Third-based MOS (was Re: periodic table/"lousy thirds")

David C Keenan TD 231.2

> >For just octaves and major thirds, x should be tuned to an
> enharmonic diesis
> >of 41.1 cents (2^7/5^3) and y should be 304.2 cents
> (5^7/2^16). Hence the
> >ideal ratio y/x is 7.4. Setting y=7 and x=1 gives 28-equal,
> and y=15, x=2
> >gives 59-equal.
> > Sure enough, both have very good major thirds.
>
> That's great! One _minor_ nit, (pun intended), according to
> Scala 2^7/5^3
> (41.1c) is called a minor diesis, and according to
> http://home.earthlink.net/~kgann/Octave.html
> an enharmonic diesis is 525/512 = 3*5^2*7/2^9 (43.4c).

I know I've seen the term "enharmonic diesis" used for different intervals.
The one I stick with is what I saw first, but I gave the numbers to avoid
ambiguity.

> Can't find a name for the 5^7/2^16 (304.2c) minor third. It
> needs one. The
> "double-diesis-flat third"? "Breed's minor third"?

It is, in fact, a major third minus two of "those" dieses. Calling it a
minor third is consistent with one of "those" dieses being a half-sharp.

I was thinking about the general cases over the weekend. You can work out
an MOS approximation for any specific interval. I've got a spreadsheet that
does the calculations, and intend to produce a Java applet sometime.
Anyway, it should be obvious that a scale with major thirds can be
represented by three intervals of a major third and a diesis to make up the
octave. I call that a 3+1 scale: 3 large intervals and 1 small. Split the
large interval into a smaller one plus the same diesis, and you get a 3+4
scale. Split it again, and you get 3+7. Or again, to get 3+10: stop
whenever you have enough notes.

So, major-third MOS's are of the pattern 3+1 -> 3+4 -> 3+7 -> 3+10

I've previously identified scales approximating the 11/9 neutral third as
3+1 -> 3+4 -> 7+3 -> 10+7

For minor thirds slightly exceeding an octave, an octave is still 3 large
intervals and one small interval. The next stage is 4 reduced minor thirds
and 3 dieses, and so on.

So, minor third scales are 3+1 -> 4+3 -> 4+7 -> 4+11 or somesuch.

[hmm. The major-third scales as 3+7 are 7 "dieses" and 3 "minor thirds".
Make those real minor thirds, and you get 3+1, the start of this cascade.]

These third-based MOS's can be represented on something like Wilson's scale
tree:

3+1
/ \
/ \
/ \
4+3 3+4
/ \ / \
4+7 7+4 7+3 3+7

4+7 is minor thirds, 3+7 is major thirds, 7+3 is neutral thirds and 7+4 are
neutral-ish thirds that may or may not double to give a fifth or approximate
11/9. Each scale can be assigned a "representative" EDI/ET/whatever:

7
/ \
/ \
/ \
11 10
/ \ / \
15 18 17 13

Why do this? Because it shows the essential properties while making the
diagram look simpler. These trees are so interesting that I'll digress to
explain why. You can make them Pascal's triangle-type structures:

4 3
7
11 10
15 18 17 13

Each number is the sum of numbers immediately to the left and right in
preceeding rows. It differs from Pascal's triangle in that you can look
more than one row back. I think this has all been mentioned on the list
before, but it can stand repeating. Anyway, I call this diagram a web
rather than a tree.

The web is bounded by 4 and 3, so a "third" could be anything between 1/4
and 1/3 octaves. The same web can be made to show the number of steps in
the generating interval for the "typical" ET:

1 1
2
3 3
4 5 5 4

You can generate "good" ETs approximating a given interval by forming an
equivalence interval from the "ideal" interval and a diesis, then finding
the best matches to the ideal ratio. Choosing each integer as the number of
steps in the small interval, and calculating the large interval, gives the
best approximations. This misses a lot of scales however, so taking the
large interval and calculating the small is how these lists were produced:

3/2: 2 5 7 10 12 14 17 19 22 24 27 29 31 34 36 39 41 43 46 48 51 43
5/4: 3 6 9 12 16 19 22 25 28 31 34 37 40 43 47 50 53
6/5: 4 8 11 15 19 23 27 30 34 38 42 46 49 53

By this method, the lowest EDIs with good representations of all 5-limit
intervals are 12, 19, 34 and 53. 65 is the next.

> So the corresponding intervals relating to fifths are 3^2/2^3
> (203.9c) the
> major whole tone (or Pythagorean whole tone), and 2^8/3^5 (90.2c) the
> Pythagorean semitone (or Pythagorean limma). The ideal t/s ratio is
> therefore 2.26 and thus we have 53-tET with excellent fifths
> where t = 9 and s = 4.

For Pythagorean scales, you'd start with 2+1 for fourths and tones. The
tree is:

2+1
/ \
/ \
/ \
/ \
/ \
/ \
3+2 2+3
/ \ / \
/ \ / \
/ \ / \
3+5 5+3 5+2 2+5
/ \ / \ / \ / \
3+8 8+3 8+5 5+8 5+7 7+5 7+2 2+7

The 5+2 branch is the one with semitones and limmas. 5+7 is schismic and
7+5 meantone scales. The web is:

3 2
5
8 7
11 13 12 9
14 19 21 18 17 19 16 11

And the "diatonic" bit:

5 7
12
17 19
22 29 31 26
27 39 46 41 43 50 45 33

So, where am I getting with all this? I don't know, I'll try and get a web
page done sometime. However, the top web is for a "fourth" between 1/3 and
1/2 octaves, and the lower web for a more realistic fourth-like fourth
between 1/7 and 1/5 octaves. The number of steps in that fourth being, for
the diatonic bit:

2 3
5
7 8
9 12 13 11
11 16 19 17 18 21 19 14

Fourths and thirds can also be united by a super-web:

1 2
3
4 5

or

0+1 1+0
1+1
1+2 2+1

Any MOS will be on this web somewhere!

> I find it interesting that the diatonic scale falls out of
> this formula for
> ET's (EDO's) with reasonable fifths, N = 5T + 2s where T >= 0
> and T/9 <= s <= 6T/7.

Er, hang on, where do these numbers come from?

> This suggests the question of whether the analogous thing for
> major thirds
> might be of interest. i.e. "that scale having the maximum
> notes, with two
> different step sizes arranged maximally evenly, which can be
> constructed in
> every equal division of the octave that has reasonable major thirds".

I think this kind of thing is very interesting. It lets us use two-interval
scales without falling into "chronic 3-ism". As I said on the list a while
back, I've been looking at 3+4 scales: where 3 large intervals and 4 small
make up an octave. If the large interval is much larger, major thirds come
out and the fifths aren't so good. I don't think these scales would work
melodically as d is so much smaller than T for good major thirds.