[Joe Monzo]

The most frequently encountered (and undoubtedly most important) tuning of the meantone family, and the only one in which the whole-tone is *exactly* the geometric mean between the two commonly encountered whole-tones in just-intonation. The earliest description of it -- not mathematical, but acoustically correct -- is by Pietro Aron (or Aaron), in his toscanello in musica of 1523. The "5th" (for example, C:G) is slightly, but audibly, smaller than the usual one familiar from the standard 12-edo tuning.

1/4-comma meantone is historically very important to the development of Western music, as the paradigms postulated by "common-practice" music-theory to a great extent imply basic 5-limit JI intervals and chord structures as a harmonic ideal, and depend on the elimination or tempering-out of the syntonic comma, which is the defining feature of all tunings in the meantone family.

In the familiar 12-edo (which also belongs to the meantone family) there is a complete set of "enharmonic equivalents", such that all 7 notes which have "flats" can also be "spelled" as 7 different notes which have "sharps". But in all other meantones the "flats" are higher in pitch than supposedly enharmonically-equivalent "sharps". This is the opposite of the case in the much older pythagorean tuning, and also the opposite of the case in the "expressive intonation" which has been widely taught in Eurocentric schools of playing since Beethoven's time (around 1800).

1/4-comma meantone was probably the closest thing to a "standard" tuning in most of Europe from approximately 1500 to 1700, and was still commonly found on keyboards (especially organs) until about 1850. It would probably be fair to say that most instrumental music of the Renaissance and Baroque periods was intended to be played in 1/4-comma meantone or a close relative, and even after the growth in popularity of well-temperaments for keyboards after 1700, some form of meantone (generally more like 1/6-comma or 55edo) was usually still intended for orchestral music.

During Mozart's lifetime (late 1700s) orchestral players began using an "expressive intonation" which veered back towards pythagorean, and Beethoven's musical language certainly encouraged the spread of 12-edo, but to some extent meantone persisted in orchestral playing until approximately around the time of Wagner (mid-to-late 1800s). After the nearly universal adoption of 12-edo, the loss of meantone was lamented by Mahler in the early 1900s. (see Monzo, A Century of New Music in Vienna.)

. . . . . . . . .

1/4-comma meantone tuning narrows each 5th in a series of 5ths by 1/4 of a syntonic comma, hence its name. This results in 'perfect 5ths' of approximately 696.6 cents or 6.97 Semitones. Ascending 5ths will be flatter than just, and descending 5ths will be sharper.

The amount of tempering in 1/4-comma meantone is:

(81/80)^{(1/4)}= (2^{-4}* 3^{4}* 5^{-1})^{(1/4)}= 2^{-1}* 3^{1}* 5^{(-1/4)}= ~5.376572399 cents = ~5 & 3/8 cents = approximately 1 degree of 223-edo.

So the 1/4-comma meantone generator -- the (perfect) 5th -- is (3/2) / ((81/80)^{(1/4)}). Using vector addition, that's:

2^ 3^ 5^ | -1 1 0 | = 3/2 - | -1 1 -1/4 | = (81/80)^{(1/4)}= (2^{-4}3^{4}5^{-1})^{(1/4)}----------------------- | 0 0 1/4 | = 1/4-comma meantone 5th = ~696.5784285 cents.

This is shown geometrically in 3,5-prime-space on the following lattice:

It is also very easy to calculate the logarithmic cents values in a straightforward way:

702 - ( (702-680) / 4 ) = ~697

This has the effect of tempering out the syntonic comma, so that it vanishes, thus making 4 5ths minus 2 octaves exactly equal to the just major-3rd:

2^ 3^ 5^ | 0 0 4/4 | = 4 meantone 5ths - | 2 0 0 | = 2 octaves ----------------------- | -2 0 1 | = just major-3rd = 5/4 ~= 386.3137139 cents.

A 2-dimensional Monzo lattice illustrating the [3,5] components of the prime-factorization of the syntonic comma itself (in grey) and of the syntonic comma lying between 3^{4}: 5^{1}(in black). The proper geometry for meantone would twist this lattice into a 4-strand helix, where 3^{4}and 5^{1}both occupy the same point.

