exotic generalized diatonics of 20-tet

Hi Robert V. (et al),

You might interested in this related 9-tone 11:13:15, 1/(15:13:11),
[7,2] index that maps remarkably well to 20-tet as well,

11 16 20 5 9 14 18 3 7

Here's the 1/5 comma meantone rotations where the tonic chord is the
optimal tetrachord position. Note that the four rotations that have
the triads that chain together to make the scale, VI-I-V-IX, are also
the four that are skewed to the right.

0 165 289 414 538 662 827 951 1076 1200
0 124 249 373 497 662 786 911 1035 1200
0 124 249 373 538 662 786 911 1076 1200
0 124 249 414 538 662 786 951 1076 1200
0 124 289 414 538 662 827 951 1076 1200
0 165 289 414 538 703 827 951 1076 1200
0 124 249 373 538 662 786 911 1035 1200
0 124 249 414 538 662 786 911 1076 1200
0 124 289 414 538 662 786 951 1076 1200

Note that 20-tet is consistent all the way through the,

11:13:15:17:19:21

The 11-tone 1/6 comma meantone 13:16:19, 1/(19:16:13), [2,9] index is
for all intents and purposes 20-tet,

9 15 20 6 11 17 2 8 13 19 4

Here's the meantone rotations whit the tonic is in the optimal
tetrachordal position. (Again note that the five triads that chain
together to make up the tonic scale, VI-I-VII-II-VIII, are also the
five that are skewed to the right.)

0 120 240 359 479 540 660 780 900 1019 1139 1200
0 120 240 359 420 540 660 780 900 1019 1080 1200
0 120 240 300 420 540 660 780 900 960 1080 1200
0 120 181 300 420 540 660 780 841 960 1080 1200
0 61 181 300 420 540 660 721 841 960 1080 1200
0 120 240 359 479 599 660 780 900 1019 1139 1200
0 120 240 359 479 540 660 780 900 1019 1080 1200
0 120 240 359 420 540 660 780 900 960 1080 1200
0 120 240 300 420 540 660 780 841 960 1080 1200
0 120 181 300 420 540 660 721 841 960 1080 1200
0 61 181 300 420 540 601 721 841 960 1080 1200

Note that 20-tet is consistent through the,

13:16:19:22:25:28:31:34:37

Does anyone happen to know if Balzano or any of the "group theoretic
description of 12-fold and microtonal pitch systems" type work touched
on any of these fascinating rational, meantone, tempering and ordering
schemes?

--Dan Stearns

I wrote,

<<Note that the four rotations that have the triads that chain
together to make the scale, VI-I-V-IX, are also the four that are
skewed to the right. [SNIP] (Again note that the five triads that
chain together to make up the tonic scale, VI-I-VII-II-VIII, are also
the five that are skewed to the right.)>>

Both of those should've read "skewed to the left".

--Dan Stearns

I wrote,

The 11-tone 1/6 comma meantone 13:16:19, 1/(19:16:13), [2,9] index is
for all intents and purposes 20-tet,

9 15 20 6 11 17 2 8 13 19 4

This would be true, though somewhat less true, of an 11-tone 1/6 comma
meantone 17:21:25, 1/(25:21:17), [2,9] index as well.

Here's the rotations where tonic scale is VI-I-VII-II-VIII

0 122 244 366 488 539 661 783 905 1027 1149 1200
0 122 244 366 417 539 661 783 905 1027 1078 1200
0 122 244 295 417 539 661 783 905 956 1078 1200
0 122 173 295 417 539 661 783 834 956 1078 1200
0 51 173 295 417 539 661 712 834 956 1078 1200
0 122 244 366 488 610 661 783 905 1027 1149 1200
0 122 244 366 488 539 661 783 905 1027 1078 1200
0 122 244 366 417 539 661 783 905 956 1078 1200
0 122 244 295 417 539 661 783 834 956 1078 1200
0 122 173 295 417 539 661 712 834 956 1078 1200
0 51 173 295 417 539 590 712 834 956 1078 1200

20-tet is consistent through the,

17:21:25:29:33:37:41:45

--Dan Stearns

> From: "D.Stearns" <STEARNS@CAPECOD.NET>
> Subject: exotic generalized diatonics of 20-tet
>
> Hi Robert V. (et al),
>
> You might interested in this related 9-tone 11:13:15, 1/(15:13:11),
> [7,2] index that maps remarkably well to 20-tet as well,
>
> 11 16 20 5 9 14 18 3 7

Its amazing how much "hidden thought" there is in a paragraph like
this, but I think I've fully decoded it. Others might describe this
as the 9/20 generator, right?

>
> Here's the 1/5 comma meantone rotations where the tonic chord is the
> optimal tetrachord position.

I'm completely lost with this statement, where is the 1/5 comma
taken from and what is an optimal tetrachord position? (Is that
another reason to buy Chalmers book?)

