Kirnberger's quasi-12-tet by JI

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Kirnberger's 10935:8192 tuning:
(Re-)"discovered" -- Again
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Hello, there, and this is happily to announce that the "16384:10935"
schisma fifth tuning discussed in one of my recent posts indeed has a
rich history of documentation going back at least to the 18th century,
when Johann Philipp Kirnberger may have been the first to discover
and propose it as a method of tuning a scale virtually identical to
12-tone equal temperament or 12-tet (1766).[1]


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1. Kirnberger's Tuning: the Lambert-Marpurg Developments
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Interestingly, Kirnberger shared his concept with the eminent
mathematician Leonhard Euler. Johann Heinrich Lambert further
developed and published it in 1774 in the _Memoires de l'academie
royale des sciences et belle lettres_ (Berlin), pp. 64ff.[2]

In turn, the notable musician and tuning theorist Friedrich Wilhelm
Marpurg reported Lambert's method and later translated this theorist's
essay into German, adding a commentary praising its accuracy.[3] Here
there may have been a bit of irony, since Kirnberger was generally
opposed to 12-tet while Marpurg favored it:

Marpurg praised Lambert's method and pointed out
that its deviations from exact equal temperament
never exceed .00001. Another advantage of this
method is that it does not require a monochord.
Marpurg, who supported equal temperament and was
violently opposed to Kirnberger's views on
temperament as well as on harmony, was only too
eager to point out that this feature of Lambert's
method invalidated one of Kirnberger's objections
to equal temperament.[4]

In other words, while it had previously seemed necessary (using
18th-century keyboard tuning technology) to tune 12-tet by reference
to a standard monochord or other instrument lending itself to a
geometric rule or the like for defining the 100-cent intervals, the
Kirnberger-Lambert-Marpurg approach in theory called only for the
tuning by ear of pure fifths and thirds to generate each 16384:10935
fifth (or 10935:8192 fourth) of the scale.

As Barbour notes in his classic history -- in a portion with which,
unfortunately, I was less familiar at the time of my first post --
Marpurg "believed that the tuning of the just intervals used in [this
method] could be made more quickly and accurately than the estimation
by ear of the tempering needed for the fourth or fifth."[5]

However, Owen Jorgensen observes in his thorough 1991 account of the
same tuning that quite apart from the large number of pure intervals
that would need to be tuned, small errors would tend to accumulate
that could make accurately closing the circle of fifths quite
problematic[6]. While Barbour and Jorgensen are known at times to
differ in their viewpoints, the latter in this case seems to agree
also with the former's observation that in addition to issues of
accuracy in practice, there is also "the labor of tuning eight pure
intervals in order to have only one tempered interval!"[7]

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2. Farey's (Re)discovery of 1807
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Both Barbour and Jorgensen additionally document how this tuning was
apparently _re_-"discovered independently" in England by a certain
John Farey, who on 21 June 1807 chronicled his discovery that a chain
of "five just fourths minus two just fifths minus one just major third
create a fifth that is almost identical to an equal temperament
fifth. Today, this type of fifth is known as a `schisma fifth,' and it
has a ratio of 16384 to 10935."[8]

As Jorgensen notes, Farey described this method as "new" because "he
thought he had discovered it" -- although, as we know, Kirnberger,
Lambert, and Marpurg actually had precedence.[9]

Having myself stumbled upon this possibly "new" concept -- although as
noted in my original post, I knew that the schisma fifth had been
documented, and was uncertain at that time only as to whether a
complete quasi-12-tet scale had been published -- I can testify to the
excitement that Farey might also have experienced.


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3. Helmholtz-Ellis and the Schisma Fifth
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Paul Erlich has mentioned the names of Hermann L. F. Helmholtz and his
translator Alexander J. Ellis as documenters of this tuning, and they
do indeed document the schisma fifth and its very close approximation
to a 12-tet fifth, Ellis giving the difference as 0.001280 cents.[10]
While I'm not sure that either would be too anxious to take credit for
promoting a tuning using just intervals to approximate 12-tet ,
this measurement of what I playfully referred to in my post as the
"scintilla of Artusi" is certainly worthy of recognition.

Additionally, the schisma fifth or fourth does actually appear in a
Ellis tuning for the harmonium designed as a practical realization of
the "Helmholtzian temperament" based theoretically on obtaining pure
major thirds by flattening each fifth by 1/8 of a schisma. However,
since Ellis finds such a small temperament impossible to tune
accurately by ear, it appears to me that a whole schisma is therefore
dumped on one fifth at certain points in the chain so that a small
number of fifths or fourths are "700 cents down or 500, in place of
702 and 498 as all the others."[11]

This procedure might be treated as a pre-digital equivalent of
nanotemperament, the problem here being not the tuning limitations of
a digital synthesizer but the limited discriminations possible for the
human ear. In either case, rounding off a few intervals further from
just than might otherwise be necessarily helps to minimize the
inaccuracies of the tuning as a whole.


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4. A small difference -- not necessarily "new"
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One rather trivial difference between what I've seen _so far_ of
Kirnberger-Lambert-Marpurg or Farey lines of quasi-12-tet just tuning
and my proposal is that the former seem to generate a 10935:8196
schisma fourth by combining seven pure fifths and a pure major third,
i.e. a Pythagorean apotome of 2187:2048 (about 113.69 cents) and a
pure 5:4 (about 386.31 cents), or almost exactly 500 cents in all.

