Apropos of Pythagorean and related tunings, I saved this article from
rec.music.early a while ago. Margo is r.m.e's resident exotic-early-
tunings wonk (she plays this way herself on a pitch-configurable
electronic keyboard). I *dare* any of you to ask her to expand on this...
>From "M. Schulter" <[EMAIL PROTECTED]> Sun Feb 18 23:00:09 2001
>Status:
>Subject: Re: temperament term???
>From: "M. Schulter" <[EMAIL PROTECTED]>
>Date: 18 Feb 2001 23:00:09 GMT
>Organization: Value Net Internetwork Services
>Newsgroups: rec.music.early
>References: <v04003a50b6add422bc5a@[128.173.232.136]> <[EMAIL PROTECTED]>
>Path:
>purr!news.demon.co.uk!demon!dispose.news.demon.net!demon!newsfeed.gamma.ru!Gamma.RU!news.maxwell.syr.edu!feed2.news.rcn.net!rcn!news2.best.com!vnetnews.value.net!not-for-mail
>Message-ID: <96pk5p$1bu$[EMAIL PROTECTED]>
>Lines: 130
>Article 9424
>
>
>Jonathan Addleman <[EMAIL PROTECTED]> wrote:
>
>: But it WAS done now and then, if only to accomodate the range of
>: various singers. Vicentino talks about this use of the archicembalo,
>: since you can play in meantone in any key. Frescobaldi at some point
>: mentioned that an organ tuned in equal temperament would be good for
>: this reason as well, though I don't know where that reference is..
>: (I got it 2nd or third hand...)
>
>Hello, there, and I must admit to being a bit confused by parts of this
>thread, which is one reason that I've preferred simply to read rather than
>to post up until now -- but maybe I can comment usefully on certain
>points, at least.
>
>First of all, I haven't really previously heard the terms "base" or
>"focus" in describing a tuning, although I might speak of range, for
>example "a 12-note meantone tuning of Eb-G#," or "a 19-note tuning, likely
>1/4-comma, of Gb-B#," or "a 17-note Pythagorean tuning of Gb-A#, evidently
>of the type described by Prosdocimus de Beldemandis and Ugolino of Orvieto
>in the earlier 15th century."
>
>In this thread, there seems to be a focus on two types of temperament: the
>regular meantone tunings of the late 15th to late 17th centuries, still in
>use in the 18th century, which might feature anything from 12 to 31 notes
>per octave; and the 12-note "well-temperaments" of the late 17th to 19th
>centuries, where the circle of fifths closes -- as it does, either
>precisely or "virtually" for musical purposes, also in a 19-note meantone
>tuning of around 1/3-comma, or in a 31-note meantone of around 1/4-comma.
>
>Indeed Vicentino promoted his _archicembalo_ and _arciorgano_ -- his
>superharpsichord and superorgan (the latter a kind of positive organ which
>could be disassembled, carried on a mule's back, and then reassembled at
>the next performance location -- as permitting free transposition. If we
>speak in "keys" in an Elizabethan sense as referring to the pitch level of
>a modal final, rather than to later major/minor concepts, then it is
>indeed correct that Vicentino's 31-note meantone tuning makes available
>all intervals on all 31 steps of the cycle.
>
>Basically Vicentino's tuning scheme of 1555 seems to combine two features
>which by the late 17th century were recognized to result in _very
>slightly_ different tunings. He describes a division of the whole-tone
>into five "minor dieses" of equal size, which would call for 31-tone equal
>temperament (31-tET), with major thirds very slightly larger than pure; he
>also suggests that major thirds are pure (1/4-comma meantone). In
>practice, the variations in a tuning by ear could be greater than the
>theoretical difference between these two models.
>
>Quite apart from accommodating singers, Vicentino's tuning makes available
>"enharmonic" steps inspired by those of Ancient Greek theory, about
>1/5-tone in size, which this composer and theorist espouses for their
>subtlety and "gentleness." Indeed, these fifthtone steps have a remarkable
>effect, and add an expressive dimension to some more typical 16th-century
>chromatic progressions also.
