Re: Digest Number 23

Dan Stearns wrote:

>Graham Breed wrote:

>>The set of scales in my 3+4 family are roughly that described as "w-h=2".

>The set (or sequence) of w-h=2 in 3 + 4 @ +t +s +s +t +s +t +s would be:

Hang on, I'm referring to your previous post on 18th Jan (digest 22) which
appears to be about "5w + 2h" scales. Try and get me to understand that
first. Am I right in thinking that these scales are built up of 5 large and
2 small steps? That would be my definition of a "5+2" scale.

In that older post of your, you give a list of numbers under the heading "w
- h =2". Those numbers are the series 3+7n where n is an integer >=0. The
equally tempered 3t+4s scales I'm considering at the moment happen to be
those where t-s is one step. Those scales have 3+7n steps to the octave.
So, they're the same scales you previously defined as "w - h =2" in a
different context.

The 3+4 scales, then, can be written as "2w + 2h"? So, in this context, "w
- h =2" is the set 6+7n. That is 6, 13, 20, 27, 34, 41, etc. These are the
examples you give, so am I right about the definitions?

>>Is this like what I'd call septimally double-positive? That is, you start
>>with a 5+2 scale, and these scales are the ones where the larger interval
is
>>2 steps bigger than the smaller one.

>No. (Unless it also happens to be.)

What's that supposed to mean? I'm talking there about the scales originally
defined as "w - h =2".

>>what's this stuff about exteriors and interiors and perimeters?

>You could use n"F" sets of n"f" to explicitly define an exterior, interior,
>and perimeter sets of equidistant divisions of the octave for the 3 + 4 @
+t
>+s +s +t +s +t +s heptad where n"F" = 7, n"f" = 3, and columns one through
>seven depict w - h = 1 through w - h = 7:

So, "F" is an octave and "f" is a fifth? What do "exterior", "interior" and
"perimeter" mean? I think I remember seeing them somewhere, but do you have
a reference? Are you trying to construct this pattern of notes from a
spiral of some kinds of fifths, then? A spiral of neutral thirds makes much
more sense, but that's by the by.

> 3 6 2 5 1 4 7
>10 13 9 12 8 11 14
>17 20 16 19 15 18 21

In this table, each row gets bigger by 7 as you go down. The numbers in the
top row get bigger by 3 as you go along, and are modulo 7, or near enough
(should 7 (mod 7) be zero?) so this presumably has something to do with
octaves and fifths. But what does it mean?

>Exterior set @ 2, 5, 1, 4, 7, 8, 11, and 14:

> Es Es Es Es Es
> Es Es Es

This means nothing to me.

>>The simplest examples of scales fitting this pattern are 7-equal, 10-equal
>(t=2 steps, s=1 step) and 17-equal (t=3, s=2). Also, 31-equal fits: t=5,
>s=4.

>Ostensibly, an interior set would delimit the simplest n-tET examples of
>the particular heptads characteristic "f" and "F" properties**

Is this the definition of "interior set" then? In the case of my 3t+4s
scales, t is equivalent to a major tone: it's the difference between a
fourth and fifth. You get s as the difference between t and a neutral
third. For the double-Pythagorean scale, you get t/s = log(3/2)/2log(9/8)-1
= 0.721. Then, s/(t-s)-0.721/(1-0/721)=2.59. So, s should be about 2.6
times the size of t-s. If t-s is 2 steps, s will ideally be 5 steps and
there will be 41 steps to the octave. That's why the simplest scales are
those where t-s is 1 step. They happen to be the ones with a lot fewer than
41 notes with good fifths and neutral thirds. 34-equal is the first to
break the pattern I think. This is all by the by, though.

>Mapped to nS sets of nt, every n-tET>ntnS/nSnt (i.e., 21tET) yields a
heptad
>were h or (using your definition) s>0 of the perimeter set, and s>-n of the
>exterior set:

Right, so s and h are equivalent in our different systems. Heptad means a
set of seven notes. The perimeter set, then, is where s is larger than a
unison? And the perimeter set means that s can be a fall in pitch where t
is a rise?

>w-h=1
>------------------------------------------------
>Es
>Is
>Is
>24
>31
>38
<snip>

Okay, we now have the series 24+7n up to 1200. I don't need these all
written out: I've got a spreadsheet. Presumably, w-h=m will be the pattern
3m+7n for 3+4 scales? The difference is that you leave out the first few.

Please try to think down to my level. If we can get speaking the same
language, maybe we'll be able to communicate something!

Graham