Yahoo Tuning Groups Ultimate Backup tuning Aristoxenus and 12-ET

> [Paul Erlich, TD 398.14]
>
> <snip...> Aristoxenus described the division of the octave
> into 12, 18, or 24 aurally equal parts in 330 B.C., <...snip>

Well, close, but not exactly. While Aristoxenus's
mathematics inevitably imply ultimately 144-ET and
all its subdivisions (see the Archives for my postings
on 144-ET), that's not quite what he was describing.

Aristoxenus grew up in Tarantum, where Archytas had
lived, and he presumably had a good knowledge of rational
music-theories as propounded by the Pythagoreans, and their
empirical modifications by Archytas, in addition to the
non-rational theories described by the Harmonists, whom
he criticizes (as far as their use of _katapyknosis_)
while saying that there is some merit in their work.

Aristoxenus lived during a time of exciting changes in
Greek music and mathematics. More strings had been added
to the lyre, the chromatic genus took its place alongside
the ancient enharmonic ('quarter-tone') and diatonic,
and all three genera blossomed with many different 'shades'
and subdivisions. It was also around this time that Euclid
wrote his _Geometry_, which is the method of explaination
invoked by Aristoxenus in his _Elementa Harmonia_ to describe
the intervals used in musical scales.

Hardly anything exists that can actually be attributed to
Aristoxenus himself, and the fragments of 'his' book that
do survive appear to be lecture-notes taken down by two
separate students of his.

In the writings attributed to Aristoxenus (either
directly or indirectly), he never actually describes any
division of the 'octave', or indeed, never uses the 'octave'
as a boundary to describe any of his systems. His basic
unit was the 'perfect 4th', which he refused to describe
as a ratio but did assume could be tuned perfectly by ear,
which would pretty much equate it with the 4:3 ratio.

(Any references to the contrary would be appreciated!
- meaning both further speculations on Aristoxenean theory,
and modern psycho-acoustical research quantifying the
'perfect 4th' thru experimentation as something other
than 4:3.)

He then 'proved' that this interval was divisible into
'2 and a half tones'. Since his method was geometrical
division, this means that a 'tone' is 2/5 of a '4th',
a 'semitone' is 1/5 of a '4th', and a 'quarter-tone'
(Aristoxenus defines the 'smallest enharmonic diesis'
as '1/4 of a tone') is 1/10 of a '4th':

Aristoxenus's 'tone' = (4/3)^(2/5) = ~199.218 cents
Aristoxenus's 'semitone' = (4/3)^(1/5) = ~99.609 cents.
Aristoxenus's 'quarter-tone' = (4/3)^(1/10) = ~49.804 cents.

~199.218 + ~199.218 + ~99.609 = ~498.045 cents
(the approximate size of a 4:3 'perfect 4th'), which
is in essence what Arixtoxenus proves.

Admittedly these figures are extremely close to 12-ET,
with Aristoxenus's 'tone' less than 4/5 of a cent smaller
than 2^(2/12), and his 'semitone' less than 2/5 of a cent
smaller than 2^(1/12) [ - stemming from the fact that
4:3 is only ~1.955 cents smaller than 2^(5/12) ].

But they *are* different.

On the other hand (but still contradicting Paul to an
extent), Aristoxenus implicity recognizes [_Elementa
Harmonia_ 1.46] the 4:3 ratio for the '4th', and the 'tone'
as the difference between a '5th' and a '4th' [= (3/2) / (4/3)
= 9/8], so it could also be argued that his division of a
'4th' into 'two and a half tones' recognizes the Pythagorean
9:8 [= 3^2] 'tone' and 256:243 [= 3^-5] 'semitone':
(9/8) * (9/8) * (256/243) = 4/3
~203.91 + ~203.91 + ~90.225 = ~498.045 cents

This would make his 'quarter-tones' either 1/4 of the former
or 1/2 of the latter:

either (9/8)^(1/4) = ~50.978 cents
or (256/243)^(1/2) = ~45.112 cents.

(It is not inconcievable that Aristoxenus had all of these
different measurements in mind, as he admitted the possibility
of combining all kinds of different-sized intervals in
his genera.)

In either case, it's not entirely accurate to give
Aristoxenus credit for describing 'the division of
the octave into 12, 18, or 24 aurally equal parts'
without further qualification.

And I didn't even go into Aristoxenus's descriptions of
'third-tones' (his 'smallest chromatic diesis' implying
18-ET), and the 'unmelodic' (i.e., not used as part of
any genera, but only involved in comparing different genera)
'sixth-tones' (implying 36-ET), 'twelfth-tones' (implying
72-ET), and '1&1/2 times the enharmonic diesis' [_Elementa
Harmonia_, 1.51] (which would be '3/8-tones', implying 48-ET).

The equal division of an 'octave' which contains all these
divisions, and thus makes it easy to compare them all to one
another in the geometrical way Aristoxenus was doing, is 144-ET.

In fact, it's not entirely clear to me why Aristoxenus
so strongly criticized the use of _katapyknosis_, which
appears to be a way of drawing all the scales in a graphical
diagram which compares their intervals together, when he
himself describes the limits of the generic ranges with
an explicit allusion to '1/12-tones' and an implicit one
to '1/24-tones', whatever he actually meant by 'tone'.

