Yahoo Tuning Groups Ultimate Backup tuning Aristoxenos and 144-EDO (was: Arabian comma dispute)

Hi monz,

On Tue, 13 Dec 2005, you wrote:
>
Hi Yahya,
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
>
> > So, if "Aristoxenus did not carry the Pythagorean
> > tuning out far enough to derive the quarter-tones
> > from it", is it reasonable to assume that he ever
> > contemplated 144-EDO? It seems to me that the
> > most likely reasons for anyone's choosing 144-EDO
> > would be to allow further rational division of 12
> > equal semitones into halves, quarters, sixths and
> > twelfths - the simplest of these being the quarter-
> > tone. Had he contemplated 144-EDO, it can hardly
> > have escaped his notice that it supports quarter-
> > tones.
> >
> > In perplexity,
> > Yahya
>
> At least part of the reason for your perplexity is
> the fact that my Aristoxenus page is very sloppy and
> badly in need of editing ... something which i've needed
> to do for years but just can't get to until i have a
> serious block of time to devote to it.
>
> For starters, i used the Latinized spelling of his name,
> but when i finally edit the page i'll probably stick to
> the Greek spelling Aristoxenos.
>
> The most important thing to point out about his theories
> is that, at the end of his process of "tuning by concords",
> that is, upon reaching the 12th note, the last concord
> derived is supposed to sound similar to all the other ones.
>
> This is impossible in true pythagorean tuning, because
> the last "concord" will be smaller by a pythagorean comma,
> i.e., a type of "wolf-5th" of ~678 cents.
>
> The last section of my page, "Amendment from 2003.07.20",
> represents my latest thoughts on the subject -- so while
> everything before it is interesting, and worth reading from
> a historical point of view, what i propose in this last
> section is what i think most likely represents Aristoxenos's
> theories.
>
> Aristoxenos used divisions as small as 1/12-tone and 1/8-tone
> in his actual descriptions, and so the smallest division
> needed to encompass all the possibilities in his various
> generic shades is 1/24-tone. Obviously, this implies 144-edo.

Obviously!

> I describe a mathematical procedure which exactly follows
> his descriptions, wherein a "ditone" (major-3rd) is tuned
> by concords but then is assumed to be a logarithmic 4/5 of
> a "perfect-4th". ...

Perhaps by a statement in the form that 5 ditones
produce the same interval as 4 fourths?

> ... Assuming the perfect-4th to be a pythagorean
> 4/3 ratio, this leads to a smallest division of (4/3)^(1/60),
> and was the division described by Cleonides, whose work is
> a main source for Aristoxenos.
>
> But one cannot assume Aristoxenos's "perfect-4th" to be
> a 4/3 ratio, because of the discrepancy of the size of
> "concords" in his "tuning by concords" method, as mentioned
> above.
>
> So ultimately, 144-edo seems to be the best tuning to assume
> for the fitting together of all the disparate and seemingly
> contradictory statements made by Aristoxenos. That's why
> i used 144-edo for the tuning of the notated illustrations
> and audio examples at the end of my webpage.

Thank you. This explains your reasoning perfectly
clearly. We are all indebted to you, both for the
quality of your scholarship and your willingness to
share its fruits.

(On Cleonides' tuning, 60 of his smallest step will
encompass a perfect fourth 4/3, so 120 will cover
two fourths 16/9, leaving but a wholetone 9/8 to
complete the octave. For this tuning to coincide
with 144-EDO, the wholetone 9/8 would have to
be exactly 24 of Cleonides' steps, ie (4/3)^(2/5),
or 199.218 cents. However, 9/8 is 203.910 cents,
almost 4.7 cents greater. As Cleonides' unit is just
over 8.3 cents, it implies an approximate 144.5-EDO
(or better, a 299-EDO). Not that he would have
cared much about the steps in the octave, since
the fourth was _the_ perfect consonance for the
Greeks, wasn't it?)

Regards,
Yahya

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🔗monz <monz@tonalsoft.com>

Hi Yahya,

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
>
> Hi monz,
>
> On Tue, 13 Dec 2005, you wrote:

> > I describe a mathematical procedure which exactly follows
> > his descriptions, wherein a "ditone" (major-3rd) is tuned
> > by concords but then is assumed to be a logarithmic 4/5 of
> > a "perfect-4th". ...
>
> Perhaps by a statement in the form that 5 ditones
> produce the same interval as 4 fourths?

No -- by his statement that a "4th" (_diatessaron_)
contains 2 & 1/2 tones.

Thus, a "tone" = 2/5 of a "4th", so therefore a
"ditone" = 4/5 of a "4th".

> > ... Assuming the perfect-4th to be a pythagorean
> > 4/3 ratio, this leads to a smallest division of (4/3)^(1/60),
> > and was the division described by Cleonides, whose work is
> > a main source for Aristoxenos.
> >
> > But one cannot assume Aristoxenos's "perfect-4th" to be
> > a 4/3 ratio, because of the discrepancy of the size of
> > "concords" in his "tuning by concords" method, as mentioned
> > above.
> >
> > So ultimately, 144-edo seems to be the best tuning to assume
> > for the fitting together of all the disparate and seemingly
> > contradictory statements made by Aristoxenos. That's why
> > i used 144-edo for the tuning of the notated illustrations
> > and audio examples at the end of my webpage.
>
>
> Thank you. This explains your reasoning perfectly
> clearly. We are all indebted to you, both for the
> quality of your scholarship and your willingness to
> share its fruits.

Thanks. I appreciate that.