> Note that the four rotations that have
> the triads that chain together to make the scale, VI-I-V-IX, are also
> the four that are skewed to the left.
^^^^<-(corrected)
>
> 0 165 289 414 538 662 827 951 1076 1200
> 0 124 249 373 497 662 786 911 1035 1200
> 0 124 249 373 538 662 786 911 1076 1200
> 0 124 249 414 538 662 786 951 1076 1200
> 0 124 289 414 538 662 827 951 1076 1200
> 0 165 289 414 538 703 827 951 1076 1200
> 0 124 249 373 538 662 786 911 1035 1200
> 0 124 249 414 538 662 786 911 1076 1200
> 0 124 289 414 538 662 786 951 1076 1200
>

I think I understand your comment about skewed
to the left

VI BaaaBaaaa
I BaaaaBaaa
V aBaaaBaaa
IX aBaaaaBaa

and playing withe the rooted utonal scales, well..,

IV aaBaaaBaa
VIII aaBaaaaBa
III aaaBaaaBa
VII aaaBaaaaB

there doesn't seem to be any similar arrangement.

What do you find to be the significance of
this observation?

For what its worth, my scale program finds a
local minimum at the value B=165 for the pattern

BaaaaBaaa

and understands it as

'1/1 '11/10 '13/11 '14/11 '15/11 '19/13 '8/5 '19/11 '13/7
'1/1 '14/13 '15/13 '5/4 '4/3 '19/13 '11/7 '22/13 '20/11
'1/1 '14/13 '15/13 '5/4 '15/11 '19/13 '11/7 '22/13 '13/7
'1/1 '14/13 '15/13 '14/11 '15/11 '19/13 '11/7 '19/11 '13/7
'1/1 '14/13 '13/11 '14/11 '15/11 '19/13 '8/5 '19/11 '13/7
'1/1 '11/10 '13/11 '14/11 '15/11 '3/2 '8/5 '19/11 '13/7
'1/1 '14/13 '15/13 '5/4 '15/11 '19/13 '11/7 '22/13 '20/11
'1/1 '14/13 '15/13 '14/11 '15/11 '19/13 '11/7 '22/13 '13/7
'1/1 '14/13 '13/11 '14/11 '15/11 '19/13 '11/7 '19/11 '13/7

I see that modes I V VI and IX have the otonal triads in
root position, and modes III IV VII and VIII have the
utonal triads in root position. (Paul, jump in and straighten
me out for good if I messed this up, I understand otonal
relations to be "fixed denominator" and utonal to be
"fixed numerator").

I also see that the full stack could be expressed by stacking
the modes

IV VI VIII I III V VII IX (II)

which seems to match your starting point if rearranged
for symmetry

7 11 16 20 5 9 14 18 3

Regarding expressing THIS scale (the 165c "L") in 20,
11c error is probably good enough for my ears at this
stage in my development.

Some more notes about how it appeared in my scale program.
It was pretty far down the list of scales when rated
according to minimal complexity and maximum accuracy
(a psudoe-HE measure), though it did appear as a local
"best". It also appears as a local best when rated
according to maximal complexity and maximal accuracy,
a strange fish indeed (either the scale or my program...)

Just to show more about a difference of approach, the
first thing that jumps out at me in this pitch set is the
presence of a '3/2. The fact that its also the
characteristic interval says something about a useage
model. Unfortunately, I'm so pedestrian, I would probably
resolve into the 3/2 and not notice the 11:13:15. On
the other hand, the '3/2 can nicely embrace the 11:13:15
for a 22:26:30:33 so maybe the two worlds come together...

> Note that 20-tet is consistent all the way through the,
>
> 11:13:15:17:19:21
>

Is this discounting 5 7 and 9? I know there was a URL
for inclusive consistency, is there one for the
concept thats come up a few times, something like
set consistency?

Cool stuff, I'll try to grok more of it (ears and mind) over
the weekend.

thanks,

Bob Valentine

Hi Robert V.,

<<Others might describe this as the 9/20 generator, right>>

Right, or its compliment, 11/20. When I say a "[7,2] index" I mean
that as well, in as much as any two-term index has to be converted
into a adjacent fractions to give the series of generators.

<<I'm completely lost with this statement, where is the 1/5 comma
taken from>>

Just think of the 5-limit lattice where 11:13:15 and the 1/(15:13:11)
are substituted for 4:5:6 and 1/(6:5:4).

So instead of the (octave reduced) difference between 3^4/2^4 and 5/4
being the comma, you'd have the (octave reduced) difference between
15^5/11^5 and 13/11 being the comma. And as the 13/11 is larger than
the 759375/644204 you'd add 1/5th of the 8374652/8353125 to the 15/11
to get the analogous "meantone" generator.

<<and what is an optimal tetrachord position? (Is that another reason
to buy Chalmers book?)>>

By "optimal" I'm using more of an Paul Erlich point of view there than
my own -- the basic idea being that the tetrachords span the nearest
4/3. But it also works well in these examples. Unfortunately I don't
know anything about John's book other than I'd bet that there must be
quite a few good reasons to buy it!

<<I think I understand your comment about skewed to the left [SNIP]
What do you find to be the significance of this observation?>>

Left and right skews coincide with "modal" rotations separated along
the lines of major and minor, and zero skew with palindromic symmetry.

--Dan Stearns