super superparticulars

Hi!

Using J Gill's formatting test, I did a graphic
repesentation of my view of superparticulars. The point is
the subdivision of intervals, so just look at the triangles,
don't look down the tree.

---------------------2/1
---------------------/--\
--------------------/----\
-------------------/------\
------------------/--------\
-----------------/----------\
----------------/------------\
---------------/--------------\
--------------/----------------\
-------------/------------------\
------------/--------------------\
-----------3/2------------------4/3
----------/---\----------------/---\
---------/-----\--------------/-----\
--------/-------\------------/-------\
-------/---------\----------/---------\
-----5/4--------6/5--------7/6--------8/7
-----/--\-------/--\------/--\-------/--\
---9/8-10\9--11/1012\1113/1214\13-15/1416\15

or

---------------------5/3
---------------------/--\
--------------------/----\
-------------------/------\
------------------/--------\
-----------------/----------\
----------------/------------\
---------------/--------------\
--------------/----------------\
-------------/------------------\
------------/--------------------\
-----------4/3------------------5/4
----------/---\----------------/---\
which subdivide as above.

Application?

Take e.g. a pythagorean sixth and divide into epimorics:
17/16, 18/17, 19/18, ..., 26/25, 27/26. 11 steps. You don't
want that many? Just combine some adjacent ones. Some stay
as as high in the harmonic series as they were, some reduce:
to superparticulars again. The lowest pair of the 11 steps
give you 9/8, the next triples 7/6, 8/7, and another 9/8.
You get (now referring to 1/1, where superparticulars cease
to be that distinguished) 9/8 21/16 3/2 27/16, a pentatonic
that I like much better than the usual 3-limit one.

klaus

Klaus,

Your post is interesting to me, but (being quite inexperienced with
these things) I wondered if you could explain a bit more about your
intermediate steps in generating your results (see questions below).

Thanks, J Gill

--- In tuning@y..., klaus schmirler <KSchmir@z...> wrote:
> Hi!
>
> Using J Gill's formatting test, I did a graphic
> repesentation of my view of superparticulars. The point is
> the subdivision of intervals, so just look at the triangles,
> don't look down the tree.
>
> ---------------------2/1
> ---------------------/--\
> --------------------/----\
> -------------------/------\
> ------------------/--------\
> -----------------/----------\
> ----------------/------------\
> ---------------/--------------\
> --------------/----------------\
> -------------/------------------\
> ------------/--------------------\
> -----------3/2------------------4/3
> ----------/---\----------------/---\
> ---------/-----\--------------/-----\
> --------/-------\------------/-------\
> -------/---------\----------/---------\
> -----5/4--------6/5--------7/6--------8/7
> -----/--\-------/--\------/--\-------/--\
> ---9/8-10\9--11/1012\1113/1214\13-15/1416\15
>
> or
>
> ---------------------5/3
> ---------------------/--\
> --------------------/----\
> -------------------/------\
> ------------------/--------\
> -----------------/----------\
> ----------------/------------\
> ---------------/--------------\
> --------------/----------------\
> -------------/------------------\
> ------------/--------------------\
> -----------4/3------------------5/4
> ----------/---\----------------/---\
> which subdivide as above.
>
> Application?
>
> Take e.g. a pythagorean sixth and divide into epimorics:

JG: So, 27/16 is "divided into epimorics" by what process to achieve
the ratios below?

> 17/16, 18/17, 19/18, ..., 26/25, 27/26. 11 steps. You don't
> want that many? Just combine some adjacent ones. Some stay
> as as high in the harmonic series as they were, some reduce:
> to superparticulars again.

JG: What exactly do you mean by,
>"Some stay
> as as high in the harmonic series as they were, some reduce:
> to superparticulars again."?

The lowest pair of the 11 steps
> give you 9/8, the next triples 7/6, 8/7, and another 9/8.

JG: By what operation?

> You get (now referring to 1/1, where superparticulars cease
> to be that distinguished) 9/8 21/16 3/2 27/16, a pentatonic
> that I like much better than the usual 3-limit one.
>
> klaus

unidala schrieb:
>
> Klaus,
>
> Your post is interesting to me, but (being quite inexperienced with
> these things) I wondered if you could explain a bit more about your
> intermediate steps in generating your results (see questions below).

