Highschool scales 2

Gene,

I had another look at your 1st and 2nd 12 note highschool scales. They are similar to my NPJ scale and one of yours has a 15/14 which you don't see too often. In both of your scales you use a 7/4 but I suspect that using a 9/5 instead would yield more good intervals (although 7/4 seems stronger than 9/5). As I said before a huge criterion for me, when building a scale, was symmetry. I think it is impossible to have a 12 key scale where 1/1 and 2/1 have equal strength unless you use sqrt(2) as the tritone. With NPJ 1/1 and 3/2 can be used as tonics with equal strength. 1/1 is better for music that ascends from and then descends to 1/1. 3/2 is better for music that descends from and then ascends to 3/2. With NPJ the notes going up from 1/1 are the exact mirror image of the notes going down from 3/2. What do you think of this symmetry business?

John.

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:

> What do you think of this symmetry business?

I don't see why it is important, and of course it's impossible to produce a JI scale with sqrt(2) in it, though you can get arbitrarily close.

The symmetrical pentatonic, 6/5, 4/3, 3/2, 5/3, 2 is not notably more successul than non-symmetrical versions. However, if you start the highschool scale generation process from it (or equivalently, from 3/2, 2 or 4/3, 3/2, 2) you can produce more symmetrical alternatives. 6/5 has a triangular denominator, and factors as 9/8 * 16/15, and we could go to either 9/8, 6/5, 4/3, 3/2, 5/3, 16/9, 2 or 16/15, 6/5, 4/3, 5/3, 15/8, 2 as the next step. Neither scale is known to the Scala archives, and they are not Fokker blocks, though their symmetry makes for some cute lattice diagram patterns. We now have the same kind of choices breaking apart 9/8 and 10/9 as before, and will not be able to preserve strict symmetry. However, we can break uo 10/9, giving us a 5-limit nine-note scale, obtaining 9/8, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 16/9, 2 and 16/15, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 15/8, 2. We now do have Fokker blocks; the first scale turned up in my survey as "mavlim1":

! mavlim1.scl
!
First 27/25&135/128 scale
9
!
9/8
6/5
5/4
4/3
3/2
8/5
5/3
16/9
2/1

and the second as "mavdie1":

! mavdie1.scl
First 128/125&135/128 scale = Dwarf(<19 14 21|) = efg3355
9
!
16/15
6/5
5/4
4/3
3/2
8/5
5/3
15/8
2

It's also a dwarf scale and an Euler genus.

I can't continue the highschool process with 9/8 and get symmetrical scales, but for mavdie1 in particular it seems worth looking at what we do get. Going once again for the obvious best choices, I get the following scale and its inversion:

! 12highschool3.scl
Third 12-note Highschool scale
12
!
16/15
8/7
6/5
5/4
4/3
7/5
3/2
8/5
5/3
7/4
15/8
2

The inversion simply replaces 7/5 with 10/7. This scale is very close to the Hahn reduction of a 12-note 7-limit epimorphic scale, which is where the least complex interval in terms of the Hahn metric is chosen:

! hahn12.scl
Hahn-reduced 12 note scale
12
!
15/14
8/7
6/5
5/4
4/3
7/5
3/2
8/5
5/3
7/4
15/8
2

This also has steps which are superparticular and have square or triangular numerators.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

Eventually this thign breaks down (for on thing, 36 is both square and triangular) but it hasn't yet, so let's go to the next step. That would be breaking 15/14 = 25/24 * 36/35.

Starting from the highschool1 scale, it isn't obvious what to do between 21/20 and 9/8, or between 7/5 and 3/2. But between 7/4 and 15/8, it's clear we should go with 9/5 = 36/35 * 7/4 rather than 175/96 = 25/24 * 7/4. Hence we pick 36/35 * 25/24 rather than the reverse in all three cases, so they correspond, giving the follwing 15-note scale:

! 15highschool1.scl
First 15-note Highschool scale
15
!
21/20
27/25
9/8
6/5
5/4
4/3
7/5
36/25
3/2
8/5
5/3
7/4
9/5
15/8
2

Similar reason applies to the second scale, giving this:

! 15highschool2.scl
Second 15-note Highschool scale
15
!
36/35
15/14
9/8
6/5
5/4
4/3
48/35
10/7
3/2
8/5
5/3
7/4
9/5
15/8
2

