source file: m1523.txt Date: Thu, 3 Sep 1998 02:35:55 -0700 (PDT) Subject: Re: Kirnberger's quasi-12-tet by JI From: "M. Schulter" --------------------------------- Kirnberger's 10935:8192 tuning: (Re-)"discovered" -- Again --------------------------------- 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] -------------------------------------------------------- 1. Kirnberger's Tuning: the Lambert-Marpurg Developments -------------------------------------------------------- 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] -------------------------------- 2. Farey's (Re)discovery of 1807 -------------------------------- 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. ---------------------------------------- 3. Helmholtz-Ellis and the Schisma Fifth ---------------------------------------- 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. ---------------------------------------------- 4. A small difference -- not necessarily "new" ---------------------------------------------- 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. ----- Notes ----- 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