Yahoo Tuning Groups Ultimate Backup mills-tuning-list Aristoxenus's + Ptolemy's enharmonics

John Chalmers wrote:

> As for the use of 5/4. Eratosthenes, who may be responsible for
> the linear misinterpretation of Aristoxenos's parts, used 19/15 as
> his enharmonic ditone.
> Frankly, I'm somewhat suspicious of Ptolemy's enharmonic and
> chromatic genera as they have a rough 1:2 division of the pyknon
> (Barbera), even if it meant that he had to reorder the intervals
> resulting from "katapyknosis" to obtain usable superparticular
> ratios. But, I do agree that the kithara and lyra tunings probably .
> represent practice

I'll deal with Eratosthenes first, then Ptolemy.

I was discounting Eratosthenes precisely because of the general view
that he tried to interpret Aristoxenus's tetrachordal figures in terms
of string-length proportions. Ptolemy provides us with a version of
the enharmonic tetrachord from five different theorists, including
himself (Bk.II, sect.14). Of the four who give ratio descriptions,
Archytas, Didymus and Ptolemy all agree on 5/4 for the large interval;
only Eratosthenes differs by giving 19/15 instead. This leaves only
Aristoxenus, who gives us a large interval of 24 parts (out of 30 for
the 4/3).

How are we to interpret Aristoxenus in ratio terms? If we call our
Aristoxenean measure "A", we can use the formula

exp(Aln(4/3)/30)

and then find the first few convergents of the continued fraction
expression of the result. In the case of A 24, the third convergent
is 5/4 (the fourth is 34/27). If this is indeed Aristoxenus's
interval, then the remaining pyknon of 16/15 is divided equally into 3
+3 parts. This would yield a tetrachord that is perceptually
indistinguishable from the enharmonic of Didymus, who is clearly
aiming for near-equal division of the pyknon (31/30 and 32/31
descending), while Archytas and Ptolemy both choose to divide the
pyknon unequally. We have good reason then to identify the enharmonic
of Aristoxenus with that of Didymus.

What of Eratosthenes then? Let's take him seriously in his own right
for a moment, rather than as a mere misinterpreter of Aristoxenus. As
it happens, his ratio of 19/15 is in fact one cent closer to 24
Aristoxenean units (if they are interpreted as a 30-fold equal
division of the 4/3). 24 parts is about 398 cents; 5/4 is about 12
cents smaller, while 19/15 is about 11 cents higher. I have no idea
how Eratosthenes made his calculation, but clearly he didn't use
logarithms and convergents, nor could he extract the 30th root of 4/3.
Nevertheless, 19/15 would seem considerably harder to arrive at for
lyra and kithara players, and it never reappears except under
Eratosthenes' name. Given that three out of our four sources for the
enharmonic agree on 5/4, I would suggest that the balance is in favour
of this ratio, rather than 5/4.

The only further option is to take 81/64 -- the ditone -- as
Aristoxenus's intended interval. At this point, I'll confess that I've
been disingenuous in the above two paragraphs, since I don't believe
for a moment that Aristoxenus's 30 parts of a 4/3 allow anything like
such precision. The ditone is in fact Aristoxenus's own _stated_
preference for the large interval of the enharmonic tetrachord, and he
laments that the trend in his time was in favour of a more intense
tuning, which "sweetened" the ditone (clearly the 5/4 of the other
theorists -- see El. Harm. 23, 12ff.). For what it's worth, then, how
close does the ditone come to the cents measurement of 24 parts of
(4/3)^(1/30) each? The improvement on 19/15 is only one cent more.
This should give us a good idea of the margins of vagueness and
precision within which we should Aristoxenus's figures. Archytas was a
near predecessor of Aristoxenus, or an older contemporary, and he, as
we have seen, was already content to give 5/4 as the width of the
large interval, so the trend was already well under way some time
before Aristoxenus wrote; we might even speculate that Aristoxenus was
stating his preference for a practice that had died out by the time he
wrote. Whatever the case may be, Barker argues persuasively that
neither Aristoxenus nor Archytas convinced themselves of the relative
sizes of ditone and 5/4 by means of the monochord (a method which
would hardly have claimed much of Aristoxenus's time!) nor by
listening alone, but rather by observation: they would have seen
players tune their string by fourths and fifths to arrive at the
ditone, after which they would have noticed the players tighten the
string until the "sweetness" of the interval between that string and
its higher neighbour satisfied them.

> Frankly, I'm somewhat suspicious of Ptolemy's enharmonic and
> chromatic genera as they have a rough 1:2 division of the pyknon
> (Barbera), even if it meant that he had to reorder the intervals
> resulting from "katapyknosis" to obtain usable superparticular
> ratios. But, I do agree that the kithara and lyra tunings probably .
> represent practice

I don't see that there is any more reason to doubt Ptolemy's
pyknomatic tetrachordal divisions than there is to doubt his kithara
and lyra tunings. Of course players would not have been able to intuit
the sizes of Ptolemy's ratios within the pyknon, but nor would they
have been able to intuit the ratios of any of the other theorists
either. No, what leads me to believe that Ptolemy's enharmonic and
chromatic divisions were trustworthy as a description of contemporary
practice (though perhaps not earlier) is the peculiar and awkard
method he devises for calculating how the pyknon was to be divided.
This has nothing to do with the constraint of finding superparticular
ratios, since the tunings Ptolemy takes issue with -- the enharmonics
of Archytas, Eratosthenes and Didymus, and the chromatics of
Eratosthenes and Didymus -- all remain within these constraints (only
Archytas's chromatic contains an epimeric ratio). Rather, he has
devised the method to arrive at a result which gives a higher interval
that is noticeably larger than the lower, indeed, as John Chalmers
says, about twice as large as the lower interval. He criticises
Archytas for reversing the order of large and small in both enharmonic
and chromatic, and Didymus for the same offence in his chromatic. The
only plausible explanation would seem to be that Ptolemy heard such
divisions of the pyknon practised by players; his task, based on this
perception, was to find the method of generating superparticular
ratios that best reflected this perception.

