the etymology of ratios

From "The American Heritage Dictionary" --
"etymology: 1. The origin and historical
development of a linguistic form as shown by
determining its basic elements, earliest known use,
and changes in form and meaning, tracing its
transmission from one language to another,
identifying its cognates in other languages, and
reconstructing its ancestral form where possible."

As a student of tuning theory, I have learned much
by tracing the etymology of ratios. Although a
person may judge a given tuning, tetrachord, or
even an entire book to have no musical
significance, there is great meaning in the origins
and historical development of ratios, as there is
great meaning in the origins and historical
development of words. Nowhere is the language of
ratios, and therefore the etymology of ratios, more
significant than between the writings of Ptolemy
and the works of Al-Farabi and Ibn Sina. Since all
of Farhat's modern dastgaha are playable on Ibn
Sina's `ud, only a study of the etymology of ratios
will illuminate the "…transmission from one
language to another…"

Cris

> As a student of tuning theory, I have learned much
> by tracing the etymology of ratios. Although a
> person may judge a given tuning, tetrachord, or
> even an entire book to have no musical
> significance, there is great meaning in the origins
> and historical development of ratios, as there is
> great meaning in the origins and historical
> development of words. Nowhere is the language of
> ratios, and therefore the etymology of ratios, more
> significant than between the writings of Ptolemy
> and the works of Al-Farabi and Ibn Sina. Since all
> of Farhat's modern dastgaha are playable on Ibn
> Sina's `ud, only a study of the etymology of ratios
> will illuminate the "…transmission from one
> language to another…"

The etymology of ratios? I'm struggling to understand
what that might mean. The application of the rational
numbers in music is very straightforward.

-Carl

Carl,

In his "Harmonics," Ptolemy describes the division
of his Even Diatonic tetrachord as consisting of
near-equal intervals. The history of dividing
intervals into nearly equal parts is deep and
complex, especially in Arabian and Persian music.
Ptolemy's Even Diatonic tetrachord consists of
ascending interval ratios 12/11, 11/10, 10/9. The
first two may be considered, in the tradition of near-
equal intervals, as "neutral seconds," consisting of
150.6 cents and 165.0 cents, respectively. When
one traces the presence of such ratios, and the
practice of near-equal division, in the writings of Al-
Farabi and Ibn Sina one begins to understand the
etymology of ratios.

As a designer and builder of acoustic instruments, I
know that a well-designed part has at least two
functions. Sometimes, it can have five or even six
equally important functions. Sometimes, I discover
that a part has more functions than I initially
intended. It is the same with the writings of
Ptolemy, Al-Farabi, and Ibn Sina. Whether or not
they consciously intended or anticipated all the
functions of their text is irrelevant.

The same holds true for the writers of this group.
None of us can intentionally predict all the functions
of our messages. However, a thread or a
connection does in time eventually emerge.

Ptolemy's tetrachords are musically relevant for
many reasons. As a musician, I am not in need of
convincing. I am offering the concept of the
etymology of ratios as another method, or another
approach for reading ancient texts. The verbal
language of numbers, and the verbal language of
the origins of numbers, especially as exemplified in
the practice of dividing canon and `ud strings, is in
my opinion essential to understanding not only
ancient music, but modern music as well.

Cris

Thank you, Margo, for your acknowledgments.
You inspired me to keep writing.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> > As a student of tuning theory, I have learned much
> > by tracing the etymology of ratios. Although a
> > person may judge a given tuning, tetrachord, or
> > even an entire book to have no musical
> > significance, there is great meaning in the origins
> > and historical development of ratios, as there is
> > great meaning in the origins and historical
> > development of words. Nowhere is the language of
> > ratios, and therefore the etymology of ratios, more
> > significant than between the writings of Ptolemy
> > and the works of Al-Farabi and Ibn Sina. Since all
> > of Farhat's modern dastgaha are playable on Ibn
> > Sina's `ud, only a study of the etymology of ratios
> > will illuminate the "…transmission from one
> > language to another…"
>
> The etymology of ratios? I'm struggling to understand
> what that might mean. The application of the rational
> numbers in music is very straightforward.
>
> -Carl
>

Cris wrote:

> Thank you, Margo, for your acknowledgments.
> You inspired me to keep writing.

Thank you, Cris, for a great post! And while I have not
experienced that etymology in the same intimate way you have as a
builder, tuner, and player of acoustical instruments, I certainly
have found what George Secor and I came generally to term the
"equable divisions" as central to at least some tempered systems
also, and to digital as well as acoustical instruments.

What immediately got me engaged was 14:13:12 or 12:13:14 as a
division of 7:6, which George pointed out was analogous precisely
to Ptolemy's 12:11:10:9 Equable Diatonic with its 12:11:10
division of 6:5.

As I temper nowadays, 14:13:12 or 128.3-138.6 cents tends to
translate 126.6-138.3 cents or the like. Another of my favorites,
although one I haven't yet encountered in a medieval context, is
13:12:11 at 138.6-150.6 cents, a favorite of mine as one flavor
of Shur not too far from Ibn Sina or Farhat, but with a slightly
narrower third at 13:11 rather than 32:27, which in a tempered
rendition comes out as 138.3-150.0, for example.

