More on Pythagorean Tuning

Hi Bob,

<<aren't these 1845 and 243 limit repsectively? (Or is prime limit rather than
odd limit the default)?>>

I seem to have been adhering to an idiosyncratic definition of odd-limit,
which when I examined it in detail was not even internally consistent. I
should have stuck to prime limit.

<<What does LCM stand for?>>

Least Common Multiple.

Hi Margo,

<<
Marion has raised the interesting issue of why Ancient Greek theorists
were apparently not concerned with polyphony, and here I would like
only to suggest that the use of the Pythagorean scale does not seem to
me a persuasive explanation. From what I understand, for whatever
reasons, the outlook at least of the Greek practice and theory for
which we have evidence was essentially monophonic. Apparently neither
the tuning of the Pythagoreans with its pure ratios of 6:8:9:12, nor
that of Ptolemy with its intervals of 5:4 and 6:5, nor that of
Aristoxenos with its equal semitones, happened to evolve in a musical
milieu oriented toward polyphony.
>>

I don't think the polyphony issue makes a lot of difference. After all, if
the consonance of tones that are not sounded together made no difference,
then there would be no need of tuning or scales for monophonic music. We all
know that isn't true. All of us have experienced people who can "sing out of
tune" totally without accompaniment. Apparently there is some sort of
short-term memory for tone that allows us to "compare" recently sounded notes
as if they were sounded simultaneously.

I think that this ability relates to speech, or even more fundamentally vocal
signaling in birds and mammals. If one signal has been identified, and
another signal comes along whose pitch is related by small-whole-number
ratios, then there is a higher probability that the new signal is not noise.

I find it constructive to try to listen to conversation as if it were music.
There are definite musical patterns. When a speaker is trying to interrupt
another, he will often make an introductory sound, establishing his pitch
reference, the famous "uh", which may actually have a communications function
other than annoying English teachers. Is music a human attempt to create
artificial speech?

What intervals did the Greeks use in their speech? There is some chance that
modern Greek uses the same tonality as such things vary slowly in language.

<<
For example, as Marion has ingeniously asked, might performers
realizing the ideal of a Pythagorean intonation on non-fixed-pitch
instruments tend to shade an 81:64 toward a 19:15, or a 729:612 toward
a 10:7?
>>

Upon reflection, it has occurred to me that even if the 729:512 is being
sounded with absolute accuracy, the listener is going to hear over 500
repetitions of the 10:7 pattern while he is hearing one repetition of the
729:512. I believe the 10:7 pattern will pretty much swamp the longer
729:512.

It is certainly your right to love Pythagorean tunings. I have found them
difficult to work because they are much more difficult to reproduce
accurately with electronics. Maybe I'll make an effort to understand them
better one of these days.

Hi Paul,

<<
You didn't, through no fault of your own. I haven't signed my messages, but
my name
is Paul Erlich.
>>

Ah. You've been on this list a long time then, and we've had other
conversations before this.

<<
Marion -- this discussion was about styles in which only 3-(odd) limit
intervals
were considered consonant. Of course, once 5-limit intervals were accepted as
consonant, the tuning of the scales needed to be tweaked in order to increase
the
total number of consonant intervals. This led to developments such as
schismatic
tuning and meantone temperament, which you can read more about in Margo
Schulter's
paper (and I recommend you do!) But before that, only fifths and fourths were
used
as consonances in the music and other intervals were used as dissonances. To
apply
your criterion of maximizing the number of 5-limit consonant intervals would
be a
terrible anachronism if applied to Western music c. 900-1200.
>>

That's your concept of what this discussion was about. My view is that I
made a remark about Pythagorean tunings in a post that was basically about
something else, and some people took exception. Not that I mind. The whole
discussion has been a good education for me, and thank you for taking the
time to analyze what I am saying.

