Perfect difference scales

Margo's "Variegated Intonation", of her Zest-24 scale, made me think
about taking that idea to a ridiculous extreme, namely perfect
difference scales. These are based on perfect difference sets, which
I'm sure have been discussed before on this list or on tuning-math.

The idea is simple enough. The smallest set of pitches in some EDO
such that every possible interval-size in that EDO can be generated
exactly once per octave (except of course the unison which occurs once
per note).

12 notes per octave is a convenient number for keyboard mapping and
the following 12 note "scale" allows you to generate every
interval-size in 133-EDO. The rule is that the number of notes in the
scale N, must be one more than a prime or prime power (in this case
11+1=12, and the EDO is then N(N-1)+1 (in this case 12*11+1 = 133).

! PerfDif12.scl
!
Perfect difference scale for 133-EDO
12
!
18.04511278
54.13533835
216.5413534
261.6541353
360.9022556
387.9699248
496.2406015
613.5338346
676.6917293
685.7142857
766.9172932
2/1

133-EDO isn't very good at approximating JI. However N=13+1 just
happens to give us 14*13+1 = 183-EDO which is very good indeed.

! PerfDif14.scl
!
Perfect difference scale for 183-EDO
14
!
32.78688525
183.6065574
249.1803279
268.852459
321.3114754
327.8688525
445.9016393
491.8032787
603.2786885
701.6393443
793.442623
806.557377
832.7868852
2/1

So the above incredibly uneven "scale" lets us play an example of
every JI or MI or in-between interval to within 3.3 cents. Of course
you're going to need a table or chart to find them.

There is sometimes more than one perfect difference scale for a given
EDO. Some are slightly more even than others. But it's hard enough to
find one.

Perfect difference sets are the circular or modular version of perfect
Golomb rulers, and that's how I finally found the data needed to
generate the scales. See
http://www.research.ibm.com/people/s/shearer/gropt.html

Apart from 183-EDO, the next lower and next higher "good" EDOs that
have perfect difference scales are 31-EDO needing 6 notes and 381-EDO
needing 20 notes.

-- Dave Keenan

Walter O' Connell worked allot with all interval sets in 12 like
C D F F#
C E F#G
and their inversions

but does this not fit your formula since
4*3 +1 =13? or is it that 13 also has 4 not all interval sets.

I wrote quite bit of music using these BTW before i started with microtones.
i have Erv's all interval set for 31here. i thought i had put it up but hadn't
http://anaphoria.com/31allintvl.PDF

I still have about a hundred pages of Erv stuff to put up which now that i got my scanner dealing with 240 now i can begin

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Dave Keenan wrote:
>
> Margo's "Variegated Intonation", of her Zest-24 scale, made me think
> about taking that idea to a ridiculous extreme, namely perfect
> difference scales. These are based on perfect difference sets, which
> I'm sure have been discussed before on this list or on tuning-math.
>
> The idea is simple enough. The smallest set of pitches in some EDO
> such that every possible interval-size in that EDO can be generated
> exactly once per octave (except of course the unison which occurs once
> per note).
>
> 12 notes per octave is a convenient number for keyboard mapping and
> the following 12 note "scale" allows you to generate every
> interval-size in 133-EDO. The rule is that the number of notes in the
> scale N, must be one more than a prime or prime power (in this case
> 11+1=12, and the EDO is then N(N-1)+1 (in this case 12*11+1 = 133).
>
> ! PerfDif12.scl
> !
> Perfect difference scale for 133-EDO
> 12
> !
> 18.04511278
> 54.13533835
> 216.5413534
> 261.6541353
> 360.9022556
> 387.9699248
> 496.2406015
> 613.5338346
> 676.6917293
> 685.7142857
> 766.9172932
> 2/1
>
> 133-EDO isn't very good at approximating JI. However N=13+1 just
> happens to give us 14*13+1 = 183-EDO which is very good indeed.
>
> ! PerfDif14.scl
> !
> Perfect difference scale for 183-EDO
> 14
> !
> 32.78688525
> 183.6065574
> 249.1803279
> 268.852459
> 321.3114754
> 327.8688525
> 445.9016393
> 491.8032787
> 603.2786885
> 701.6393443
> 793.442623
> 806.557377
> 832.7868852
> 2/1
>
> So the above incredibly uneven "scale" lets us play an example of
> every JI or MI or in-between interval to within 3.3 cents. Of course
> you're going to need a table or chart to find them.
>
> There is sometimes more than one perfect difference scale for a given
> EDO. Some are slightly more even than others. But it's hard enough to
> find one.
>
> Perfect difference sets are the circular or modular version of perfect
> Golomb rulers, and that's how I finally found the data needed to
> generate the scales. See
> http://www.research.ibm.com/people/s/shearer/gropt.html > <http://www.research.ibm.com/people/s/shearer/gropt.html>
>
> Apart from 183-EDO, the next lower and next higher "good" EDOs that
> have perfect difference scales are 31-EDO needing 6 notes and 381-EDO
> needing 20 notes.
>
> -- Dave Keenan
>
>

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> Walter O' Connell worked allot with all interval sets in 12 like
> C D F F#
> C E F#G
> and their inversions

"All-interval set" is a better name for them in this context. Thanks.

> but does this not fit your formula since
> 4*3 +1 =13? or is it that 13 also has 4 not all interval sets.

Good point. I was speaking of the "perfect" cases, where each interval
appears exactly once (per octave). In the case of 4 notes for 12 ET,
one of the intervals (the tritone) occurs twice. Yes, we can get all
intervals of 13-EDO with only 4 notes as well.

