Early Music, Temperaments, Logs

I saw this on rec.music.early. Does anyone know anything more about
this and the music played? (Johnny?).

rec.music.early #16829 (0 + 17 more)
From: mccomb@best.com (Todd Michel McComb)
[2] Re: alternative tunings in 16th century
+ music
Date: Fri Jan 03 13:49:47 PST 1997


In article <2184@purr.demon.co.uk>, Jack Campin
wrote:
>A few years ago I heard an astonishing broadcast (on BBC Radio 3) of 16th
>century Neapolitan harpsichord music written for microtonal insytruments -
>the concert was part of an International Festival of Microtonal Music mainly
>centred around contemporary works. Does anybody out there know what I'm
>talking about? Who was the performer, and has he or she toured this stuff
>or put it on a published recording?

I don't know anything about this particular performance, but the repertory
is fairly well-known to EM keyboardists. Definitely, as far as recordings
go, the one to hear is Alan Curtis' on Nuova Era 7177
http://www.medieval.org/emfaq/cds/nuo7177.htm

Todd McComb
mccomb@medieval.org


BTW, I was re-reading Bosanquet's 1876 book "An Elementary Treatist on
Musical Intervals and Temperament" in which he describes his famous
generalized keyboard. What caught my attention was a section where he
calculates musical logarithms, in this case the semitone measure of
ratios (equivalent to cents divided by 100). Bosanquet computes his own
logs from a series expansion approximation rather than using tables
and implies that the proofs of such expansions were commonly given in
19th century trig books. Anyway, a longer discussion which illuminated
the method of Ellis, which I posted to the tuning list last Spring (or
so). The method appears less to have been pulled out of thin air than
it does in Ellis's exposition in Helmholtz.

I, for one, would like to see David Feldman's table of "isotempered"
interval pairs. Alas, I have an old, small screen Mac and would need
the reformatted or a hardcopy version.

By using Brun's algorithm, it was easy to show that Feldman is correct
in that the series of relations of the degrees of the fifth Q to those
in the octave T: Q\T ~3\5, 4\7, 7\12, 10\17, 17\29, 24\41, 31\53,
55\94 corresponds to the number of degrees of the 13/9 in the
equal divisions of the 15/8.

One might describe this as series of compressed octaves which in
which the relative size of the 13/9 corresponds to the relative
size of the 3/2 in the original series of octaval ET's.

I also played around with another, but related, problem of finding
optimal ET's for sets of all 4 numbers, in this case 2/1, 15/8, 3/2
and 13/9. Using Viggo Brun's version of the euclidean algorithm, I got
the following N-tone ET's: 10, 19, 24, 43, 53, 72, 125, 178...(only
those which distinguish all 4 ratios are listed). Selmer's version
oscillates a bit, but finds 20, 21, 12, 26, 34, 46, 33, 41, 54, 75,
62, 87, 162. Pipping's ramified version gave erratic results, though
it works fine for 3 terms.



--John


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Manuel is correct the flat fifths are: C-G, G-D, D-A, B-F#

(Circling in the back of my mind was the fact that F Major is the most
"just" of the major keys.)

Johnny Reinhard
American Festival of Microtonal Music
318 East 70th Street, Suite 5FW
New York, New York 10021 USA
(212)517-3550/fax (212) 517-5495
reinhard@ios.com

On Sat, 4 Jan 1997, Manuel Op de Coul wrote:

> > Translation of the Werckmeister treatise of 1691
> > remains available only in archaic Thuringian German (and in an unpublished
> No that's the original.
>
> Johnny's description of Werckmeister III is incorrect. It's this:
> Eb Bb F C G D A E B F# C# G#
> 0 0 0 1/4 1/4 1/4 0 0 1/4 0 0 syntonic commas flat.
>
> Manuel Op de Coul coul@ezh.nl
>


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> Johnny's description of Werckmeister III is incorrect. It's this:
> Eb Bb F C G D A E B F# C# G#
> 0 0 0 1/4 1/4 1/4 0 0 1/4 0 0 syntonic commas flat.

Actually those would have to be in Pythagorean commas (flat) or the
circle would not close.

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