41 ET 11-tone diatonic

Where am I these days?

Well, some of the conversation is just about understandable, but I don't
have the Rotheberg definition in front of me, nor do I have Scala.

Anyway, looking again at C41 11-diatonic.

Scale:
0,4,8,11,15,19,23,26,30,38(,0 etc
44344434443

(all cent values approximate to 0 or 1dp)
Tetrachord is 4,4,3
4 of 41 = 117.1cent
3 of 41 = 87.8cent

Triad:
0,8,15 = 0cent, 234.2cent, 439cent
0,7,15 = 0cent, 204.9cent, 439cent

Now compare C19 11-diatonic:

Scale:
0,2,4,5,7,9,11,12,14,16,18(,0 etc
22122212221

Tetrachord is 2,2,1
2 of 19 = 126cent approx
1 of 19 = 63cent approx

Triad:
0,4,7 = 0cent, 253cent, 442cent
0,3,7 = 0cent,189cent, 442cent

and C30 11-diatonic (in Balzano)

Scale:
0,3,6,8,11,14,17,19,22,25,28(,0
33233323332

Triad:
0,6,11 = 0cent, 240cent, 440cent
0,5,11 = 0cent, 200cent, 440cent

Looking at all three together:

19 30 41
--- --- ---
Major 253 240 234
Minor 189 200 205

scale patterns

19 22122212221
30 33233323332
41 44344434443

The general pattern I get is thus:

aabaaabaaab

where 2b >= a > b
and tentatively:
if b>1 the relation is: 2b > a > b
and
a Mod b > 0 (so we don't get a=6 and b=3 and suchlike)

The chromatic comes out as 8a+3b

so:

a b chrom
-- -- -----
2 1 19
3 2 30
4 3 41
5 3 49
5 4 52
and so forth.

For comparison, here is the 7 diatonic

5a+2b = chromatic

a b chrom
-- -- -----
2 1 12
3 2 19
4 3 26
5 3 31
5 4 33
6 5 40
7 4 43
and so forth

From this, I make the grand assumption that these are batter and better
approximations to the 'ratio-background' structure. p99 in my article:
"I believe that the diatonic scales derived from different Cn scales conform
to background formations existing independently of the number of tones in
the Cn scale."

Therefore, the Cn samples the pitch space, and so the diatonic scales
presented in the little tables above indicate that when the Cn samples the
music space appositely, a diatonic will appear that approximates to this
background formation. Therefore the comment about 'universes' of different
numbers of tones is a misnomer in some way, as really it's the scale itself
that is important. This corresponds to C31 and C19 diatonic scales: we don't
really appreciate having 19 or 31 'major' and minor keys, as most normal
tonal relations are via intervals to be found in Ptolemy's 'intense'
diatonic, so the other intervals approximated in these two ET scales are
unlikely to be used as relations between tone centres. Like modulating an
augmented second upward or diminished fifth downward, for example, where in
the scales stated there are probably no common tones to pivot about.

[Aside to Margo: Harry Partch says the Pythagorean Dorian tone sequence:
1/1 256/243 32/27 4/3 3/2 128/81 16/9 2/1
is 'virtually unsingable' (GOAM, 2d ed, p168). Is this true?]

12TET (my C12) just enharmonizes until everything is much the same:
augmented seconds, minor thirds and double-diminished 4ths are all 3 C12
semitones wide, etc, etc. This is what HP moans about a lot:
overstandardisation to the point where the 'senses' of the tones are made
pretty much 'senseless, and the listener becomes 'desensitised' to them.

Let us assume that the background formation is 7:8:9 for the 11 diatonic
under consideration:

From my own calculations, I get this comma in the 7:8:9 11-diatonic:

537824
------
531441

which I work out to be: 20.66952cent to 5 dp.

so, it's of similar size to the Pythagorean and Didymean commas, which would
indicate that this scale is not unreasonable. Its chromatic semitone is

8/7*8/9 = 64/63 = 27.26, which is close to the above 20.67cent comma, which
does imply that the 11-diatonic as 7:8:9 could have problems. The
4:5:6-diatonic has no such confusibiliy:

25/24 >> 81/80

8:9:10 25-tone diatonic appears to be worse:

the (tentative) comma I get is

1220703125
----------
1207959552

which is 18.168cent, I think.

but the 8:9:10 chromatic semitone is 81/80 = 21.5cent, so these two are very
confusible.

My 'hunch' is that the 'comma' of the given diatonic must be audibly smaller
than the chromatic semitone, otherwise the ear could confuse the
displacement of a comma to make the 'fifths' true as a chromatic alteration,
and so distort the sense of key.

But this is purely subjective. Especially if listeners can distinguish
smaller intervals, then there is room for scales whose 'commas' can be made
out from their 'chromas'.