See particularly 1.23-27 of _Elementa Harmonia_, where
Aristoxenus says that the moveable notes within the tetrachord
can be *any* of the infinite variety of intervals within
the characteristic ranges of the genera.

At 1.47 he argues emphatically for a functional interpretation
of note-names wherein intervals that span various ranges
can be described by the same name, qualified by their
genera and shade; and argues against a naming convention
which provides a different name for each interval. He
says this is an impossiblity anyway, because 'an infinite
number of names would be necessary to describe an infinite
number of intervals'.

What is pretty clear is that Aristoxenus didn't really mean
to provide an accurate numerical measurement of any kind
for his musical intervals, but rather sought to describe
ranges of relative aural perception into which intervals
could be categorized.

This reminds me somewhat of Dan Stearns's recent musings
on Ives's tunings contra Johnny Reinhard's acceptance of
Pythagorean tuning in Ives (is it ironic that Aristoxenus
implies 144-ET, the tuning Dan likes for his notation?),
and what Brian Ferneyhough wrote about some of his own work
(forwarded to the List by Daniel Wolf recently).

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
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🔗Joe Monzo <monz@xxxx.xxxx>

In my last post, I wrote:

>> [Paul Erlich, TD 398.14]
>>
>> <snip...> Aristoxenus described the division of the octave
>> into 12, 18, or 24 aurally equal parts in 330 B.C., <...snip>
>
>
> Well, close, but not exactly. <...etc. - snip>

and then went on to mathematically quantify a number of
Aristoxenus's small divisions of the 'tone' in various
different ways, according to his somewhat vague specifications.

Among them were these interval-sizes for Aristoxenus's
'semitone':

(4/3)^(1/5) = ~99.609 cents.
and 256:243 = ~90.225 cents

It wasn't until I re-read my post tonight that I realized
I must take foot out of mouth and apologize to Paul.
He did say '*aurally* (and not mathematically) equal parts'...

So for at least some of the ratios I posted, Paul's
statement is correct: the 'octave' would have been
divided into 12 *aurally* equal parts.

Of course, some of the ratios, such as the Pythagorean
'semitone' quantified above, are far enough away from
12-eq that they may not properly be described as '12
aurally equal parts': a sensitive ear *might* not hear
the intervals of his genera as equal divisions.

The main point behind my last post is that it is somewhat
futile to quantify Aristoxenus's intervals numerically
in the first place. But in the main, Paul's statement
is correct.

An 'octave' divided according to Aristoxenus's various
genera descriptions would be more-or-less aurally perceived
as subsets of various ET scales as follows:

12-ET Tense Diatonic
24-ET Soft Diatonic

12-ET Tonic Chromatic
16-ET Hemiolic Chromatic
18-ET Soft Chromatic

24-ET Enharmonic

but of course keeping in mind that Aristoxenus used these
particular divisions only as illustrations of the various
'shades' of the different genera, and insisted that the
actual notes were quite flexible in pitch-placement, within
the *ranges* he specified, which is where he used more
accurate descriptions.

-monz

🔗Joe Monzo <monz@xxxx.xxxx>

ANOTHER CORRECTION:

Just as with my previous analysis of Aristoxenus's
'Hemiolic Chromatic', his 'Soft Chromatic' also requires
a larger ET in order to approximate the 'perfect 4th':
this time 36-ET. 18-ET won't work, as it has 'perfect 4ths'
of 466&2/3 and 533&1/3 cents, neither one coming close enough
to represent Aristoxenus's basic implied interval of 4:3.

The table I gave in my original posting should read
as follows:

> An 'octave' divided according to Aristoxenus's various
> genera descriptions would be more-or-less aurally perceived
> as subsets of various ET scales as follows:
>
> 12-ET Tense Diatonic
> 24-ET Soft Diatonic
>
> 12-ET Tonic Chromatic
> 48-ET Hemiolic Chromatic
> 36-ET Soft Chromatic
>
> 24-ET Enharmonic

As I said in the very first post on this thread, 144-ET
would be the scale which contains all these divisions,
and would be the only ET that could give an overview of
all of Aristoxenus's examples together.

(But keep in mind that those examples are just selections
from the infinite number possible, for the purpose of
illustrating the *characteristic* intervals of the various
shades of the genera.)

Every one of these has a 500-cent 'perfect 4th' substituting
for the 4:3 that was probably intended by Aristoxenus in his
descriptions, but the ~2-cent difference is small enough that
the ear would be likely to perceive his genera in terms of
these ETs.

But that still doesn't mean that he intended to imply ETs
- he was using *geometry* to make his measurements, and
his basic intervals almost certainly came from the Pythagorean
scale.

-monz

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

🔗Joe Monzo <monz@xxxx.xxxx>

YET ANOTHER CORRECTION:
I left a letter out of the title of Aristoxenus's book
in my citations: it should be _Elementa Harmonica_
['the Elements of Harmonics'].

