Keenan4

From: David C Keenan (1999-07-27)
Subject: Chain-of-quarter-4ths tuning

I thought I'd wait for the dust to settle on the chains of minor 3rds
thread before posting this.

Some weeks ago I posted the results of a computer search for MOS scale
generators that produce a useful number of 7-limit harmonies. You can see
it at
http://uq.net.au/~zzdkeena/Music/7LimitGenerators.txt

Yesterday I extended the search to triple-chains 1/3 of an octave apart,
and found no good 7-limit generators.

So far, (and I expect forever), the winner is Paul Erlich's symmetrical
decatonic. The chain of 11 minor thirds (6th root of 3) comes second, and
third is this chain of nine or ten 124.5c "quarter-fourths" (4th root of 4/3).

!
! Keenan4.scl
!
Chain of quarter-4ths MOS, 6 tetrads, max abs err 18c
 10
!
 124.51125
 249.02250
 373.53375
 4/3
 622.55625 ! or 577.44375 or none
 3/2
 826.46625
 950.97750
 1075.48875
 2/1

9-tone subsets appears in the Scala archive as:
! quasi_9.scl
!
Quasi-Equal Enneatonic, Each "tetrachord" has 125 + 125 + 125 + 125 cents

Also as a 9-tone mode of 19-tET:
! 09-19.scl
!
9 out of 19-tET

Here are some of its harmonic properties.
                              

Single chain:                               No. generators in
                        Min        Min      interval
Generator  No. tetrads  7-limit    7-limit  2  4  5  4  5  6  4  5  7
(+-0.5c)   in 10 notes  RMS error  MA err.  3  5  6  7  7  7  9  9  9
---------------------------------------------------------------------
125c           6        12.2c      17.9c   -4  3 -7 -2 -5  2 -8 11 -6

It is similar to the chain of minor thirds in that it uses the same
interval to approximate 6:7 and 7:8 and so it has the same minimum
max-absolute error and is likewise (barely) acceptable as a mode of 19-tET.
The higher ETs in which it exists are completely unrelated (except of
course for multiples of 19-tET). It exists to various degrees of accuracy
in 29, 38, 48, 58, 67 and 77-tET.

It is unlike the other 7-limit MOSs I've looked at, in that the generating
interval is not itself a consonant interval.

Here are the errors in the intervals with various sizes of generator.

Generator                     Errors in intervals (cents)
(cents)           2:3   4:5   5:6   4:7   5:7   6:7   4:9   5:9   7:9
124.14 (29-tET)  +1.5 -13.9 +15.4 -17.1  -3.2 -18.6  +3.0 +16.9 +20.1
124.51 (min MA)     0 -12.8 +12.8 -17.8  -5.1 -17.8     0 +12.8 +17.8
125.53 (min RMS) -4.1  -9.7  +5.7 -19.9 -10.2 -15.8  -8.1  +1.6 +11.7
126.32 (19-tET)  -7.2  -7.4  +0.1 -21.5 -14.1 -14.2 -14.4  -7.1  +7.0

Considering chains with from 3 to 18 tones; all those with 10 or fewer
tones have 2 step sizes, Myhill's property, and are MOS', but only 9 and 10
are proper, in fact they are strictly proper.

Regards,
-- Dave Keenan
http://uq.net.au/~zzdkeena

From: David C Keenan (1999-07-27)
Subject: Chain-of-quarter-4ths tuning

I thought I'd wait for the dust to settle on the chains of minor 3rds
thread before posting this.

Some weeks ago I posted the results of a computer search for MOS scale
generators that produce a useful number of 7-limit harmonies. You can see
it at
http://uq.net.au/~zzdkeena/Music/7LimitGenerators.txt

Yesterday I extended the search to triple-chains 1/3 of an octave apart,
and found no good 7-limit generators.

So far, (and I expect forever), the winner is Paul Erlich's symmetrical
decatonic. The chain of 11 minor thirds (6th root of 3) comes second, and
third is this chain of nine or ten 124.5c "quarter-fourths" (4th root of 4/3).

!
! Keenan4.scl
!
Chain of quarter-4ths MOS, 6 tetrads, max abs err 18c
 10
!
 124.51125
 249.02250
 373.53375
 4/3
 622.55625 ! or 577.44375 or none
 3/2
 826.46625
 950.97750
 1075.48875
 2/1

9-tone subsets appears in the Scala archive as:
! quasi_9.scl
!
Quasi-Equal Enneatonic, Each "tetrachord" has 125 + 125 + 125 + 125 cents

Also as a 9-tone mode of 19-tET:
! 09-19.scl
!
9 out of 19-tET

Here are some of its harmonic properties.
                              

Single chain:                               No. generators in
                        Min        Min      interval
Generator  No. tetrads  7-limit    7-limit  2  4  5  4  5  6  4  5  7
(+-0.5c)   in 10 notes  RMS error  MA err.  3  5  6  7  7  7  9  9  9
---------------------------------------------------------------------
125c           6        12.2c      17.9c   -4  3 -7 -2 -5  2 -8 11 -6

It is similar to the chain of minor thirds in that it uses the same
interval to approximate 6:7 and 7:8 and so it has the same minimum
max-absolute error and is likewise (barely) acceptable as a mode of 19-tET.
The higher ETs in which it exists are completely unrelated (except of
course for multiples of 19-tET). It exists to various degrees of accuracy
in 29, 38, 48, 58, 67 and 77-tET.

It is unlike the other 7-limit MOSs I've looked at, in that the generating
interval is not itself a consonant interval.

Here are the errors in the intervals with various sizes of generator.

Generator                     Errors in intervals (cents)
(cents)           2:3   4:5   5:6   4:7   5:7   6:7   4:9   5:9   7:9
124.14 (29-tET)  +1.5 -13.9 +15.4 -17.1  -3.2 -18.6  +3.0 +16.9 +20.1
124.51 (min MA)     0 -12.8 +12.8 -17.8  -5.1 -17.8     0 +12.8 +17.8
125.53 (min RMS) -4.1  -9.7  +5.7 -19.9 -10.2 -15.8  -8.1  +1.6 +11.7
126.32 (19-tET)  -7.2  -7.4  +0.1 -21.5 -14.1 -14.2 -14.4  -7.1  +7.0

Considering chains with from 3 to 18 tones; all those with 10 or fewer
tones have 2 step sizes, Myhill's property, and are MOS', but only 9 and 10
are proper, in fact they are strictly proper.

Regards,
-- Dave Keenan
http://uq.net.au/~zzdkeena