If we ascend thru the positive numbers of generators, the next note in the cycle after the 5th is that of +2 generators, the major-9th of 5^{(2/4)} ~= 1393.156857 cents. This interval is larger than an octave (ratio 2:1 = 1200 cents), which is the interval generally assumed to be the equivalence-interval in all meantones. Since scales are generally described in terms of one particular reference octave, we may narrow the size of the resulting ratio of the "major-9th" by 1/2, (and subtract one octave from the interval measurement) to give the major-2nd. For the calculation of the ratio of this major-2nd (which functions as a whole-tone in this scale), we add 2^{-1} to the vector, and obtain 2^{(-1)}5^{(2/4)} ~= 193.1568569 cents. If we compare this to the two just-intonation whole-tones, subtracting the meantone from the larger pythagorean 9/8 and subtracting the smaller 5-limit 10/9 from the meantone, we find that it is indeed the exact logarithmic mean between them:

2^ 3^ 5^ | -3 2 0 | = 9/8 - | -1 0 2/4 | = meantone ----------------------- | -2 2 -2/4 | = ~10.7531448 cents. 2^ 3^ 5^ | -1 0 2/4 | = meantone - | 1 -2 1 | = 10/9 ----------------------- | -2 2 -2/4 | = ~10.7531448 cents. |

Below is a simple ASCII text diagram showing exactly how the tempering of 1/4-comma meantone works. The reference pitch at the 1:1 ratio is labeled as "C" and the ratios which include prime-factor 5 use the HEWM notation of a minus sign as an accidental, to indicate that they are a syntonic-comma (ratio 81:80) lower in pitch than the pythagorean notes with the same letters:

lattice showing the tempering of 1/4-comma meantone first, the ji lattice showing the syntonic-commas: 5-axis / 40:27 10:9 5:3 5:4 G- ---- D- ---- A- ---- E- . . . / . . . . / . . . . / . . . / . . . / . . . . / . .-- syntonic-comma 81:80 . . / . . . / . . . / . . . . / . . . . / . . . / . . . / . . . . / . . . . / . . . . 3-axis -- C ----- G ----- D ----- A ----- E --- 3-axis 1:1 3:2 9:8 27:16 81:64 / 5-axis next, the syntonic-commas have each been divided into 4 parts, and each note from the 3-axis moved by minus 1/4 of a comma successively: 5-axis / 40:27 10:9 5:3 5:4 x ----- x ----- x ----- E . . . / . . . . / . . . . / . minus + + A + ^ . . / . . ^ . . / . .-- syntonic-comma 81:80 . . / . . v + D + + v . / . . . plus . / . . . . / . . . G + + + / . . . . / . . . . / . . . . 3-axis -- C ----- x ----- x ----- x ----- x --- 3-axis 1:1 3:2 9:8 27:16 81:64 / 5-axis

Note these measurements, which hold for both lattice coordinates and pitch:

- C is the reference origin-point along both the 3 and 5 axes;
- G is 1/4 comma lower than 3:2 and 3/4 comma higher than 40:27, and = 5^(1/4);
- D, the exact "mean tone", is 1/2 comma lower than 9:8 and 1/2 comma higher than 10:9, and = 5^(1/2);
- A is 3/4 comma lower than 27:16 and 1/4 comma higher than 5:3, and = 5^(3/4);
- E is 1 full comma lower than 81:64 and is exactly 5:4, and = 5^1.

. . . . . . . . .

It was noted in 1691 by Christiaan Huygens (and again in the 1940s by Adriaan Fokker) that 31-edo is audibly indistinguishable from 1/4-comma meantone. The 31-edo 5th is 2^{(18/31)} ~= 696.7741935 cents.

Using vector addition again to compare the 1/4-comma meantone 5th with the 31-EDO 5th, we get a difference between the two of:

2^ 3^ 5^ | 18/31 0 0 | = 31-EDO 5th - | 0 0 1/4 | = 1/4-comma meantone 5th ------------------------- | 18/31 0 -1/4 | = 31-EDO 5th "-" 1/4-comma meantone 5th = ~0.195765082 cent = ~1/5 cent = approximately the superparticular ratio 8844:8843 = as Huygens noted, about 1/110 of the syntonic comma = almost exactly 1/10 grad or 6 tuning units.

Higher cardinality EDOs which approximate 1/4-comma meantone more closely than 31-edo, are 174-edo and especially 205-edo.

. . . . . . . . .

The 7-tone diatonic scale in 1/4-comma meantone contains only two step sizes: the ~193.1568569-cent whole-tone described above, between C:D, D:E, F:G, G:A, and A:B, and the ~117.1078577-cent "diatonic semitone" between E:F and B:C:

generator 8ves 5 ~cents -1 1 * [-1/4] 503.4215715 "F" 4 - -2 * [ 4/4] 386.3137139 "E" --------------------------- -5 3 * [-5/4] 117.1078577 1/4-comma meantone diatonic semitone ("minor-2nd")

The Blackwood R value is ~1.649392797.

