Yahoo Tuning Groups Ultimate Backup tuning Zarlino 1588

In Lindley/Turner-Smith 1993, p 34, footnote 42, is
presented a copy of a musical illustration from Zarlino 1588,
in which I have found two errors.

This illustration gives string-length ratios for a diatonic scale
by Ptolemy, with Latin names for "major semitone", "minor tone",
and "major tone" for the intervals between scale-degrees.

The original Zarlino text and diagrams (Zarlino 1588, book 4) are here:
(Cut and paste to remove line-break; non-members may view the material
by signing in as "guest" with the password "password".)

http://pcm1671.cs.uu.nl:6334/dynaweb/tmiweb/z/zarsop/@Generic__BookTextView/
23007;ts=default;pt=23007;uf=0#X

The ratios are given by Ptolemy in his Book 2, Chapter 14
(Barker 1989, p. 349-350).

Below is an analysis of Zarlino's string-length numbers,
for both the systems of Didymus and Ptolemy, with a comparison
to the equivalent string-length numbers given by Ptolemy.
Zarlino's numbers are based on a modal transposition of
Ptolemy's scale -- Ptolemy's "8ve" was from E=60 to E=120.

Cents values for the pitches and the intervals between are
also given, with lattice diagrams (Monzo formula) below that.

The Ptolemy scale includes the cents values for the two
intervals which Zarlino called "dissonante". The diagram of
Didymus's scale includes 6 such intervals (not indicated on
my diagram). Zarlino thus argues that the Ptolemy system is
the one normally used in singing, because of the greater amount
of consonant intervals.

Zarlino's ultimate point was that even in the Ptolemy scale, which
has many more consonant harmonies than that of Didymus, there
are still unavoidably two basic intervals which are given as
dissonances, and it is upon this basis that Zarlino cites the
need for temperament.

Didymus scale:

Zarlino Ptolemy ~cents
note str-len str-len ratio ~cents between

B 64 80 2/1 1200
> 204
A 72 90 16/9 996
> 204
G 81 101+1/4 128/81 792+1/6
> 182+2/5
F 90 112+1/2 64/45 609+7/9
> 111+3/4
E 96 60 & 120 4/3 498
> 204
D 108 67+1/2 32/27 294+1/7
> 182+2/5
C 120 75 16/15 111+3/4
> 111+3/4
B 128 80 1/1 0

G
/81
/
/
D
/108
F+ /
90'-._ /
/ ' A
/ /72
C+ /
120'-._ /
' E
/96
/
/
B
128 and 64

Zarlino Ptolemy ~cents
note str-len str-len ratio ~cents between

Ptolemy scale:

B 64 80 2/1 1200
> 204
A 72 90 16/9 996 --
> 182+2/5 |
G 80 100 8/5 813+2/3 |
> 204 |
F 90 112+1/2 64/45 609+7/9 -- 680+4/9
> 111+3/4 | |
E 96 60 & 120 4/3 498 294+1/7 |
> 182+2/5 | |
D 106+2/3 66+2/3 6/5 315+2/3 -- --
> 204
C 120 75 16/15 111+3/4
> 111+3/4
B 128 80 1/1 0

F+
90'-._
/ '- A
/ /72
C+ /
120'-._ /
/ '- E
/ / 96
G+ /
80'-._ /
/ ' B
/ 128 and 64
D+
106+2/3

This illustration of a "Ptolemy" scale has a few
errors:

1)
The string-length given by Zarlino as 116+2/3 should be 106+2/3.

2)
Zarlino's diagram has the names of the top two intervals
reversed: it should be a "major tone" at the top (64:72)
then a "minor tone" under it (72:80).

REFERENCES
----------

Ptolemaeus, Claudius. c.125-150 AD.
_Klaudiou Ptolemaiou harmoniko vivlia iii Claudii Ptolemaei
harmonicorum libri tres_.
John Wallis (ed.), Oxford, e theatro Scheldoniano, 1682. London, 1699.
Reprint Ingemar Dï¿½ring (ed.), Elanders, Gï¿½teborg, 1930.
English translation _The Harmonics_ by A. Barker in
_Greek Musical Writings vol. 2_,
Cambridge University Press, Cambridge, 1989.

Zarlino, Gioseffe. 1588.
_Sopplementi musicali del Rev. M. Gioseffo Zarlino da Chioggia,
Maestro di Cappella della Sereniss. Signoria di Venetia_.
Francesco de' Franceschi, Venezia.
Reprint Broude Brothers, New York, c.1980

Lindley, Mark & Ronald Turner-Smith. 1994.
_Mathematical Models of Musical Scales: A New Approach_.
Orpheus-Schriftenreihe zu Grundfragen der Musik vol. 66,
Verlag fï¿½r systematische Musikwissenschaft, Bonn-Bad Godesberg.

love / peace / harmony ...

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

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Zarlino 1588

🔗monz <joemonz@yahoo.com>

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