And I thought I was describing a plain procedure in
jargonless language... Besides, I think it is easy stuff. It
used to be referred to as division of the octave, of the
fifth, etc., which of course was originally done on a
monochord.

Numerically, you have to expand the octave ration to divide
it: 2/1 -> 4/2. Then you fill the gap in the latter term
with superparticular ratios: 3/2 and 4/3; dividing 3/2 (6/4)
gets you the thirds 5/4 and 6/5.

> JG: So, 27/16 is "divided into epimorics" by what process to achieve
> the ratios below?

In general terms (now straining hard to be taken serious),
superparticular ratios divide a ratio n/d into n-d
harmonically related parts (d+1)/d .... n/(n-1). (and I sure
hope I didn't botch this up anywhere.)

>
> > 17/16, 18/17, 19/18, ..., 26/25, 27/26. 11 steps. You don't
> > want that many? Just combine some adjacent ones. Some stay
> > as as high in the harmonic series as they were, some reduce:
> > to superparticulars again.
>
> JG: What exactly do you mean by,
> >"Some stay
> > as as high in the harmonic series as they were, some reduce:
> > to superparticulars again."?

17/16*18/17=9/8; reduced.
18/17*19/18=19/17; this stays complex.

> The lowest pair of the 11 steps
> > give you 9/8, the next triples 7/6, 8/7, and another 9/8.
>
> JG: By what operation?

17/16*18/17=9/8
19/18*20/19*21/20=7/6
22/21*23/22*24/23=8/7
25/24*26/25*27/26=9/8
And you are left with a 32/27 to complete the octave.

For me and my misguided imagination, this must be the way my
ears
work. The successive superparticulars relate everything to a
common tone, whereas e.g. playing a pythagorean third (which
I much prefer to call ditonus for this reason) involves
redefining the tonic.

Being of Paul Erlich's "benign cult", I do have some facts
in history on my side (the Greeks' [non-Pythagorean]
involvement with string divisions and the very term
"harmonic division", the 15c invasion of
the 5/4 third and the undisputed definition of the major
triad as 4:5:6 as compared to the hassles about minor), but
psychologists and, of course, scale designers (by definition
occupied with limiting the number of usable tones), have
looked the other way.

Something like a year ago, I think, someone here complained
about the bias of harmony and the neglect of melody...

klaus

--- In tuning@y..., klaus schmirler <KSchmir@z...> wrote:

> Being of Paul Erlich's "benign cult", I do have some facts
> in history on my side (the Greeks' [non-Pythagorean]
> involvement with string divisions

At least as much with arithmetic as with "harmonic" ones; and anyhow
this, it seems to me, was an essentially _cultural_ phenomenon.

> and the very term
> "harmonic division",

When did this term first become associated with the type of division
opposite from the arithmetic divisions?

> the 15c invasion of
> the 5/4 third and the undisputed definition of the major
> triad as 4:5:6 as compared to the hassles about minor),

Hmm . . . well yes, harmonic series chords are special, and if
they're "full", they'll be built of superparticular intervals (the
chord 1:2:3:....:n is built of the intervals 1:2, 2:3, 3:4, . . . n-
1:n).

> but
> psychologists and, of course, scale designers (by definition
> occupied with limiting the number of usable tones), have
> looked the other way.

On this latter point, no, not at all, I feel.

> Something like a year ago, I think, someone here complained
> about the bias of harmony and the neglect of melody...

In cultures where harmony (i.e., the simultaneous sounding of tones)
is not found, one finds virtually no propensity for melodies to mimic
the harmonic series -- except for those rare cases (Tuva) where the
melodies are produced _by_ harmonics.

paulerlich schrieb:
>
> --- In tuning@y..., klaus schmirler <KSchmir@z...> wrote:
>
> > Being of Paul Erlich's "benign cult", I do have some facts
> > in history on my side (the Greeks' [non-Pythagorean]
> > involvement with string divisions
>
> At least as much with arithmetic as with "harmonic" ones;

But the resulting intervals are the same, just their order
is reversed.

and anyhow
> this, it seems to me, was an essentially _cultural_ phenomenon.

