in set theory this situation is represented by the prime form:

i like eliminating the 0, which seems redundant to me if we're always referencing the first tone at noon; therefore, i call the intervals represented by any dyad simply as:

1, 2, 3, 4, 5, or 6

while in set theory land, or "what is the closest distance on the clock between the two tones"

and no, i don't talk like this in front of my students - well, most of them anyway

Dog, that's a good idea in eliminating zero to identify the six basic intervals, but I will emphasize that with sets, we keep the zero, because we are identifying **distances**, not note names. Am I correct?

For me, this was an important distinction, between using number as identity, and number as quantity.

It's *not* like the Sudoku game, which has nothing to do with math at all. There, the numbers are just place-holders or identifiers; names. They could have made them apples and oranges.

Whereas with a set, we are defining intervals, or *quantities* of pitch-distance. This is number as quantity.

So when we look at a set such as 02479, we should remember that we are looking at a series of relationships or distances, not "number-names."

Earlier, when you said "...so, yes it's been changed in the form of rearranging the "set" as in a collection of objects - in set theory the order doesn't matter as opposed to a tone row where order is everything," I can see how this might be misunderstood to mean "discrete objects" with "number names." Aren't they really a series of distances between things? Correct me if I err.

i believe that it is important to have clear and concise definitions of terms. the elimination of the "0" may lead to ambiguity if the observer forgets that he's

*standing on the zero* thus he cannot see it. then during a simple act of transposition, say up a minor third that "4" becomes a "7" and if we forget that there was a "0" involved, two notes (0 and 4) become one (7).

i just observed in the listing of the prime forms that every one of them started with a zero and it seemed redundant so i took 'em out in order to see what they looked like and here is what i discovered:

by using 1) the prime form itself, 2) the inverted prime form, 3) the complement of the prime form, and 4) the inverted complement of the prime form, all of the sets may be described with a maximum of five digits and by adding leading zeros, they all look like zip codes.

02479 represents not only the pentatonic scale but also the complementary major scale

00073 represents not only the minor triad but also its inversion, the major triad as well the complementary sets of two distinct nine tone scales, and

the blues scale is the complement of 13568...

but, what a minute! the reader is going to say, "why do we need to do all this? isn't music theory perfectly well described by the tools already given us???"

and he'd be right. a major scale is 1 2 3 4 5 6 7 and a dorian scale is 2 3 4 5 6 7 1 [2] or 1 2 b3 4 5 6 b7

and a major triad is 1 3 5 and minor is 1 b3 5, etc...

so why all the fuss? in an attempt to describe

*all* possible sets the classical language of using the major scale to describe tones and intervals is just too cumbersome. there are just too many possibilities [something like 4096 total tonal combinations reduced to 2048 by establishing a reference tone (root) reduced to 352 scale/arpeggio types reduced to 200+ prime forms through inversion...]

argh!

i don't really want to think of 352 of anything let alone 4096, although i was surprised that forte was able to reduce the problem to just over two hundred...

but it's still too much. i wanted to see all the possibilites in as simple terms as possible and what i did was to use:

122 prime forms

inverted forms of the ones that turned out to be unique

complementary forms (again, the ones that turned out to be unique)

inverted complementary forms of the ones that turned out to be unique

in order to get

*all* of them

forte's list already used inversions and at least listed them according to complementary sets of all except the six-tone groups and this is what i had to go back and figure out: the sixes.

so who are these studies designed for? those who have spent enough time with dyads and triads, seventh and ninth chords, major and minor scales (natural, harmonic and melodic too), diminished and whole-tone and says:

"i want more"

have i mastered them? nope. not by a long shot. these studies were written for me, out of curiosity, and fun, and because i could; and for anyone else who wants to see. i would not presume to replace the rich and diverse history of music practice, theory and instruction - all i can do is add my 2 cents worth (or 352, or 122, or however you want to count them).

but you are correct, millions, in that we must state whether we are writing of

*locations* or

*distances* when specifying a prime form or any other of the related forms, because there

*is* danger of misunderstanding, confusion, and the occasional food fight!

ps - this reminds me of a conversation i had with a wonderful GIT instructor who convinced me that a b3 is not equivalent to a minor third because a b3 is a

*location* and a minor third is a

*distance...*i hope that this

*sets* well with y'all