But you're right that there's no point in limiting ourselves to the
"perfect" cases, we only want them to be "optimal", i.e. using the
minimum number of notes for a given ET. So there may well be
all-interval sets having 12 notes and giving all intervals for some
good EDOs a little smaller than 133, such as 118 or 130.

I note that the largest all-interval set known to be optimal, has only
24 notes (and gets us all intervals of 553-EDO). It took more than
41,000 computers 4 years to prove that it was optimal (and was the
only optimal one of that size, excluding inversions and rotations)!
This was completed in 2004. The set was actually found in 1967, but it
wasn't known for sure whether it was optimal.

It can be found here
http://www.maa.org/editorial/mathgames/mathgames_11_15_04.html
0 9 33 37 38 97 122 129 140 142 152 191 205 208 252 278 286 326
332 353 368 384 403 425

> I wrote quite bit of music using these BTW before i started with
microtones.
> i have Erv's all interval set for 31here. i thought i had put it up
but
> hadn't
> http://anaphoria.com/31allintvl.PDF

Wow. Beautifully laid out.

It's interesting that for smaller sets like 6 notes giving
all-intervals for 31-EDO (a "perfect" case) there are several
different optimal sets, but for larger ones there is usually only one
(ignoring rotations and inversions). A real needle in a factorial
haystack.

If you have one all-interval set, you can find others for the same EDO
(if they exist) by multiplying all the degree numbers by a fixed
integer, modulo the EDO size. That's presumably what the x1, x2, x3
etc. mean on Erv's diagrams. The fixed integers must be relatively
prime with respect to the EDO size.

For example, for 12-EDO you gave
C D F F#
C E F#G
which as degree numbers are
0 2 5 6 with step sizes 2 3 1 6
0 4 6 7 with step sizes 4 2 1 5

If you take the first one and multiply each degree number by 5 you get
0 10 25 30
Then reduce that modulo 12 and you get
0 10 1 6
Then put them in pitch order and you get
0 1 6 10 with step sizes 1 5 4 2
and you can see from the sequence of step sizes that this is just a
rotation of the second set you gave above.

I occurs to me that, like the noble numbers, these difference sets
should have an application to the linear frequency domain as well as
the logarithmic pitch domain, in which case they would surely relate
to difference tones.

-- Dave Keenan

one could apply these to rhythm if one wanted to work with a 553 beat pattern.:)
cleaver method of finding the other sets i will say!

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Dave Keenan wrote:
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, Kraig > Grady <kraiggrady@...> wrote:
> >
> > Walter O' Connell worked allot with all interval sets in 12 like
> > C D F F#
> > C E F#G
> > and their inversions
>
> "All-interval set" is a better name for them in this context. Thanks.
>
> > but does this not fit your formula since
> > 4*3 +1 =13? or is it that 13 also has 4 not all interval sets.
>
> Good point. I was speaking of the "perfect" cases, where each interval
> appears exactly once (per octave). In the case of 4 notes for 12 ET,
> one of the intervals (the tritone) occurs twice. Yes, we can get all
> intervals of 13-EDO with only 4 notes as well.
>
> But you're right that there's no point in limiting ourselves to the
> "perfect" cases, we only want them to be "optimal", i.e. using the
> minimum number of notes for a given ET. So there may well be
> all-interval sets having 12 notes and giving all intervals for some
> good EDOs a little smaller than 133, such as 118 or 130.
>
> I note that the largest all-interval set known to be optimal, has only
> 24 notes (and gets us all intervals of 553-EDO). It took more than
> 41,000 computers 4 years to prove that it was optimal (and was the
> only optimal one of that size, excluding inversions and rotations)!
> This was completed in 2004. The set was actually found in 1967, but it
> wasn't known for sure whether it was optimal.
>
> It can be found here
> http://www.maa.org/editorial/mathgames/mathgames_11_15_04.html > <http://www.maa.org/editorial/mathgames/mathgames_11_15_04.html>
> 0 9 33 37 38 97 122 129 140 142 152 191 205 208 252 278 286 326
> 332 353 368 384 403 425
>
> > I wrote quite bit of music using these BTW before i started with
> microtones.
> > i have Erv's all interval set for 31here. i thought i had put it up
> but
> > hadn't
> > http://anaphoria.com/31allintvl.PDF > <http://anaphoria.com/31allintvl.PDF>
>
> Wow. Beautifully laid out.
>
> It's interesting that for smaller sets like 6 notes giving
> all-intervals for 31-EDO (a "perfect" case) there are several
> different optimal sets, but for larger ones there is usually only one
> (ignoring rotations and inversions). A real needle in a factorial
> haystack.
>
> If you have one all-interval set, you can find others for the same EDO
> (if they exist) by multiplying all the degree numbers by a fixed
> integer, modulo the EDO size. That's presumably what the x1, x2, x3
> etc. mean on Erv's diagrams. The fixed integers must be relatively
> prime with respect to the EDO size.
>
> For example, for 12-EDO you gave
> C D F F#
> C E F#G
> which as degree numbers are
> 0 2 5 6 with step sizes 2 3 1 6
> 0 4 6 7 with step sizes 4 2 1 5
>
> If you take the first one and multiply each degree number by 5 you get
> 0 10 25 30
> Then reduce that modulo 12 and you get
> 0 10 1 6
> Then put them in pitch order and you get
> 0 1 6 10 with step sizes 1 5 4 2
> and you can see from the sequence of step sizes that this is just a
> rotation of the second set you gave above.
>
> I occurs to me that, like the noble numbers, these difference sets
> should have an application to the linear frequency domain as well as
> the logarithmic pitch domain, in which case they would surely relate
> to difference tones.
>
> -- Dave Keenan
>
>