/4 }}
|-
|-
| 1123.95
| 1123.95
| -7
| −7
| C'b
| C'b
| Diminished octave
| Diminished octave
| d8
| d8
| [5 0 -7/4&gt;
| {{monzo| 5 0 -7/4 }}
|-
|-
| 1158.94
| 1158.94
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| Augmented seventh
| Augmented seventh
| A7
| A7
| [-6 0 3&gt;
| {{monzo| -6 0 3 }}
|-
|-
| 1200.00
| 1200.00
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| Perfect octave
| Perfect octave
| P8
| P8
| [1 0 0&gt;
| {{monzo| 1 0 0 }}
|}
|}


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; {{W|Johann Sebastian Bach}}
; {{W|Johann Sebastian Bach}}
* [https://web.archive.org/web/20201127013537/http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Ricercar%20a%206/Quarter-comma-meantone.mp3 ''Bach's Ricercar a 6 from A Musical Offering''] rendered by Claudi Meneghin
* [https://web.archive.org/web/20201127013537/http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Ricercar%20a%206/Quarter-comma-meantone.mp3 ''Bach's Ricercar a 6 from A Musical Offering''] rendered by Claudi Meneghin
* [https://www.youtube.com/watch?v=2osCf4AQ_0k ''Praeludium et fuga in G, BWV 541'' - J S Bach, in meantone temperament] (late 1500s &ndash; early 1600s; performed on quarter-comma meantone pipe organ by Helena Holmlund, recorded 2013)


; {{W|Chuck Berry}}
; {{W|Chuck Berry}}
* [https://web.archive.org/web/20201127012202/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/reelrock.mp3 ''Reelin' and Rockin'''] (1957)
* [https://web.archive.org/web/20201127012202/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/reelrock.mp3 ''Reelin' and Rockin'''] (1957)
; {{W|John Bull (composer)|John Bull}}
* [https://www.youtube.com/watch?v=hpDkbRpxwKw ''Fantasia «Ut Re Mi Fa Sol La»''] (late 1500s/early 1600s, from ''Fitzwilliam Virginal Book Vol.1 No.51'') – performed on Cembalo Universale (harpsichord having 19 notes per octave for extended meantone) by {{W|Robert_Hill_(musician)|Robert Hill}} (2015)


; [[Giovanni Pietro Del Buono]]
; [[Giovanni Pietro Del Buono]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/quartercomma/Giovanni%20Pietro%20Del%20Buono-%20Tre%20Sonate%20Sopra%20Ave%20Maris%20Stella%20(V,%20VI,%20VII).mp3 ''Tre Sonate Sopra Ave Maris Stella'' (V, VI, VII)] (''c''. 1641–1644)
* [http://micro.soonlabel.com/gene_ward_smith/Others/quartercomma/Giovanni%20Pietro%20Del%20Buono-%20Tre%20Sonate%20Sopra%20Ave%20Maris%20Stella%20(V,%20VI,%20VII).mp3 ''Tre Sonate Sopra Ave Maris Stella'' (V, VI, VII)] (''c''. 1641–1644)
; {{W|Dieterich Buxtehude}}
* [https://www.youtube.com/watch?v=SR0gWBjdzsA ''Magnificat primi toni'' BuxWv 203] (c. 1690; performed on quarter-comma meantone pipe organ by Helena Holmlund, recorded 2013)


; {{W|François Couperin}}
; {{W|François Couperin}}
* [https://web.archive.org/web/20201127015607/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Le%20Tic-Toc%20Choc.mp3 ''Le Tic-Toc Choc''] (1722)
* [https://web.archive.org/web/20201127015607/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Le%20Tic-Toc%20Choc.mp3 ''Le Tic-Toc Choc''] (1722)
; {{W|Ferenc Farkas}}
* [https://www.youtube.com/watch?v=SkZ2FDXMzlA ''The 1699 Meantone Organ from Rupea at the Black Church in Brasov | Farkas Ferenc Hungarian Dances''] (1961; performed in 2024 by {{W|Bálint Karosi}})


; {{W|Sergei Rachmaninoff}}
; {{W|Sergei Rachmaninoff}}
Line 311: Line 324:
; {{W|Arthur Singer}}, {{w|John Medora}} and {{w|David White %28musician%29|David White}}
; {{W|Arthur Singer}}, {{w|John Medora}} and {{w|David White %28musician%29|David White}}
* [https://web.archive.org/web/20201127014551/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/At%20the%20Hop.mp3 ''At the Hop''] (1957)
* [https://web.archive.org/web/20201127014551/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/At%20the%20Hop.mp3 ''At the Hop''] (1957)
; {{W|Jan Pieterszoon Sweelinck}}
* [https://www.youtube.com/watch?v=-_z6lhgLNJI&list=WL&index=238 ''Fantasia chromatica by Sweelinck'', in meantone temperament] (late 1500s &ndash; early 1600s; performed on quarter-comma meantone pipe organ by Helena Holmlund, recorded 2013)