Introducing "Bb" into the scale causes a new between-degree interval to appear: the ~76.04899926-cent chromatic-semitone between Bb:B :

generator 8ves 5 ~cents 5 -2 * [ 5/4] 1082.892142 "B" -2 - 2 * [-2/4] 1006.843143 "Bb" --------------------------- 7 -4 * [ 7/4] 76.04899926 1/4-comma meantone chromatic semitone ("augmented prime")

Continuing to add pitches at either end of the chain, we eventually come to the typical 12-tone chromatic scale used in Europe during the meantone era, from Eb to G#. This scale has as between-degree intervals only the two sizes of semitones, the chromatic-semitone and diatonic-semitone:

Adding one more note to either end results in another new between-degree interval, of ~41.05885841 cents, as between G#:Ab in the example here:

generator 8ves 5 ~cents -4 3 * [ -4/4] 813.6862861 "Ab" 8 - -4 * [ 8/4] 772.6274277 "G#" --------------------------- -12 7 * [-12/4] 41.05885841 2/7-comma meantone "great" (enharmonic) diesis (diminished-2nd)

It was fairly common during the 1600s and 1700s to find keyboards which had "split keys", so that some or all "black keys" were actually pairs of keys giving both flats and sharps, which are separated by a diesis:

One may continue to add many more notes in this manner without encountering a new step-size; thus, the 19-tone chain of 1/4-comma meantone has between-degree intervals of ~41.05885841 and ~76.04899926 cents:

If one continues to add notes, eventually at the 31st note there will occur another smaller interval, as described below. Extend the chain, for example, to 14 pitches on the "flat" side of the origin and 17 on the "sharp" side, so that the -14th generator = "Cbb" and +17th generator = "Ax" :

generator 8ves 5 ~cents -14 9 * [-14/4] 1047.902001 "Cbb" 17 - -9 * [ 17/4] 1041.833284 "Ax" ------------------------------ -31 18 * [-31/4] 6.068717548 (= ~6^{ 1}/_{15}cents) = 1/4-comma meantone "quadruply-diminished 3rd"

This is the 1/4-comma meantone interval of enharmonicity: in the example here, I assumed as system of -13 ...+17, and have added the -14th generator (it makes no difference to which side, positive or negative, the extra generator is added -- the result is the same).

Thus, 2^{18} * 5^{-(31/4)} acts as a unison-vector which is not tempered out in 1/4-comma meantone, and it acts as a unison-vector which *is* tempered out in 31edo.

Below is a graph showing the pitch-height of this 32-tone chain of 1/4-comma meantone. The red line connects the two pitches which are close together.

6.068717548 / 31 = 0.195765082 cents --> compare with above: this is the amount each 1/4-comma meantone generator must be tempered in order to acheive 31edo.

Below is a 2-dimensional 5-limit bingo-card lattice-diagram, showing the periodicity of the 31edo representations of 5-limit ratios, with a typical spelling in a chain from Gbb at -13 generators to Ax at +17 generators where C=n^{0} -- there is hardly any music in the "common-practice (c. 1600-1900) repertoire which has notes falling outside this range.

Exponents of 3 run across the top row and exponents of 5 run in a column down the left side; C n^{0} is outlined in heavy black; syntonic-comma equivalents (which have the same spelling) are in a light shade of grey; enharmonically equivalent pitches (which have a different spelling but are the same pitch as those in the central block) are in a darker shade of grey. Integers designate the degrees of 31edo.

For more commentary on 1/4-comma meantone and 31-EDO, see

- Monzo, 31edo as approximation of 1/4-comma meantone
- Margo Schulter, Yahoo Tuning List message 29985 (Wed Nov 7, 2001 6:49 pm, Subject: Re: 31-EDO and 1/4-comma meantone compared)

. . . . . . . . .

Meantone tunings can be and have been extended to as many notes per octave as desired, but when used for fretted strings or, especially, keyboards, they are usually limited to 12 discrete pitches per 'octave'.

Regardless of how many notes the meantone has, comparing any two notes which are inclusively 12 steps apart in the chain of "5ths" results in a "wolf 5th" and its complementary "wolf 4th", as can be seen in the following two tables. The "wolves" occur between the 5th/4th complementary pair which bound the notes at the end of the series of 12 5ths.

In the tables below, the reference pitch (1/1) is 'A', and the series of '5ths' runs arbitrarily from 3^{-6} (= 'Eb') to 3^{5} (= G#). The first table is an interval matrix of all possible intervals between any two pitches in this tuning. The second table lists all available intervals and the notes between which they can be found.

interval sizes given in Semitones; "3^{x}" indicates the *implied* '5th' in the chain-of-5ths, *not* the actual tuning

This is the most common mapping of meantone to a set of 12 pitch-classes, extending from Eb on the flat side to G# on the sharp side.

