Aristoxenos and 72et

Claiming 72 for Aristoxenos is a bit controversial, but here's the argument in a nutshell:

(1) Aristoxenos rejects the idea of specifying musical intervals by ratios, and had available the sophisticated Eudoxian definition of ratios. In modern terms, he knew the positive real numbers.

(2) Aristoxenos defines the tone as the difference between a fifth and a fourth, and says it can be divided into 12 parts.

(3) He says the fourth is 2 1/2 tones, which would mean 30 parts.

(4) This gives two equations in two unknowns for the 3-limit mapping if we assume there is one, and that gives 72et up to the 3-limit. And if we assume Aristoxenos is talking in a precise and consistent way, we are stuck with having a mapping and hence stuck with 72.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> Claiming 72 for Aristoxenos is a bit controversial,

You can say that twice and mean it. -C.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> >
> > Claiming 72 for Aristoxenos is a bit controversial,
>
> You can say that twice and mean it. -C.
>

Doesn't it come down to whether he really meant what he said?

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> > > Claiming 72 for Aristoxenos is a bit controversial,
> >
> > You can say that twice and mean it. -C.
>
> Doesn't it come down to whether he really meant what he said?

It comes down to whether he conceived of 72 equal steps
to an octave period, and I'll bake a pie if he did.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> >
> > > > Claiming 72 for Aristoxenos is a bit controversial,
> > >
> > > You can say that twice and mean it. -C.
> >
> > Doesn't it come down to whether he really meant what he said?
>
> It comes down to whether he conceived of 72 equal steps
> to an octave period, and I'll bake a pie if he did.

It comes down to what he would have said if someone had asked him how many of his parts there were in an octave, and I think he would have said 72, on the grounds that he wasn't an idiot and moreover knew how to add. What he conceived of hardly matters if his answer to that question would have been "72".

What's YOUR theory of what he would have said?

I think he would have said he was a Turk, not a Greek.

This is how ridiculous I find the argument at hand.

Oz.

✩ ✩ ✩
www.ozanyarman.com

On Jun 10, 2010, at 4:00 AM, genewardsmith wrote:

>
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>>
>> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@>
>> wrote:
>>>
>>>>> Claiming 72 for Aristoxenos is a bit controversial,
>>>>
>>>> You can say that twice and mean it. -C.
>>>
>>> Doesn't it come down to whether he really meant what he said?
>>
>> It comes down to whether he conceived of 72 equal steps
>> to an octave period, and I'll bake a pie if he did.
>
> It comes down to what he would have said if someone had asked him
> how many of his parts there were in an octave, and I think he would
> have said 72, on the grounds that he wasn't an idiot and moreover
> knew how to add. What he conceived of hardly matters if his answer
> to that question would have been "72".
>
> What's YOUR theory of what he would have said?
>

The only problem with this is that 72-edo is missing
anything close to one of the intervals in one of
the divisions described by Aristoxenus, namely
that for the hemiolic chromatic genus.

In the reference tetrachord, his hemiolic chromatic parhypate
can be approximated neatly by using 144-edo, and in
that case it would be 51 degrees of 144-edo, which is
exactly 425 cents.

See my detailed (and unfortunately, still somewhat sloppy)
exploration of Aristoxenus's tuning theories:

http://tonalsoft.com/monzo/aristoxenus/aristoxenus.aspx

The description of 144-edo is near the bottom, where i
give the illustrations in HEWM staff-notation.

Note also that above that section, i have another
interpretation which suggests that 318-edo might
give a good explanation of Aristoxenus's tunings.

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

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> Claiming 72 for Aristoxenos is a bit controversial, but here's the argument in a nutshell:
>
> (1) Aristoxenos rejects the idea of specifying musical intervals by ratios, and had available the sophisticated Eudoxian definition of ratios. In modern terms, he knew the positive real numbers.
>
> (2) Aristoxenos defines the tone as the difference between a fifth and a fourth, and says it can be divided into 12 parts.
>
> (3) He says the fourth is 2 1/2 tones, which would mean 30 parts.
>
> (4) This gives two equations in two unknowns for the 3-limit mapping if we assume there is one, and that gives 72et up to the 3-limit. And if we assume Aristoxenos is talking in a precise and consistent way, we are stuck with having a mapping and hence stuck with 72.
>

--- In tuning@yahoogroups.com, "monz" <joemonz@...> wrote:
>
> The only problem with this is that 72-edo is missing
> anything close to one of the intervals in one of
> the divisions described by Aristoxenus, namely
> that for the hemiolic chromatic genus.
>
> In the reference tetrachord, his hemiolic chromatic parhypate
> can be approximated neatly by using 144-edo, and in
> that case it would be 51 degrees of 144-edo, which is
> exactly 425 cents.

When last sighted, 144 was a multiple of 72, and hence if he did this it in no way contradicts anything I said. As I pointed out to Carl, the actual bottom line is whether the parts he described were intended to be equal parts. Despite Ozan's belief to the contrary, it would not have required logarithms for Aristoxenos to have conceptualized such a thing, though they certainly would have helped any attempt to use actual computations.