Mozart's tuning: 55EDO
and its close relative, 1/6comma meantone
Text and diagrams © 2001 by Joseph L. Monzo
with helpful comments by Paul Erlich
(You're listening to the beginning of my version of Mozart's 40th Symphony in Gminor, K. 550, tuned in a subset of 55EDO.)
It is known that Leopold Mozart taught his son Wolfgang to use a meantone tuning where "flats" would be higher in pitch than "sharps". There is no documentation as to exactly what this tuning was, but there are a few clues.
Paul Erlich quoted in Yahoo Tuning List message 24125 (Thu May 31, 2001 9:03 pm) from Chesnut 1977 as follows:
Leopold Mozart refers to Tosi in general terms as an authoritative source in a letter to Wolfgang from Salzburg dated June 11, 1778. Tosi, in 1723, considered the correct tuning system to be what we would today call a form of regular meantone temperament ... according to Tosi, the large diatonic half step is theoretically equal to five ninths of a whole step, and the small chromatic half step is theoretically fourninths of a whole step. Tosi thereby divides the octave into fiftyfive equal parts. This is equivalent to tempering the perfect fifth by approximately onesixth of a 'comma,' ...Leopold Mozart, in his violin method of 1756  which happens to be the year of Wolfgang's birth  also describes what we have called 'extended regular meantone temperament' as the correct intonation for the violin; he tells us that keyboard instruments of his time were played with some form of tempered [i.e., welltempered] tuning, but that in the "right ratio" [i.e., meantone] tuning that he recommends for the violin, flats are higher by a comma than enharmonically equivalent sharps. It can be shown that for whichever of the standard commas we choose, the perfect fifths in Leopold Mozart's system were theoretically flattened by about onesixth of that comma . . . Leopold Mozart wrote down a couple of scales specifically intended for practice in intonation, one leading through the flats, the other through the sharps. In practicing these scales, the student is supposed to learn to distinguish between the large diatonic half steps and the small chromatic half steps. It is important to emphasize that these scales are not abstractions but exercises to be mastered . . .
Thus it is apparent that for instruments other than keyboards (which both Mozart and his predecessors tuned in various welltemperaments, typically resembling 1/6comma meantone for the "natural" notes, and Pythagorean for the "chromatic" keys), Mozart's tuning would be based on 55EDO, or something very closely approximating it. It would have the following sizes for the basic intervals:
ratio Semitones cents 1 55EDO degree 2^{(1/55)} ^{12}/_{55} 21 ^{9}/_{11} smaller (chromatic) semitone 2^{(4/55)} ^{48}/_{55} 87 ^{3}/_{11} larger (diatonic) semitone 2^{(5/55)} 1 ^{1}/_{11} 109 ^{1}/_{11} whole tone 2^{(9/55)} 1 ^{53}/_{55} 196 ^{4}/_{11}
55EDO renders two differentsized semitones but only one size wholetone; thus, it is a meantone. (Compare with the 50EDO tuning recommended as a very close approximation to an "optimal meantone" by Woolhouse in 1835; and contrast with 53EDO, which is so close to both Pythagorean and 5limit JI that it gives two differentsized wholetones.)
On this webpage I will first explore a 12tone subset of 55EDO, as I had mistakenly thought that it was the most likely intended tuning for Mozart's music. I have since learned that Mozart actually intended an interesting 20tone subset of 55EDO, and I give an update below on that. I decided to keep the info about the 12tone subset because many people with ordinary Halberstadt keyboards may wish to explore that subset of 55EDO, as it still gives a nice meantone system.
Note that for nonkeyboard instruments, Mozart's actual conception of this tuning extended in a meantone cycle from Ebb to A#, with Cb omitted, for a total of 20 notes, thus giving two different pitches for the pairs of "sharp and "flat" notes (which are each one "black key" on the Halberstadt keyboard), and alternates for several of the "natural" notes too. So there is no evidence that Mozart ever actually used or advocated a 12tone subset of this tuning (which is what I present here).