G# | G | F# | F | E | Eb | D | C# | C | B | Bb | A | |||

3^{x} |
5 |
-2 |
3 |
-4 |
1 |
-6 |
-1 |
4 |
-3 |
2 |
-5 |
0 |
||

G# |
5 |
0.00 | 11.24 | 10.07 | 9.31 | 8.14 | 7.38 | 6.21 | 5.03 | 4.27 | 3.10 | 2.34 | 1.17 | |

G |
-2 |
0.76 | 0.00 | 10.83 | 10.07 | 8.90 | 8.14 | 6.97 | 5.79 | 5.03 | 3.86 | 3.10 | 1.93 | |

F# |
3 |
1.93 | 1.17 | 0.00 | 11.24 | 10.07 | 9.31 | 8.14 | 6.97 | 6.21 | 5.03 | 4.27 | 3.10 | |

F |
-4 |
2.69 | 1.93 | 0.76 | 0.00 | 10.83 | 10.07 | 8.90 | 7.73 | 6.97 | 5.79 | 5.03 | 3.86 | |

E |
1 |
3.86 | 3.10 | 1.93 | 1.17 | 0.00 | 11.24 | 10.07 | 8.90 | 8.14 | 6.97 | 6.21 | 5.03 | |

Eb |
-6 |
4.62 | 3.86 | 2.69 | 1.93 | 0.76 | 0.00 | 10.83 | 9.66 | 8.90 | 7.73 | 6.97 | 5.79 | |

D |
-1 |
5.79 | 5.03 | 3.86 | 3.10 | 1.93 | 1.17 | 0.00 | 10.83 | 10.07 | 8.90 | 8.14 | 6.97 | |

C# |
4 |
6.97 | 6.21 | 5.03 | 4.27 | 3.10 | 2.34 | 1.17 | 0.00 | 11.24 | 10.07 | 9.31 | 8.14 | |

C |
-3 |
7.73 | 6.97 | 5.79 | 5.03 | 3.86 | 3.10 | 1.93 | 0.76 | 0.00 | 10.83 | 10.07 | 8.90 | |

B |
2 |
8.90 | 8.14 | 6.97 | 6.21 | 5.03 | 4.27 | 3.10 | 1.93 | 1.17 | 0.00 | 11.24 | 10.07 | |

Bb |
-5 |
9.66 | 8.90 | 7.73 | 6.97 | 5.79 | 5.03 | 3.86 | 2.69 | 1.93 | 0.76 | 0.00 | 10.83 | |

A |
0 |
10.83 | 10.07 | 8.90 | 8.14 | 6.97 | 6.21 | 5.03 | 3.86 | 3.10 | 1.93 | 1.17 | 0.00 |

. . . . . . . . .

(Same mapping as that used above.) The "algebra" column defines the following relationships:

s' = t - s = chromatic semitone (augmented-prime) s = t - s' = diatonic semitone (minor-2nd) t = s + s' = tone (major-2nd, whole-tone)

interval | generators | Semitones | 31edo degrees | algebra | instances | |||||||||||

8ve | 0 |
12.00 |
31
| 5t + 2s | G#:G# | G:G | F#:F# | F:F | E:E | Eb:Eb | D:D | C#:C# | C:C | B:B | Bb:Bb | A:A |

diminished 8ve | -7 |
11.24 |
29
| 4t + 3s | G#:G | F#:F | E:Eb | C#:C | B:Bb | |||||||

major 7th | +5 |
10.83 |
28
| 5t + s | G:F# | F:E | Eb:D | D:C# | C:B | Bb:A | A:G# | |||||

minor 7th | -2 |
10.07 |
26
| 4t + 2s | G#:F# | G:F | F#:E | F:Eb | E:D | D:C | C#:B | C:Bb | B:A | A:G | ||

augmented 6th | +10 |
9.66 |
25
| 5t | Eb:C# | Bb:G# | ||||||||||

diminshed 7th | -9 |
9.31 |
24
| 3t + 3s | G#:F | F#:Eb | C#:Bb | |||||||||

major 6th | +3 |
8.90 |
23
| 4t + s | G:E | F:D | E:C# | Eb:C | D:B | C:A | B:G# | Bb:G | A:F# | |||

minor 6th | -4 |
8.14 |
21
| 3t + 2s | G#:E | G:Eb | F#:D | E:C | D:B | C#:A | B:G | A:F | ||||

augmented 5th | +8 |
7.73 |
20
| 4t | F:C# | Eb:B | C:G# | Bb:F# | ||||||||

wolf 5th | -11 |
7.38 |
19
| 2t + 3s | G#:Eb | |||||||||||

'perfect' 5th | +1 |
6.97 |
18
| 3t + s | G:D | F#:C# | F:C | E:B | Eb:Bb | D:A | C#:G# | C:G | B:F# | Bb:F | A:E | |

diminished 5th | -6 |
6.21 |
16
| 2t + 2s | G#:D | F#:C | E:Bb | C#:G | B:F | A:Eb | ||||||

augmented 4th | +6 |
5.79 |
15
| 3t | G:C# | F:B | Eb:A | D:G# | C:F# | Bb:E | ||||||