> (On Cleonides' tuning, 60 of his smallest step will
> encompass a perfect fourth 4/3, so 120 will cover
> two fourths 16/9, leaving but a wholetone 9/8 to
> complete the octave. For this tuning to coincide
> with 144-EDO, the wholetone 9/8 would have to
> be exactly 24 of Cleonides' steps, ie (4/3)^(2/5),
> or 199.218 cents. However, 9/8 is 203.910 cents,
> almost 4.7 cents greater. As Cleonides' unit is just
> over 8.3 cents, it implies an approximate 144.5-EDO
> (or better, a 299-EDO). Not that he would have
> cared much about the steps in the octave, since
> the fourth was _the_ perfect consonance for the
> Greeks, wasn't it?)

Yes. Cleonides was a neo-Pythagorean who came along
a few centuries after Aristoxenos, so for Cleonides,
a _diatessaron_ ("4th") could never be anything but
a 4/3 ratio. He tried to shoehorn Aristoxenos's theory
into the pythagorean mold by using divisions of a 4/3.

(After the Romans conquered Greece, there was a whole
spate of neo-pythagoreans who surfaced around 100 AD
-- notably including Nicomachus -- who tried to re-entrench
pythagorean tuning after such luminaries as Archytas,
Eratosthenes, and Didymus had advocated prime-factors
higher than 3. Ptolemy, shortly after, wiped the
neo-pythagoreans off the board with his huge compendium
of various rational tunings of not only all of Aristoxenos's
different generic shades, but also of other scales "commonly
used by musicians" ... but then the Germans conquered Rome,
leaving only Boethius to translate the Greek theories into
Latin, the "dark ages" ensued, and when Charlemagne began
to bring Europe out of them, all that apparently was left
of ancient musical tunings was the pythagorean, and so it
remained until at least 1300. But i digress ...)

But as i said, Aristoxenos's "tuning by concords" with
12 different pitches which encompass a series of "5ths"
(_diapente_) which are *all* supposed to be "perfect",
rules out using a "pure" 4/3 for the "4th", and hence
the whole system must be tempered.

I hasten to point out that Aristoxenos *emphatically*
avoided using string-lengths or any other type of pitch
measurement which would result in the use of ratios,
preferring instead to speak of string *tension*.

Hence, Aristoxenos's _syntonon_ should be translated as
"tense" and his _malakon_ as "relaxed", instead of the
"intense" and "soft" which Partch and most other theorists
use when discussing his theories in English.

Aristoxenos insisted that "musicians know what a concord
sounds like", and refused to quantify it -- presumably
because without being able to calculate powers and
logarithms, he couldn't pinpoint the measurements of
his tempered system any more precisely than speaking of
"parts of a tone".

Note, however, that i don't agree with Partch that
Aristoxenos deserves the credit (dubious credit, according
to Partch) for being "the father of temperament".
I speculated that the Sumerians, with their base-60
mathematics, did have the ability to derive a very good
approximation of 12-edo (or indeed any temperament)
as early as c.2500 BC -- and it has been satisfactorily
documented that the Greeks got a lot of their cultural
and scientific knowledge from the Babylonians, who got
theirs from the Sumerians.

http://tonalsoft.com/monzo/sumerian/simplified-sumerian-tuning.htm

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

🔗monz <monz@tonalsoft.com>

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Hi monz,

On Wed, 14 Dec 2005, you wrote:
> ...
> > > I describe a mathematical procedure which exactly follows
> > > his descriptions, wherein a "ditone" (major-3rd) is tuned
> > > by concords but then is assumed to be a logarithmic 4/5 of
> > > a "perfect-4th". ...
> >
> > Perhaps by a statement in the form that 5 ditones
> > produce the same interval as 4 fourths?
>
> No -- by his statement that a "4th" (_diatessaron_)
> contains 2 & 1/2 tones.
>
> Thus, a "tone" = 2/5 of a "4th", so therefore a
> "ditone" = 4/5 of a "4th".

That's fine, if he meant that 1/2 to be exact. It was
still quite common practice, up to at least the middle
of last century here, in Australia for people to mean
by "a half", in most contexts, not "one of two exactly
equal divisions of a whole" - as we mostly do now, but
rather, "one of two (probably unequal) divisions of a
whole". This sense still lingers in several common
expressions, such as "You took the bigger half!" and
"You don't know the half of it". Or in the saying that
"Half (some part of) a loaf is better than none." If
your friend asked you for half your chocolate, and
you took out a ruler before cutting it exactly in half,
he would think you either a pedant or just plain silly!
I think that, in most cultures, "half" has only quite
recently come to mean an exact half, as a result
mostly of the swing to greater use of measurement
in all spheres, especially in time and money. Historically
recently, I mean.

Did Aristoxenos always mean an exact 0.5 tone when he
wrote of a half-tone? Or did he perhaps sometimes
mean one kind of semitone, and sometimes another?
The Greeks had already distinguished the large and
small semitones by his time, hadn't they?

Without the context of his writings, I can't be sure,
but at first glance, I wouldn't take that statement too
literally.

> > > ... Assuming the perfect-4th to be a pythagorean
> > > 4/3 ratio, this leads to a smallest division of (4/3)^(1/60),
> > > and was the division described by Cleonides, whose work is
> > > a main source for Aristoxenos.
> > >
> > > But one cannot assume Aristoxenos's "perfect-4th" to be
> > > a 4/3 ratio, because of the discrepancy of the size of
> > > "concords" in his "tuning by concords" method, as mentioned
> > > above.
> > >
> > > So ultimately, 144-edo seems to be the best tuning to assume
> > > for the fitting together of all the disparate and seemingly
> > > contradictory statements made by Aristoxenos. That's why
> > > i used 144-edo for the tuning of the notated illustrations
> > > and audio examples at the end of my webpage.
> >
> >
> > Thank you. This explains your reasoning perfectly
> > clearly. We are all indebted to you, both for the
> > quality of your scholarship and your willingness to
> > share its fruits.
>
>
> Thanks. I appreciate that.