The mapping uses "D" as the symmetrical pitch center, and renders the 12tone subset as a meantone "cycle of 5ths" from Eb to G#, with the "wolf" appearing between those two pitches.
Note that 55EDO is audibly identical to 1/6comma meantone. The "5th" in 55EDO,
2^{(32/55)} = ~698.181818... cents,is equivalent to that of ~0.175445544comma meantone. To describe that in terms of lowinteger fractionofacomma meantones, that's
a little less than 1/5 (just slightly more than 3/16, extremely close to 7/37, and almost exactly 10/53) of a cent narrower than the 1/6comma meantone "5th" = ~698.3706193 cents,
even closer (less than 1/7 of a cent wider) to the 2/11comma meantone "5th" = ~698.0447664 cents,
much closer (~1/46cent wider) to the 3/17comma meantone "5th" = ~698.1597733 cents,
closer still (less than 1/100cent narrower) to the 7/40comma meantone "5th" = ~698.1914002 cents,
and almost bang on (~1/6692cent wider) the 10/57comma meantone "5th" = ~698.1819676 cents.
The amount of tempering in 1/6comma meantone is:
(81/80)^{(1/6)} = (2^{4} * 3^{4} * 5^{1})^{(1/6)} = 2^{(2/3)} * 3^{(2/3)} * 5^{(1/6)} = ~3.584381599 cents = ~3 & 3/5 cents.
So the 1/6comma meantone "5th" is (3/2) / ((81/80)^{(1/6)}). Using vector addition, that's:
2^ 3^ 5^  1 1 0  = 3/2   2/3 2/3 1/6  = (81/80)^{(1/6)}   1/3 1/3 1/6  = 1/6comma meantone "5th" = ~698.3706193 cents.
Using vector addition again to compare the 1/6comma meantone "5th" with the 55EDO "5th", we get:
2^ 3^ 5^  1/3 1/3 1/6  = 1/6comma meantone "5th"   32/55 0 0  = 55EDO "5th"   151/165 1/3 1/6  = 1/6comma "5th" "" 55EDO "5th" = ~0.188801084 cent = ~1/5 or ~10/53 cent, as stated above.
Below I present a sidebyside comparison of 55EDO and 1/6comma meantone.
First, the 12outof55EDO subset and the 12tone 1/6comma meantone presented as cycles of "5ths" from Eb to G#:


Next, both of the scales presented as 12tone chromatic scales, with their cycleof"5ths" and Semitone values, the Semitones and 55EDO degrees between notes in the scale, and the 55EDO degree representing the scale notes:


And last, the interval matrices for both tunings, showing the 55EDO degrees for the 12oo55 subset and the "cycleof5ths" value for the 1/6comma meantone, with the Semitone values of all possible intervals given in the body of the table. Some important intervals are colorcoded for easy recognition:


Here is the above table reproduced with fractional cents values instead of decimal Semitones:
Note also the following comment by Paul Erlich (from a Yahoo Tuning List post), echoing what Hind wrote at the end of his article:
There is no real evidence that Mozart intended 1/6 comma meantone rather than 1/5comma or some other meantone system. It wouldn't make much sense for one to be that specific in regard to Mozart anyway, since he clearly assumes enharmonic equivalence even in a few passages for strings unaccompanied by keyboard, which violates his known violin teaching altogether.
Chesnut, John Hind. 1977.