As I said elsewhere, introspection tells me that my ears
divide harmonically. And whether this is cultural or not
seems to be question not interesting to psychophysicists.

>
> > and the very term
> > "harmonic division",
>
> When did this term first become associated with the type of division
> opposite from the arithmetic divisions?

Very good question. All the books I have at hand speak
uncritically about harmonic, arithmetic, and geometric
means, including the one specifically concerned with
antiquity which presumably would use Greek letters here if
the terms were original.

BTW and on topic, under "proportion" there was a remark that
"proportional series" is usually translated "geometrical
series", and there is an epimoric proportion with a
reference to Archytas. Archytas proved that there is no
"rational middle proportion" for epimoric ratios; his
special concern here was the division of 9/8. As he was
aware of a geometric solution to this, it might be that at
least the terms arithemtic and geometric mean were in use
then.

>
> > the 15c invasion of
> > the 5/4 third and the undisputed definition of the major
> > triad as 4:5:6 as compared to the hassles about minor),
>
> Hmm . . . well yes, harmonic series chords are special, and if
> they're "full", they'll be built of superparticular intervals (the
> chord 1:2:3:....:n is built of the intervals 1:2, 2:3, 3:4, . . . n-
> 1:n).
>
> > but
> > psychologists and, of course, scale designers (by definition
> > occupied with limiting the number of usable tones), have
> > looked the other way.
>
> On this latter point, no, not at all, I feel.

How?

>
> > Something like a year ago, I think, someone here complained
> > about the bias of harmony and the neglect of melody...
>
> In cultures where harmony (i.e., the simultaneous sounding of tones)
> is not found, one finds virtually no propensity for melodies to mimic
> the harmonic series -- except for those rare cases (Tuva) where the
> melodies are produced _by_ harmonics.

... or where instruments using harmonics are around.
Admittedly I can only think musics that use harmony now, but
I don't think harmony came _before_ the tonal system.

and now good night

klaus

--- In tuning@y..., klaus schmirler <KSchmir@z...> wrote:
>
>
> paulerlich schrieb:
> >
> > --- In tuning@y..., klaus schmirler <KSchmir@z...> wrote:
> >
> > > Being of Paul Erlich's "benign cult", I do have some facts
> > > in history on my side (the Greeks' [non-Pythagorean]
> > > involvement with string divisions
> >
> > At least as much with arithmetic as with "harmonic" ones;
>
> But the resulting intervals are the same, just their order
> is reversed.

But the resulting sets of three or more notes, whether viewed as
harmonies or as melodies, are different.
>
> and anyhow
> > this, it seems to me, was an essentially _cultural_ phenomenon.
>
> As I said elsewhere, introspection tells me that my ears
> divide harmonically. And whether this is cultural or not
> seems to be question not interesting to psychophysicists.

Hmm . . . the psychophysical phenomenon of virtual pitch is
inextricably linked to the proportions in the harmonic series.
Parncutt founded a whole theory of harmony upon this, although he
rounded everything to 12-tET from the outset. My harmonic entropy
theory is, in part, an attempt to remedy that.

> > > and the very term
> > > "harmonic division",
> >
> > When did this term first become associated with the type of
division
> > opposite from the arithmetic divisions?
>
> Very good question. All the books I have at hand speak
> uncritically about harmonic, arithmetic, and geometric
> means, including the one specifically concerned with
> antiquity which presumably would use Greek letters here if
> the terms were original.

I suspect maybe the terms were not original.

> BTW and on topic, under "proportion" there was a remark that
> "proportional series" is usually translated "geometrical
> series", and there is an epimoric proportion with a
> reference to Archytas. Archytas proved that there is no
> "rational middle proportion" for epimoric ratios; his
> special concern here was the division of 9/8. As he was
> aware of a geometric solution to this, it might be that at
> least the terms arithemtic and geometric mean were in use
> then.

Yes, I believe they were.