After reading your post, I looked for and found a version of
Ptolemy's Equable Diatonic on my keyboard -- a good exercise.
While this wasn't part of my deliberate design for the tuning,
it's maybe not so surprising. As you discuss, 12:11:10 is
150.6-165.0 cents; I get 151.2-161.7 cents as the best
approximation. While the 14:13:12 and 13:12:11 divisions (also
available in their harmonic rather than arithmetic forms) are
mainly what the tuning is about, it's neat to know that I have a
reasonable Equable Diatonic approximation.

Anyway, while I wrote a long post for Carl to show that it's
possible to discuss Near Eastern intonation without using integer
ratios and divisions like these, that may be a bit like
navigating a road without benefit of the natural signposts.

And for explaining the continuum of neutral intervals, for
example between a semitone at 16:15 or 15:14 and a small tone at
10:9, I can think of no signposts more simple or intuitive than
the superparticular ratios of 14:13, 13:12, 12:11, and 11:10.

If I understand your "etymology" correctly, we're indeed talking
of the origins and evolution of ideas such as the Equable
Diatonic and how it can get expanded from Ptolemy's 12:11:10:9 to
a division like Ibn Sina's 12:13:14 or 14:13:12, or to 13:12:11
or 11:12:13 (certainly used, as I learned, in the harmoniai of
Kathleen Schlesinger, if not in some medieval Islamic source).

And there's also the kind of etymology where a set of step sizes
in a tetrachord undergo a permutation. For example, we have
Ptolemy's Intense Chromatic at 22:21-12:11-7:6 (80.5-150.6-266.9
cents) evidently lead to Qutb al-Din al-Shirazi's version of
Hijaz around 1300 as 12:11-7:6-22:21, a classic form that remains
one of my all-time favorites.

If, as Lou Harrison said, just intonation is the best intonation,
then the next best may be a tempered system which observes as
closely as possible some approximation of the distinctions and
relations between superparticular steps, and thus some modicum of
a connection with the rich etymologies we're discussing.

I hope that I have you right for the most part, and in any event
that you for your kind words while encouraging you indeed to keep
writing!

Best,

Margo

>If I understand your "etymology" correctly, we're
>indeed talking of the origins and evolution of ideas
>such as the Equable Diatonic and how it can get
>expanded from Ptolemy's 12:11:10:9 to a division
>like Ibn Sina's 12:13:14 or 14:13:12, or to
>13:12:11 or 11:12:13 […]

Margo,

It's a rare occurrence for me not to have potentially
complicated or controversial messages shredded
beyond recognition and repair.

In the quotation above, you understood my usage
of the term `etymology' perfectly. I thank you for the
time and effort required to think this idea through.
And I thank you for your own insights and those of
George Secor that parallel my observations on
independent tracks.

Words as composites of different alphabets and
different languages make tracing the historical
sources and historical evolution of words infinitely
easier than numbers. The only method by which an
etymology of ratios can be established is by
understanding the contexts from which certain
ratios emerge, and thereby, the groups to which
certain ratios belong.

From my text:

******************************

Section 11.66

Finally, and most importantly, Figure 11.68 verifies
that the first six ascending interval ratios of Ibn
Sina's Nawa in Table 11.32 are identical to the six
ascending interval ratios of the modern Persian
mode called Dastgah-e Nava in Table 11.34.
Examine the intervals encompassed by the curly
brackets in Table 11.32, and notice that these
markers indicate the division of the Pythagorean
"minor third," interval ratio 32/27, into two near-
equal "neutral seconds," ratios 13/12 and 128/117.
(See Figure 11.64.) Table 11.32 shows that Ibn
Sina divided this "minor third" into two "neutral
seconds" in four different locations on the ud:
between D–F, or 4/3 ÷ 9/8 = 32/27; between F–Ab,
or 128/81 ÷ 4/3 = 32/27; between G–Bb, or 16/9 ÷
3/2 = 32/27, and between C'–Eb', or 64/27 ÷ 2/1 =
32/27. Now, recall Farhat's observation that one
divides this "minor third" so that "…the lower of the
two ["neutral seconds"] tends to be the smaller…"
In seven of Ibn Sina's 11 Melodic Modes, and in
eight of the twelve modern dastgaha: Shur, Abuta,
Dashti, Bayat-e Tork, Afshari, Segah, Homayun,
and Nava, this division of the "minor third," and the
hierarchical order of the "neutral seconds," defines
one of the most essential melodic features of
Persian music.

