Shape of things to come?

[Richard Moody]

Here is another example of the blending of temperaments Ed Foote
mentioned. 

I don't seem to have a url execpt for the email address.  If ucsb is
University of Calif. Santa Barbra don't we have a tech from there on this
list? One could probably enter MTO in a search engine such as Yahoo, or
Alta Vista. I would be happy to forward the complete article to those
interested. 




       M U S I C          T H E O R Y         O N L I N E

                     A Publication of the
                   Society for Music Theory
         Copyright (c) 1998 Society for Music Theory
+-------------------------------------------------------------+
| Volume 4, Number 4      July, 1998       ISSN:  1067-3040   |
+-------------------------------------------------------------+

  General Editor                          Lee Rothfarb


All queries to: mto-editor@smt.ucsb.edu  or to
                  mto-manager@smt.ucsb.edu
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1. Target Articles

AUTHOR: Scholtz, Kenneth P.
TITLE: Algorithms for Mapping Diatonic Keyboard
  Tunings and Temperaments
KEYWORDS: Algorithm, chain of fifths, diatonic
  scale, equal temperament, enharmonic, just
  intonation, meantone, Pythagorean tuning,
  schisma, syntonic comma

Kenneth P. Scholtz
2821 Anchor Ave.
Los Angeles, CA 90064-4605
kscholtz@earthlink.net

ABSTRACT: Diatonic keyboard tunings in equal temperament, just
intonation, meantone and well tempered scales are derived from
Pythagorean tuning using algorithms whose operations involve
combinations of pure fifths and syntonic commas.  Graphic
diagrams of the line of fifths are used to show the harmonic and
mathematical relationships between common tunings and
temperaments.  Four modes of just intonation are derived from
Pythagorean tuning by an algorithm that narrows each major third
by one syntonic comma.  Equal temperament is approximated with
imperceptible error by algorithms that narrow Pythagorean and
justly tuned enharmonic intervals by one or more syntonic commas.

Buried in this article (its not as complicated as as the abstract sounds)
is the suggeston that since Meantone has four very narrow fifths, (in
order to acheive a pure third with C to begin with) the remaining fifths
could be tuned pure, which would eliminate the wolf  (in this case C#--Ab)
That though defeats the objective of Meantone of having the most possible
pure thirds and the fifths as  equal (narrow) as possible, execpt one. The
one they call the Wolf.....

The beginning of the article expresses its objective better than the
abstract ........

1. The syntonic comma is defined as the difference between the
Pythagorean tuning and just tuning of the major third.  The
difference between the just third (5/4) and the Pythagorean third
(81/64) is 81/80, calculated as follows: 81/64 x 4/5 = 81/16 x
1/5 = 81/80.  The syntonic comma is also the difference between
the Pythagorean and just tunings for all diatonic intervals other
than the fourth and fifth, which are the same in both tunings.
The reason for the repeated appearance of the syntonic comma will
be apparent from the discussion of the four modes of just
intonation in section 5.

If you don't follow the  "calculated as follows......"  don't worry, the
fractions are the most complicated math, and the 1/x commas (that come
later on) make sense in a general way, as this is how he compares various
temperaments to just intonation. 

5.  Intervals measured by rational fractions can be converted
into cents using the following approximate values: octave = 1200
cents, perfect fifth = 702 cents; Pythagorean comma = 24 cents;
syntonic comma = 22 cents.


Anyhow here is how he explains the merging of Meantone with pure
fifths.......

[7.1] The term well temperament includes a family of temperaments
that modified meantone temperament to eliminate wolves and to
expand the range of playable keys by taking advantage of the
small difference between the Pythagorean comma and the syntonic
comma.  The difference between the two commas is an interval of
32805/32786 (2 cents), which is called the schisma.  If only four
links of a chromatic keyboard scale are tempered by a
quarter-comma, with the remainder {of the fifths}being tuned pure, the
chromaticscale will exceed an acoustic octave by only a schisma and the
wolf fifth will be thereby minimized to the point of
nonexistence.

Whether Scholtz's "four links of a chromatic keyboard" are the same as the
first four fifths narrowed that produce a pure third, is to be determined.
 Never the less, the idea of blending two temperaments is as interesting
as the various means and methods.  

Richard Moody 


> From: A440A@AOL.COM
> To: pianotech@ptg.org
> Subject: Shape of things to come?
> Date: Wednesday, April 14, 1999 8:03 AM
> 
>
> 
 to merge two tunings together, and he had combined ET with 
> various historical tunings, resulting in some, (at least to me!), very 
> interesting looking tunings.  
> Ed Foote