Not as much as I do! :-)

> > (On Cleonides' tuning, 60 of his smallest step will
> > encompass a perfect fourth 4/3, so 120 will cover
> > two fourths 16/9, leaving but a wholetone 9/8 to
> > complete the octave. For this tuning to coincide
> > with 144-EDO, the wholetone 9/8 would have to
> > be exactly 24 of Cleonides' steps, ie (4/3)^(2/5),
> > or 199.218 cents. However, 9/8 is 203.910 cents,
> > almost 4.7 cents greater. As Cleonides' unit is just
> > over 8.3 cents, it implies an approximate 144.5-EDO
> > (or better, a 299-EDO). Not that he would have
> > cared much about the steps in the octave, since
> > the fourth was _the_ perfect consonance for the
> > Greeks, wasn't it?)
>
>
> Yes. Cleonides was a neo-Pythagorean who came along
> a few centuries after Aristoxenos, so for Cleonides,
> a _diatessaron_ ("4th") could never be anything but
> a 4/3 ratio. He tried to shoehorn Aristoxenos's theory
> into the pythagorean mold by using divisions of a 4/3.
>
> (After the Romans conquered Greece, there was a whole
> spate of neo-pythagoreans who surfaced around 100 AD
> -- notably including Nicomachus -- who tried to re-entrench
> pythagorean tuning after such luminaries as Archytas,
> Eratosthenes, and Didymus had advocated prime-factors
> higher than 3. Ptolemy, shortly after, wiped the
> neo-pythagoreans off the board with his huge compendium
> of various rational tunings of not only all of Aristoxenos's
> different generic shades, but also of other scales "commonly
> used by musicians" ... but then the Germans conquered Rome,
> leaving only Boethius to translate the Greek theories into
> Latin, the "dark ages" ensued, and when Charlemagne began
> to bring Europe out of them, all that apparently was left
> of ancient musical tunings was the pythagorean, and so it
> remained until at least 1300. But i digress ...)
>
> But as i said, Aristoxenos's "tuning by concords" with
> 12 different pitches which encompass a series of "5ths"
> (_diapente_) which are *all* supposed to be "perfect",
> rules out using a "pure" 4/3 for the "4th", and hence
> the whole system must be tempered.
>
> I hasten to point out that Aristoxenos *emphatically*
> avoided using string-lengths or any other type of pitch
> measurement which would result in the use of ratios,
> preferring instead to speak of string *tension*.
>
> Hence, Aristoxenos's _syntonon_ should be translated as
> "tense" and his _malakon_ as "relaxed", instead of the
> "intense" and "soft" which Partch and most other theorists
> use when discussing his theories in English.
>
> Aristoxenos insisted that "musicians know what a concord
> sounds like", and refused to quantify it -- presumably
> because without being able to calculate powers and
> logarithms, he couldn't pinpoint the measurements of
> his tempered system any more precisely than speaking of
> "parts of a tone".

That's a reasonable theory. To it you might add
that he found the previous attempts to quantify
it unable to produce the sound of a concord that
every musician recognised.

However, a bias in favour of perception rather
than measurement could also have lead him to
ignore the difference in size of the two tones,
and to use the term "half a tone" less precisely,
to mean "a part of a tone". So that in writing of
"two and a half tones" making the diatessaron,
perhaps he meant, from the starting note, "one
tone higher than that, then another, then part
of another"? I'm disinclined to see "two and a
half tones" as meaning "2.5 tones", unless we see
him doing arithmetic with that number. What do
you think?

> Note, however, that i don't agree with Partch that
> Aristoxenos deserves the credit (dubious credit, according
> to Partch) for being "the father of temperament".
> I speculated that the Sumerians, with their base-60
> mathematics, ...

(It occurred to me that this was perhaps a
factor in Cleonides' choice - dividing the
classically perfect consonance, the diatessaron
or fourth 4/3, into the classically perfect
number of parts, 60, which the Greeks also
inherited and perpetuated in geometry.)

> ... did have the ability to derive a very good
> approximation of 12-edo (or indeed any temperament)
> as early as c.2500 BC -- and it has been satisfactorily
> documented that the Greeks got a lot of their cultural
> and scientific knowledge from the Babylonians, who got
> theirs from the Sumerians.

> http://tonalsoft.com/monzo/sumerian/simplified-sumerian-tuning.htm

Tho the Greeks got much from the
Babylonians, they built on it. Did the
Babylonians themselves invent nothing? :-)
The inheritors of the Greeks, the Arabs and
other savants of early Islam, in their turn
added much that he Greeks had not thought
of.

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Free Edition.
Version: 7.1.371 / Virus Database: 267.14.1/204 - Release Date: 15/12/05

🔗monz <monz@tonalsoft.com>

Hi Yahya,

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> Did Aristoxenos always mean an exact 0.5 tone when he
> wrote of a half-tone? Or did he perhaps sometimes
> mean one kind of semitone, and sometimes another?
> The Greeks had already distinguished the large and
> small semitones by his time, hadn't they?
>
> Without the context of his writings, I can't be sure,
> but at first glance, I wouldn't take that statement too
> literally.

You *have* the context of a good portion of his writings
if you have the patience to wade thru the sloppiness of my
Aristoxenos webpage. I've quoted quite a bit of his
discussion of tuning.

Anyway, yes, the pythagoreans had already distinguished
between the large and small semitones:

.. 2,3-monzo ...... ratio .......... ~cents ...... name

.... [-11 7> .... 2187 / 2048 .... 113.6850061 ... apotome
.... [8 -5> ...... 256 / 243 ...... 90.22499567 .. limma

In fact it was the pythagoreans who pointed out that
the tone could not be equally divided into two identical
semitones -- i'll bring this up again shortly ...

