Re: temperament in "Ad chlvichordium faciendum"

Replying to both Margo and Roland:

Greetings, and I have been lurking here for far too long.  I'm finally 
back on a system in which I can post.

But here is (yet) another way to display the tuning Roland got from the 
document (this should be viewed in a fixed-width font):

                  Db
     G#   D#   Bb    F   C   G   D
                       A   E   B   F#  C#

using Roland's parenthetical names for the black notes, except for Db 
for the note in step 3, to distinguish it from C# in step 9.

This kind of diagram shows (pure) fifths in horizontal rows, and (pure) 
thirds as adjacent notes in adjacent rows:  major thirds going from 
upper left to lower right (like F-A) and minor thirds going the other 
way (like A-C).  So pure triads show up as triangles.

Also, the string of fifths in the bottom row are all 1 syntonic comma 
lower than the string in the main row, so in Roland's follow-up article 
notation, they would appear as A-1, E-1, etc.  And the lone Db is Db+1.

I like this kind of diagram (which is adapted from one in a couple of 
Mark Lindley's articles) because it makes *all* the pure fifths and pure 
thirds obvious, and also makes it clear that D-A is *not* a pure fifth.

Margo:
 
> As has been discussed, this tuning is an unlikely choice for any practical 
> keyboard because of the "Wolf" fifth D-A a syntonic comma narrow (40:27 
> rather than 3:2, a full 81:80 or ~21.51 cents from just). This is the same 
> problem which one encounters in a 5-limit just intonation system based on 
> Ptolemy's syntonic diatonic, and which Zarlino solves on an experimental 
> keyboard with 16 notes per octave by providing two keys for D, one forming  
> a 9/8 above C (as in this tuning) and another a 10/9 providing a just 
> fifth with A at 5/3.

Salinas (De musica) also published a monochord division with two D's 
serving the same purposes (a fifth above G and a fifth below A).  Did 
anyone advocate building a keyboard instrument with split or different 
keys for a white note?  There are lots of examples of keyboards with 
split keys on black notes, but none that I know of on white notes 
(Vicentino's archicembalo is really a different case).

> From a theoretical point of view, this tuning is of interest because it 
> uses simple integer ratios such as 5:4, as does also the Ramos monochord 
> of 1482. 

The monochord part of Ramos' treatise comes out this way:

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

with the wolf fifth between D and G.  But Ramos fudges the issue later 
in the treatise by stating that the only "bad" fifth is that between Ab 
and C#, which in this tuning is only a schisma off (about 2 cents), 
while G-D, a "good" fifth, is a syntonic comma off, about 21 cents.  
That's what led Lindley to the conclusion that Ramos "must have" 
tempered the fifths, to make G-D good and Ab-C# bad.

> It is interesting that the Italian philosopher Ficino, possibly 
> influenced by Ramos, writes of the major third at 5:4 as one of the 
> principal concords of music. 

Ramos (Practica musica) is the earliest describer of 5:4 thirds (as part 
of stable concords) that I know of; it would be interesting to know if 
this manuscript is earlier than 1482.  One might note that Ramos took a 
lot of flak for daring to say that tuning was anything other than 
pythagorean.

The problem is that if Ramos actually adjusted his intervals from his 
prescribed ratios, he can't claim to be an advocate of 5:4 thirds or any 
other nice-ratio intervals.  Ramos never specifically mentions adjusting 
intervals at all; the earliest person to describe tempering intervals 
(that I know of) was Gaffurius, in 1496.

-- 
Neal Plotkin
e-mail:  neal dot plotkin at nyu dot edu