> > > the 15c invasion of
> > > the 5/4 third and the undisputed definition of the major
> > > triad as 4:5:6 as compared to the hassles about minor),
> >
> > Hmm . . . well yes, harmonic series chords are special, and if
> > they're "full", they'll be built of superparticular intervals (the
> > chord 1:2:3:....:n is built of the intervals 1:2, 2:3, 3:4, . . .
n-
> > 1:n).
> >
> > > but
> > > psychologists and, of course, scale designers (by definition
> > > occupied with limiting the number of usable tones), have
> > > looked the other way.
> >
> > On this latter point, no, not at all, I feel.
>
> How?

For example, as a scale designer, I try to put as many harmonic
series chords into my scales as possible.

> Admittedly I can only think musics that use harmony now,

What do you mean by "think musics"?

> but
> I don't think harmony came _before_ the tonal system.

If you mean the common practice major/minor tonal system, I think
you'll find that it most certainly did -- and, as Jeremy pointed out,
the modes had very different emotional flavors in the era of 3-limit
harmony, and then also in the pre-tonal 5-limit era.

paulerlich schrieb:
>
> --- In tuning@y..., klaus schmirler <KSchmir@z...> wrote:
> >
> >
> > paulerlich schrieb:
> > >
> > > --- In tuning@y..., klaus schmirler <KSchmir@z...> wrote:
> > >
> > > > Being of Paul Erlich's "benign cult", I do have some facts
> > > > in history on my side (the Greeks' [non-Pythagorean]
> > > > involvement with string divisions
> > >
> > > At least as much with arithmetic as with "harmonic" ones;
> >
> > But the resulting intervals are the same, just their order
> > is reversed.
>
> But the resulting sets of three or more notes, whether viewed as
> harmonies or as melodies, are different.

For me, arithmetic division is more difficult than harmonic,
but I feel that I know what I am doing when I heard the
intervals themselves first (after dividing harmonically).
It's still superparticulars. With even, geometric division,
I can't do better than say, "it should be somewhere around
there". (If some people, like Johnny Reinhardt, have been
able to learn that, I would certainly agree that it is a
cultural phenomenon.)

> >
> > and anyhow
> > > this, it seems to me, was an essentially _cultural_ phenomenon.
> >
> > As I said elsewhere, introspection tells me that my ears
> > divide harmonically. And whether this is cultural or not
> > seems to be question not interesting to psychophysicists.
>
> Hmm . . . the psychophysical phenomenon of virtual pitch is
> inextricably linked to the proportions in the harmonic series.
> Parncutt founded a whole theory of harmony upon this, although he
> rounded everything to 12-tET from the outset. My harmonic entropy
> theory is, in part, an attempt to remedy that.

But I wasn't talking about pitch perception, but about
picking tones from the continuum. My view of
superparticulars and their specialness is that they offer
subdivisions of larger intervals that sound logical without
recourse to tetrachords and without too many equal step
sizes.

>
> > > > and the very term
> > > > "harmonic division",
> > >
> > > When did this term first become associated with the type of
> division
> > > opposite from the arithmetic divisions?
> >
> > Very good question. All the books I have at hand speak
> > uncritically about harmonic, arithmetic, and geometric
> > means, including the one specifically concerned with
> > antiquity which presumably would use Greek letters here if
> > the terms were original.
>
> I suspect maybe the terms were not original.
>
> > BTW and on topic, under "proportion" there was a remark that
> > "proportional series" is usually translated "geometrical
> > series", and there is an epimoric proportion with a
> > reference to Archytas. Archytas proved that there is no
> > "rational middle proportion" for epimoric ratios; his
> > special concern here was the division of 9/8. As he was
> > aware of a geometric solution to this, it might be that at
> > least the terms arithemtic and geometric mean were in use
> > then.
>
> Yes, I believe they were.
>
> > > > the 15c invasion of
> > > > the 5/4 third and the undisputed definition of the major
> > > > triad as 4:5:6 as compared to the hassles about minor),
> > >
> > > Hmm . . . well yes, harmonic series chords are special, and if
> > > they're "full", they'll be built of superparticular intervals (the
> > > chord 1:2:3:....:n is built of the intervals 1:2, 2:3, 3:4, . . .
> n-
> > > 1:n).
> > >
> > > > but
> > > > psychologists and, of course, scale designers (by definition
> > > > occupied with limiting the number of usable tones), have
> > > > looked the other way.
> > >
> > > On this latter point, no, not at all, I feel.
> >
> > How?
>
> For example, as a scale designer, I try to put as many harmonic
> series chords into my scales as possible.