"Mozart's teaching of intonation",
Journal of the American Musicological Society
vol. 30 no. 2 [summer], pp. 254271.
section above updated: 2001.7.8, 2001.6.3
Here is a graph of a 56tone cycle of 1/6comma meantone, centered on "C" as the reference (= generator 0):
As seen above, extending the cycle beyond 12 pitches results in pairs of pitches separated by approximately a comma. Upon passing beyond the 27..+27 cycle, note that the pitches represented as Cbbbb and Fxx are closer than that:
Fxx = ( (3/2)^{27} / ( (81/80)^{(27/5)} ) ) / 2^{15} = 2^{(144/6)} * 3^{(54/6)} * 5^{(27/6)} = ~856.0067202 cents Cbbbb = ( (3/2)^{28} / ( (81/80)^{(28/5)} ) ) / 2^{12} = 2^{(158/6)} * 3^{(56/6)} * 5^{(28/6)} = ~845.6226606 cents 2^{(144/6)} * 3^{(54/6)} * 5^{(27/6)} Fxx  2^{(158/6)} * 3^{(56/6)} * 5^{(28/6)} Cbbbb  2^{(302/6)} * 3^{(110/6)} * 5^{(55/6)} = ~10.38405963 (= 10 ^{~3}/_{8}) cents. 10.38405963 / 55 = 0.188801084 > compare with above.
Thus, limiting the meantone cycle to 55 pitches and distributing this difference equally among them, results in 55EDO. If we call the 21 ^{9}/_{11} stepsize of 55EDO a "comma", then the "chromatic semitone" = 4 commas, the "diatonic semitone" = 5 commas, and the "wholetone" = 9 commas, exactly as Mozart taught his students.
Other EDOs which approximate 1/6comma meantone more closely than 55EDO are 67 and 122EDO:
2^{(39/67)} = ~698.5074627 cents 2^{(71/122)} = ~698.3606557 cents
(This section added 2001.11.8)
I'm adding a new section here which explores Mozart's actual intended tuning. Knowledge of this tuning is based on notes written down by Thomas Atwood, who studied with Mozart during the mid1780s. My source for this is the Chesnut article cited above.
Mozart used the same process as given above, but extended it to a cycle of 21 meantone "5ths", an elegantly symmetrical system from implied 3^{10} = Ebb to implied 3^{10} = A#. He then destroyed the symmetry of the scale by limiting himself to 20 notes and eliminating the implied 3^{7} = Cb.
It's curious to me that Mozart decided to do this, rather than keep the symmetrical tuning. It's interesting that the very note he leaves out is the one which would become the most characteristic feature of the later development (just after Mozart's death) of "expressive" Pythagoreanbased intonation: the "raised" (sharpened or narrowed) "leadingtone" [again, see the Chesnut article for details].
Chesnut notes that Tosi advocated 55EDO, and that one degrees or step in this tuning (i.e., the ratio 2^{(1/55)} = 21 & 9/11 cents) is so close to both the syntonic and Pythagorean commas that it could be called a "Tosi comma". There are exactly 4 & 7/12 of these "Tosi commas" in one 12EDO Semitone.
Below is the mapping of the basic
prime
intervals in 55EDO:
 "per" = number of instances of the period of equivalence needed for "correction"  in this case, "octaves"  to put the generated interval into the proper register to represent the basic prime interval.
 "gen" = number of 2^{(32/55)} generators from 0. In this temperament, the generator is considered to be the "5th".
This notation thus gives the 55EDO analogue of a vector notation using 2 (the "octave") and 3/2 (the "5th") as factors.
32/55EDO mapping prime (per, gen) ~cents error ~ % error 2 ( 1, 0 ) 0 0 3 ( 1, 1 ) 3.773182684 0.198384435 5 ( 0, 4 ) 6.413558862 0.230180788 7 (3, 10) 12.99227535 0.385661821 11 (7, 18) 15.95478491 0.384330594
(For a detailed explanation of this type of matrix notation, see Graham Breed's matrix webpage. Graham would define 1/6comma meantone as follows:
factors: 2 3 5 implied ratio ( 1 0 0 ) (1200) 2:1 H' = ( 2/3 1/3 1/6)H = (1898) cents ~3:1 (4/3 4/3 2/3) (2793) ~5:1
And 55EDO as follows:
)factors: 2 3 5 implied ratio ( 1 0 0) (1200) 2:1 H' = ( 87/55 0 0)H = (1898) cents ~3:1 (128/55 0 0) (2793) ~5:1
Note that the generator which represents 11 lies outside the 21tone cycle given in the above graph, thus, this particular scale only provides three approximate 11:8s, between the pairs of notes (expressed as generators and pitches where C = 0): (10, 8) = Ebb:G#, (9, 9) = Bbb:D#, and (8, 10) = Fb:A#. This can be seen in the interval matrix further below.