--
Jonathan Walker
Queen's University Belfast
mailto:kollos@cavehill.dnet.co.uk
http://www.music.qub.ac.uk/~walker/

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Paul Hahn wrote:
"Ray: have you read "The Year of the Jackpot" by Heinlein?
If not, I strongly recommend it."

I have read a few Heinlein but not that one. I will have a look for it
thanks.

To continue the story about nature's music, after visiting the Cycles
Foundation and finding that others before me had also find that cycles
in many things seemed to have periods that were harmonically related I
gradually developed a theory which would explain this. It also turned
out to make a lot of valid predictions about the universe at large.

On my WWW site (at the URL in sig block) I explain the harmonics theory
which lead to explaining these observations. I will give a brief
outline here which is oriented towards music and which will tell us
something about tuning systems.

Imagine an instrument which when it is plucked (or struck or blown or
whatever) produces some fundamental frequency plus all of its harmonics.
The higher order harmonics will have less energy in them. Now imagine
that over time each harmonic loses energy to its harmonics. For the
curious, this is caused by a non-linearity in the wave equation and only
happens in a 3D wave so will not happen in a string for example).

So what goes on is this, energy is transferred as follows:

Frequency --> harmonics

1 --> 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
2 --> 4 6 8 10 12 14 ...
3 --> 6 9 12 15 ...
4 --> 8 12 ...
5 --> 10 15 ...
6 --> 12 ...
7 --> 14 ...
.. etc etc

So what is happening here is that some harmonics are receiving energy
from multiple different paths, in particular 12 gets lots of energy,
while other harmonics such as 11 and 13 receive very little. The
resulting energy pattern depends totally on how many ways each harmonic
number can be factorised.

When calculations are carried out to higher numbered harmonics an
interesting pattern develops. These calculations can be carried out by
a very simple BASIC program:

n = (highest harmonic number)
dim x(n): x(1)=1
for i=1 to n/2
for j=i*2 to n
x(j)=x(j)+x(i)
next j
next i

and the results are now in the x array and may be printed.

The accurate details of the patterns generated can be seen at:
http://www.kcbbs.gen.nz/users/rtomes/rt-ha-no.gif Harmonics to 1000000
http://www.kcbbs.gen.nz/users/rtomes/rt-hanox.gif Harmonics 2 to 288
http://www.kcbbs.gen.nz/users/rtomes/rt-hanoy.gif Harmonics 288 to 69120
but for those who don't have WWW access here is a rough ascii
reconstruction of the relative harmonic energy for the strong harmonics
between 48 and 96:

--------------------------------------------- 48
----
--------------------- 54
-------------------- 56
------------------------------------ 60
-------------------------- 64
---
------------------------------------------ 72
---
--------------------------- 80
------------------------- 84
----------------------- 90
------------------------------------------- 96

Now a tuning person will instantly recognise the relationships present.
The 4 strongest harmonics are 48-60-72-90 or a major chord with ratios
4:5:6:8 and the just intonation scale is present as double the
traditional 24:27:30:32:36:40:45:48 ratios. However there are two
additional values which are quite strong and these 56 and 84 which put
the ratios 28 and 42 in the just scale so as to allow a couple of good
dominant 7th chords with ratios 4:5:6:7:8.

There are additional weaker harmonics between these strong ones and I
have investigated indian music which has extra notes and found that
these extra notes are where they would be expected by the harmonics
theory.

At higher harmonics this pattern is repeated with variations.
I was going to write "minor variations" but realised that "minor" would
have a different connotation. There are also examples of minor keys to
be found, such as the range 1440 to 2880:

------------------------------------- 1440
----
----
-------------------------------- 1728
---
--------------------------- 1920
----------------------- 2016
------------------------------ 2160
---------------------------- 2304
----
----------------------- 2688
---
----------------------------------------- 2880

Here the strongest harmonics are 1440-1728-2160-2880 which is a
10:12:15:20 tuning.

The predicted pattern of harmonics by this very simple rule is found to
very rich and to have many interesting musical meanings present. The
pattern is characterised by many ratios of 2, lots of ratios of 3 and
lesser still of 5 and 7. Although the other primes do appear as ratios
they do so very seldom.

Also, as we move up the harmonics scale we find that the key keeps
changing. After every 2 or 3 ratios of 2 (2.38 on average) the key
shifts by a ratio of 3 and so that means that every about every 4
octaves (including the 2.38 and 1.585 from the 3) there is a shift to
the dominant. At still larger intervals there is a key shift by a major
third. It is at these points that minors appear.

The above patterns also match closely to the pattern of cycles found by
Dewey. From this pattern I have calculated what periods should exist
for longer and shorter cycles and found that I get the right answers
even for long geological cycles of ~600 million years. I am quite
convinced that the universe is a giant musical instrument oscillating in
the pattern that I described.

-- Ray Tomes -- rtomes@kcbbs.gen.nz -- Harmonics Theory --
http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm

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Aristoxenus's + Ptolemy's enharmonics

🔗kollos@cavehill.dnet.co.uk (Jonathan Walker)

🔗Paul Hahn <Paul-Hahn@...>

🔗rtomes@kcbbs.gen.nz (Ray Tomes)

🔗rtomes@kcbbs.gen.nz (Ray Tomes)