
© 2004 Tonalsoft Inc. 
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
(This page should open with audio of the beginning of my version
of Mozart's 40th Symphony in Gminor, K. 550, tuned in a subset of 55EDO.)
Download:
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 Chesnut 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 above updated: 2001.6.5, 2001.6.28
Yahoo Tuning Group, Message 45865 From:Date: Sat Jul 26, 2003 3:56 am Subject: smoking gun: W.A. Mozart's writings on intonation i wrote: http://groups.yahoo.com/group/tuning/message/44784 > From: "monz" > Date: Tue Jun 17, 2003 5:54 pm > Subject: Re: [tuning] Re: Mozart 31equal?? > > > > > ... right now i'm busy tracking down a copy of > the Attwood studies to see what W.A. Mozart himself > wrote about intonation. and now i've gotten it, and can share what i found. (thank the gods for interlibrary loan!) basically there's nothing new that i didn't already glean from Chesnut's article, and which i've already included on my 55edo webpage: http://sonicarts.org/monzo/55edo/55edo.htm The largest set of pitches actually written in Mozart's handwriting is a 21tone meantone chain of 5ths Ebb ... A#, but without Cb : Ebb Bbb Fb [Cb] Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# Attwood apparently understood Italian better than German, because Mozart wrote in Italian for his benefit. It is clearly stated several times in both of their handwritings that the 8ve contains 5 tones and 2 large semitones. The notes begin with ascending and descending (one 8ve) Cmajor and Amelodicminor scales, in Mozart's handwriting, with notes explaining where the semitones occur. then (p I/1 and I/2) Mozart wrote the first note of several other major scales and their relative minors, and Attwood filled in all the other notes correctly, except for the leadingtone of G#minor, which he wrote as F# ascending (corrected by Mozart to Fx) and Fnatural descending (corrected by Mozart to F#). the list of major scales is, in Mozart's order, C F Bb Eb Ab Db Gb B E A. the relative minors are thus A D G C F Bb Eb G# C# F#. the 21note meantone chain of 5ths which contains all the notes included in these scales is thus: Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# Fx page I/4, entirely in Mozart's handwriting except for one sentence by Attwood, is a list of large and small semitones and then a table of intervals arranged in 8vecomplementary pairs (and continued on page I/5). Mozart's list of semitones is: large semitones: CDb, C#D, DEb, D#E, EF, FGb, F#G small semitones: CC#, DbD, DD#, EbE, EE#, FF#, GbG (the complete text is quoted below) page I/10 contains Attwood's attempt at making a table of all the meantone intervals from a reference pitch of A#, and it is remarkable for the number of errors in what is otherwise a rather welldone set of studies  i have marked them all and indicated the corrections, which Mozart didn't even bother to do. Mozart instead preferred to write a new table of intervals on the next page, again using a reference pitch of C, as on pages I/45, giving Attwood only one of each pair of 8vecomplementary intervals and having Attwood write the complement. Attwood finally got all of information