... to be used in as many ways as possible. Not the aspect I
was talking about (unique rootedness in a melodic
succession).

>
> > Admittedly I can only think musics that use harmony now,
>
> What do you mean by "think musics"?

think of musics
>
> > but
> > I don't think harmony came _before_ the tonal system.

intervallic system

I hope not too many letters and words got lost between my
brain and fingers this time. (hope yes, believe no)

good night, really

klaus

--- In tuning@y..., klaus schmirler <KSchmir@z...> wrote:

> > But the resulting sets of three or more notes, whether viewed as
> > harmonies or as melodies, are different.
>
> For me, arithmetic division is more difficult than harmonic,

Even in pure melody? If so, I would be tempted to say that you're
constantly listening to everything in terms of the harmonic series,
and that you've taught yourself to do so due to constant proximity
with an "overtone machine" of some sort or another. Listen to those
of other cultures as they sing. That's all I can ask of you.

> With even, geometric division,
> I can't do better than say, "it should be somewhere around
> there".

Right -- but it _is_ interesting that if you _try_ to melodically
bisect the interval dozens of times, the statistical average of your
results will be _right at_ the geometric mean.

> > > and anyhow
> > > > this, it seems to me, was an essentially _cultural_
phenomenon.
> > >
> > > As I said elsewhere, introspection tells me that my ears
> > > divide harmonically. And whether this is cultural or not
> > > seems to be question not interesting to psychophysicists.
> >
> > Hmm . . . the psychophysical phenomenon of virtual pitch is
> > inextricably linked to the proportions in the harmonic series.
> > Parncutt founded a whole theory of harmony upon this, although he
> > rounded everything to 12-tET from the outset. My harmonic entropy
> > theory is, in part, an attempt to remedy that.
>
> But I wasn't talking about pitch perception, but about
> picking tones from the continuum.

The mechanism of pitch perception will affect _the musical meaning_
of note combinations, by describing how much or how little the heard
combination corresponds to the sensation of a _single pitch_.

> My view of
> superparticulars and their specialness is that they offer
> subdivisions of larger intervals that sound logical without
> recourse to tetrachords and without too many equal step
> sizes.

That's perfectly fine, and "benign" as I say, but I think one can go
further, both in describing other options, and in considering the
potential desire of a composer to have heirarchical harmonic
organization in their music, and/or one or more degrees of complete
modulational freedom.

> ... to be used in as many ways as possible. Not the aspect I
> was talking about (unique rootedness in a melodic
> succession).

I think the weight of the evidence from world cultures is that
straight pentatonic would be the most likely candidate for a melodic
succession to root anything in . . . (?)

paulerlich schrieb:
>
> --- In tuning@y..., klaus schmirler <KSchmir@z...> wrote:
>
> > > But the resulting sets of three or more notes, whether viewed as
> > > harmonies or as melodies, are different.
> >
> > For me, arithmetic division is more difficult than harmonic,
>
> Even in pure melody? If so, I would be tempted to say that you're
> constantly listening to everything in terms of the harmonic series,
> and that you've taught yourself to do so due to constant proximity
> with an "overtone machine" of some sort or another.

Trombone.
I'm actully thinking of quite artificial procedures too.
How can you get a 12-tET tone? Find the harmonic mean
between 5/4 and 9/8 of a tone--melodically. That's the 19/16
of your reference. Now play the 4/3. Between them you have
200.53198c :O). So far I didn't make any music with this.

Listen to those
> of other cultures as they sing. That's all I can ask of you.

They might not think about superparticulars. I never said
this was the only method to get a collection of tones
(should I say "notes" here?).

