Dudon scales

I have added the possibility to create Dudon scales in Scala version 2.34a. See the menu behind File:New or look at CPS/DUDON.

Manuel

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote (10-2-2011):

> I'll be interested in what Jacques has to say, but I can tell you it has some properties.

When I investigated those, it had several purposes. First it was mainly didactic, as you very well just pointed in your example of Dudon(81/80) in the sense that it lights up a relation between some of our most common traditional scales (here the anhemitonic pentatonic scale) and some archetypal coïncidences between harmonics (such as between 3 and 5 here with the syntonic comma). So the idea was to see if I could find other examples, and I did find many indeed. Then it was tempting to compare the musicality of all possible commas through this very basic traduction, and to compare musicalities produced by different limits, etc. Also while doing this I started quickly to extend this idea to groups of two or three commas together, and to use it as a tool for scale creation.
Another purpose was more directly related with a specific photosonic disk technique that I call omission, where I replace one black arc every one given number of them, by a blank space. This generates a subharmonic drone in polyphony. And commas are simply the kind of notes that will generate the more rich polyphonies.

> (1) One thing it does is to find simple intervals between which there is the comma. For instance, Dudon(81/80) is 9/8-5/4-81/64-3/2-27/16-2, and between 5/4 and 81/64 there appears 81/80. Transposing this scale to its various modes, we find 10/9-9/8-4/3-3/2-16/9-2, and thus we see 81/80 appears between 10/9 and 9/8.
>
> (2) Tempering out the comma gives a scale tempered from the subgroup defined by the prime divisors of the numerator or denominator of the comma which is small but which actually does make use of the tempering. For instance, from Dudon(81/80) above we get Meantone[5].

--- In tuning@yahoogroups.com, "kleisma" wrote:
>
> I have added the possibility to create Dudon scales in Scala version 2.34a. See the menu behind File:New or look at CPS/DUDON.

I'm getting an error in 2.24a: "The procedure entry point cairo_region_contains_point could not be located in the dynamic link library libcairo-2.dll".

--- In tuning@yahoogroups.com, "kleisma" wrote:
>
> I have added the possibility to create Dudon scales in Scala version 2.34a. See the menu behind File:New or look at CPS/DUDON.
>
> Manuel
>
> --- In tuning@yahoogroups.com, "genewardsmith" wrote (10-2-2011):
>
> > I'll be interested in what Jacques has to say, but I can tell you it has some properties.
>
> When I investigated those, it had several purposes. First it was mainly didactic, as you very well just pointed in your example of Dudon(81/80) in the sense that it lights up a relation between some of our most common traditional scales (here the anhemitonic pentatonic scale) and some archetypal coïncidences between harmonics (such as between 3 and 5 here with the syntonic comma). So the idea was to see if I could find other examples, and I did find many indeed. Then it was tempting to compare the musicality of all possible commas through this very basic traduction, and to compare musicalities produced by different limits, etc. Also while doing this I started quickly to extend this idea to groups of two or three commas together, and to use it as a tool for scale creation.

It's also possible to use things which are not commas, but commatic approximations. For instance, dudon(35/34)=dudon(48/35), 35/32, 5/4, 3/2, 7/4, 2. 48/35 is 385/384 flat of 11/8, and assuming 385/384 tempering you get that the scale can also be written 12/11, 5/4, 3/2, 7/4, 2. This pentatonic scale looks quite interesting to me, and doesn't seem to be known. It's epimorphic, strictly proper, and (assuming tempering) full 11-limit. That's pretty impressive for a pentatonic!

--- In tuning@yahoogroups.com, "genewardsmith" wrote:
This pentatonic scale looks quite interesting to me, and doesn't seem to be known. It's epimorphic, strictly proper, and (assuming tempering) full 11-limit. That's pretty impressive for a pentatonic!
>

It's not known as a pentatonic scale, but it is listed as a keenanismic chord on the Xenwiki. The six keenanismic pentads/penatonics are the 385/384 temperings of these:

"Finally, there are six keenanismic pentads coming in three inverse pairs. These are 1-5/4-11/8-3/2-12/7 with steps 5/4-11/10-12/11-8/7-7/6 and 1-8/7-5/4-11/8-12/7 with steps 8/7-12/11-11/10-5/4-7/6; and the pair of pairs 1-6/5-11/8-3/2-7/4 with steps 6/5-8/7-12/11-7/6-8/7 and 1-12/11-5/4-3/2-12/7 with steps 12/11-8/7-6/5-8/7-7/6 plus 1-6/5-11/8-3/2-12/7 with steps 6/5-8/7-12/11-8/7-7/6 and 1-12/11-5/4-3/2-7/4 with steps 12/11-8/7-6/5-7/6-8/7."

! keen1.scl
!
Keenanismic tempering of [5/4, 11/8, 3/2, 12/7, 2], 284et tuning
5
!
384.50704
549.29577
701.40845
933.80282
2/1

! keen2.scl
!
Keenanismic tempering of [8/7, 5/4, 11/8, 12/7, 2], 284et tuning
5
!
232.39437
384.50704
549.29577
933.80282
2/1

! keen3.scl
!
Keenanismic tempering of [6/5, 11/8, 3/2, 7/4, 2], 284et tuning
5
!
316.90141
549.29577
701.40845
967.60563
2/1

! keen4.scl
!
Keenanismic tempering of [12/11, 5/4, 3/2, 12/7, 2], 284et tuning
5
!
152.11268
384.50704
701.40845
933.80282
2/1

! keen5.scl
!
Keenanismic tempering of [6/5, 11/8, 3/2, 12/7, 2], 284et tuning
5
!
316.90141
549.29577
701.40845
933.80282
2/1

! keen6.scl
!
Keenanismic tempering of [12/11, 5/4, 3/2, 7/4, 2], 284et tuning
5
!
152.11268
384.50704
701.40845
967.60563
2/1

If you like 7 and 11 in your pentatonic flavors, these could be what you've been looking for.

Dear Gene and Manuel,

The topic of pentatonics including lots of primes, just or
tempered, led me to this 2-3-7-11-13 JI scale which is
strictly proper as well as JI epimorphic with prime-degree
mapping, although I'm not sure if it's a Dudon scale:

! pentatonic-proper_5-prime.scl
!
Strictly proper 2-3-7-11-13 pentatonic
5
!
63/52
11/8
3/2
7/4
2/1

One tempered form would be:

! met24-pentatonic-proper_5-prime_F.scl
!
Approximate 63/52-11/8-3/2-7/4-2/1
5
!
332.81250
553.12500
703.12500
967.96875
2/1

However, if we are also considering improper scales, then
this 2-3-7-11-13 tuning seems quite attractive:

! pentatonic-2_3_7_11_13.scl
!
Pentatonic, primes 2-3-7-11-13
5
!
7/6
11/8
3/2
13/8
2/1

! met24-pentatonic-5prime_A.scl
!
Approximate 7/6-11/8-3/2-13/8-2/1
5
!
264.84375
553.12500
703.12500
842.57813
2/1

Curiously there didn't seem to be any close match in a
not-so-up-to-date version of the Scala scale archive.
But from a harmonic point of view, the availability of both
6:7:9 and 8:11:13 might be interesting.

Best,

Margo