Aristoxenos himself was the first music-theorist to
break the chromatic genus (the one featuring semitones)
into further subdivisions which he called "shades",
thus resulting in several other sizes of semitones:

(assuming my interpretation of Aristoxenos's tuning
as 144-edo)

chromatic shade ...... cent sizes of semitones

... relaxed ............... 66 & (2/3)
... hemiolic .............. 75
... tonic ................ 100

As i pointed out regarding Aristoxenos's use of the
concept of string tension instead of length, he was
deliberately distancing himself from the pythagoreans.

His description of "tuning by concords" can only be
achieved by some form of temperament, and he impudently
states that "musicians know what a concord sounds like"
when he deliberately avoids using a ratio to measure one.

So it seems to me that when he speaks of a "4th" containing
"two and a half tones", he is again deliberately throwing
sand in the face of the pythagoreans. IOW, regardless of
what numerical complexities the pythagoreans weave in order
to prove that there is no such thing as precisely 1/2-tone,
Aristoxenos states that musicians deal with quasi-equal
1/2-tones as an established fact.

> That's a reasonable theory. To it you might add
> that he found the previous attempts to quantify
> it unable to produce the sound of a concord that
> every musician recognised.

To the extent that pythagorean mathematics and the
"tuning by concords" process produces a wolf-4th at
the end of the procedure, that's true. But it also
needs to be pointed out that the 144-edo tempered 4th
is the familiar 12-edo one of 500 cents, which is a
largely inaudible discrepancy from the ~498 cents of
the pythagorean 4/3 ratio 4th.

> However, a bias in favour of perception rather
> than measurement could also have lead him to
> ignore the difference in size of the two tones,
> and to use the term "half a tone" less precisely,
> to mean "a part of a tone". So that in writing of
> "two and a half tones" making the diatessaron,
> perhaps he meant, from the starting note, "one
> tone higher than that, then another, then part
> of another"? I'm disinclined to see "two and a
> half tones" as meaning "2.5 tones", unless we see
> him doing arithmetic with that number. What do
> you think?

Aristoxenos was certainly dealing in approximations,
as Partch so indignantly accused him of doing.

But apparently what's going on here is that Aristoxenos
wanted to employ concepts derived from the then-new
geometry expounded by Euclid. He saw Euclidean geometry
as a way to measure divisions of irrational musical intervals,
which the pythagorean mathematics before him were unable
to deal with.

So i'm not convinced that he intended his measurements
to be as vague as they might seem to us moderns. Having
thought about this for years, and done a lot of research
into it, i'm quite convinced that 144-edo is pretty much
what Aristoxenos had in mind when writing his theories.

(BTW, the character of Aristoxenos's two suriviving books
seems to indicate that they were not written down by him,
but that they are notes written down by students attending
his lectures.)

> > Note, however, that i don't agree with Partch that
> > Aristoxenos deserves the credit (dubious credit, according
> > to Partch) for being "the father of temperament".
> > I speculated that the Sumerians, with their base-60
> > mathematics, ...
>
> (It occurred to me that this was perhaps a
> factor in Cleonides' choice - dividing the
> classically perfect consonance, the diatessaron
> or fourth 4/3, into the classically perfect
> number of parts, 60, which the Greeks also
> inherited and perpetuated in geometry.)

The Sumerians, Babylonians, and Greeks were all quite
fond of base-60 mathematics.

> > ... did have the ability to derive a very good
> > approximation of 12-edo (or indeed any temperament)
> > as early as c.2500 BC -- and it has been satisfactorily
> > documented that the Greeks got a lot of their cultural
> > and scientific knowledge from the Babylonians, who got
> > theirs from the Sumerians.
>
> > http://tonalsoft.com/monzo/sumerian/simplified-sumerian-tuning.htm
>
> Tho the Greeks got much from the
> Babylonians, they built on it. Did the
> Babylonians themselves invent nothing? :-)
> The inheritors of the Greeks, the Arabs and
> other savants of early Islam, in their turn
> added much that he Greeks had not thought
> of.

Your assessment is pretty accurate. The Babylonians
did add a few new twists, especially in mathematics,
but culturally their legacy was basically that of
adapting the Sumerian innovations to their own society.

The truth is that there have not been many Sumerian
mathematical tablets deciphered and/or published.

But the Babylonians retained the Sumerian logograms
in their own words for the "root" concept of the word
and added their own written symbols to represent the
phonics of the Akkadian syllables -- very similar to
the way modern Japanese writing retains the Chinese
logograms and adds further symbols for the Japanese
syllables.

So by stripping away the Akkadian syllables, as i have
done on my webpage, one can see the underlying Sumerian
basis of all of the Babylonian mathematical texts.
And the essence of the math problems is already there
in the Sumerian.

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

Thank you for this excellent summary on Aristoxenos and Sumerians monz. I
enjoyed reading it thru.

Cordially,
Oz.

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 18 Aralï¿½k 2005 Pazar 1:09
Subject: [tuning] Re: Aristoxenos and 144-EDO (was: Arabian comma dispute)