>
> > With even, geometric division,
> > I can't do better than say, "it should be somewhere around
> > there".
>
> Right -- but it _is_ interesting that if you _try_ to melodically
> bisect the interval dozens of times, the statistical average of your
> results will be _right at_ the geometric mean.
>
> > > > and anyhow
> > > > > this, it seems to me, was an essentially _cultural_
> phenomenon.
> > > >
> > > > As I said elsewhere, introspection tells me that my ears
> > > > divide harmonically. And whether this is cultural or not
> > > > seems to be question not interesting to psychophysicists.
> > >
> > > Hmm . . . the psychophysical phenomenon of virtual pitch is
> > > inextricably linked to the proportions in the harmonic series.
> > > Parncutt founded a whole theory of harmony upon this, although he
> > > rounded everything to 12-tET from the outset. My harmonic entropy
> > > theory is, in part, an attempt to remedy that.
> >
> > But I wasn't talking about pitch perception, but about
> > picking tones from the continuum.
>
> The mechanism of pitch perception will affect _the musical meaning_
> of note combinations, by describing how much or how little the heard
> combination corresponds to the sensation of a _single pitch_.
>
> > My view of
> > superparticulars and their specialness is that they offer
> > subdivisions of larger intervals that sound logical without
> > recourse to tetrachords and without too many equal step
> > sizes.
>
> That's perfectly fine, and "benign" as I say, but I think one can go
> further, both in describing other options, and in considering the
> potential desire of a composer to have heirarchical harmonic
> organization in their music, and/or one or more degrees of complete
> modulational freedom.

If you have an open pitch set and are not afraid of comma
shifts (somehow this seems to go together with me) nothing
keeps you from modulating.

Division by superparticulars also lets you do other things
like dividing a frame interval into different numbers of
steps (e.g. for reasons of text metrics). (Reach a 4/3 in
two steps: 7/6, 8/6. In three steps: 10/9, 11/10, 12/11. In
four steps: 31/12, 14/13, 15/14, 16/15.) My hunch is that
these intervals can be played by intuition, unlike a
culturally acquired collection of pitches generated by
cycling some interval through the octaves or dividing into
equal steps.

>
> > ... to be used in as many ways as possible. Not the aspect I
> > was talking about (unique rootedness in a melodic
> > succession).
>
> I think the weight of the evidence from world cultures is that
> straight pentatonic would be the most likely candidate for a melodic
> succession to root anything in . . . (?)

> From: klaus schmirler <KSchmir@z.zgs.de>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, December 12, 2001 6:08 PM
> Subject: Re: [tuning] Re: super superparticulars
>
>
> > > [Klaus]
> > > and the very term
> > > "harmonic division",
> >
> > [Paul Erlich]
> > When did this term first become associated with the type
> > of division opposite from the arithmetic divisions?
>
> [Klaus]
> Very good question. All the books I have at hand speak
> uncritically about harmonic, arithmetic, and geometric
> means, including the one specifically concerned with
> antiquity which presumably would use Greek letters here if
> the terms were original.
>
> BTW and on topic, under "proportion" there was a remark that
> "proportional series" is usually translated "geometrical
> series", and there is an epimoric proportion with a
> reference to Archytas. Archytas proved that there is no
> "rational middle proportion" for epimoric ratios; his
> special concern here was the division of 9/8. As he was
> aware of a geometric solution to this, it might be that at
> least the terms arithemtic and geometric mean were in use
> then.

AFAIK, the ancient Greeks were entirely conversant with
the three different types of mean: arithmetic, geometric,
and harmonic. Archytas seems to be the first music-theorist
to make use of them in anything written down.

But naturally I would guess that the Sumerians knew about
all three. No, I don't have any hard evidence... but I'm
always looking. :)

At any rate, Bablyonian mathematics of c. 1600 BC have a
level of sophistication that is not seen again until Renaissance
Europe (c. 1600 AD), and all the evidence I've seen strongly
suggests that the Sumerians, as early as 3000 BC or thereabouts,
were aware of the geometrical sublteties described on Babylonian
tablets.

Archytas's geometrical methods appear chronologically right after
Euclid, but the Babylonians knew it all at least 1000 years earlier.