for semitone sizes correct on page I/25, and the interval sizes on pages I/26 and I/27, none of which i have included as they simply recap what Mozart wrote. however, he does note that in the cases of Ebb, Fb, and Bbb, "these tones the Harpsichord has not, but all other Instruments have ". the numerous other pages of the book contain harmony and counterpoint exercises. now, it is true that the relative sizes of the two different semitones it is not stated anywhere in this book. however, we know that Mozart's father Leopold taught that the flats were a comma higher than the sharps, which indicates 55edo or another meantone similar to it.  begin pages quoted from Attwood's notes  * I/1, p. 3 * Nella scala maggiore il 1mo mezzo tuono è dalla terza alla quarta. ed il 2do dalla 7ma all' octava. il lmo dall' octava all 7ma ed il 2do dalla quarta alla terza. * I/4, p. 6 * [see I/10, p. 10] [in Attwood's hand:] A tone ought to have a little tone & half a great one [Attwood mangled this: he meant to say "A tone ought to have one little halftone and one great one".] [all music examples on this page and the next in Mozart's hand:] Mezzi tuoni grandi CDb, C#D, DEb, D#E, EF, FGb, F#G mezzi tuoni piccoli. Cc#, DbD, DD#, EbE, EE#, FF#, GbG unisono CC mezzo tuono piccolo. unisono superfluo CC# 8tava diminiuta C#C mezzo tuono grande. seconda minore CDb 7ma maggiore DbC un tuono. seconda maggiore CD 7ma minore DC 1 tuono ed un mezzo tuono piccolo. seconda superflua CD# 7ma diminuta D#C 1 tuono, ed un mezzo tuono gr. terza min: CEb 6ta mag: EbC 2 tuoni terza mag: CE 6ta min: EC 1 tuono, e 2 mezzi tuoni gr: quarta diminuta CFb 5ta superflua FbC 2 tuoni, ed un semitonio gr: quarta minore CF 5ta reale FC 3 tuoni quarta mag: CF# 5ta falsa: F#C 2 tuoni, e 2 mezzi tuoni gr: quinta falsa CGb 4ta mag: GbC 3 tuoni, ed un semit: gr: quinta vera o 5ta reale CG 4ta minore GC 3 tuoni, un semit: gr. ed uno pic: quinta superflua CG# 4ta diminuta: G#C * I/5, p. 7 * ... 3 tuoni, e 2 semit: gr: sesta minore CAb terza mag: AbC 4 tuoni, ed un semit: gr: sesta mag: CA terza min: AC 4 tuoni, un semit: gr: ed uno pic: sesta superflua CA# terza diminuta: A#C 4 tuoni, e 2 semit: gr: septima min: CBb 2da mag: BbC 5 tuoni, ed un semit: gr: septima mag: CB 2da min: BC 5 tuoni, e 2 semit: grandi 8tava CC (high) unisono: CC [in Attwood's hand:] An Octave must have 5 tones & two Great half tones. * I/6, p. 8 * C C#,Db D D#,Eb E E#,F F#,Gb G G#,Ab A A#,Bb B B# c [meantone chain: Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B#] * I/10, P. 10 * [see I/4, p. 6] [entire page in Attwood's hand, and filled with errors:] Unis: Mez. t Gr A#B, BC, B#C#, C#D, CxD#, D#E, DxE#, E#F#, ExFx, FxG# Mez. t p: A#Ax, BB#, CC#, C#Cx, DD#, D#Dx, EE#, E#Ex, F#Fx, GG# [meantone chain: C G D [A] E B F# C# G# D# A# E# B# Fx Cx [Gx] Dx Ax Ex] Antonio Sdadler [?