<<
> With a lattice approach you might, for example use the familiar:
>
> 6 9
> 5 8 15
> 4 6 16

What is that? It doen't remind me of any lattice approach I've ever seen. A
lattice is an infinite
arrangement of points such that the configuration of every point relative to
its neighbors is the same as
that of any other point relative to the second point's neighbors.
>>

In my original post, I said:

<<
I have written a program that generates scales from aliquot parts. It also
does lattices if my definition of lattices is the same as yours. To find
out, you can download the file from my web site at
http://members/aol.com/mckyyy.
>>

I was in some doubt as to whether my concept of lattice was the same as that
used on the list, and it now seems that doubt was justified. In view of what
I have learned since then, I am going to change my terminology and call the
system I was using chord multiplication. In other words, I just multiply all
those numbers together, 4*6*16, 4*6*15, 4*8*16, etc. and sort the products,
eliminate duplications, and octave adjust the results, and I have a scale.

From my current understanding of "lattice" as used in the list, select a
subset of points from an "infinite" lattice, and use a procedure much like
the above to generate the scale? I realize it's highly probable that my
understanding of lattice is about to undergo some evolution.

Since I have been a professional programmer for so long, I tend to view these
things in terms of combinations, sorting, searching, and so forth. The idea
of a graphical representation of these problems seems to be taking the long
way around to me, so my interest in lattices has been minimal.

<<What does electronics have to do with it?>>

IMHO, electronics is the main reason we are here today talking about tuning.
The invention of electronic frequency divider instruments that can reproduce
tunings economically and with unprecedented levels of accuracy has
fundamentally changed the art of music and renewed the debate about tuning
standards that was "settled" by the near universal adoption of 12-et in the
Western World.

A lot of it has to do with exact JI, which, as far as I can tell was invented
in the 1930's by someone at the Wurlitzer Organ Co. The Wurlitzer organ had
"tone wheels" which were all connected mechanically to the same shaft, but
with different numbers of holes in the wheels. The ratio between the tones
was mechanically fixed and could never vary. Exact Just Intonation tuning.
In the late 50's Freeman filed a patent on using a frequency divider network
in exactly the same way. In 1966, I did a patent search on the same idea,
and it did not uncover Freeman's work because his patent hadn't issued yet, I
think. I negotiated with several musical instrument companies to promote the
idea, and eventually found out about the Freeman/Wurlitzer work. Wurlitzer
successfully broke Freeman's patent on the grounds that it was obvious in the
light of their prior art. I had the opportunity to discuss this with the
Wurlitzer people when I was trying to sell them some of my work on automatic
harmony. They referred to Freeman informally and indirectly as a "parasitic
inventor." In my opinion, Freeman's work is both brilliant and
unappreciated, not that I mean to slight the groundbreaking efforts of the
Wurlitzer Organ Co.

Anyway, you can have exact JI using either tone wheels or frequency dividers,
and since frequency dividers are a lot cheaper and more flexible, they
commonly used in electronic musical instruments today. The potential for
exact JI is there, but the programming doesn't support it

<<
> Of course, these considerations only apply to exact JI. If you allow
> approximation to come into the picture, then you are in a different
universe,
> which in my opinion is not a JI universe but a tempered universe.

Approximations exist in JI as well. It is unavoidable that one will hear a
very complex ratio as a simple
one if it the former is very near to the latter, e.g., 30001:20001 will be
heard as 3:2.
>>

That is why I coined the term exact JI to differentiate it from ordinary JI.
Actually, there is some error, even in frequency divider instruments, but it
is usually measured in picoseconds.

The LCM of 3:2 is 6, which means you can produce the exact ratio with 3
flip-flops and a master clock frequency which is only twice the lowest
desired frequency. The LCM of 30001:20001 is 600050001. To generate this
interval with a frequency divider would require about 30 flip-flops and a
master clock 20001 times higher than the lowest frequency. That's only a
little over eight megacycles at 440HZ, so it could be done.

<<
Do you mean that the fact that she made contributions to the list made you
stop thinking her paper was
authoritative?
>>

No, I was convinced because she corroborated what you were saying, and she
had hard numbers to back it up. I try to look beyond credentials when I can.
I try not to be prejudiced for or against the tuning list.