> Hi Yahya,
>
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
> > Did Aristoxenos always mean an exact 0.5 tone when he
> > wrote of a half-tone? Or did he perhaps sometimes
> > mean one kind of semitone, and sometimes another?
> > The Greeks had already distinguished the large and
> > small semitones by his time, hadn't they?
> >
> > Without the context of his writings, I can't be sure,
> > but at first glance, I wouldn't take that statement too
> > literally.
>
>
>
> You *have* the context of a good portion of his writings
> if you have the patience to wade thru the sloppiness of my
> Aristoxenos webpage. I've quoted quite a bit of his
> discussion of tuning.
>
>
> Anyway, yes, the pythagoreans had already distinguished
> between the large and small semitones:
>
>
> .. 2,3-monzo ...... ratio .......... ~cents ...... name
>
> .... [-11 7> .... 2187 / 2048 .... 113.6850061 ... apotome
> .... [8 -5> ...... 256 / 243 ...... 90.22499567 .. limma
>
>
> In fact it was the pythagoreans who pointed out that
> the tone could not be equally divided into two identical
> semitones -- i'll bring this up again shortly ...
>
>
>
> Aristoxenos himself was the first music-theorist to
> break the chromatic genus (the one featuring semitones)
> into further subdivisions which he called "shades",
> thus resulting in several other sizes of semitones:
>
> (assuming my interpretation of Aristoxenos's tuning
> as 144-edo)
>
>
> chromatic shade ...... cent sizes of semitones
>
> ... relaxed ............... 66 & (2/3)
> ... hemiolic .............. 75
> ... tonic ................ 100
>
>
>
> As i pointed out regarding Aristoxenos's use of the
> concept of string tension instead of length, he was
> deliberately distancing himself from the pythagoreans.
>
> His description of "tuning by concords" can only be
> achieved by some form of temperament, and he impudently
> states that "musicians know what a concord sounds like"
> when he deliberately avoids using a ratio to measure one.
>
> So it seems to me that when he speaks of a "4th" containing
> "two and a half tones", he is again deliberately throwing
> sand in the face of the pythagoreans. IOW, regardless of
> what numerical complexities the pythagoreans weave in order
> to prove that there is no such thing as precisely 1/2-tone,
> Aristoxenos states that musicians deal with quasi-equal
> 1/2-tones as an established fact.
>
>
>
> > That's a reasonable theory. To it you might add
> > that he found the previous attempts to quantify
> > it unable to produce the sound of a concord that
> > every musician recognised.
>
>
> To the extent that pythagorean mathematics and the
> "tuning by concords" process produces a wolf-4th at
> the end of the procedure, that's true. But it also
> needs to be pointed out that the 144-edo tempered 4th
> is the familiar 12-edo one of 500 cents, which is a
> largely inaudible discrepancy from the ~498 cents of
> the pythagorean 4/3 ratio 4th.
>
>
>
> > However, a bias in favour of perception rather
> > than measurement could also have lead him to
> > ignore the difference in size of the two tones,
> > and to use the term "half a tone" less precisely,
> > to mean "a part of a tone". So that in writing of
> > "two and a half tones" making the diatessaron,
> > perhaps he meant, from the starting note, "one
> > tone higher than that, then another, then part
> > of another"? I'm disinclined to see "two and a
> > half tones" as meaning "2.5 tones", unless we see
> > him doing arithmetic with that number. What do
> > you think?
>
>
> Aristoxenos was certainly dealing in approximations,
> as Partch so indignantly accused him of doing.
>
> But apparently what's going on here is that Aristoxenos
> wanted to employ concepts derived from the then-new
> geometry expounded by Euclid. He saw Euclidean geometry
> as a way to measure divisions of irrational musical intervals,
> which the pythagorean mathematics before him were unable
> to deal with.
>
> So i'm not convinced that he intended his measurements
> to be as vague as they might seem to us moderns. Having
> thought about this for years, and done a lot of research
> into it, i'm quite convinced that 144-edo is pretty much
> what Aristoxenos had in mind when writing his theories.
>
> (BTW, the character of Aristoxenos's two suriviving books
> seems to indicate that they were not written down by him,
> but that they are notes written down by students attending
> his lectures.)
>
>
>
> > > Note, however, that i don't agree with Partch that
> > > Aristoxenos deserves the credit (dubious credit, according
> > > to Partch) for being "the father of temperament".
> > > I speculated that the Sumerians, with their base-60
> > > mathematics, ...
> >
> > (It occurred to me that this was perhaps a
> > factor in Cleonides' choice - dividing the
> > classically perfect consonance, the diatessaron
> > or fourth 4/3, into the classically perfect
> > number of parts, 60, which the Greeks also
> > inherited and perpetuated in geometry.)
>
>
> The Sumerians, Babylonians, and Greeks were all quite
> fond of base-60 mathematics.
>
>
> > > ... did have the ability to derive a very good
> > > approximation of 12-edo (or indeed any temperament)
> > > as early as c.2500 BC -- and it has been satisfactorily
> > > documented that the Greeks got a lot of their cultural
> > > and scientific knowledge from the Babylonians, who got
> > > theirs from the Sumerians.
> >
> > > http://tonalsoft.com/monzo/sumerian/simplified-sumerian-tuning.htm
> >
> > Tho the Greeks got much from the
> > Babylonians, they built on it. Did the
> > Babylonians themselves invent nothing? :-)
> > The inheritors of the Greeks, the Arabs and
> > other savants of early Islam, in their turn
> > added much that he Greeks had not thought
> > of.
>
>
> Your assessment is pretty accurate. The Babylonians
> did add a few new twists, especially in mathematics,
> but culturally their legacy was basically that of
> adapting the Sumerian innovations to their own society.
>
>
> The truth is that there have not been many Sumerian
> mathematical tablets deciphered and/or published.
>
> But the Babylonians retained the Sumerian logograms
> in their own words for the "root" concept of the word
> and added their own written symbols to represent the
> phonics of the Akkadian syllables -- very similar to
> the way modern Japanese writing retains the Chinese
> logograms and adds further symbols for the Japanese
> syllables.
>
> So by stripping away the Akkadian syllables, as i have
> done on my webpage, one can see the underlying Sumerian
> basis of all of the Babylonian mathematical texts.
> And the essence of the math problems is already there
> in the Sumerian.
>
>
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Hi monz,

On Sat, 17 Dec 2005 you wrote:
>
> Hi Yahya,
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
>
> > Did Aristoxenos always mean an exact 0.5 tone when he
> > wrote of a half-tone? Or did he perhaps sometimes
> > mean one kind of semitone, and sometimes another?
> > The Greeks had already distinguished the large and
> > small semitones by his time, hadn't they?
> >
> > Without the context of his writings, I can't be sure,
> > but at first glance, I wouldn't take that statement too
> > literally.
>
> You *have* the context of a good portion of his writings
> if you have the patience to wade thru the sloppiness of my
> Aristoxenos webpage. I've quoted quite a bit of his
> discussion of tuning.

