Yahoo Tuning Groups Ultimate Backup tuning a question about limit of pitch perception , FW: [tuning] Re: Aristoxenos and 144-EDO (was: Arabian comma dispute)

Dear yahya

As I'm searching for jnd of pitch perception and you wrote about 2 cent
, and as it is mentioned in theoretical texts about jnd of 5 cent , may
you guide me with your refrence about jnd of 2 cent?

Thanks very much

Shaahin Mohaajeri

Tombak Player & Researcher , Composer

www.geocities.com/acousticsoftombak

My tombak musics : www.rhythmweb.com/gdg

My articles in ''Harmonytalk'':

www.harmonytalk.com/archives/000296.html
<http://www.harmonytalk.com/archives/000296.html>

www.harmonytalk.com/archives/000288.html
<http://www.harmonytalk.com/archives/000288.html>

My article in DrumDojo:

www.drumdojo.com/world/persia/tonbak_acoustics.htm
<http://www.drumdojo.com/world/persia/tonbak_acoustics.htm>

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Yahya Abdal-Aziz
Sent: Monday, December 19, 2005 4:29 AM
To: tuning@yahoogroups.com
Subject: [tuning] Re: Aristoxenos and 144-EDO (was: Arabian comma
dispute)

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

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If I may jump in, Mohajeri, I'm familiar with the psychoacoustic
literature, and 5 cents typically refers to the *melodic* or
*horizontal* jnd. The *harmonic* or *vertical* jnd cite is often much
smaller, depending on the particular experimental conditions. I
suspect the latter was at least of some relevant to Yahya and Monz
since the idea of *concord* usually involves *simultaneous*, not
successive, sounding of the notes.

--- In tuning@yahoogroups.com, "Mohajeri Shahin" <shahinm@k...> wrote:
>
> Dear yahya
>
>
>
> As I'm searching for jnd of pitch perception and you wrote about 2
cent
> , and as it is mentioned in theoretical texts about jnd of 5 cent ,
may
> you guide me with your refrence about jnd of 2 cent?
>
>
>
> Thanks very much
>
>
>
> Shaahin Mohaajeri
>
>
>
> Tombak Player & Researcher , Composer
>
> www.geocities.com/acousticsoftombak
>
> My tombak musics : www.rhythmweb.com/gdg
>
> My articles in ''Harmonytalk'':
>
> www.harmonytalk.com/archives/000296.html
> <http://www.harmonytalk.com/archives/000296.html>
>
> www.harmonytalk.com/archives/000288.html
> <http://www.harmonytalk.com/archives/000288.html>
>
> My article in DrumDojo:
>
> www.drumdojo.com/world/persia/tonbak_acoustics.htm
> <http://www.drumdojo.com/world/persia/tonbak_acoustics.htm>
>
>
>
>
>
> ________________________________
>
> From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On
Behalf
> Of Yahya Abdal-Aziz
> Sent: Monday, December 19, 2005 4:29 AM
> To: tuning@yahoogroups.com
> Subject: [tuning] Re: Aristoxenos and 144-EDO (was: Arabian comma
> dispute)
>
>
>
>
> 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
>
>
> --
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>
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t=ms&k=Music+production+education&w1=Music
>
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De
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gree+education+music+online&w5=Music+business+education&w6=Music+educa
ti
> on+online&c=6&s=174&.sig=xvNPUlceIGAdVcjAsHH8JA>
>
> Music education degree
> <http://groups.yahoo.com/gads?
t=ms&k=Music+education+degree&w1=Music+edu
>
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ee
>
+education+music+online&w5=Music+business+education&w6=Music+education
+o
> nline&c=6&s=174&.sig=lrDYh_-yrdu524mpql-csg>
>
> Degree education music online
> <http://groups.yahoo.com/gads?
t=ms&k=Degree+education+music+online&w1=Mu
>
sic+education&w2=Music+production+education&w3=Music+education+degree&
w4
>
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uc
> ation+online&c=6&s=174&.sig=SVm8lC0-Q2tryy6Hv14ihQ>
>
> Music business education
> <http://groups.yahoo.com/gads?
t=ms&k=Music+business+education&w1=Music+e
>
ducation&w2=Music+production+education&w3=Music+education+degree&w4=De
gr
>
ee+education+music+online&w5=Music+business+education&w6=Music+educati
on
> +online&c=6&s=174&.sig=pCxzd_uR0dRaYO3MB1Fpvw>
>
> Music education online
> <http://groups.yahoo.com/gads?
t=ms&k=Music+education+online&w1=Music+edu
>
cation&w2=Music+production+education&w3=Music+education+degree&w4=Degr
ee
>
+education+music+online&w5=Music+business+education&w6=Music+education
+o
> nline&c=6&s=174&.sig=DaqBb8P8ErSmB8MxxA-ePw>
>
>
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a question about limit of pitch perception , FW: [tuning] Re: Aristoxenos and 144-EDO (was: Arabian comma dispute)

🔗Mohajeri Shahin <shahinm@kayson-ir.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗monz <monz@tonalsoft.com>