'perfect' 4th | -1 |
5.03 |
13
| 2t + s | G#:C# | G:C | F#:B | F:Bb | E:A | D:G | C#:F# | C:F | B:E | Bb:Eb | A:D | |

wolf 4th | +11 |
4.62 |
12
| 2t + s' | Eb:G# | |||||||||||

diminished 4th | -8 |
4.27 |
11
| t + 2s | G#:C | F#:Bb | C#:F | B:Eb | ||||||||

major 3rd | +4 |
3.86 |
10
| 2t | G:B | F:A | E:G# | Eb:G | D:F# | C:E | Bb:D | A:C# | ||||

minor 3rd | -3 |
3.10 |
8
| t + s | G#:B | G:Bb | F#:A | E:G | D:F | C#:E | C:Eb | B:D | A:C | |||

augmented 2nd | +9 |
2.69 |
7
| t + s' | F:G# | Eb:F# | Bb:C# | |||||||||

diminished 3rd | -10 |
2.34 |
6
| 2s | G#:Bb | C#:Eb | ||||||||||

major 2nd | +2 |
1.93 |
5
| t | G:A | F#:G# | F:G | E:F# | Eb:F | D:E | C:D | B:C# | Bb:C | A:B | ||

minor 2nd | -5 |
1.17 |
3
| s | G#:A | F#:G | E:F | D:Eb | C#:D | B:C | A:Bb | |||||

augmented unison | +7 |
0.76 |
2
| s' | G:G# | F:F# | Eb:E | C:C# | Bb:B | |||||||

unison (prime) | 0 |
0.00 |
0
| 0 | G#:G# | G:G | F#:F# | F:F | E:E | Eb:Eb | D:D | C#:C# | C:C | B:B | Bb:Bb | A:A |

It can be seen that a 12-tone subset of meantone contains at least one instance of all intervals from -11 to +11 generators.

. . . . . . . . .

Use of certain meantone chords may have been the result of their resemblance to certain basic JI sonorities, for example, the "augmented-6th" (+10 generators) is a close approximation of the 7th harmonic (7:4 ratio).

Below is a table and graph showing how closely the intervals in the 12-tone subset of 1/4-comma meantone given above approximate JI chord identities.

Below is a table and graph showing how closely a 31-tone set of 1/4-comma meantone approximates JI chord identities.

. . . . . . . . .

Below is a graph comparing 1/4-comma meantone with Werckmeister III tuning:

. . . . . . . . .

Aaron, writing an introduction to music in Italian (Toscanello is a title honoring his native Tuscany), advises the not necessarily experienced reader to start by making the octave C-C "just," and then the major third C-E "sonorous and just, as united as possible." While Aaron does not specify a pure 5:4 for the major third, this seems to me a natural reading of "sonorous and just." Then the fifths C-G-D-A-E are tuned so that each is slightly "flat" or "lacking," and each by the same amount. For example, A should be the same "distance" from D as from E -- in other words, the fifths D-A and A-E should be tempered from "perfection" by the same quantity.

As far as I'm concerned, this is enough to justify listing Aaron's instructions as a description [of] the procedure for obtaining 1/4-comma meantone -- as various authors have done, and Paul [Erlich] does in his table. Further, Aaron adds after his C-G-D-A-E series of fifths that F should be tuned by a similar but opposite procedure, making F in the fifth F-C "a little high, passing a bit beyond perfection." If one assumes symmetry, then 1/4-comma meantone indeed results.

Correctly pointing out that Aaron's instructions do not give a mathematical description of 1/4-comma meantone, or explicitly direct that all major thirds be made pure, Lindley raises the question of what Aaron means by his statement that "thirds and sixths are blunted or diminished" in this temperament.

One interpretation which occurs to me is that Aaron is comparing meantone *major* thirds and sixths with the same intervals in a pythagorean tuning with pure fifths -- where they are indeed somewhat larger. Lindley emphasizes such ambiguous statements to argue that Aaron's instructions do specify 1/4-comma for the first notes C-G-D-A-E, but might lend themselves to various slightly irregular tunings for the other fifths and thirds.

