Re: schismatic tuning

Please let me respond to some of the remarks about European tuning during
the era around 1400-1450 that the usual 12-note Pythagorean keyboard
tuning often described as "schismatic" is Gb-B, with the "Wolf" fourth
between these notes.

While systems with Wolf fifths or fourths between notes of the regular
gamut (Bb-B), with Bb along with the diatonic notes counting as part of
this gamut (_musica recta_), can be found in some theoretical sources,
Mark Lindley regards them as likely of little practical relevance to
keyboard tuning as applied to most pieces of the era, although experiments
with some of these systems might have helped lead to meantone temperament.

Thus I would recommend that the Gb-B tuning be given as the standard
15th-century European "schismatic" tuning; this system, unlike schemes
with Wolves among the diatonic notes, fits much of the known keyboard
music of the era, some of which seems ideally to exploit its features.

This means that C-E, F-A, and G-B, as well as Bb-D, are usual Pythagorean
thirds; while D-Gb, E-Ab, and A-Db are near-5-limit or schismatic thirds,
along with the less important B-Eb (where adding the Wolf fifth would
produce B-Eb-Gb -- although on a 13-note per octave instrment with a true
F# added, one could play B-Eb-F# as a schismatic sonority with a pure
fifth).

In the Gb-B tuning, as in a later well-temperament or temperament
ordinaire, usual thirds (from the viewpoint of the keyboard player) thus
have contrasting sizes, providing an element of "modal color" as Lindley
nicely calls it. For example, in this type of written cadence:

D4 C#4 D4
A3 G#3 A3
F3 E3 D3

the first sixth sonority F3-A3-D4 has usual Pythagorean intervals
(~0-408-906 cents), while E3-Ab3-Db4 (as actually tuned on the
keyboard) has a near-5-limit sonority (~0-384-882 cents). The precise
ratios are 81:64 and 27:16 for the usual Pythagorean major third and
sixth, and 8192:6561 and 32768:19683 for the schismatic counterparts
(actually diminished fourths and sevenths).

While one can also give the Ramos monochord scheme (I agree with Lindley
that his practical 12-note keyboard tuning seems to be some kind of
meantone in Ab-C#), or other schemes with a Wolf fifth among the diatonic
notes or at Bb-F, I would suggest that the Gb-B scheme was likely typical
for the earlier 15th century, with more or less regular meantone coming
into vogue around 1450-1480.

Most appreciatively,

Margo Schulter
mschulter@value.net

hello Margo,

i've been hoping to see your intelligent and
inventive comments on the latest few of my
webpages. this post would make a great addendum
to my latest page, on schismatic tuning

http://sonic-arts.org/dict/schismatic-tuning.htm

may i use it?

-monz

> From: "M. Schulter" <MSCHULTER@VALUE.NET>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, March 25, 2003 10:05 PM
> Subject: [tuning] Re: schismatic tuning
>
>
> Please let me respond to some of the remarks about European tuning during
> the era around 1400-1450 that the usual 12-note Pythagorean keyboard
> tuning often described as "schismatic" is Gb-B, with the "Wolf" fourth
> between these notes.
>
><etc. -- snip>

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> hello Margo,
>
>
> i've been hoping to see your intelligent and
> inventive comments on the latest few of my
> webpages. this post would make a great addendum
> to my latest page, on schismatic tuning
>
> http://sonic-arts.org/dict/schismatic-tuning.htm
>
>
> may i use it?
>
>
>
> -monz

i don't understand your coloring scheme on this page. you're using
the same color for notes like Ab, which were in the tuning, and notes
like F#-, which were not (though Gb, a schisma away, was in the
tuning and yet is colored differently).

also, you should discuss the fact that the wolf fifth appears right
in the middle of the diatonic scale with the most 5-limit
approximations, as well as the connection to all the tunings you
currently list as "schismic", as well as the 17-tone medieval
arabic / erv wilson tuning, and maybe even the 29-tone justin white
tuning . . .