Re: [tuning] Digest Number 2324--All Interval Rows

Mats et al. Back in 1983, I used an algorithm devised by Morris and
Starr
(J. Mus. Theory 18: 364-389)to find all-interval rows. After
translating
it from FORTRAN into BASIC, I ran it on an Apple II until 1 row had been
found for each of the even-numbered ET's from 4 to 22 tones. As I
recall,
for 24-tet,the algorithm was so slow that I never had a large enough
block of time on the computer to find one (I think 26 hours before a
power
failure stopped the machine was my record). My results appeared in
EAR Magazine, vol 8(3), June/July 1983 and are listed below:

ET All-Interval Row Generator
4 0 1 3 2
6 0 1 5 2 4 3
8 0 1 3 6 2 7 5 4
10 0 1 3 2 7 4 8 6 9 5
12 0 1 3 2 7 10 8 4 11 5 9 6
14 0 1 3 2 5 10 8 4 11 6 12 9 13 7
16 0 1 3 2 5 9 4 13 7 14 6 12 10 15 11 8
18 0 1 3 2 5 10 4 8 15 7 16 11 17 13 6 14 12 9
20 0 1 3 2 5 9 4 11 7 16 13 19 12 17 15 6 14 8 18 10
22 0 1 3 2 5 9 4 10 6 16 7 18 8 17 15 20 12 19 13 21 14 11

The number of all-interval row generators goes up very fast
with the number of tones per octave.

ET Number
4 2
6 4
8 24
10 288
12 3856
(including R forms)

There is also an algorithm in A.D.Fokker's "Les Mathematiques
et La Musique," Arch. Teyler's Museum 10: 1-32 (1947). Using
this algorithm, I found the following row for 10-tet:
0 1 3 7 4 9 2 8 6 5. This result appeared in Ear in the Feb/March
1983 issue (vol 8 (1-2).

Row generators in N-tet have 4N derived forms (Prime, Inverted,
Retrograde,
and Retrograde-Inverted) and may be transposed to N different starting
notes. There are also the muliplicative transforms where each ordinal
note number is multiplied by M modulo N where M is relatively prime to
N. In
the case of 10-tet, M may be 3 or 7. The M3 and M7 transforms of the
above
row are 0 3 9 1 2 7 6 4 8 5 and 0 7 1 9 8 3 4 6 2 5.

--John