Another point made by Lindley is that Aaron uses the adjective _giusta_ ("just") to refer not only to his initial octave and major third (where "pure" is an attractive reading), but also for the temperament as a whole -- _participatione & acordo giusto & buono_, "a just and good temperament and tuning." However, Lindley himself is ready to accept Aaron's opening C-G-D-A-E with its "sonorous and just" major third C-E as a description of 1/4-comma. His statement about the octave C-C and major third C-E being as sonorous or "united" as possible would support this conclusion even we assume that "just" may mean simply "euphonious" or "pleasing."

In his instructions for the final stage of the temperament, the tuning of the sharps, Aaron directs that C#, tuned in relation to the fifth A-E, should be a major third from A and a minor third from E, and likewise with F# in relation to the fifth D-A, etc.

From one viewpoint, this language may simply be reminding the student of the locations of major and minor thirds involving accidentals. Lindley, however, pursuing his argument that a regular 1/4-comma temperament is not *necessarily* implied, argues that one *could* read this language to suggest something like Zarlino's 2/7-comma meantone for the temperament of the sharps, with major and minor thirds compromised by about the same amount from pure.

In arguing that Aaron's tuning is not *necessarily* a regular 1/4-comma meantone, Lindley may have two motivations. First, he wishes to emphasize that Aaron's instructions are not mathematically precise; and indeed, I would agree that Zarlino (1571) and Salinas (1577) may be the first known theorists to give such mathematical definitions of 1/4-comma and other meantone temperaments.

Secondly, Lindley wants to correct the view that 1/4-comma was a universal standard in the 16th century. Since Aaron's instructions are often taken as the paradigm case of this tuning, showing that they are actually open to more than one interpretation would fit with his larger campaign against "1/4-comma hegemony."

However, I find it noteworthy that without invoking any complex mathematical concepts or even defining a syntonic comma, Aaron has described in what I find beautiful as well as musicianly terms the idea of tuning a pure major third and then dividing it into four equally tempered fifths. Aaron's remaining instructions, including his suggestion of a similar but opposite tempering of fifths in the flat direction (F-C, and then Bb-F and Eb-Bb), permit a regular 1/4-comma temperament, even if they do not compel it or define it in mathematical terms.

If I were making a table like Paul's, I might list Aaron for 1/4-comma and add a footnote or annotation like this:

"Aaron evidently describes a temperament with a pure major third C-E and equally narrowed fifths for his first five notes C-G-D-A-E, with further instructions permitting but not explicitly specifying that other major thirds are pure; Zarlino (1571) and Salinas (1577) give mathematically precise descriptions of a regular 1/4-comma tuning."

. . . . . . . . .

[Joe Monzo]

Below is a table showing data to be used for tuning a piano in 1/4-comma meantone.

