Aristoxenos, 144-EDO and electronic energy levels of the atom

monz, thank you for your clarifications on Aristoxenos et al. This reminds
me of calculations by me and my father showing how electronic
energy levels produce the syntonic comma among other things. Here is how:

Delta E is the difference between two electronic energy levels of the
hydrogen atom. This is proportional to

Delta E (hi level minus low level) ~ (1 / low^2)-(1 / hi^2)

If hi is 2 and low is 1, the interval is 1- 1/4 = 3/4
If hi is 3 and low is 2, the interval is 1/4- 1/9 = 5/36
If hi is 3 and low is 1, the interval is 1- 1/9 = 8/9
If hi is 4 and low is 1, the interval is 1- 1/16= 15/16
...
If hi is 9 and low is 1, the interval is 1- 1/81 = 80/81

Interesting, no?
Oz.

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 19 Aral�k 2005 Pazartesi 10:35
Subject: [tuning] Re: Aristoxenos and 144-EDO (was: Arabian comma dispute)

> 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
>
>
>
>