>
>More conventional theorists also address the matter of transpositions to
>accommodate singers, but within an apparent framework of 12-note meantone,
>where transpositions by fifths or fourths, or by a major second up or down
>(two fifths on the tuning chain), are most typical.
>
>In 1570, Guillaume Costeley describes a 19-note keyboard arranged in
>thirdtones dividing the octave into equal parts -- this, like Vicentino's
>31-note tuning in or around 1/4-comma, is a circular scheme, which would
>permit free transposition.
>
>In 1618, Fabio Colonna describes his 31-note meantone keyboard, with a
>tuning scheme similar to Vicentino's (likely 1/4-comma), but a keyboard
>arranged in five groups of seven notes, with each rank tuned 1/5-tone
>apart (resulting in some replications of notes). To demonstrate the closed
>nature of this system, he provides a composition giving an "Example of
>Circulation" which moves through a circle of cadences on all 31 steps of
>the instrument, each featuring motion of the bass by a fifth down or a
>fourth up.
>
>He also shows how various modes can be transposed to various remote steps
>of the system.
>
>If the Vicentino-Colonna line of development had prevailed, a system of
>372 finals might have developed: 12 modes transposable to each of the 31
>steps of this circular system. If one prefers to regard authentic and
>plagal modes as equivalent -- and there remain some interesting
>differences in formulas in the early 17th century -- then we would have
>186 finals (31 steps x 6 modal octave-species). The potentials of such a
>modal system still remain largely unexplored today.
>
>By the late 17th century, major/minor tonality was becoming established as
>the new prevailing paradigm, as noted by Werckmeister, who described
>12-note well-temperaments where the fifths and thirds take on variable
>sizes in order to permit a closed tuning circle in only the 12 notes
>found on usual keyboards.
>
>For such well-temperaments, the general convention is that the near
>keys should have pure or near-pure thirds ("pure" meaning here, as in
>meantone tunings, ratios approaching the Renaissance ideal of 5:4 and
>6:5), and the most remote keys Pythagorean or near-Pythagorean thirds
>(around 81:64 and 32:27).
>
>The Pythagorean ratios, routine in the music of the Gothic era in
>Continental Europe, here represent at once a kind of mathematical
>necessity and an artistic distortion in a setting where 5:4 and 6:5 are
>the "ideal forms." Although originally they may have been accepted by
>musicians of the Werckmeister epoch (c. 1680-1700) simply as the
>unavoidable price for a 12-note circulating system with the near keys
>reasonably close to meantone, musicians of the 18th and early 19th
>centuries often exploited the extra tension of the remote keys as a
>musical virtue, and theorists such as Kirnberger (1771) and the Earl of
>Stanhope (c. 1800) viewed the differences between keys as a virtue which
>would be removed by 12-note equal temperament.
>
>Incidentally, while unequal well-temperaments tend be taken as
>"absolute" systems -- that is, with a conventional ordering of thirds more
>or less from the most frequent (e.g. C-E) to the most remote
>(e.g. F#-A#) -- it's possible to transpose such a system for special
>effects, for example a "reverse well-temperament" where the most frequent
>thirds are Pythagorean (in a typical Gothic fashion) and the most remote
>ones somewhat like meantone. There's a discussion about this in the
>well-temperament portion of my Pythagorean Tuning FAQ:
>
>http://www.medieval.org/emfaq/harmony/pyth.html
>
>Having described a scheme like this not quite three years ago, I'd add
>that for a "neo-medieval" style, a kind of "well-temperament" in 17 notes
>rather than 12 might be more stylistically conducive for many purposes.
>
>I'm not sure if these comments might provide any basis for feedback on the
>focus of this thread, and I'd welcome more discussion.
>
>Most respectfully,
>
>Margo Schulter
>[EMAIL PROTECTED]
>
=================== <http://www.purr.demon.co.uk/jack/> ===================
To subscribe/unsubscribe, point your browser to: http://www.tullochgorm.com/lists.html