---------------

ANOTHER RESPONSE TO PAUL ERLICH:

> [Paul Erlich, TD 400.9]
>
> Joe Monzo wrote,
>
>> Aristoxenus used these particular divisions only as
>> illustrations of the various 'shades' of the different
>> genera, and insisted that the actual notes were quite
>> flexible in pitch-placement, within the *ranges* he
>> specified, which is where he used more accurate descriptions.
>
> Interesting. I didn't know that. How wide were the ranges
> that he accurately described?

OK, first please note that I said 'more accurately' and
not just plain 'accurately'. As I pointed out in my original
post responding to your statement, there are many problems
with trying to determine precise numerical measurements
of Aristoxenus's intervals. (much more on this below,
in my response to your second question)

That said, let me put off the description of the ranges
of Aristoxenus's intervals until a future post. I've
gone back and re-read his book, and took notes, so that
I can quote his measurements here with his exact wording,
but I can't take the time to put it here right now.

> [Paul Erlich, TD 400.10]
>
> Joe Monzo wrote,
>
>> his basic intervals almost certainly came from the Pythagorean
>> scale.
>
> Um, so he disagreed with Pythagoras about _what_, exactly?

The first thing to note in my response is that *Pythagorean*
theory is not necessarily *by* Pythagoras himself. The ideas
attributed to Pythagoras amounted to a religion in Classical
times, both during the 400s BC in Greece and its colonies
(especially in Sicily) and during the neo-Pythagorean revival
around 100 AD in the Roman Empire (which included Greece).
See Burkert 1972 for a fascinating study of this.

To answer your question, I'll let the sources speak for
themselves:

> [Crocker 1966:]
>
> As Macran [editor of an English translation of Aristoxenus
> put it, Aristoxenus was concerned with a system "each member
> of which *is* essentially what it *does*". This was a new
> kind of music theory, as different from Pythagorean theory
> as the new geometry was from Pythagorean arithmetic - and in
> exactly the same way. The construction of a purely functional
> system in which elements were related only to each other, not
> to external units of measurement, was made possible by the
> new-found ability to deal with all types of musical intervals
> on the same basis without reference to number.

> [Barker 1978:]
>
> Aristoxenus was an innovator... His objective was to claw
> back the study of music from the hands of physicists,
> mathematicians, and mere recorders of low-level empirical
> fact, and to establish it as an independent science having
> its own laws and principles...

> [Litchfield 1988:]
>
> Aristoxenus was a philosopher whose basic purpose was to
> study all science (including but not restricted to music).
> He was not, however, necessarily interested in the actual
> phenomena of each science; rather, he was interested in the
> theory.
> ... Aristoxenus's _Harmonics_... is not primarily concerned
> with practical music, although this is the foundation of the
> study. Aristoxenus used an empirical method for evolving and
> explaining his theories, but the conclusions remain abstract,
> ideal, and conceptual. It is ironic that the Pythagoreans'
> theories with the constant recourse to physical sound turn
> out to be more empirical than Aristoxenus's theories.
> His proof that a perfect fourth is made up of two-and-one-half
> tones cannot be based on practice. Yet the concept is perfectly
> successful and tremendously advanced in its application of
> geometric logic to music. His demonstration of the _loci_
> of the genera could not be readily performed or empirically
> derived. Yet the concept is very clear and makes a fundamental
> advance in its observation of function over the earlier theorists.

I'm not sure why Litchfield says that Aristoxenus's proof
that '2&1/2 tones make up a perfect 4th' is 'perfectly
successful', when he himself goes to such length to show
how variable the aural results are when Aristoxenus's proof
is tried out in practice. I suppose it merely underlines
Litchfield's main point that Aristoxenus was not at all
concerned with actual musical practice, but was attempting
to ground musical science in a speculative Aristotelian
philosophical foundation.

It should also be noted that Aristoxenus's 'treatise'
is the first one to systematize Greek musical scales
into the 'Greater Perfect System' and 'Lesser Perfect System'.
Pythagorean theorists had contented themselves with describing
only individual scales, and the later neo-Pythagoreans
(notably Nicomachus) made use of Aristoxenus's systems
in their own Pythagorean descriptions.

REFERENCES
----------

Crocker, Richard L. 1966. 'Aristoxenus and Greek Mathematics'.
_Aspects of Medieval and Renaissance Music: A Birthday Offering
to Gustave Reese_, ed. Jan LaRue, New York, Norton, p 96-110.

Burkert, Walter. 1972. _Lore and Science in Ancient
Pythagoreanism_. English translation by Edwin L. Minar, Jr.
Cambridge, Harvard University Press. (original German
edition 1962)

Barker, Andrew. 1978. 'Music and Perception: A Study in
Aristoxenus'. _Journal of Hellenistic Studies_, v 98, p 9-16.

Litchfield, Malcolm. 1988. 'Aristoxenus and Empiricism:
A Reevalutation Based on his Theories'. _Journal of Music
Theory_, v 32 # 1, p 51-73.

Barker, Andrew. 1989. _Greek Musical Writings, volume 2:
Harmonic and Acoustic Theory_. Cambridge University Press.
(Contains a complete annotated translation of Aristoxenus's
_Elementa Harmonica_.)

-monz