; {{W|Richard Wagner}}
; {{W|Richard Wagner}}
Line 322: Line 338:
* [https://youtu.be/6t5PdeC1Trw ''Country Bumpkin''] (2010) [https://web.archive.org/web/20201127012811/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Country%20Bumpkin%20Live%20At%20The%20Turks%2023,9,2010ce.mp3 play]
* [https://youtu.be/6t5PdeC1Trw ''Country Bumpkin''] (2010) [https://web.archive.org/web/20201127012811/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Country%20Bumpkin%20Live%20At%20The%20Turks%2023,9,2010ce.mp3 play]
* [https://www.youtube.com/watch?v=HVwXUwK5_Vg ''Ozone Layer''] (2012) [https://web.archive.org/web/20201127013628/http://micro.soonlabel.com/gene_ward_smith/Others/quartercomma/Ozone%20Layer.mp3 play]
* [https://www.youtube.com/watch?v=HVwXUwK5_Vg ''Ozone Layer''] (2012) [https://web.archive.org/web/20201127013628/http://micro.soonlabel.com/gene_ward_smith/Others/quartercomma/Ozone%20Layer.mp3 play]
; [[Eufalesio]]
* [https://soundcloud.com/eufalesio/procession-for-the-death-of-a-star?in=eufalesio/sets/microtonal-stuff "Procession for the death of a star"] from [https://soundcloud.com/eufalesio/sets/microtonal-stuff ''Microtonal stuff''] (2022)


; [[Francium]]
; [[Francium]]
Line 336: Line 355:
** "Canon at the minor second on an ancient Padanian folk theme" [https://www.youtube.com/watch?v=mEeKltGny04 YouTube] | [https://web.archive.org/web/20201127014141/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Meneghin/Claudi%20Meneghin_%20Canon%20at%20the%20minor%20second%20on%20an%20ancient%20Pad.mp3 play]
** "Canon at the minor second on an ancient Padanian folk theme" [https://www.youtube.com/watch?v=mEeKltGny04 YouTube] | [https://web.archive.org/web/20201127014141/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Meneghin/Claudi%20Meneghin_%20Canon%20at%20the%20minor%20second%20on%20an%20ancient%20Pad.mp3 play]
** "Canon 2 in 1 on "Ah! vous dirai-je, maman" at the diminished fifth" [https://www.youtube.com/watch?v=6_YVhwArGTI YouTube] | [https://web.archive.org/web/20201127013943/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Meneghin/Claudi%20Meneghin_%20Canon%202%20in%201%20on%20_Ah_%20vous%20dirai-je,%20maman_%20.mp3 play]
** "Canon 2 in 1 on "Ah! vous dirai-je, maman" at the diminished fifth" [https://www.youtube.com/watch?v=6_YVhwArGTI YouTube] | [https://web.archive.org/web/20201127013943/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Meneghin/Claudi%20Meneghin_%20Canon%202%20in%201%20on%20_Ah_%20vous%20dirai-je,%20maman_%20.mp3 play]
; [[Norokusi]]
* [https://www.youtube.com/watch?v=ZjJSa9brfSY ''Suite No.6 (2025) for Piano in Quarter-comma Meantone Temperament''] (2025)
* [https://www.youtube.com/watch?v=OZriKGVcXDM ''Ballade No.5 (2025) in C Major for Piano in Quarter-comma Meantone Temperament''] (2025)
; [[Juhani Nuorvala]]
* [https://soundcloud.com/juhani-nuorvala/enharmoninen-sarja-enharmonic-suite ''Enharmoninen sarja (Enharmonic Suite)''] (2017, [[Matias Häkkinen]] on harpsichord with 2 split keys per octave, tuned in quarter-comma meantone (C C♯ D D♯ Eb E F F♯ G G♯ A♭ A=415Hz B♭ B)
; [[Cam Taylor]]
* [https://www.youtube.com/watch?v=bQcDISy6e4M ''Free jam in 1/4-comma E♭-G♯''] (2024)


; [[Chris Vaisvil]]
; [[Chris Vaisvil]]
Retrieved from "https://en.xen.wiki/w/Quarter-comma_meantone"