I trust your good intentions, and I think you have
an unsurpassed reputation on this list for musically
relevant accuracy. So I hope you'll pardon me if
I still feel that I'd only really "have the context of
anyone's writings" if I could read the darned thing
myself! However, absent the time to learn and
master the Greek of his day, the next-best thing
must always be a good translation by a subject-
matter expert who has those language skills I lack,
and I'm guessing that expert would be yourself.

So I will certainly write a proper study of that page
on my to-do list.

> Anyway, yes, the pythagoreans had already distinguished
> between the large and small semitones:
>
>
> .. 2,3-monzo ...... ratio .......... ~cents ...... name
>
> .... [-11 7> .... 2187 / 2048 .... 113.6850061 ... apotome
> .... [8 -5> ...... 256 / 243 ...... 90.22499567 .. limma
>
>
> In fact it was the pythagoreans who pointed out that
> the tone could not be equally divided into two identical
> semitones -- i'll bring this up again shortly ...
>
>
> Aristoxenos himself was the first music-theorist to
> break the chromatic genus (the one featuring semitones)
> into further subdivisions which he called "shades",
> thus resulting in several other sizes of semitones:
>
> (assuming my interpretation of Aristoxenos's tuning
> as 144-edo)
>
>
> chromatic shade ...... cent sizes of semitones
>
> ... relaxed ............... 66 & (2/3)
> ... hemiolic .............. 75
> ... tonic ................ 100

Which he used only in the chromatic genus? Two wide
tones, descending, followed by a relaxed shade, would
certainly make an expressive tetrachord.

> As i pointed out regarding Aristoxenos's use of the
> concept of string tension instead of length, he was
> deliberately distancing himself from the pythagoreans.
>
> His description of "tuning by concords" can only be
> achieved by some form of temperament, and he impudently
> states that "musicians know what a concord sounds like"
> when he deliberately avoids using a ratio to measure one.
>
> So it seems to me that when he speaks of a "4th" containing
> "two and a half tones", he is again deliberately throwing
> sand in the face of the pythagoreans. IOW, regardless of
> what numerical complexities the pythagoreans weave in order
> to prove that there is no such thing as precisely 1/2-tone,
> Aristoxenos states that musicians deal with quasi-equal
> 1/2-tones as an established fact.

A very natural and reasonable thing to do, from
a performer's point of view!

> > That's a reasonable theory. To it you might add
> > that he found the previous attempts to quantify
> > it unable to produce the sound of a concord that
> > every musician recognised.
>
> To the extent that pythagorean mathematics and the
> "tuning by concords" process produces a wolf-4th at
> the end of the procedure, that's true. But it also
> needs to be pointed out that the 144-edo tempered 4th
> is the familiar 12-edo one of 500 cents, which is a
> largely inaudible discrepancy from the ~498 cents of
> the pythagorean 4/3 ratio 4th.

The difference, 2 cents, is about the agreed
limit of our ability to perceive. However, I think
your previous paragraph confuses the issue. Let's
disentangle the threads:
1. You agreed that the pythagorean maths produces
a wolf-4th, which I was guessing Aristoxenos was
objecting to.
2. You state, as justification for your theory that
Aristoxenos meant 144-EDO, that its 4th is almost
just, as far as our ears can tell.
3. But you also state that Aristoxenos' "tuning
by concords" process produces a wolf-4th!

Points 2. and 3. are not compatible; either the last
4th is a wolf, or it isn't; if it's (almost) just, it isn't
a wolf.

I'm really going to have to read your page on
Aristoxenos, aren't I?

> > However, a bias in favour of perception rather
> > than measurement could also have lead him to
> > ignore the difference in size of the two tones,
> > and to use the term "half a tone" less precisely,
> > to mean "a part of a tone". So that in writing of
> > "two and a half tones" making the diatessaron,
> > perhaps he meant, from the starting note, "one
> > tone higher than that, then another, then part
> > of another"? I'm disinclined to see "two and a
> > half tones" as meaning "2.5 tones", unless we see
> > him doing arithmetic with that number. What do
> > you think?
>
> Aristoxenos was certainly dealing in approximations,
> as Partch so indignantly accused him of doing.

Good on him! ... if that served his musical purposes
well enough. But then how would he be thinking of
an unprecedentedly fine division of the fourth?

> But apparently what's going on here is that Aristoxenos
> wanted to employ concepts derived from the then-new
> geometry expounded by Euclid. He saw Euclidean geometry
> as a way to measure divisions of irrational musical intervals,
> which the pythagorean mathematics before him were unable
> to deal with.

Now that is interesting! And those geometric
methods permit, in principle, of arbitrary accuracy
of measurement - simply draw your diagrams large
enough, and you get the number of decimal places
you want. Of course, there ARE practical limits.

But given that Aristoxenos was impressed by the
mensurational powers of the new geometry, I'm
surprised you say he was dealing in approximations.

> So i'm not convinced that he intended his measurements
> to be as vague as they might seem to us moderns. Having
> thought about this for years, and done a lot of research
> into it, i'm quite convinced that 144-edo is pretty much
> what Aristoxenos had in mind when writing his theories.
>
> (BTW, the character of Aristoxenos's two suriviving books
> seems to indicate that they were not written down by him,
> but that they are notes written down by students attending
> his lectures.)

