Re: comma and chroma (for Joe Pehrson)

Hello, there, Joe Pehrson and everyone, and please let me try to
answer at least part of your question about the term _chroma_,
specifically this term as applied to the 25:24.

In 16th-century just intonation theory based on ratios of 2-3-5, this
interval represents the chromatic semitone, or rather a typical size
of chromatic semitone equal to the difference between the pure minor
third (e.g. C-Eb) at 6:5 and the major third at 5:4. For example, this
chromatic semitone would occur as Eb-E.

As I recall, the term _chroma_ around 1540 also refers to the note
value of half of a semiminim or crotchet, or in modern terms a quaver
or eighth note. Thus the term "chromatic madrigal" in that epoch often
referred not to chromaticism (although became a feature of the
emerging Manneristic style as used by composers such as Rore and the
young Lasso), but to the use of smaller note values or "black notes"
(_note nere_).

Of course, as people such as Dave Keenan have recognized in their
efforts to devise consistent and descriptive interval naming systems,
the concept of 25:24 as a _chromatic_ semitone assumes certain
stylistic patterns, here a 5-limit intonation.

As it happens, Zarlino's 2/7-comma meantone has a chromatic semitone
at precisely 25:24, while in 1/4-comma meantone with pure 5:4 major
thirds, this semitone is slightly larger (~76.05 cents).

However, in neo-Gothic music based on 17-tET, for example, the
_diatonic_ semitone (e.g. E-F, B-C) is equal to almost exactly this
same ratio of 25:24 or ~70.67 cents -- it's 1/17 octave (~70.59
cents).

While I'm not sure what role 33:32 plays in 11-limit music, I would
call it very nice size for a small neo-Gothic semitone.

Anyway, on the chroma vs. comma distinction, I might describe it like
this.

The syntonic comma of 81:80 (~21.51 cents) is in 16th-century
technique not an interval which would explicitly occur in a notated
piece, although it is the theoretical difference in size between the
two whole-tones of Ptolemy's syntonic diatonic as adapted by Zarlino,
at 9:8 and 10:9 (together forming a pure 5:4 third).

In contrast, the chromatic semitone may occur as a direct melodic
interval in more adventurous pieces, and would ideally be 25:24 if the
singers are approximating 5-limit JI.

Note that Zarlino himself takes the position that while the syntonic
diatonic defines the natural ratios toward which voices tend, singers
can and will make adjustments to avoid dissonances such as a 40:27
"Wolf fifth." While this is suggested by a statement that he makes in
his treatise of 1558, he deals with the point at more length in his
_Musical Supplements_ of 1588.

In other words, what Zarlino apparently sees happening in practice is
some kind of adaptive JI, with the flexibility of voices making
possible a close _approximation_ of the syntonic diatonic, and more
specifically its pure vertical intervals.

One way of expressing the difference between the chroma and the comma
is that the difference of a chroma affects the rules of counterpoint;
the difference of a comma does not have this effect. In other words,
Zarlino makes a distinction between a major third and a minor third
(e.g. closest-approach progressions m3-1 and M3-5), but not, at least
in the same way, between a theoretical 9:8 or 10:9 major second.

One partial exception to this statement might be a passage in 1558
about the issue of parallel thirds, where Zarlino says that some of
these movements may have more variety than others because of the
differences in the sizes of melodic steps.

However, I would generally agree with scholars such as Carl Dahlhaus
and Karol Berger that the chromatic semitone, but not the syntonic
comma, makes a difference in musical grammar, so to speak, as well as
in the tuning system.

In other words, I agree with Paul Erlich that if we realize this music
in some appropriate shading of meantone, where the comma is dispersed,
the musical grammar and meaning is accurately communicated, with what
might be considered either the compromise or the sonorous pleasantries
of temperament. Mark Lindley, for example, prefers meantone to 12-tET
for some of Milan's pieces in part because he likes the "warmth" of
the distinctly beating fifths.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:

/tuning/topicId_26318.html#26318

> Hello, there, Joe Pehrson and everyone, and please let me try to
> answer at least part of your question about the term _chroma_,
> specifically this term as applied to the 25:24.
>

Thank you *very* much, Margo, for your response to this. Using the
term "chroma" for this 25:24 semitone makes sense... curiously
enough, in my introduction to Ben Johnston's _String Quartet #6_ this
interval is called a "diatonic comma..."

That seems a bit ambiguous...

>
> The syntonic comma of 81:80 (~21.51 cents) is in 16th-century
> technique not an interval which would explicitly occur in a notated
> piece, although it is the theoretical difference in size between the
> two whole-tones of Ptolemy's syntonic diatonic as adapted by
Zarlino, at 9:8 and 10:9 (together forming a pure 5:4 third).
>
> In contrast, the chromatic semitone may occur as a direct melodic
> interval in more adventurous pieces, and would ideally be 25:24 if
the singers are approximating 5-limit JI.
>

I see... so the "comma" is generally considered to be a
more "theoretical" construct that is even "eliminated" in certain
optimal cases, rather than a physical interval that is to be utilized
in a piece...

Thanks so much for the help!

Joe

________ ________ _____
Joseph Pehrson

> From: <jpehrson@rcn.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, July 19, 2001 7:48 PM
> Subject: [tuning] Re: comma and chroma (for Joe Pehrson)
>
>
> --- In tuning@y..., mschulter <MSCHULTER@V...> wrote:
>
> /tuning/topicId_26318.html#26318
>
> > Hello, there, Joe Pehrson and everyone, and please let me try to
> > answer at least part of your question about the term _chroma_,
> > specifically this term as applied to the 25:24.
> >
>
> Thank you *very* much, Margo, for your response to this. Using the
> term "chroma" for this 25:24 semitone makes sense... curiously
> enough, in my introduction to Ben Johnston's _String Quartet #6_ this
> interval is called a "diatonic comma..."
>
> That seems a bit ambiguous...

Hi Joe,

Paul has already posted along these lines. I'm pretty sure
that that must be a misprint, as all of Johnston's other
characteristic intervals are called "chroma".

Is Andy here? Can we find out directly from Ben?
(I could always email him directly, if necessary...)

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com