note ... cents ... cents from 12-edo .. Hz C .... 310.2647146 . +10.2647146 ... 4210.902168 B .... 193.1568569 .. -6.843143068 . 3935.47964 Bb ... 117.1078577 . +17.10785767 .. 3766.345398 A ...... 0 ........... 0 ........... 3520.0 G# .. 1082.892142 .. -17.10785767 .. 3289.767319 G ... 1006.843143 ... +6.843143068 . 3148.383712 F# ... 889.7352854 . -10.2647146 ... 2942.457342 F .... 813.6862861 . +13.68628614 .. 2816.0 E .... 696.5784285 .. -3.421571534 . 2631.813855 Eb ... 620.5294292 . +20.5294292 ... 2518.70697 D .... 503.4215715 .. +3.421571534 . 2353.965874 C# ... 386.3137139 . -13.68628614 .. 2200.0 C .... 310.2647146 . +10.2647146 ... 2105.451084 B .... 193.1568569 .. -6.843143068 . 1967.73982 Bb ... 117.1078577 . +17.10785767 .. 1883.172699 A ...... 0 ........... 0 ........... 1760.0 G# .. 1082.892142 .. -17.10785767 .. 1644.883659 G ... 1006.843143 ... +6.843143068 . 1574.191856 F# ... 889.7352854 . -10.2647146 ... 1471.228671 F .... 813.6862861 . +13.68628614 .. 1408.0 E .... 696.5784285 .. -3.421571534 . 1315.906927 Eb ... 620.5294292 . +20.5294292 ... 1259.353485 D .... 503.4215715 .. +3.421571534 . 1176.982937 C# ... 386.3137139 . -13.68628614 .. 1100.0 C .... 310.2647146 . +10.2647146 ... 1052.725542 B .... 193.1568569 .. -6.843143068 . 983.8699101 Bb ... 117.1078577 . +17.10785767 .. 941.5863494 A ...... 0 ........... 0 ........... 880.0 G# .. 1082.892142 .. -17.10785767 ... 822.4418297 G ... 1006.843143 ... +6.843143068 .. 787.0959281 F# ... 889.7352854 . -10.2647146 .... 735.6143355 F .... 813.6862861 . +13.68628614 ... 704.0 E .... 696.5784285 .. -3.421571534 .. 657.9534637 Eb ... 620.5294292 . +20.5294292 .... 629.6767425 D .... 503.4215715 .. +3.421571534 .. 588.4914684 C# ... 386.3137139 . -13.68628614 ... 550.0 C .... 310.2647146 . +10.2647146 .... 526.362771 B .... 193.1568569 .. -6.843143068 .. 491.934955 Bb ... 117.1078577 . +17.10785767 ... 470.7931747 A ...... 0 ........... 0 ............ 440.0 G# .. 1082.892142 .. -17.10785767 ... 411.2209148 G ... 1006.843143 ... +6.843143068 .. 393.547964 F# ... 889.7352854 . -10.2647146 .... 367.8071677 F .... 813.6862861 . +13.68628614 ... 352.0 E .... 696.5784285 .. -3.421571534 .. 328.9767319 Eb ... 620.5294292 . +20.5294292 .... 314.8383712 D .... 503.4215715 .. +3.421571534 .. 294.2457342 C# ... 386.3137139 . -13.68628614 ... 275.0 C .... 310.2647146 . +10.2647146 .... 263.1813855 B .... 193.1568569 .. -6.843143068 .. 245.9674775 Bb ... 117.1078577 . +17.10785767 ... 235.3965874 A ...... 0 ........... 0 ............ 220.0 G# .. 1082.892142 .. -17.10785767 ... 205.6104574 G ... 1006.843143 ... +6.843143068 .. 196.773982 F# ... 889.7352854 . -10.2647146 .... 183.9035839 F .... 813.6862861 . +13.68628614 ... 176.0 E .... 696.5784285 .. -3.421571534 .. 164.4883659 Eb ... 620.5294292 . +20.5294292 .... 157.4191856 D .... 503.4215715 .. +3.421571534 .. 147.1228671 C# ... 386.3137139 . -13.68628614 ... 137.5 C .... 310.2647146 . +10.2647146 .... 131.5906927 B .... 193.1568569 .. -6.843143068 .. 122.9837388 Bb ... 117.1078577 . +17.10785767 ... 117.6982937 A ...... 0 ........... 0 ............ 110.0 G# .. 1082.892142 .. -17.10785767 ... 102.8052287 G ... 1006.843143 ... +6.843143068 ... 98.38699101 F# ... 889.7352854 . -10.2647146 ..... 91.95179193 F .... 813.6862861 . +13.68628614 .... 88.0 E .... 696.5784285 .. -3.421571534 ... 82.24418297 Eb ... 620.5294292 . +20.5294292 ..... 78.70959281 D .... 503.4215715 .. +3.421571534 ... 73.56143355 C# ... 386.3137139 . -13.68628614 .... 68.75 C .... 310.2647146 . +10.2647146 ..... 65.79534637 B .... 193.1568569 .. -6.843143068 ... 61.49186938 Bb ... 117.1078577 . +17.10785767 .... 58.84914684 A ...... 0 ........... 0 ............. 55.0 G# .. 1082.892142 .. -17.10785767 .... 51.40261435 G ... 1006.843143 ... +6.843143068 ... 49.1934955 F# ... 889.7352854 . -10.2647146 ..... 45.97589597 F .... 813.6862861 . +13.68628614 .... 44.0 E .... 696.5784285 .. -3.421571534 ... 41.12209148 Eb ... 620.5294292 . +20.5294292 ..... 39.3547964 D .... 503.4215715 .. +3.421571534 ... 36.78071677 C# ... 386.3137139 . -13.68628614 .... 34.375 C .... 310.2647146 . +10.2647146 ..... 32.89767319 B .... 193.1568569 .. -6.843143068 ... 30.74593469 Bb ... 117.1078577 . +17.10785767 .... 29.42457342 A ...... 0 ........... 0 ............. 27.5

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The tonalsoft.com website is almost entirely the work of one person: me, Joe Monzo. Please reward me for my knowledge and effort by choosing your preferred level of financial support. Thank you.

a&b temperament [a&b are numbers]

55-edo (comma) (Mozart's tuning)

1/1 or 1:1 (ratio)