At this point we might take note of the fact that highschool1 is a much better scale from the point of view of harmony than highschool2, and is actually a good scale for someone to try who is interested in such things.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

Continuing on, the next stage is to break apart 16/15 inside 15highschool1 by 16/15 = 28/27 * 36/35. Putting a 7/6 between 9/8 and 6/5, a 9/7 between 5/4 and 4/3 and a 14/9 between 3/2 and 8/5 are all obvious, leaving the question of what to do between 15/8 and 2. The two choices lead to two scales given below, but 19highschool1 has better harmony and so gets to go on to the next round.

! 19highschool1.scl
First 19-note Highschool scale
19
!
21/20
27/25
9/8
7/6
6/5
5/4
9/7
4/3
7/5
36/25
3/2
14/9
8/5
5/3
7/4
9/5
15/8
27/14
2

! 19highschool2.scl
Second 19-note Highschool scale
19
!
21/20
27/25
9/8
7/6
6/5
5/4
9/7
4/3
7/5
36/25
3/2
14/9
8/5
5/3
7/4
9/5
15/8
35/18
2

One last highschool scale. Now I have three 21/20 intervals to break as 21/20 = 36/35 * 49/48 in 19highschool1 and get a 22-note scale. Two of them are not obvious, but it's clear we put 12/7 between 5/4 and 7/4, and this tells us how to break the other two, giving this:

! 22highschool.scl
22-note Highschool scale
22
!
36/35
21/20
27/25
9/8
7/6
6/5
5/4
9/7
4/3
48/35
7/5
36/25
3/2
14/9
8/5
5/3
12/7
7/4
9/5
15/8
27/14
2

I'm not sure that I understand your method although it's probably quite simple. What does triangular mean? I'm sticking with 12 keys per octave. 12 to me is the golden mean between simplicity and complexity.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> One last highschool scale. Now I have three 21/20 intervals to break as 21/20 = 36/35 * 49/48 in 19highschool1 and get a 22-note scale. Two of them are not obvious, but it's clear we put 12/7 between 5/4 and 7/4, and this tells us how to break the other two, giving this:
>
> ! 22highschool.scl
> 22-note Highschool scale
> 22
> !
> 36/35
> 21/20
> 27/25
> 9/8
> 7/6
> 6/5
> 5/4
> 9/7
> 4/3
> 48/35
> 7/5
> 36/25
> 3/2
> 14/9
> 8/5
> 5/3
> 12/7
> 7/4
> 9/5
> 15/8
> 27/14
> 2
>

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
>
> I'm not sure that I understand your method although it's probably quite simple. What does triangular mean?

You can arrange 1, 4, 9, 16 ... pennies in a square; these are square numbers. Similarly you can arrange 1, 3, 6, 10 ... things in a triangle. The nth triangular number is n*(n+1)/2. Superparticular ratios with square or triangular numbers in the numerator turn up in this business. Aside from being scale steps, they also turn up as commas.

Why do square or triangular numerators have any significance?

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "john777music" <jfos777@> wrote:
> >
> > I'm not sure that I understand your method although it's probably quite simple. What does triangular mean?
>
> You can arrange 1, 4, 9, 16 ... pennies in a square; these are square numbers. Similarly you can arrange 1, 3, 6, 10 ... things in a triangle. The nth triangular number is n*(n+1)/2. Superparticular ratios with square or triangular numbers in the numerator turn up in this business. Aside from being scale steps, they also turn up as commas.
>

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
>
> Why do square or triangular numerators have any significance?

If you look at intervals approaching 1 of the form 2/1, 3/2, 4/3, 5/4, 6/5, 7/6 ... then the ratios are 4/3, 9/8, 16/15, 25/24, 36/35 .... If we approach 2 by 1/1, 3/2, 5/3, 7/4 ... then the ratios are 3/2, 10/9, 21/20, triangular. From above, 3/1, 5/2, 7/3, 9/4 ... give 6/5, 15/14, 28/27 ... the rest of the triangulars. Of course you can divide by 2 and approach 1 by 3/2, 5/4, 7/6, 9/8, 11/10 ... if that seems more natural.