******************************

Section 11.62

In preparation for a discussion on Ibn Sina's 11
Melodic Modes, let us first reexamine the theory
and practice of near-equal divisions of intervals in
the tetrachords of Ptolemy, Al-Farabi, and Ibn Sina.
As discussed in Sections 10.15–10.16 and 11.56,
music theorists of antiquity preferred near-equal
arithmetic divisions over exactly-equal geometric
divisions for two reasons. (1) The musical necessity
to resolve various kinds of commas by means of
exact geometric divisions did not exist. (2) During a
musical performance, it is extremely difficult to
perceive tuning discrepancies less than 10–15 ¢ on
low-tension gut strings. In short, for Ptolemy, Al-
Farabi, and Ibn Sina, two intervals that are
mathematically different, but that sound nearly the
same are for all practical purposes
interchangeable. In his dissertation, Farhat
observes

"The difference between the two [neutral seconds],
however, in a practical sense is too small and too
much subject to fluctuation to warrant separate
recognition of each one. [We] will therefore refer to
all sizes of this interval as neutral 2nd." (Text in
brackets mine.)

Obviously, music theorists of antiquity were
perfectly capable of perceiving and calculating very
small discrepancies between intervals. Even so,
had they dwelled on such purely acoustical and
mathematical minutiae, the musical art of
tetrachord classification would not have developed.

In support of this interpretation of near-equal
divisions, consider the next passage from the "Kitab
al-shifa," where Ibn Sina classifies the most
important consonant intervals, and defines the
limits of human aural perception and musical
instrument construction with respect to very small
and very large intervals…

******************************

Cris

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> Cris wrote:
>
> > Thank you, Margo, for your acknowledgments.
> > You inspired me to keep writing.
>
> Thank you, Cris, for a great post! And while I have not
> experienced that etymology in the same intimate way you have as a
> builder, tuner, and player of acoustical instruments, I certainly
> have found what George Secor and I came generally to term the
> "equable divisions" as central to at least some tempered systems
> also, and to digital as well as acoustical instruments.
>
> What immediately got me engaged was 14:13:12 or 12:13:14 as a
> division of 7:6, which George pointed out was analogous precisely
> to Ptolemy's 12:11:10:9 Equable Diatonic with its 12:11:10
> division of 6:5.
>
> As I temper nowadays, 14:13:12 or 128.3-138.6 cents tends to
> translate 126.6-138.3 cents or the like. Another of my favorites,
> although one I haven't yet encountered in a medieval context, is
> 13:12:11 at 138.6-150.6 cents, a favorite of mine as one flavor
> of Shur not too far from Ibn Sina or Farhat, but with a slightly
> narrower third at 13:11 rather than 32:27, which in a tempered
> rendition comes out as 138.3-150.0, for example.
>
> After reading your post, I looked for and found a version of
> Ptolemy's Equable Diatonic on my keyboard -- a good exercise.
> While this wasn't part of my deliberate design for the tuning,
> it's maybe not so surprising. As you discuss, 12:11:10 is
> 150.6-165.0 cents; I get 151.2-161.7 cents as the best
> approximation. While the 14:13:12 and 13:12:11 divisions (also
> available in their harmonic rather than arithmetic forms) are
> mainly what the tuning is about, it's neat to know that I have a
> reasonable Equable Diatonic approximation.
>
> Anyway, while I wrote a long post for Carl to show that it's
> possible to discuss Near Eastern intonation without using integer
> ratios and divisions like these, that may be a bit like
> navigating a road without benefit of the natural signposts.
>
> And for explaining the continuum of neutral intervals, for
> example between a semitone at 16:15 or 15:14 and a small tone at
> 10:9, I can think of no signposts more simple or intuitive than
> the superparticular ratios of 14:13, 13:12, 12:11, and 11:10.
>
> If I understand your "etymology" correctly, we're indeed talking
> of the origins and evolution of ideas such as the Equable
> Diatonic and how it can get expanded from Ptolemy's 12:11:10:9 to
> a division like Ibn Sina's 12:13:14 or 14:13:12, or to 13:12:11
> or 11:12:13 (certainly used, as I learned, in the harmoniai of
> Kathleen Schlesinger, if not in some medieval Islamic source).
>
> And there's also the kind of etymology where a set of step sizes
> in a tetrachord undergo a permutation. For example, we have
> Ptolemy's Intense Chromatic at 22:21-12:11-7:6 (80.5-150.6-266.9
> cents) evidently lead to Qutb al-Din al-Shirazi's version of
> Hijaz around 1300 as 12:11-7:6-22:21, a classic form that remains
> one of my all-time favorites.
>
> If, as Lou Harrison said, just intonation is the best intonation,
> then the next best may be a tempered system which observes as
> closely as possible some approximation of the distinctions and
> relations between superparticular steps, and thus some modicum of
> a connection with the rich etymologies we're discussing.
>
> I hope that I have you right for the most part, and in any event
> that you for your kind words while encouraging you indeed to keep
> writing!
>
> Best,
>
> Margo
>

--- In tuning@yahoogroups.com, "c_ml_forster" <cris.forster@...> wrote:

> It's a rare occurrence for me not to have potentially
> complicated or controversial messages shredded
> beyond recognition and repair.

I didn't see anything being shredded, I saw a response based on not instantly understanding what you meant by the peculiar phrase "etymology of numbers", which is hardly surprising. If you want to toss out a phrase like that, you might consider emulating Gandhi and King in being less thin-skinned about the response.