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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'''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 ==
[[Projection map]]:
 
{| class="right-all"
|-
| [⟨ || 1 || 1 || 0 || -3 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]
|-
| ⟨ || 0 || 1/4 || 1 || 5/2 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]]
|}
 
[[Tuning map]]: {{val| 1200 1896.5784 2786.3137 3365.7843 }}
 
[[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
Line 26: Line 47:
| 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
Line 40: Line 61:
| Augmented unison, chromatic semitone
| Augmented unison, chromatic semitone
| A1
| A1
| [-4 0 7/4&gt;
| {{monzo| -4 0 7/4 }}
|-
|-
| 117.108
| 117.108
Line 47: Line 68:
| 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
Line 68: Line 89:
| 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
Line 110: Line 131:
| 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
Line 124: Line 145:
| 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
Line 145: Line 166:
| 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
Line 180: Line 201:
| 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
Line 208: Line 229:
| Major sixth
| Major sixth
| M6
| M6
| [-1 0 3/4&gt;
| {{monzo| -1 0 3/4 }}
|-
|-
| 930.794
| 930.794
Line 215: Line 236:
| Diminished seventh
| Diminished seventh
| d7
| d7
| [6 0 -9/4&gt;
| {{monzo| 6 0 -9/4 }}
|-
|-
| 965.784
| 965.784
Line 222: Line 243:
| 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
Line 250: Line 271:
| 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
Line 264: Line 285:
| Augmented seventh
| Augmented seventh
| A7
| A7
| [-6 0 3&gt;
| {{monzo| -6 0 3 }}
|-
|-
| 1200.00
| 1200.00
Line 271: Line 292:
| Perfect octave
| Perfect octave
| P8
| P8
| [1 0 0&gt;
| {{monzo| 1 0 0 }}
|}
|}


== Music ==
== Music ==
=== Modern renderings ===
; {{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://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}}
* [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]]
* [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}}
* [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}}
* [https://web.archive.org/web/20201127014732/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Prelude%20in%20g%20Op%2023%20%235.mp3 ''Prelude in g minor'' Op. 23 #5] (1901)
; {{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)
; {{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}}
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Meistersinger%20Prelude.mp3 "Prelude" to ''Die Meistersinger von Nürnberg''] (1868)
; {{W|Nicola Vicentino}}
* [https://www.youtube.com/watch?v=b68Ek5hDgDo ''Laura che il verde lauro'']{{dead link}} [https://web.archive.org/web/20201127014948/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Nicola%20Vicentino%20(1511-%20c.1576)%20-%20Laura%20che%20il%20verde%20lauro.mp3 play]
=== 21st century ===
; [[Mark Allan Barnes]]
* [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]
; [[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]]
* [https://www.youtube.com/watch?v=5S3NCRNHFzc ''A Meantone Prelude''] (2023)
; [[Adam Freese]]
* [https://www.youtube.com/watch?v=S3gnksLkjRw ''Unnamed fugue''] (2023)
; [[Claudi Meneghin]]
* ''Five canons'' (2011)
** "Canon 2 in 1 in F at the octave" [https://www.youtube.com/watch?v=AyEpCWkxSsM YouTube] | [https://web.archive.org/web/20201127013925/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Meneghin/Claudi%20Meneghin_%20Canon%202%20in%201%20in%20F%20at%20the%20octave.mp3 play]
** "Canon 2 in 1 in A- upon a ground" [https://web.archive.org/web/20190902004518/http://soonlabel.com/xenharmonic/archives/2089/cm-canon-2-in-1-upon-a-ground-4th-version-edited blog] | [https://web.archive.org/web/20190902003004/http://soonlabel.com/xenharmonic/wp-content/uploads/2013/11/CM-Canon-2-in-1-upon-a-ground-4th-version-edited.mp3 play]
** "Canon 2 in 1 in G- at the major seventh" [https://www.youtube.com/watch?v=zwKeFLgbngY YouTube] | [https://web.archive.org/web/20201127013440/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Meneghin/Claudi%20Meneghin_%20Canon%202%20in%201%20in%20G-%20at%20the%20major%20seventh.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]
; [[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)


* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/reelrock.mp3 Reelin' and Rockin'] by [https://en.wikipedia.org/wiki/Chuck_Berry Chuck Berry]
; [[Chris Vaisvil]]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Le%20Tic-Toc%20Choc.mp3 Le Tic-Toc Choc] by [https://en.wikipedia.org/wiki/Fran%C3%A7ois_Couperin François Couperin]
* [http://micro.soonlabel.com/quarter-comma-meantone/its-a-quarter-of-a-mean-tone.mp3 ''A Quarter of a Mean Tone'']
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Prelude%20in%20g%20Op%2023%20%235.mp3 Prelude in g minor Op 23 #5] by [https://en.wikipedia.org/wiki/Sergei_Rachmaninoff Sergei Rachmaninoff]
* ''No Voice Leading'' (2012) [https://chrisvaisvil.com/harpsichord-in-d-minor-in-quarter-comma-meantone-tuning/ blog] | [http://micro.soonlabel.com/quarter-comma-meantone/20120704-harpsichord-quarter-meantone.mp3 play]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/At%20the%20Hop.mp3 At the Hop] by [https://en.wikipedia.org/wiki/Arthur_Singer Singer], [https://en.wikipedia.org/wiki/John_Medora Medora] and [https://en.wikipedia.org/wiki/David_White_%28musician%29 White]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Meistersinger%20Prelude.mp3 Prelude to Die Meistersinger von Nürnberg] by [https://en.wikipedia.org/wiki/Richard_Wagner Richard Wagner]
* [https://www.youtube.com/watch?v=b68Ek5hDgDo Laura che il verde lauro] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Nicola%20Vicentino%20%281511-%20c.1576%29%20-%20Laura%20che%20il%20verde%20lauro.mp3 play] by [https://en.wikipedia.org/wiki/Nicola_Vicentino Nicola Vicentino]
* [https://youtu.be/6t5PdeC1Trw Country Bumpkin][http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/quartercomma/Country%20Bumpkin%20Live%20At%20The%20Turks%2023,9,2010ce.mp3 play] by [[Mark Allan Barnes]]
* [https://www.youtube.com/watch?v=HVwXUwK5_Vg Ozone Layer] [http://micro.soonlabel.com/gene_ward_smith/Others/quartercomma/Ozone%20Layer.mp3 play] by Mark Allan Barnes
* Five canons by [[Claudi Meneghin]]:
**[https://www.youtube.com/watch?v=AyEpCWkxSsM Canon 2 in 1 in F at the octave] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Meneghin/Claudi%20Meneghin_%20Canon%202%20in%201%20in%20F%20at%20the%20octave.mp3 play]
** [http://soonlabel.com/xenharmonic/?attachment_id=1409 Canon 2 in 1 in A- upon a ground] [http://soonlabel.com/xenharmonic/wp-content/uploads/2013/11/CM-Canon-2-in-1-upon-a-ground-4th-version-edited.mp3 play]
** [https://www.youtube.com/watch?v=zwKeFLgbngY Canon 2 in 1 in G- at the major seventh] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Meneghin/Claudi%20Meneghin_%20Canon%202%20in%201%20in%20G-%20at%20the%20major%20seventh.mp3 play]
** [https://www.youtube.com/watch?v=mEeKltGny04 Canon at the minor second on an ancient Padanian folk theme] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Meneghin/Claudi%20Meneghin_%20Canon%20at%20the%20minor%20second%20on%20an%20ancient%20Pad.mp3 play]
** [https://www.youtube.com/watch?v=6_YVhwArGTI Canon 2 in 1 on "Ah! vous dirai-je, maman" at the diminished fifth] [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]
* [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
* [http://micro.soonlabel.com/quarter-comma-meantone/its-a-quarter-of-a-mean-tone.mp3 A Quarter of a Mean Tone] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/quarter-comma-meantone/20120704-harpsichord-quarter-meantone.mp3 No Voice Leading] by Chris Vaisvil
* [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)] by [[Giovanni Pietro Del Buono]]


[[Category:Meantone]]
[[Category:Meantone]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Historical]]
[[Category:Historical]]

Latest revision as of 18:39, 4 June 2026

English Wikipedia has an article on:

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 cents. This means the fifth is flattened by 1⁄4 of the syntonic comma (81/80 ratio) of 21.506 ¢, which is to say by 5.377 ¢, hence the name.

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 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- 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 ¢ apart. In 31edo, these two intervals are enharmonically equivalent.

Tuning profile

Projection map:

[⟨ 1 1 0 -3 ]
⟨ 0 0 0 0 ]
⟨ 0 1/4 1 5/2 ]
⟨ 0 0 0 0 ]]

Tuning map: ⟨1200 1896.5784 2786.3137 3365.7843]

Error map: ⟨0 -5.3766 0 -3.0416]

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 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.

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

Music

Modern renderings

Johann Sebastian Bach
Chuck Berry
John Bull
  • 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)
Giovanni Pietro Del Buono
Dieterich Buxtehude
François Couperin
Ferenc Farkas
Sergei Rachmaninoff
Arthur Singer, John Medora and David White
Jan Pieterszoon Sweelinck
Richard Wagner
Nicola Vicentino

21st century

Mark Allan Barnes
Eufalesio
Francium
Adam Freese
Claudi Meneghin
  • Five canons (2011)
    • "Canon 2 in 1 in F at the octave" YouTube | play
    • "Canon 2 in 1 in A- upon a ground" blog | play
    • "Canon 2 in 1 in G- at the major seventh" YouTube | play
    • "Canon at the minor second on an ancient Padanian folk theme" YouTube | play
    • "Canon 2 in 1 on "Ah! vous dirai-je, maman" at the diminished fifth" YouTube | play
Norokusi
Juhani Nuorvala
Cam Taylor
Chris Vaisvil
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