So the question regarding the origin of the method of division
by harmonic mean is indeed an intriguing one, but I'm quite
certain that it was known before the apex of Greek culture.

-monz

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--- In tuning@y..., "monz" <joemonz@y...> wrote:

> At any rate, Bablyonian mathematics of c. 1600 BC have a
> level of sophistication that is not seen again until Renaissance
> Europe (c. 1600 AD), and all the evidence I've seen strongly
> suggests that the Sumerians, as early as 3000 BC or thereabouts,
> were aware of the geometrical sublteties described on Babylonian
> tablets.

So far as I know, they had not gotten to theorems, definitions, and
proofs, so I would hardly say this about them. Did they deal with the
theory of plane curves, regular solids, ruler and compass
constructions, the definition of irrational numbers, or the
beginnings of calculus? Could they give an upper and lower bound for
pi?

> Archytas's geometrical methods appear chronologically right after
> Euclid, but the Babylonians knew it all at least 1000 years earlier.

So far as I've ever heard, no one before the Greeks had a clue about
proofs.

--- In tuning@y..., "monz" <joemonz@y...> wrote:

> AFAIK, the ancient Greeks were entirely conversant with
> the three different types of mean: arithmetic, geometric,
> and harmonic.

Right -- but we were wondering about the etymology of the
term "harmonic", and also whether they had used this term.

> At any rate, Bablyonian mathematics of c. 1600 BC have a
> level of sophistication that is not seen again until Renaissance
> Europe (c. 1600 AD),

Are you serious? They were beyond Euclid, Archimedes, Pappus . . . ?
I'd love to hear more -- tuning-math might be a good place. I'm
always struck how advanced the later books of Euclid were . . .

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > At any rate, Bablyonian mathematics of c. 1600 BC have a
> > level of sophistication that is not seen again until Renaissance
> > Europe (c. 1600 AD), and all the evidence I've seen strongly
> > suggests that the Sumerians, as early as 3000 BC or thereabouts,
> > were aware of the geometrical sublteties described on Babylonian
> > tablets.
>
> So far as I know, they had not gotten to theorems, definitions, and
> proofs, so I would hardly say this about them. Did they deal with
the
> theory of plane curves, regular solids, ruler and compass
> constructions, the definition of irrational numbers, or the
> beginnings of calculus? Could they give an upper and lower bound
for
> pi?

Yeah, I was kind of wondering . . .

Hi Gene,

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, December 13, 2001 8:47 PM
> Subject: [tuning] Re: super superparticulars
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > At any rate, Bablyonian mathematics of c. 1600 BC have a
> > level of sophistication that is not seen again until Renaissance
> > Europe (c. 1600 AD), and all the evidence I've seen strongly
> > suggests that the Sumerians, as early as 3000 BC or thereabouts,
> > were aware of the geometrical sublteties described on Babylonian
> > tablets.
>
> So far as I know, they had not gotten to theorems, definitions, and
> proofs, so I would hardly say this about them. Did they deal with the
> theory of plane curves, regular solids, ruler and compass
> constructions, the definition of irrational numbers, or the
> beginnings of calculus? Could they give an upper and lower bound for
> pi?
>
> > Archytas's geometrical methods appear chronologically right after
> > Euclid, but the Babylonians knew it all at least 1000 years earlier.
>
> So far as I've ever heard, no one before the Greeks had a clue about
> proofs.

You're absolutely right to ask these questions and make these points.

The Sumerian and Babylonian math is purely practical, consisting of
methods for problem-solving. The Greeks's original contribution to it
was in adding the theoretical underpinning.

So I should have been more specific in what I posted earlier.
But this still doesn't detract from the level of sophistication of
the Babylonian math if viewed only from the practical perspective.

-monz

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--- In tuning@y..., "monz" <joemonz@y...> wrote:

> You're absolutely right to ask these questions and make these
points.
>
> The Sumerian and Babylonian math is purely practical, consisting of
> methods for problem-solving. The Greeks's original contribution to
it
> was in adding the theoretical underpinning.
>
> So I should have been more specific in what I posted earlier.
> But this still doesn't detract from the level of sophistication of
> the Babylonian math if viewed only from the practical perspective.

