Eight of everything

To go with the Type Q (19, 9, 4) cyclic difference set scale, which has five of everything, below there's a Type Q (31, 15, 7) scale, which has eight of everything. By that I mean it has eight subminor thirds, eight minor thirds, eight major thirds, eight fourths, eight 11/8s etc. A natural for two-part harmony. There's also a Type H8 cyclic difference set, whose definition is a little more complicated; the Type Qs are just the quadratic residues. I'm not sure why Golomb rulers are in the Scala directgory and not Type Q difference sets, which seem more interesting.

! diff31-q.scl
!
(31, 15, 7) type Q cyclic difference set, 31edo
16
!
38.70968
77.41935
154.83871
193.54839
270.96774
309.67742
348.38710
387.09677
541.93548
619.35484
696.77419
735.48387
774.19355
967.74194
1083.87097
1200.00000

! diff31-h8.scl
!
(31, 15, 7) type H8 cyclic difference set, 31edo
16
!
38.70968
77.41935
116.12903
154.83871
232.25806
309.67742
464.51613
580.64516
619.35484
658.06452
890.32258
929.03226
1045.16129
1122.58065
1161.29032
1200.00000

! diff19-9-4.scl
!
Scale derived from (19,9,4) Type Q cyclic difference set, 19edo
10
!
63.15789
252.63158
315.78947
378.94737
442.10526
568.42105
694.73684
1010.52632
1073.68421
1200.00000

I got them - thank you!

On Wed, Feb 15, 2012 at 11:25 PM, genewardsmith <genewardsmith@...
> wrote:

> **
>
>
> To go with the Type Q (19, 9, 4) cyclic difference set scale, which has
> five of everything, below there's a Type Q (31, 15, 7) scale, which has
> eight of everything. By that I mean it has eight subminor thirds, eight
> minor thirds, eight major thirds, eight fourths, eight 11/8s etc. A natural
> for two-part harmony. There's also a Type H8 cyclic difference set, whose
> definition is a little more complicated; the Type Qs are just the quadratic
> residues. I'm not sure why Golomb rulers are in the Scala directgory and
> not Type Q difference sets, which seem more interesting.
>
> ! diff31-q.scl
> !
> (31, 15, 7) type Q cyclic difference set, 31edo
> 16
> !
> 38.70968
> 77.41935
> 154.83871
> 193.54839
> 270.96774
> 309.67742
> 348.38710
> 387.09677
> 541.93548
> 619.35484
> 696.77419
> 735.48387
> 774.19355
> 967.74194
> 1083.87097
> 1200.00000
>
> ! diff31-h8.scl
> !
> (31, 15, 7) type H8 cyclic difference set, 31edo
> 16
> !
> 38.70968
> 77.41935
> 116.12903
> 154.83871
> 232.25806
> 309.67742
> 464.51613
> 580.64516
> 619.35484
> 658.06452
> 890.32258
> 929.03226
> 1045.16129
> 1122.58065
> 1161.29032
> 1200.00000
>
> ! diff19-9-4.scl
> !
> Scale derived from (19,9,4) Type Q cyclic difference set, 19edo
> 10
> !
> 63.15789
> 252.63158
> 315.78947
> 378.94737
> 442.10526
> 568.42105
> 694.73684
> 1010.52632
> 1073.68421
> 1200.00000
>
>
>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> To go with the Type Q (19, 9, 4) cyclic difference set scale, which has five of everything, below there's a Type Q (31, 15, 7) scale, which has eight of everything. By that I mean it has eight subminor thirds, eight minor thirds, eight major thirds, eight fourths, eight 11/8s etc. A natural for two-part harmony. There's also a Type H8 cyclic difference set, whose definition is a little more complicated; the Type Qs are just the quadratic residues. I'm not sure why Golomb rulers are in the Scala directgory and not Type Q difference sets, which seem more interesting.

So trivially, since these scales have 8 of everything, they can't be rank-2 MOS. What is the minimum number of generators required to make one of these scales?

Ryan