Yahoo Tuning Groups Ultimate Backup mills-tuning-list TUNING digest 970

Gary Morrison wrote (2/2/97):

> Or relative to the tonic anyway; 15:8 is of course is mediant
> of the dominant, so perhaps one could argue it that way. Perhaps
> one could argue that fifth comma meantone was devised to make the
> third of V-chord just in the same sense that third-comma meantone
> makes the IV-chord just. That combined with the fact that the
> total error between the third of I (the mediant) and the third of
> V (the leading tone) is smaller that way. In the quarter-comma
> meantone case, the mediant's error is 0 and the leading tone's is
> 1/4 comma, whereas in fifth-comma meantone a total error of 1/4
> comma, the mediant's is 1/5 comma, and the leading tone's is 0, a
> total of 1/5 comma.

I've been trying to catch up with a lot of list messages, and among
them I noticed the above passage; I've looked for a subsequent
retraction/correction but can't find any; I've also tried squinting
at it in various ways in the hope that the words will somehow make
sense, but with no success.

First, an ad hoc definition: let's call any key (or mode) that is
in keeping with the chosen set of pitch classes in a meantone
system an "available" key (or mode). In this context, I am
excluding the notion of enharmonic equivalence from "pitch class".
On a keyboard of twelve notes to the octave, for example, the most
common choice of pitch classes (in the 15th/16th centuries) was:

C, C#, D, Eb, E, F, F#, G, G#, A Bb, B

Given this choice, the major keys of Bb, F, C, G, D and A are all
"available", in the given sense (I am speaking only of the
availability of pitch classes for precisely these keys -- of course
even a simple piece in Bb is likely at some stage to require an Ab,
and similarly for the rest). No other major keys are available,
since they must utilise wolf intervals, e.g. the dominant triad of
E major will consist of a diminished fourth plus an augmented
second in the given meantone tuning, rather than a major third plus
minor third. I know perfectly well that this is all common
knowledge, but I have to get the issue of enharmonic equivalence
out of the way to avoid confusion.

Now in any available key, thus defined, a given interval type is
uniform in size, not only in this key, but also in all other
available keys. Thus, in a given meantone tuning, the third from C
to E will be the same as the third from F to A and the third from G
to B. In 1/4-comma meantone, all these major thirds will be just,
as will any major third in any available key. In any other meantone
tuning, the major thirds in available keys will not, of course, be
just; for any meantone tuning based on a rational power of the
syntonic comma, there will, of course, be at least one just
interval class, but aside from 1/4-comma, where maj. thirds and
min. sixths are just, the only other meantone which was chosen (in
the period from 15th - 17th centuries) to produce just intervals of
musical interest was 1/3-comma, with just min. thirds and maj.
sixths.

What on earth, then, is the meaning of the passage quoted at the
top? The fact that 1/5-comma meantone has just, i.e. 15/8, major
sevenths in no way distinguishes the tuning of the dominant chord
in an available key from the tuning of the tonic chord; likewise,
1/3-comma doesn't distinguish between the tuning of subdominants
and tonics in available keys.

In any available key, a given meantone system will no more favour
one triad over another than it will favour one presidential
candidate over another.

--
Jonathan Walker
Queen's University Belfast
mailto:kollos@cavehill.dnet.co.uk
http://www.music.qub.ac.uk/~walker/

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TUNING digest 970

🔗David Malkin <dmmalkin@...>

🔗kollos@cavehill.dnet.co.uk (Jonathan Walker)