Ouch! But better than nothing. I wonder if they
were GOOD students?

> > > Note, however, that i don't agree with Partch that
> > > Aristoxenos deserves the credit (dubious credit, according
> > > to Partch) for being "the father of temperament".
> > > I speculated that the Sumerians, with their base-60
> > > mathematics, ...
> >
> > (It occurred to me that this was perhaps a
> > factor in Cleonides' choice - dividing the
> > classically perfect consonance, the diatessaron
> > or fourth 4/3, into the classically perfect
> > number of parts, 60, which the Greeks also
> > inherited and perpetuated in geometry.)
>
> The Sumerians, Babylonians, and Greeks were all quite
> fond of base-60 mathematics.
>
> > > ... did have the ability to derive a very good
> > > approximation of 12-edo (or indeed any temperament)
> > > as early as c.2500 BC -- and it has been satisfactorily
> > > documented that the Greeks got a lot of their cultural
> > > and scientific knowledge from the Babylonians, who got
> > > theirs from the Sumerians.
> >
> > > http://tonalsoft.com/monzo/sumerian/simplified-sumerian-tuning.htm
> >
> > Tho the Greeks got much from the
> > Babylonians, they built on it. Did the
> > Babylonians themselves invent nothing? :-)
> > The inheritors of the Greeks, the Arabs and
> > other savants of early Islam, in their turn
> > added much that he Greeks had not thought
> > of.
>
> Your assessment is pretty accurate. The Babylonians
> did add a few new twists, especially in mathematics,
> but culturally their legacy was basically that of
> adapting the Sumerian innovations to their own society.
>
> The truth is that there have not been many Sumerian
> mathematical tablets deciphered and/or published.
>
> But the Babylonians retained the Sumerian logograms
> in their own words for the "root" concept of the word
> and added their own written symbols to represent the
> phonics of the Akkadian syllables -- very similar to
> the way modern Japanese writing retains the Chinese
> logograms and adds further symbols for the Japanese
> syllables.
>
> So by stripping away the Akkadian syllables, as i have
> done on my webpage, one can see the underlying Sumerian
> basis of all of the Babylonian mathematical texts.
> And the essence of the math problems is already there
> in the Sumerian.

A very nice deduction!

I won't bother you further on this topic until
I've done at least the basic reading - your page.

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Free Edition.
Version: 7.1.371 / Virus Database: 267.14.1/206 - Release Date: 16/12/05

🔗monz <monz@tonalsoft.com>

Hi Yahya,

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> > --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> > <yahya@m...> wrote:

> I trust your good intentions, and I think you have
> an unsurpassed reputation on this list for musically
> relevant accuracy. So I hope you'll pardon me if
> I still feel that I'd only really "have the context of
> anyone's writings" if I could read the darned thing
> myself! However, absent the time to learn and
> master the Greek of his day, the next-best thing
> must always be a good translation by a subject-
> matter expert who has those language skills I lack,
> and I'm guessing that expert would be yourself.
>
> So I will certainly write a proper study of that page
> on my to-do list.

Of course i'm going to tell you that my Aristoxenos
webpage is worth the effort of "a proper study". :)
I just have to apologize again that it *will* take
so much effort, because i really need to do a lot of
serious editing of that page.

Anyway, besides the bits and pieces on my webpage,
there are two full English translations of Aristoxenos
available:

Barker, Andrew. 1989.
_Greek Musical Writings_,
volume 2: 'Harmonic and Acoustic Theory'.
Translated and edited.
Cambridge University Press, New York.
[Contains complete English translation of Aristoxenus
_Elementa harmonica_ in vol 2, p 126-184.]

Macran, Henry Stewart. 1902.
_The Harmonics of Aristonexus_.
Edited with translation, notes, introduction, and index of words.
Clarendon Press, Oxford.
Reprinted 1974, Hildesheim; New York: G. Olms Verlag.
[Contains complete English translation.]

The Barker is the newer, more modern translation, and
has the added benefit that along with Aristoxenos you
also get *all* of the other ancient Greek musical writings.
The drawback is that you won't buy it, because it's $180.
Try a good university library.

The Macran is the "classic" English edition, and also
includes the complete Greek text. This should be quite
easy to find in a library ... i've even seen it in a
few public libraries, and certainly every university
will have it. I was going to say that you might even
find a used copy online, but i see from Amazon that
there's a new reprint of it, so now it goes for $65.

> > To the extent that pythagorean mathematics and the
> > "tuning by concords" process produces a wolf-4th at
> > the end of the procedure, that's true. But it also
> > needs to be pointed out that the 144-edo tempered 4th
> > is the familiar 12-edo one of 500 cents, which is a
> > largely inaudible discrepancy from the ~498 cents of
> > the pythagorean 4/3 ratio 4th.
>
> The difference, 2 cents, is about the agreed
> limit of our ability to perceive. However, I think
> your previous paragraph confuses the issue. Let's
> disentangle the threads:
> 1. You agreed that the pythagorean maths produces
> a wolf-4th, which I was guessing Aristoxenos was
> objecting to.
> 2. You state, as justification for your theory that
> Aristoxenos meant 144-EDO, that its 4th is almost
> just, as far as our ears can tell.
> 3. But you also state that Aristoxenos' "tuning
> by concords" process produces a wolf-4th!
>
> Points 2. and 3. are not compatible; either the last
> 4th is a wolf, or it isn't; if it's (almost) just, it isn't
> a wolf.
>
> I'm really going to have to read your page on
> Aristoxenos, aren't I?

You got point 3 wrong.

*Aristoxenos's* "tuning by concords" produces a series
of 11 4ths and 5ths which are all considered to be concords.
That can only be accomplished with a slight tempering of
each 4th and 5th.