} Unis A#A# 8tava A#(low)A# mez t pic. unis sup: A#Ax 8ta diminuta AxA# mez Gr: 2do Minore A#B 7ma Mag: BbA# [incorrect: should be BA#] 1 t. 2da Mag: A#B# 7ma min. BA# [incorrect: should be B#A#] 1 t e un mez p: 2 sup: A#Bx 7ma dim: B#A# [incorrect: should be BxA#] 2 mag gr 3za dimi: A#C [Cbb crossed out] 6ta Super CbbA# [incorrect: should be CA#] 1 t e mag gr 3za Minore A#C# 6ta Mag. CbA# [incorrect: should be C#A#] 2 t p: 3 Mag: A#Cx 6ta Min CA# [incorrect: should be CxA#] 1 t: e dui mez. gr: 4ta dimi A#D [Dbb crossed out] 5ta Super DbbA# [incorrect: should be DA#] 2 t e mez Gr 4 Minore A#D# [Db crossed out] 5ta Mag DbA# [incorrect: should be D#A#] why can't the fourth have a Superflua [Attwood's marginal note] 3 t: 4ta Mag: A#D [natural sign crossed out] [incorrect: should be A#Dx] 5 Mag. DA# [incorrect: should be DxA#] 2 t e 2 mez Gr 5ta falsa A#Ebb [incorrect: should be A#E] 4 Mag EbbA# [incorrect: should be EA#] 3 t e mez Gr 5ta vera A#Eb [incorrect: should be A#E#] 4 Minore EbA# [incorrect: should be E#A#] 3 t e mag Gr e mez p: 5ta Super A#E [incorrect: should be A#Ex] 4ta dimi EA# [incorrect: should be ExA#] 3 t e 2 mag Gr 6ta minore A#Fb [incorrect: should be A#F#] 3za Mag: FbA# [incorrect: should be F#A#] 4 t e mez Gr 6ta Mag. A#F [incorrect: should be A#Fx] 3za Minore FA# [incorrect: should be FxA#] 4 t e 2 Gr mag p: [incorrect: should be 4 t e mez Gr: e mez p:] 6ta Super A#F# [incorrect: should be A#FX#] 3za dimi F#A# [incorrect: should be Fx#A#] 4 t e mez Gr: [incorrect: should be 3 t e 3 mez Gr:] 7ma dim: A#Gbb [incorrect: should be A#G] 2 Super GbbA# [incorrect: should be GA#] 4 t e dui Mez Gr. 7 minore A#Gb [incorrect: should be A#G#] 2 Mag GbA# [incorrect: should be G#A#] 5ta e mez Gr 7ma Mag. A#G [incorrect: should be A#Gx] 2 Super GA# [incorrect: should be GxA#] 5ta e 2 mez Gr 8tava A#G# [incorrect: should be A#A#(high)] unis. G#A# [incorrect: should be A#A#] [meantone chain for Attwood's incorrect notes: Cbb Gbb Dbb [Abb] Ebb [Bbb] Fb Cb Gb Db [Ab] Eb Bb F C G D [A] E B F# C# G# D# A# [E#] B# [Fx] Cx [Gx] [Dx] Ax [Ex] Bx  clearly incorrect, no corrections given by Mozart  Mozart wrote out a new example on next page.] [the correct meantone chain for the intervals given by Attwood from A# is: C G D [A] E B F# C# G# D# A# E# B# Fx Cx Gx Dx Ax Ex Bx Fx#] * I/11, p. 11 * [all top intervals in Mozart's hand, all bottom intervals in Attwood's:] unis: CC [Mozart] otava C(low)C [Attwood, etc.] unis: superfl: CC# 8tava diminuta C#C 2da min: CDb 7 mag: DbC 2da mag: CD 7ma minore DC 2da sup: CD# 7ma diminuta D#C 3: dim: CEbb 6ta super: EbbC 3: min: CEb 6ta mag: EbC 3 mag: CE 6ta minore EC 4: dim: CFb 5ta super: FbC 4ta min: CF 5ta vera FC 4: mag: CF# 5ta falsa F#C 5: falsa CGb 4 mag: GbC 5: vera CG 4 minore GC 5: sup: CG# 4 diminuta G#C 6: min: CAb 3za mag: AbC 6: mag: CA 3 minore AC 6: sup: CA# 3za dim: A#C 7ma dim: CBbb 2da super BbbC 7: min: CBb 2 Mag: BbC 7: mag: CB 2 minore BC Octava CC (high) unis CC [i.e., thus this is the same as I/45, except that it also adds diminished3rd/superfluous6th and diminished7th/superfluous2nd pairs.] [meantone chain: Ebb Bbb Fb [Cb] Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A#] [on I/26,p.28 and I/27,p.29, Attwood rewrites exactly this same diagram, with English names for the intervals.]  end pages quoted from Attwood's notes  REFERENCE  Heartz, Daniel; Mann, Alfred; Oldman, Cecil B.; Hertzmann, Erich. 1965. _Thomas Attwoods Theorie und Kompositionsstudien bei Mozart_. Wolfgang Amadeus Mozart: Neue Ausgabe sämtlicher Werke, Ser. X, Werkgruppe 30, Bd. 1. Kassel. monz