Below is the interval matrix for Mozart's 20outof55EDO tuning, with intervals given in cents:
Below is a lattice illustrating the relationship of 1/6comma meantone with the 5limit JI pitchclasses it implies. (This particular example illustrates a symmetrical 27tone chain of 1/6comma meantone "5th"s; it could be extended in either direction.)
(lattice added 2001.12.3)
Disclaimer: My intention in presenting the following two sections was to show how 55EDO compares to 72EDO and Pythagorean tuning. Upon further reflection, however, it turns out that neither of these two tunings approximates 55EDO very well, since 55EDO acts as a meantone and those two do not. Perhaps this material should be removed... but I'll leave it in for now; just take note of this disclaimer.
The table and graph below show the deviation of 72EDO, which is gathering enthusiasm for adoption as a new tuning standard, from Mozart's 55EDO subset. The table also shows the nearest 12EDO pitch and the cawapus necessary to make Cakewalk^{TM} supply the proper tuning using pitchbend in a MIDIfile.
Paul Erlich has criticized this comparison, since I originally claimed that 72EDO provided a good approximation to Mozart's scale, and I did not take into consideration an examination of all the intervals. Note that in several cases there are two 72EDO pitches which give roughly the same amount of error from 55EDO, so I arbitrarily chose one where the other could have been chosen, and these choices will affect the interval matrix. In light of this, consider these graphs and tables to show the error for 72EDO approximation of 55EDO, as support for 72EDO as a standard continues to grow.


Paul's criticism emphasized that one must consider not just the proximity of pitches in the two scales, but rather a comparison of all intervals occurring in the two tunings. So here is the interval matrix for this 20tone subset of 72EDO... compare it with the 20tone subset of 55EDO above. Paul doesn't consider 72EDO to give a good approximation of any meantone or meantonelike tuning.
Mozart's 20tone subset of 55EDO can also be approximated by an extended Pythagorean system which is very similar to one proposed by Prosdocimus c. 1425, with the important difference that the meaning of the "flat" and "sharp" accidentals with respect to type of semitone is reversed. It is precisely because both meantone and Pythagorean tunings provide 2 differentsized semitones that they can spawn similar subsets like this one. But in the Pythagorean tuning the diatonic semitone (where the accidental sign doesn't change but the nominal does) is the smaller one, and the chromatic semitone (where the accidental sign changes but the nominal doesn't) is the larger one, whereas in the meantone tunings it is exactly the opposite.
Below is a table and graph showing the comparison between the 21tone system from which Mozart's extracted his 20tone subset of 55EDO, and a 21tone Pythgorean tuning which closely approximates it. Note that 19 tones of the Pythagorean system form a complete selfcontained chain of "5ths", from 3^{9} to 3^{8}, and that a big skip in the chain ocurrs at both ends, to include 3^{14} and 3^{14}. I have more to say on this 19tone subset below. Note that on the graph, if the blue plotline for 55EDO can't be seen, that's because the Pythgorean pitch approximates it so closely that it's covered.
Paul Erlich also added in a private email to me:
19 is an MOS of the meantonefifth generator. Hence it will be a CS  any given specific interval will always span the same number of degrees.
Paul suggested that I make an interval matrix of the 19tone subset, which I would spell with D again at the center for symmetry, and which would thus extend from 3^{9} = Cb to 3^{9} = E#... hopefully I'll get around to it. It was more important for me to explore the historical ramifications of Mozart's actual tuning.
last section updated: 2001.6.5, 2001.6.28
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