As i stated, if the "tuning by concords" is done with
*pythagorean* 4ths and 5ths (exact 4/3 and 3/2 ratios,
respectively), then the 11th interval in the series is
a wolf ... and it's a wolf-5th, not 4th.

The "Amendment from 2003.07.20" section of my webpage
opens with diagrams which show graphically the "tuning
by concords" process. This link takes you right to it:

http://sonic-arts.org/monzo/aristoxenus/318tet.htm#144edo

The brilliant insight in Aristoxenos's "tuning by concords"
-- if you accept the 144-edo interpretation or something
close to it -- is that by tempering each 4th and 5th by
only 2 cents, they still sound essentially the same as
the pythagorean concords, but do not produce the wolf
at the end, instead producing a final concord just like
all the rest.

As i've said, i hesitate to give Aristoxenos the credit
for this, because i am convinced that the Sumerians
discovered it 2000 years before him, and that it filtered
into Greek culture via Babylon.

> > Aristoxenos was certainly dealing in approximations,
> > as Partch so indignantly accused him of doing.
>
> Good on him! ... if that served his musical purposes
> well enough. But then how would he be thinking of
> an unprecedentedly fine division of the fourth?

A large part of Aristoxenos's program was to refute
the traditional, well-established music-theory of the
pythagoreans, in part because their mathematics were
unable to describe the more subtle divisions of
generic shades which Aristoxenos perceived to be in
use among musicians.

Archytas had been the first music-theorist to write
about the 3 different genera (diatonic, chromatic,
and enharmonic). But Archytas made only the second
note from the top of the tetrachord a moveable one
-- i.e., all 3 of his genera have same pitch at the
top, bottom, and the note above the bottom.

Aristoxenos criticized this and said that the
enharmonic genus has its own small interval at the
bottom of the tetrachord. He gives a nod of recognition
to the harmonists, a school of non-pythagorean
music-theorists who used _katapyknosis_ to divide
larger intervals.

> > But apparently what's going on here is that Aristoxenos
> > wanted to employ concepts derived from the then-new
> > geometry expounded by Euclid. He saw Euclidean geometry
> > as a way to measure divisions of irrational musical intervals,
> > which the pythagorean mathematics before him were unable
> > to deal with.
>
> Now that is interesting! And those geometric
> methods permit, in principle, of arbitrary accuracy
> of measurement - simply draw your diagrams large
> enough, and you get the number of decimal places
> you want. Of course, there ARE practical limits.
>
> But given that Aristoxenos was impressed by the
> mensurational powers of the new geometry, I'm
> surprised you say he was dealing in approximations.

Well, he had to be, if he intended his musical system
to be tempered, because the mathematics of his time
was not able to deal with the irrational intervals
of temperament in any way other than the use of geometry.
Tempered intervals can only be represented exactly in
numbers with the use of base-plus-exponent notation,
which didn't come along until the 1600s (and was refined
only a few years ago by yours truly and Gene Ward Smith
into the "monzo" notation).

AFAIK, the scholar who first pointed out the geometrical
aspect of Aristoxenos's theories is Malcolm Litchfield:

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

> > (BTW, the character of Aristoxenos's two suriviving books
> > seems to indicate that they were not written down by him,
> > but that they are notes written down by students attending
> > his lectures.)
>
> Ouch! But better than nothing. I wonder if they
> were GOOD students?

Apparently so ... the surviving texts are very detailed.
And for the parts that are missing, we have Cleonides to
fill in the gaps.

The existing texts of Aristoxenos appear to be from
two different periods in his career, one early and one
late. They cover much of the same material, but the
second book (the presumed later text) is a good deal
more sophisticated.

I should also point out here that Aristoxenos's theories
carried a great deal of weight in the ancient world:
*every* subsequent Greek and Roman music-theorist whose
work survives speaks of musical material in the Aristoxenean
terms of the Lesser Perfect System, Greater Perfect System,
Perfect Immutable System, and tetrachords with two fixed
bounding notes and two moveable inner notes.

This is true all the way up to Boethius, who lived more
than 800 years after Aristoxenos -- and of course,
Boethius's theories were perpetuated first by the
Carolingian Franks c.800-1100, and then again by the
Renaissance Italians in the 1500s.

> > So by stripping away the Akkadian syllables, as i have
> > done on my webpage, one can see the underlying Sumerian
> > basis of all of the Babylonian mathematical texts.
> > And the essence of the math problems is already there
> > in the Sumerian.
>
> A very nice deduction!

But the link you have is for my webpage about the
*simplified* Sumerian version of 12-edo. The older
webpage i made about this has a 3-sexagesimal-place
approximation, which obviously is more accurate still,
and that page is the one where i have translations of
an actual Babylonian mathematical tablet.

http://tonalsoft.com/monzo/sumerian/sumerian-tuning.htm

I should also mention that someone, quite a while back,
sent me an email about this page explaining that these
math problems were not about tuning, but were about
farming acreage or something like that. I've been really
busy and just filed that away for later perusal, but
thought that it should be mentioned here.

Robson, whose book has the translations of these and
hundreds of other Babylonian math texts, surmises that
they deal with the measurement of bricks. I'm inclined
to agree with that, as the Sumerian and Babylonian
civilizations were entirely based on their ability
to use the clay-mud at their disposal to make bricks,
which they then used to build dams, canals, roads,
temples, and cities. There were no other natural
resources available to them -- neither stone nor wood.

I find it *very* interesting that all of the common
small musical intervals ("commas") are also found in
Babylonian texts concerning weights and measures.
Ratios such as 81/80 (the syntonic-comma) and 128/125
(the diesis) pop up as discrepancies between different
systems of measurement for such things as silver, barley,
or standard brick sizes for different city-states.

-monz
http://tonalsoft.com
Tonescape microtonal music software