Re: Ramos (1482): A division of arithmetic convenience?

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The monochord division of Ramos (1482)
Some notes and observations
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Hello, everyone, and here are some remarks on the division of the
monochord suggested by Bartolome Ramos in 1482 from a certain
historical viewpoint.

Specifically, while I notice that many people here tend to use
conventions such as lattice diagrams, another approach would be to
diagram the Ramos tuning in terms of string lengths and ratios.

Incidentally, the concept of a given "base note" or "1/1" for a tuning
seems to me mainly a convenience, since a tuning might be used with
various modes and finals. Often I tend to ask for a 12-note tuning:
"Where's the Wolf?" in order to get my bearing (e.g. Eb-G# in one
14th-century Pythagorean tuning, and typically in 16th-century
meantone).

While Ramos's tuning is famed for its introduction of 5-based ratios
for major and minor thirds, Ramos himself advocates it especially as
an easy method for students who may not be conversant with the more
complex arithmetic used in traditional Pythagorean tunings such as
those of Boethius and Guido d'Arezzo.

At the same time, pedagogical convenience quite aside, Ramos both in
this monochord tuning and in his discussion of "good and bad
semitones" (likely on a meantone keyboard instrument, argues Mark
Lindley) reflects the orientation of his era toward a texture pervaded
by thirds and sixths, which approach stability.

In 1477, Tinctoris had reported the view of the time that there was no
music worth hearing save that written in the previous 40 years. What
he is reporting is not, of course, an objective judgment of the
beauties of the previous Gothic styles, but an impatience with the
"old" common to many eras.

Part of the "new" was an altered approach toward concord and discord,
one which not only treated thirds and sixths as more and more favored
intervals, but restricted the often bold role in Gothic music of major
seconds and minor sevenths. By around 1500, composers such as Josquin
and Isaac were ending pieces on sonorities including a third,
reflecting the same changes in taste suggested by the Ramos monochord
tuning.

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1. The Ramos monochord of 1482: string-ratios and divisions
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Following the suggestion of Oliver Strunk, translator of an extract
from Ramos describing his tuning, I am diagramming this tuning based
on a string length of 288, so that all ratios may be expressed with
integers:

1 1
2 2
80 96 0 8 160 192 240 256
q p o n m l k # i h g f e d c b a
|--------------|--|--|-|--|--|-|-|--|---|---|--|----|-----|--|-------|
0 72 90 108 135 144 180 216 288
| | | | | | | | |
a g f e d c # bb a

Note that while the upper B-natural between "i" and "k" should
properly be marked with a "square-B" sign -- our natural sign --
medieval usage often treats this and "#" sign (available in ASCII) as
equivalent. Either sign indicates the solmization syllable _mi_, with
a semitone above (here b-c).

One of the very striking things I realized while reading through
Ramos's division and diagramming it above is that his tuning, a
version of Ptolemy's syntonic diatonic, seems to result naturally from
a convenient arithmetic division or "bisection" of the fifth into two
thirds. Quite apart from matters of tertian just intonation, it _is_
an easy division to perform from a practical point of view.

After dividing the entire string length a-q (288) in half, with h as
the division point (144, 2:1 to a), Ramos uses an arithmetic division
(bisection into two equal lengths) of a-h to find point d (216). This
note is a fourth 288:216 (4:3) above a and a fifth 216:144 (3:2) below
h (or aa). In other words, the basic division is 4:3:2.

Now comes another arithmetic division, this time of d-h, to find f
(180). The fifth d-aa is thus divided into intervals of d-f or 216:180
(6:5), and f-aa, 180:144 (5:4). Here the division is 6:5:4.

Note that with such arithmetic divisions of an interval, the smaller
resulting interval is located below, and the larger interval above --
a-d-aa for the octave a-aa, or d-f-aa for the fifth d-aa.

Ramos then bisects h-q to find p or aaa (72), the double octave to a,
with a ratio of 288:72 or 4:1. Then he arithmetically divides h-p to
find l or dd (108), a 144:108 (4:3) fourth above aa and a 108:72 (3:2)
fifth below aaa.

Next l-p is bisected to find n of ff (90), arithmetically dividing the
fifth dd-aaa into a lower minor third dd-ff of 108:90 (6:5), and an
upper major third ff-aaa of 90:72 (5:4).

Making an arithmetic division of f-n, Ramos finds i (135), or Bb, the
note dividing the octave f-ff into a lower 180:135 (4:3) fourth and an
upper 135:90 (3:2) fifth.

These divisions so far are shown by the numbers below the line
representing the monochord in the diagram. Subsequent divisions are
shown by numbers above this line.

The next step is to divide the whole length a-q into three equal
parts, thus finding m or ee at one-third length (96), and e at
two-thirds length (192), respectively the 288:96 (3:1) twelfth a-ee
and the 288:192 (3:2) fifth a-e.