You said it was more sophisticated than any until the Renaissance.
_From a practical perspective_, you don't think Archimedes was more
sophisticated that the Babylonians?

monz schrieb:

> Archytas's geometrical methods appear chronologically right after
> Euclid, but the Babylonians knew it all at least 1000 years earlier.
>
> So the question regarding the origin of the method of division
> by harmonic mean is indeed an intriguing one, but I'm quite
> certain that it was known before the apex of Greek culture.

My "Lexikon der alten Welt" suggested that Archytas was just
putting more or less common Pythagorean lore to whatever
they used to write on in those days.

klaus

In message: /tuning/topicId_31250.html#31452

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., klaus schmirler <KSchmir@z...> wrote:

> >KS: With even, geometric division,
> > I can't do better than say, "it should be somewhere around
> > there".
>
>PE: Right -- but it _is_ interesting that if you _try_ to
>melodically
> bisect the interval dozens of times, the statistical average of
>your
> results will be _right at_ the geometric mean.

J Gill: Paul, by "melodic bisection", do you intend to mean "division
of an interval into two other intervals (where the *product* of those
two resulting intervals is equal to the original interval)"? When you
say "dozens of times", are you referring to an "iterative" process,
or to a "non-iterated" process (where the interval would be, in the
above defined manner, subdivided into a number of separate pairs of
intervals from one single original interval?

Curiously, J Gill

--- In tuning@y..., "unidala" <JGill99@i...> wrote:

> J Gill: Paul, by "melodic bisection", do you intend to
mean "division
> of an interval into two other intervals (where the *product* of
those
> two resulting intervals is equal to the original interval)"?

Well, the subject is not expected to understand "product", etc., so
the goal is simply to make the two intervals sound like they're _the
same size_.

> When you
> say "dozens of times", are you referring to an "iterative" process,
> or to a "non-iterated" process (where the interval would be, in the
> above defined manner, subdivided into a number of separate pairs of
> intervals from one single original interval?

Non-iterated.

/tuning/topicId_21894.html#22059
> message 22059
> From: paul@s...
> Date: Thu May 3, 2001 5:06 pm
> Subject: Re: DAVE KEENAN'S MIRACLE SCALE
>
> > [me, monz]
> > There's no other way I know of to represent the harmonic
> > relationships of *all* of them [the primes] uniquely as
> > subsets of one huge all-inclusive set, other than my
> > lattice formula.

(Paul later pointed out that there are infinitely many
variations that could be made in my formula, so here I'll
simply acknowledge knowledge *that*. But *this* below is
what I want to comment on...)

> First of all, that rules out all non-JI systems. Non-JI systems
> (which I believe have been more important than JI systems, except for
> Pythagorean) are not uniquely represented by your lattice, because
> the fundamental theorem of arithmetic only applies when the exponents
> of the primes are integers.

Hmmm... this is interesting. I don't fully understand it, because
ratios with exponents made of integer fractions (as in fraction-
of-a-comma meantones) seem to me to be as incommensurable as
those with plain integers. Please explain more fully.

-monz

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--- In tuning@y..., "monz" <joemonz@y...> wrote:
> /tuning/topicId_21894.html#22059
> > message 22059
> > From: paul@s...
> > Date: Thu May 3, 2001 5:06 pm

Wow, we're really doing the time-warp now!

> > First of all, that rules out all non-JI systems. Non-JI systems
> > (which I believe have been more important than JI systems, except
for
> > Pythagorean) are not uniquely represented by your lattice,
because
> > the fundamental theorem of arithmetic only applies when the
exponents
> > of the primes are integers.
>
>
> Hmmm... this is interesting. I don't fully understand it, because
> ratios with exponents made of integer fractions (as in fraction-
> of-a-comma meantones) seem to me to be as incommensurable as
> those with plain integers. Please explain more fully.

It's just that you use (correctly) the fudamental theorem of
arithmetic as a justification for latticing JI with prime axes. But
once the exponents can be fractions rather than simply integers, the
analogy to the fundamental theorem of arithmetic fails.