It was this comma problem (not mentioned in article, but I did make notes on
it for myself when compiling most of the scales sometime in 1993-94), that
prompted me to assume that scales could be formed from 'segments' of the
alternating ratio sequence (like 5/4 and 6/5). A scale consists of one
'segment', but there was no compunction to make the segment long enough to
'run into' the comma, like the 5/4 6/5 grid distonic segment runs into 10/9
at one end and 9/8 at the other. Of course, when I started thinking about
n-dimensional structures, I decided that some other criteria was going to
have to take its place (comma boundaries). The questions I still have are:

When do the scales 'run out'? When does the ear decide that it can no longer
differentiate between 'tonalities' from different Cn scales or higher order
lattices?

Mark Gould

In-Reply-To: <B8E24EED.3AAA%mark.gould@argonet.co.uk>
Mark Gould wrote:

> Well, some of the conversation is just about understandable, but I don't
> have the Rotheberg definition in front of me, nor do I have Scala.

Definitions are all at <http://www.ixpres.com/interval/dict/>. Propriety
and stability are easy enough. Efficiency is hard to calculate, but tends
to be high for an MOS anyway. I don't know of anything that can do the
calculations on a Macintosh. If I ever understand it myself, I'll try
implementing it in Python.

> scale patterns
>
> 19 22122212221
> 30 33233323332
> 41 44344434443
>
> The general pattern I get is thus:
>
> aabaaabaaab

So your scale is really a different tuning of Balzano's 11 from 30?

Graham

--- In tuning-math@y..., Mark Gould <mark.gould@a...> wrote:

> My 'hunch' is that the 'comma' of the given diatonic must be
audibly smaller
> than the chromatic semitone,

now you're speaking my language (i think)! these are what i call the
commatic unison vector and the chromatic unison vector of periodicity
blocks. of course, we've discussed not only two-dimensional but also
three- and four-dimensional periodicity blocks quite a bit -- we've
focused more attention on the ones where there's only one chromatic
unison vector and all the rest are commatic, but there's no 'rule'
that says there shouldn't be more than one chromatic unison vector.

> otherwise the ear could confuse the
> displacement of a comma to make the 'fifths' true as a chromatic
alteration,
> and so distort the sense of key.

what if the music were using a tuning system where the comma were
tempered out, distributed among the consonant intervals making it up?

> It was this comma problem (not mentioned in article, but I did make
notes on
> it for myself when compiling most of the scales sometime in 1993-
94), that
> prompted me to assume that scales could be formed from 'segments'
of the
> alternating ratio sequence (like 5/4 and 6/5). A scale consists of
one
> 'segment', but there was no compunction to make the segment long
enough to
> 'run into' the comma, like the 5/4 6/5 grid distonic segment runs
into 10/9
> at one end and 9/8 at the other. Of course, when I started thinking
about
> n-dimensional structures, I decided that some other criteria was
going to
> have to take its place (comma boundaries). The questions I still
have are:
>
> When do the scales 'run out'? When does the ear decide that it can
no longer
> differentiate between 'tonalities' from different Cn scales or
higher order
> lattices?
>
>
> Mark Gould

i hope to gain a better understanding of what your questions mean in
the coming months. it sounds like we may have independently covered a
lot of the same intellectual ground, but haven't yet found a common
language with which to discuss it.

best,
paul

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:

> of course, we've discussed not only two-dimensional but also
> three- and four-dimensional periodicity blocks quite a bit -- we've
> focused more attention on the ones where there's only one chromatic
> unison vector and all the rest are commatic, but there's no 'rule'
> that says there shouldn't be more than one chromatic unison vector.

How would you work it?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
>
> > of course, we've discussed not only two-dimensional but also
> > three- and four-dimensional periodicity blocks quite a bit --
we've
> > focused more attention on the ones where there's only one
chromatic
> > unison vector and all the rest are commatic, but there's
no 'rule'
> > that says there shouldn't be more than one chromatic unison
vector.
>
> How would you work it?

what do you mean, how would you work it? and btw, didn't you produce
a 9-tone scale that is a perfect example of this, having two
chromatic unison vectors?

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:

> what do you mean, how would you work it? and btw, didn't you produce
> a 9-tone scale that is a perfect example of this, having two
> chromatic unison vectors?

You tell me, starting with the scale and passing on to the unison vectors.

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Kwick Pick? They're carrying advertisements for the
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I'm in favour of moving to the Columbia list too.

Manuel

--- In tuning-math@y..., manuel.op.de.coul@e... wrote:
> Yahoo wrote:
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>
> Kwick Pick? They're carrying advertisements for the
> burglar's guild now?

ha -- i just made the same comment at metatuning!

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
>
> > what do you mean, how would you work it? and btw, didn't you
produce
> > a 9-tone scale that is a perfect example of this, having two
> > chromatic unison vectors?
>
> You tell me, starting with the scale and passing on to the unison
>vectors.

you just gave the same example again, yesterday!!! a 2-dimensional JI
periodicity block, where both of the unison vectors are too large to
be ignored and not tempered out, is a perfect example of this.