Then e-q is similarly trisected to find the two-thirds point #
or square-B (128), forming the 192:128 (3:2) fifth e-#. Taking twice
the length #-q, we find b (256), forming a 256:128 or 2:1 octave b-#.

Next we bisect m-h to find k or cc (120), dividing the fifth aa-ee
into a 144:120 (6:5) minor third aa-cc and a 120:96 (5:4) major third
cc-ee. Taking twice the length k-q, we find c (240), forming the
240:120 (2:1) octave c-cc. This note, in turn, divides the fifth a-e
into the 288:240 (6:5) minor third a-c and the 240:192 (5:4) major
third c-e.

Finally, bisecting e-#, we find g (160), dividing the fifth e-# into a
192:160 (6:5) minor third e-g and a 160:128 (5:4) major third g-#.
Finding half of the length g-q gives us o or gg (80), forming the
160:80 (2:1) octave g-gg.

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2. The syntonic diatonic: a division of arithmetic convenience?
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As promised by Ramos, this division of the monochord indeed is based
on a few simple processes: division or multiplication by two to find
octaves and double octaves; division by three to find fifths;
arithmetic divisions of octaves into lower fourths and upper fifths;
and arithmetic divisions of fifths into lower minor thirds and upper
major thirds.

Curiously, looking into the monochord of Ramos has suddenly impressed
me that Ptolemy's syntonic diatonic could arise as a simple bisection
of the fifth -- 6:5:4. Being able to quote this string-ratio is one
thing; realizing how easily it could arise on a monochord is another.

The analogous arithmetic division of the 9:8 whole-tone, 18:17:16, is
presented by Boethius and later medieval theorists of his tradition
not as a practical tuning, but as a demonstration that the whole-tone
cannot be _equally_ divided (we might add, "by using integer ratios").
However, Marchettus of Padua seems to present this division as one
possible practice, while an English treatise on organ pipe lengths
from around 1373 recommends this 18:17:16 rule for dividing a
Pythagorean whole-tone into two semitones (e.g. with the length for c#
an average of the lengths for c and d).

From this point of view, while Ramos's arithmetic division of the
fifth into a 6:5 minor third and 5:4 major third is indeed the first
Renaissance formulation of Ptolemy's syntonic diatonic, it may also be
(like an 18:17:16 scheme for organ pipes) partially motivated by sheer
simplicity.

Incidentally, if we take the lowest note of Ramos's monochord as A re
(A2, the note immediately above Gamma ut or G2, the usual starting
point of the hexachord system), then his scheme follows the medieval
tradition of the integral gamut or _musica recta_, including Bb in the
upper octave but not in its lower octave, where the first hexachord on
G2 has only B-mi, not any Bb or B-fa (which would be _musica ficta_,
for example the _fa_ of a hexachord on F2, the step below gamma ut).

As a practical tuning scheme for fixed-pitch instruments, of course,
the Ramos division would have the obvious "bug," as Mark Lindley
points out also for some comparable 15th-century schemes, of quite
unobligingly placing a Wolf fourth or fifth at D-G. In the diagram, we
thus have d-g at 216:160 (27:20) rather than 4:3, and g-dd at 160:108
(40:27) rather than 3:2, etc. These Wolf fourths and fifths, as is
characteristic of 5-limit just intonation, are respectively a syntonic
comma (81:80) larger or smaller than pure.

In fact, as Ramos suggests in his own discussion of "good" and "bad"
semitones as interpreted by Lindley, keyboardists were already solving
this problem in practice by using meantone, where fifths were slightly
narrowed to achieve pure or near-pure thirds without the complications
of the syntonic comma. The practice may have started by around 1450,
and Gaffurius more explicitly confirms it in 1496.

To conclude, the Ramos division of the monochord has two appealing
aspects from a Renaissance standpoint: it is simple, and it typifies
some of the musical trends of the epoch, codified in the next century
by Fogliano and Zarlino in their advocacy of the syntonic diatonic.

From a medieval viewpoint, of course, such a tuning might be seen in
some ways as an _oversimplification_ of a subtle balance between the
intervals, and in others as a scheme with unwelcome complications such
as a Wolf at D-G in a tuning with only eight notes per octave. A
traditional Pythagorean tuning in eight notes, or in 12 notes with a
Wolf at some less prominent location such as Eb-G# or even Gb-B, might
seem more elegant, not to mention a 17-note tuning of Gb-A#, as
proposed earlier in the century by Prosdocimus and Ugolino, with its
many resources.

For music up to around 1420 or 1450, indeed, such a Pythagorean scheme
would nicely fit practice as well as theory; the late 15th century is
a time of new practices and theories, not better but different. Ramos
nicely reflects the spirit of his age.

Most respectfully,

Margo Schulter
mschulter@value.net