why EDO

I can't be the only one who is bothered by the use of the term "octave," esp.
in terms like EDO. We are always putting it in quotes because we are
apologising for its meaning as derived from "eight". Why not just avoid it
and use something better that can stand for the 13th root of 3 just as well
as the 72nd root of 2? Why not n-EDk ? 144-EDO would be 144-ED2 and the
17th root of 5 would be 17-ED5. Or if need be, n-EDk/m? So a 7/5 equally
divided into 7 parts would be 7-ED7/5. Go for maximum flexibilty in the
formulas.
Aaron

Hi Aaron,

--- In tuning@y..., Pitchcolor@a... wrote:

/tuning/topicId_22998.html#22998

> I can't be the only one who is bothered by the use of
> the term "octave," esp. in terms like EDO. We are always
> putting it in quotes because we are apologising for its
> meaning as derived from "eight".

Basically, I agree with you and would go even further.
But there were reasons why EDO was adopted.

As background: EDO is part of a system of classification
of different categories of tunings:

EDO = equally-divided "octave" ex: equal temperaments
UDO = unequally-divided "octave" ex: meantones
EDNO = equally-divided non-"octave" ex: BP; 88-cet
UDNO = unequally-divided non-"octave" ex: stretch tunings

(Note that the MIRACLE temperaments except the full 72-EDO
can be either EDO or EDNO, at the user's discretion.)

The list archive has several posts in the last week of 1999
and the following January with EDO in the subject line.
There was a fair amount of discussion of it then.

I had at that time good reasons for wanting to use
that classification of EDOs, UDOs, EDNOs, and UDNOs
that several of us had hammered out here on the list.
It would be interesting for me to go back and read those
old posts and see if I still feel the same way.

Now, to "go even further":

Back on Sun Dec 26, 1999 1:51 pm, I wrote:

/tuning/topicId_7273.html#7273

> Subject: tET vs. EDO (or was that e?...)
>
> But if we're going to get *that* cryptic, why not just
> use for the name the mathematical notation which provides
> a completely lucid description of what's going on?
> (3/2)^(16/16). I hereby nominate this kind of notation
> to describe any equal-temperament of any interval.

The purely mathematical description may be the most
flexible and descriptive of all.

An interesting argument Brian McLaren made with me
(disclaimer: I'm paraphrasing... not a quote) was that
"mathematics has absolutely nothing to do with music,
except that it can be used to devise interesting tunings".

So if we recognize this as valid, then the one thing
math is good for in music-theory is to describe the
nature of the tuning. So...

> Why not just avoid it and use something better that can
> stand for the 13th root of 3 just as well as the 72nd root
> of 2?

I propose 3^(i/13) and 2^(i/72).

i stands for "integer", so that it defines those certain
particular categories of tunings that you would prefer
I don't call EDOs and EDNOs.

(Or should I be using z? See
<http://www.iwu.edu/~lstout/BasicTerminology/node2.html>.

And I admit with ^() it's much more awkward,
but I still think it's best of all.

> Why not n-EDk ? 144-EDO would be 144-ED2

Or 2^(i/144).

> and the 17th root of 5 would be 17-ED5.

Or 5^(i/17).

Or if need be, n-EDk/m? So a 7/5 equally
> divided into 7 parts would be 7-ED7/5.

Or (7/5)^(i/7).

> Go for maximum flexibilty in the formulas.
> Aaron

Ah, now we come to the real power of my new idea.
How about this?:

The beauty of the format I'm proposing here
is that you can have a whole system of
alphabetical abbreviations to specify what
kind of structure the tuning has, which give
a lot more flexibility and specificity to the
categories we had labeled UDO and UDNO,
especially the latter. That always seemed
to me like such a gigantic catch-all category.
This system breaks it up into several different
subcategories.

So, EDOs are always of the form 2^(i/i). (right?)

And if I'm not mistaken, n is for "natural number"
which is the big superset of number categories.

Would the mathematicians be kind enough to drop on
us a list of standard algebraic abbreviations for
different kinds of numbers? I thought i was
integer, but I'm guessing at all of this. Also
please say whatever needs to be said about the
information on that webpage
<http://www.iwu.edu/~lstout/BasicTerminology/node2.html>.

And EDNOs could have a number of different
formulas to describe them.

For example, 2^(PI/38) creates an interesting
meantone-like tuning in which the generator
is ~99.2081891 cents and the "5th" is
~694.4573234 cents. Under the 1999 classification
this would be an EDNO.

PI is a transcendental number, so I suppose we
could call it t, if indeed we need anything shorter
than PI itself. Under this new classification
it would be a 2^(t/38), or in even more general
terms, a 2^(t/i), or even an i^(t/i). Or perhaps
even more generally, i^(r), an integer to a rational
power.

So isn't it strange how it looks like an
"octave"-based tuning in the new notation,
but is specified as NO in the other one.
(I'm sure that the mathematicians can step
in and explain what's happening here.)

-monz
http://www.monz.org
"All roads lead to n^0"

Hi Aaron,

Just for the record I came up with the EDO acronym as way to try and
distinguish between tempering and the intentionally free use of equal
tunings. So it was not an attempt at any sort of precise generalized
terminology.

It's kind of taken on a bit of a life of its own for whatever reason,
but I more or less stopped using it preferring "equal" and "equal
temperament" to draw the kinds of distinctions I was originally
interested in.

Anyway, I don't mean to put the kibosh on the terminology discussion
because I think it is an interesting problem -- how to find accurate
and flexible terminology that also sounds "cool" (more important than
you might think when it comes to acronyms) -- but I did want to say
what it was that I was thinking at the time.

--Dan Stearns