Thanks to all for your comments,

Marion

>Marion has raised the interesting issue of why Ancient Greek theorists
>were apparently not concerned with polyphony, and here I would like
>only to suggest that the use of the Pythagorean scale does not seem to
>me a persuasive explanation. From what I understand, for whatever
>reasons, the outlook at least of the Greek practice and theory for
>which we have evidence was essentially monophonic.
>
I recall reading in West's book on ancient Greek music that there is
evidence that kithara (lyre) players rapidly strummed the strings with a
plectrum at the ends of phrases. He further speculates that the players
pressed their fingers on certain strings, dampening them so that only
certain pitches would sound, creating chords.

In fact, without dampening monophonically plucked strings, the decays would
overlap, like a lot of celtic harp players I've heard, also creating, as
Dudley Duncan pointed out, harmonies. Further, there is evidence of drone
pitches used in some contexts.

Clearly, the Greeks were not unaware of simultaneous sonorities, despite
the monophonic basis of much of their music. Much of Chinese and Indian
music is also "monophonically based," and yet simultaneous sonorities have
an influence on tuning there.

While many ancient writers in the Pythagorean tradition justify tunings
through numerology/astrology, Ptolemy at least made reference to tunings
used in practice as whole number ratios. Aristoxenus' appeal to the
musician's ear certainly suggests that, at least for some, tuning was more
than a numerological exercise.

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
^ Bill Alves email: alves@hmc.edu ^
^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
^ 301 E. Twelfth St. (909)607-4170 (office) ^
^ Claremont CA 91711 USA (909)607-7600 (fax) ^
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Thanks for reminding us of the crux of the bicuit.
While I cannot be 100% sur was'nt the Pythag Cult
divided into the Akoustimakoi and the Mathematikoi?
or was there three divisions.
anyone?

Bill Alves wrote:

> Aristoxenus' appeal to the
> musician's ear certainly suggests that, at least for some, tuning was more
> than a numerological exercise.
>
> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> ^ Bill Alves email: alves@hmc.edu ^
> ^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
> ^ 301 E. Twelfth St. (909)607-4170 (office) ^
> ^ Claremont CA 91711 USA (909)607-7600 (fax) ^
> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
>
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Marion wrote:

>I was in some doubt as to whether my concept of lattice was the same as
that
>used on the list, and it now seems that doubt was justified. In view of
what
>I have learned since then, I am going to change my terminology and call
the
>system I was using chord multiplication. In other words, I just multiply
all
>those numbers together, 4*6*16, 4*6*15, 4*8*16, etc. and sort the products,
>eliminate duplications, and octave adjust the results, and I have a scale.

Are you multiplying the numbers 3 at a time, or are you taking products of
all possible subsets of the numbers? The former is a generalization of the
combination product set idea, while the latter is a generalization of the
Euler genus idea. I say "generalization" because usually only odd or prime
(respectively) numbers are used in the source sets for these procedures.

>From my current understanding of "lattice" as used in the list, select a
>subset of points from an "infinite" lattice,

Yes.

>and use a procedure much like
>the above to generate the scale?

No -- once you've selected a subset of points, you're done. However, it can
be very informative and visually beautiful to look at the subset of points
that represents a combination product set or an Euler genus.

>> Of course, these considerations only apply to exact JI. If you allow
>> approximation to come into the picture, then you are in a different
>universe,
>> which in my opinion is not a JI universe but a tempered universe.
>
>Approximations exist in JI as well. It is unavoidable that one will hear a
>very complex ratio as a simple
>one if it the former is very near to the latter, e.g., 30001:20001 will be
>heard as 3:2.
>>>
>
>That is why I coined the term exact JI to differentiate it from ordinary
JI.
>Actually, there is some error, even in frequency divider instruments, but
it
>is usually measured in picoseconds.
>
>The LCM of 3:2 is 6, which means you can produce the exact ratio with 3
>flip-flops and a master clock frequency which is only twice the lowest
>desired frequency. The LCM of 30001:20001 is 600050001. To generate this
>interval with a frequency divider would require about 30 flip-flops and a
>master clock 20001 times higher than the lowest frequency. That's only a
>little over eight megacycles at 440HZ, so it could be done.

My point is that tempered versions of intervals exist and cannot be ignored
even in exact JI (schismatic tuning comes to mind).