2/1 or 2:1 (ratio)

a&b temperament [a&b are numbers]

apotome (Greek interval)

aristoxenean (temperament family)

atomic (temperament family)

augmented / diesic (temperament family)

augmented-2nd / aug-2 / #2 (interval)

augmented-4th / aug-4 / #4 (interval)

augmented-5th / aug-5 / #5 (interval)

augmented-6th / aug-6 / #6 (interval)

augmented-9th / aug-9 / #9 (interval)

blackjack (tuning)

cent / ¢ (unit of interval measurement)

centitone / iring (unit of interval measurement)

chromatic-semitone / augmented-prime (interval)

daseian (musical notation)

dekamu / 10mu (MIDI-unit)

diapason (Greek interval)

diapente (Greek interval)

diatessaron (Greek interval)

diatonic semitone (minor-2nd) (interval)

diesic (temperament family)

diezeugmenon (Greek tetrachord)

diminished-5th / dim5 / -5 / b5 (interval)

diminished-7th / dim7 / o7 (interval)

doamu / 2mu (MIDI-unit)

dodekamu / 12mu (MIDI-unit)

dominant-7th (dom-7, x7) (chord)

dorian (mode)

eleventh / 11th (interval)

enamu / 1mu (MIDI-unit)

endekamu / 11mu (MIDI-unit)

enharmonic semitone (interval)

ennealimmal (temperament family)

enneamu / 9mu (MIDI-unit)

farab (unit of interval measurement)

fifth / 5th (interval)

flu (unit of interval measurement)

Gentle introduction to Fokker periodicity-blocks (by Paul Erlich)

fourth / 4th (interval)

Gentle introduction to Fokker periodicity-blocks (by Paul Erlich)

grad (unit of interval measurement)

hexamu / 6mu (MIDI-unit)

Hurrian Hymn (Monzo reconstruction)

hypate (Greek note)

hypaton (Greek tetrachord)

hyperbolaion / hyperboleon (Greek tetrachord)

hypophrygian (Greek mode)

imperfect (interval quality)

iring / centitone (unit of interval measurement)

1/1 or 1:1 (ratio)

2/1 or 2:1 (ratio)

jot (unit of interval measurement)

JustMusic: A New Harmony [Monzo's book]

JustMusic prime-factor notation [Monzo essay]

kwazy (temperament family)

leimma / limma (Greek interval)

lichanos (Greek note)

limma / leimma (Greek interval)

locrian (mode)

lydian (mode)

magic (temperament family)

Mahler 7th/1 [Monzo score and analysis]

marvel (temperament family)

meantone (temperament family)

mem (unit of interval measurement)

meride (unit of interval measurement)

mese (Greek note)

meson (Greek tetrachord)

millioctave / m8ve (unit of interval measurement)

mina (unit of interval measurement)

minerva (temperament family)

miracle (temperament family)

mixolydian (mode)

monzo (prime-exponent vector)

Monzo, Joe (music-theorist)

morion / moria (unit of interval measurement)

mutt (temperament family)

mystery (temperament family)

octamu / oktamu / 8mu (MIDI-unit)

octave (interval)

oktamu / octamu / 8mu (MIDI-unit)

orwell (temperament family)

p4, perfect 4th, perfect fourth (interval)

p5, perfect 5th, perfect fifth (interval)

pantonality of Schoenberg [Monzo essay]

paramese (Greek note)

paranete (Greek note)

parhypate (Greek note)

pentamu / 5mu (MIDI-unit)

prime-factor notation (JustMusic) [Monzo essay]

proslambanomenos (Greek note)

savart (unit of interval measurement)

schismic / skhismic (temperament family)

Schoenberg's pantonality [Monzo essay]

second / 2nd (interval)

semisixths (temperament family)

semitone (unit of interval measurement)

seventh / 7th (interval)

sixth / 6th (interval)

sk (unit of interval measurement)

skhismic / schismic (temperament family)

sruti tuning [Monzo essay]

studloco (tuning)

subminor 3rd (interval)

Sumerian tuning [speculations by Monzo]

synemmenon (Greek tetrachord)

temperament-unit / tu (unit of interval measurement)

tenth / 10th (interval)

tetrachord-theory tutorial [by Monzo]

tetradekamu / 14mu (MIDI-unit)

tetramu / 4mu (MIDI-unit)

third / 3rd (interval)

thirteenth / 13th (interval)

tina (unit of interval measurement)

tone (interval, and other definitions)

tredek (unit of interval measurement)

triamu / 3mu (MIDI-unit)

tridekamu / 13mu (MIDI-unit)

trihemitone (Greek interval)

trite (Greek note)

tu / temperament-unit (unit of interval measurement)

Türk sent (unit of interval measurement)

twelfth / 12th (interval)

whole-tone (interval)

woolhouse-unit (unit of interval measurement)