Mailing list post
From: Gene Ward Smith (2005-07-15)
Subject: A 19-note breed tempered diamond scale
I find four {2,5,7} diamond-type scales with no more than 19 notes to
the octave and at least six breed-tempered otonal tetrads. Only one of
these is convex, the scale resulting from taking all ratios of
[25, 12,5 175, 245, 1715, 2401] and reducing to the octave. If we TM
reduce this by 2401/2400, we get
1 49/48 15/14 8/7 7/6 6/5 49/40 5/4 4/3 7/5 10/7
3/2 8/5 49/30 5/3 12/7 7/4 28/15 49/25 2
This has five JI tetrads of each kind, plus one each of a
breed-tempered tetrad and a marvel tempered tetrad; putting that all
together gives
seven miracle tetrads of each kind. The miracle tempered version
is -13, -10, -8, -7, -6 -5, -3, -2, -1, 0, 1, 2, 3, 5, 6, 7, 8, 10, 13.
Here it is in both 72 and 612 edo:
! diab19_72
diab19a in 72-et
19
!
33.333333
116.666667
233.333333
266.666667
316.666667
350.000000
383.333333
500.000000
583.333333
616.666667
700.000000
816.666667
850.000000
883.333333
933.333333
966.666667
1083.333333
1166.666667
1200.000000
! diab19_612.scl
diab19a in 612 et tuning
19
!
35.294118
119.607843
231.372549
266.666667
315.686275
350.980392
386.274510
498.039216
582.352941
617.647059
701.960784
813.725490
849.019608
884.313725
933.333333
968.627451
1080.392157
1164.705882
1200.000000
Full thread (2 messages)
From: Gene Ward Smith (2005-07-15)
Subject: A 19-note breed tempered diamond scale
I find four {2,5,7} diamond-type scales with no more than 19 notes to
the octave and at least six breed-tempered otonal tetrads. Only one of
these is convex, the scale resulting from taking all ratios of
[25, 12,5 175, 245, 1715, 2401] and reducing to the octave. If we TM
reduce this by 2401/2400, we get
1 49/48 15/14 8/7 7/6 6/5 49/40 5/4 4/3 7/5 10/7
3/2 8/5 49/30 5/3 12/7 7/4 28/15 49/25 2
This has five JI tetrads of each kind, plus one each of a
breed-tempered tetrad and a marvel tempered tetrad; putting that all
together gives
seven miracle tetrads of each kind. The miracle tempered version
is -13, -10, -8, -7, -6 -5, -3, -2, -1, 0, 1, 2, 3, 5, 6, 7, 8, 10, 13.
Here it is in both 72 and 612 edo:
! diab19_72
diab19a in 72-et
19
!
33.333333
116.666667
233.333333
266.666667
316.666667
350.000000
383.333333
500.000000
583.333333
616.666667
700.000000
816.666667
850.000000
883.333333
933.333333
966.666667
1083.333333
1166.666667
1200.000000
! diab19_612.scl
diab19a in 612 et tuning
19
!
35.294118
119.607843
231.372549
266.666667
315.686275
350.980392
386.274510
498.039216
582.352941
617.647059
701.960784
813.725490
849.019608
884.313725
933.333333
968.627451
1080.392157
1164.705882
1200.000000
From: Carl Lumma (2006-08-26)
Subject: Re: A 19-note breed tempered diamond scale
> I find four {2,5,7} diamond-type scales with no more than
> 19 notes to the octave and at least six breed-tempered otonal
> tetrads. Only one of these is convex, the scale resulting
> from taking all ratios of [25, 12,5 175, 245, 1715, 2401] and
> reducing to the octave. If we TM reduce this by 2401/2400, we
> get
>
> 1 49/48 15/14 8/7 7/6 6/5 49/40 5/4 4/3 7/5 10/7
> 3/2 8/5 49/30 5/3 12/7 7/4 28/15 49/25 2
>
> This has five JI tetrads of each kind, plus one each of a
> breed-tempered tetrad and a marvel tempered tetrad; putting that
> all together gives seven miracle tetrads of each kind. The
> miracle tempered version is
> -13, -10, -8, -7, -6 -5, -3, -2, -1, 0, 1, 2, 3, 5, 6, 7, 8, 10, 13.
Interesting -- this isn't a continuous chain.
-Carl