Maybe because of my fascination with the schisma major third of
8192:6561 (about 384.36 cents) and its role in early 15th-century
keyboard music, I approached the problem as one of building a schisma
fifth from this familiar interval plus an appropriate type of minor
third. An m3 at a pure 6:5 (about 315.64 cents) neatly fills this
role, and provides another (and longer) road to the same "12-tet by
just intonation" destination as the apotome-plus-pure-M3 solution.

In fact, from an even marginally "practical" point of view, the
well-documented Kirnberger and Farey approach generates each schisma
fourth with only seven pure fifths plus a pure M3, as opposed to the
_eight_ pure fifths plus a pure m3 required for the approach described
in my post.

However, I would be anything but surprised to see this minor variant
on Kirnberger's scheme also documented in the published literature,
possibly, like the basic scheme itself, early and often.


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Notes
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1. Helpful modern sources of documentation include a note in an
English translation by David Beach and Jurgen Thym of Kirnberger's
treatise _The Art of Strict Musical Composition_ (New Haven and
London: Yale University Press, 1982), p. 20 n. h; J. Murray Barbour's
classic _Tuning and Temperament: A Historical Survey_ (East Lansing:
Michigan State College Press (1953), pp. 64-65; and Owen Jorgensen's
Jorgensen, Owen H.., 1991. _Tuning: Containing The Perfection of
Eighteenth-Century Temperament, The Lost Art of Nineteeth-Century
Temperament, and the Science of Equal Temperament, Complete with
Instructions for Aural and Electronic Tuning_ (East Lansing: Michigan
State University Press, 1991), pp. 312-313.

2. Barbour, op. cit., pp. 64-65; Beach and Thym, eds. and trs.,
op. cit., p. 20 n. h; Jorgensen, op. cit., p. 312.

3. Barbour, ibid.; Beach and Thym, ibid. One primary source I have
found on microcard (similar to microfiche) is Marpurg's translation of
Lambert and commentary in his _Historisch-kritisch Beytrage zur
Aufnahme der Musik_, vol. 5. part 6 (Berlin, 1778), pp. 417-450.

4. Beach and Thym, ibid.

5. Barbour, see n. 1, p. 65.

6. Jorgensen, see n. 1, p. 312. Unfortunately, while I was familiar
when posting my original article with a brief description of the
"schisma fifth" in the glossary of Jorgensen's earlier _Tuning the
Historical Temperaments by Ear_ (Marquette: Northern Michigan
University Press, 1977), I was not aware of his full account of the
Kirnberger and Farey "quasi-12-tet" in this latter work. The heading
is "Tuning Equal Temperament by Using Just Intonation Techniques in
1807," referring to the date of Farey's account.

7. Jorgensen (1991), ibid., p. 312; Barbour, see n. 1, p. 65.
Jorgensen remarks on "the degree of perfection" required in
tuning so many pure intervals makes it "more difficult" than "modern
tempering methods" for obtaining 12-tet, although "[u]sually, just
intonation is considered easier for amateurs." Barbour focuses more
specifically on the traditional view that tuning a pure major third is
a difficult task. "If this be true, a type of tuning in which the
essential feature is a pure major third could not be very
accurate..."

7. Barbour, ibid.

8. Jorgensen, see n. 1, p. 312, the source of the quote; and also
Barber, ibid., who notes that Farey's version was communicated across
the Atlantic, appearing in "Dr. Rees's _New Cyclopedia_" (1st American
edition, Vol. 14, Part 1, article on "Equal Temperament"), in which
"we are shown how Farey's method `differs only in an insensible
degree' from correct equal temperament."

9. Jorgensen, ibid.

10. Hermann L. F. Helmholtz, with translation, notes, and Appendices
by Alexander J. Ellis, _On the Sensations of Tone as a Physiological
Basis for the Theory of Music_, 6th ed. (New York: Peter Smith, 1948),
p. 316, n. at *. Helmholtz having noted that difference between a
perfect fifth and what is now known as a schisma fifth is "about the
same as that between a perfect and an equally tempered fifth," Ellis
offers a more precise calculation in order to show "the extreme
closeness of the result..." He takes a schisma of 32805:32768 as
1.953721 cents, and the narrowing of a 12-tet fifth as 1.955001 cents:
"Difference .001280 cents. Human ears, however much assisted by human
contrivances, could never hear the difference." Ibid., p. 432 at
"Art. 10," Ellis again emphasizes the virtual identity of these two
intervals: "The Skhisma will therefore be considered as the twelfth
part of a Pythagorean Comma, and also as the error of an equal Fifth."
Jorgensen, ibid., observes that "the ear cannot distinguish between
just intonation quasi-equal temperament and equal temperament."

11. Helmholtz and Ellis, ibid., pp. 316-317, last note beginning at
bottom of 316. In the table included in this note, schisma fourths of
virtually 500 cents are found in moving from the bottom of column I to
the top of II; and likewise with columns II and III; IV and V; and V
and VI. Kirnberger also originally arrived at the schisma fourth or
fifth "in the process of developing his own temperament" (i.e. an
unequal well-temperament), Beach and Thym, see n. 1, 20 n. h;
Jorgensen, see n. 1, p. 312, refers to the account elsewhere in his
book of this Kirnberger well- temperament demonstrating how "the
schisma fifth G-flat D-flat" arises from a chain of seven pure fifths
and fourths plus a pure major third.

Most respectfully,

Margo Schulter
mschulter@value.net