Reply to Graham Breed

>I'm still working with 3+4 scales, to which I will now attach the cute
name
>"neutronic" after their neutral thirds. I'll also call "my" scale the
>"symmetrical" version and "Paul's" scale (which he says is also Arabic)
I
>will term "tetrachordal".

>I have Manuel's scale list, but these are all dry numbers with nothing
of
>how the scales are used in real music.

Too bad, eh?

>The relevant scales are presumably
>those in 12-equal with two quarter-tone additions. There aren't any
>symmetrical neutronic scales here as you need three notes from the
>alternative spiral of fifths.

Don't know what you mean. Look at Manuel's list (have you really found
it? Where?) under 24-tone. You'll see 3 4 3 4 3 4 3 there, as well as
some modes of "my" Arabic scale.

>>Well, I'm not a big fan of maximal evenness, as my paper makes clear.
If
>>I could rewrite the contest, I'd eliminate that, but maybe allow for a
>>structure that spans a 5:4 and occurs 3 times in every octave span (as
>>in the ME 22-out-of-41).

>Ah yes, if in doubt, change the rules! The version of your paper I've
got
>(the original HTML one) doesn't make clear any dislike of maximal
evenness
>that I can see.

Uh-uh. From the HTML version of my paper:

"The author hears the standard pentachordal modes as most stable and
most likely to define key centers and modulatory practice. The tonic
chords of the alternate pentachordal modes may simply serve as points of
intermediate harmonic stability within the standard pentachordal mode.
The symmetrical modes have a weird, bitonal quality due to their
symmetry at the half-octave"

>I don't think a melodic rule should presuppose both a 3/2 _and_ a 5/4.

Neither do I! I meant allowing for a 5/4 structure _instead_ of maximal
evenness, which started out as an _alternative_ to a 3/2 structure.

> Don't know what you mean. Look at Manuel's list (have you really found
> it? Where?) under 24-tone. You'll see 3 4 3 4 3 4 3 there, as well as
> some modes of "my" Arabic scale.

There are two different lists, I think this is giving the confusion. The
list Paul refers to I have called "list of modes" (scales, modes, the
distinction is artificial). This list also disappeared with the Mills
ftp-site. I'll send it to Graham. If anyone else wants it, I can send it
as an HTML attachment (60 Kb). I don't want to post it to the list,
because of the line breaking by the list server. And it's a bit long,
870 lines.
Interpret the dry numbers at your own risk. Though I think it's a nice
source if you use the names as an index for searching more information.

Manuel Op de Coul coul@ezh.nl

>I'm still working with 3+4 scales, to which I will now attach the cute
name
>"neutronic" after their neutral thirds. I'll also call "my" scale the
>"symmetrical" version and "Paul's" scale (which he says is also Arabic)
I
>will term "tetrachordal".

The term "symmetrical" in music theory refers to translational symmetry
at some interval other than the octave. Traditionally (in 12-tET), the
diminished, augmented, and whole-tone scales are the most common
symmetrical scales, as they contain 7+/- 1 notes. Blackwood used the
term in describing his 10-note scale in 15-tET, which repeats five times
per octave. You'd be better off using another name for your scale than
"symmetrical", as it is not symmetrical in the traditional
music-theoretic sense, and the inversional symmetry that it does have,
it shares with the "tetrachordal" scale.

>>Don't know what you mean. Look at Manuel's list (have you really found
>>it? Where?) under 24-tone. You'll see 3 4 3 4 3 4 3 there, as well as
>>some modes of "my" Arabic scale.

>I downloaded the list a few months ago, presumably from the FTP site.
It
>may not be the very latest one, but it might be. I was looking under
>"arabic" so I will try "24-tone" tonight.

If this is the list I'm thinking of, it's organized by number of tones
per octave (or tritave). I think the heading for all these arabic scales
is "24 tones".

Graham Breed wrote,

>By 'eck, you're right! I added the 11-limit stuff nearly two weeks ago,
>but obviously forgot to upload it. I have done so now, so let's try
>again.

>http://x31eq.com/lattice.htm#11limit

Nice work! I note a few corrections:

>[In a temperament where 121:120 vanishes,] we can stick the 11 midway
between the 8 and >the 15. That gives an 11-limit diamond looking like this:

> 5
> / \
> / \
> / 11
> / 7 \
> / \
>1-----------3-----------9

Of course, that's not the diamond, it's just the otonality on 1/1. The
diamond would be:

10/9---------5/3---------5/4
/ \ / \ / \
/ \ / \ 10/7 / \
/ 11/9----+-----&-----+---11/8
/ 14/9---------7/6---------7/4/ \
/ \ / \ / \ /***/ / \
16/9---------4/3------+--1/1--/-+----3/2---------9/8
\ / * / \ / / / \ /
\ 8/7--\---/-/12/7-/---\--9/7 /
16/11---+-\-/-@-----+---18/11 /
\ / 7/5 \ / \ /
\ / \ / \ /
8/5---------6/5---------9/5

* 14/11 goes here but there's no room to write it
*** 11/7 goes here but there's no room to write it or any of the lines it's
connected to
@ 12/11 and 11/10 both go here since they're equivalent
& 11/6 and 20/11 both go here since they're equivalent

>The other approximation is to set the neutral third of 11/9 to be exactly
half a fifth of >3/2. This gives the comma (-1 5 0 0 -2)H or 243/242. The
11-limit diamond isn't so compact >as above.

> 5
> / \
> / \
> / \
> / 7 \
> / \
>1-----------3-----------9-----11

Again, that's an otonality. The diamond would be:

10/11-10/9---------5/3---------5/4
/ \ / \ / \ / \
/ \ / \ / \ 10/7 / \
/ X \ / \ / \ / \
/ 14/11-\14/9--------7/6---------7/4 \
/ / \ \ / \ / \ / \ / \
16/11-16/9--12/11--4/3-18/11\--1/1--/11/9--3/2--11/6---9/8--11/8
\ / \ / \ / \ / \ \ / /
\ 8/7--\---/-12/7--/---\-9/7-\-/11/7 /
\ / \ / \ / \ X /
\ / 7/5 \ / \ / \ /
\ / \ / \ / \ /
8/5---------6/5---------9/5--20/11

>A lot of scales don't work with either approximation. The most notable is
72, because it's >very good in the 11-limit. So, it's an accurate
representation of just intonation where >all commas are finite.

Actually, 72 works with the second approximation. 243:242 vanishes in 72.