Quarter-comma meantone: Difference between revisions

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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.  
'''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.  


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]], 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-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–Gbb) 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.
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]], 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 six cents apart. If we want a tuning which will equate the two, we need to use [[31edo]] instead.
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 ==
Line 31: Line 31:


== 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 49: Line 50:
|-
|-
| 41.0590
| 41.0590
| &minus;12
| −12
| Dbb
| Dbb
| Diminished second, diesis
| Diminished second, diesis
Line 77: Line 78:
|-
|-
| 158.167
| 158.167
| &minus;17
| −17
| Ebbb
| Ebbb
| Doubly diminished third
| Doubly diminished third
Line 91: Line 92:
|-
|-
| 234.216
| 234.216
| &minus;10
| −10
| Ebb
| Ebb
| Diminished third
| Diminished third
Line 105: Line 106:
|-
|-
| 310.265
| 310.265
| &minus;3
| −3
| Eb
| Eb
| Minor third
| Minor third
Line 119: Line 120:
|-
|-
| 351.324
| 351.324
| &minus;15
| −15
| Fbb
| Fbb
| Doubly diminished fourth
| Doubly diminished fourth
Line 133: Line 134:
|-
|-
| 427.373
| 427.373
| &minus;8
| −8
| Fb
| Fb
| Diminished fourth
| Diminished fourth
Line 147: Line 148:
|-
|-
| 503.422
| 503.422
| &minus;1
| −1
| F
| F
| Perfect fourth
| Perfect fourth
Line 154: Line 155:
|-
|-
| 544.480
| 544.480
| &minus;13
| −13
| Gbb
| Gbb
| Doubly diminished fifth
| Doubly diminished fifth
Line 168: Line 169:
|-
|-
| 620.529
| 620.529
| &minus;6
| −6
| Gb
| Gb
| Diminished fifth
| Diminished fifth
Line 189: Line 190:
|-
|-
| 737.637
| 737.637
| &minus;11
| −11
| Abb
| Abb
| Diminished sixth
| Diminished sixth
Line 203: Line 204:
|-
|-
| 813.686
| 813.686
| &minus;4
| −4
| Ab
| Ab
| Minor sixth
| Minor sixth
Line 217: Line 218:
|-
|-
| 854.745
| 854.745
| &minus;16
| −16
| Bbbb
| Bbbb
| Doubly diminished seventh
| Doubly diminished seventh
Line 245: Line 246:
|-
|-
| 1006.84
| 1006.84
| &minus;2
| −2
| Bb
| Bb
| Minor seventh
| Minor seventh
Line 259: Line 260:
|-
|-
| 1047.90
| 1047.90
| &minus;14
| −14
| C'bb
| C'bb
| Doubly diminished octave
| Doubly diminished octave
Line 273: Line 274:
|-
|-
| 1123.95
| 1123.95
| &minus;7
| −7
| C'b
| C'b
| Diminished octave
| Diminished octave

Revision as of 14:29, 25 February 2025

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, 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
Giovanni Pietro Del Buono
François Couperin
Sergei Rachmaninoff
Arthur Singer, John Medora and David White
Richard Wagner
Nicola Vicentino

21st century

Mark Allan Barnes
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
Cam Taylor
Chris Vaisvil
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