Mailing list post
From: Gene Ward Smith (2006-07-21)
Subject: CS vs Proper
Here are two interesting and unusual scales, both are built from
11-note cycles of (with apologies to Carl) major, minor, and subminor
thirds; that is, 4 5/4s, 4 6/5s, and 3 7/6s, which close due to
tempering out 1029/1024.
One is constant structure but not proper, and the other is proper but
not constant structure. If M is a major third, m a minor third, and s
a septimal minor or subminor third, then the first has the cycle
MmsMmsMmsMm and the second has the cycle MmMsMmMsmms. You can see just
by examining these cycles how various assorted triads and tetrads will
appear.
Here they both are in 252-et tuning:
! septicyc.scl
septicyclic 1029/1024-tempered scale, 252 et
11
!
152.380952
233.333333
385.714286
466.666667
500.000000
700.000000
733.333333
885.714286
966.666667
1119.047619
1200.000000
! MmsMmsMmsMm
! propsep.scl
proper septicyclic 1029/1024-tempered scale in 252 et
11
!
38.095238
152.380952
304.761905
385.714286
538.095238
619.047619
700.000000
852.380952
933.333333
1085.714286
1200.000000
! MmMsMmMsmms
Full thread (4 messages)
From: Gene Ward Smith (2006-07-21)
Subject: CS vs Proper
Here are two interesting and unusual scales, both are built from
11-note cycles of (with apologies to Carl) major, minor, and subminor
thirds; that is, 4 5/4s, 4 6/5s, and 3 7/6s, which close due to
tempering out 1029/1024.
One is constant structure but not proper, and the other is proper but
not constant structure. If M is a major third, m a minor third, and s
a septimal minor or subminor third, then the first has the cycle
MmsMmsMmsMm and the second has the cycle MmMsMmMsmms. You can see just
by examining these cycles how various assorted triads and tetrads will
appear.
Here they both are in 252-et tuning:
! septicyc.scl
septicyclic 1029/1024-tempered scale, 252 et
11
!
152.380952
233.333333
385.714286
466.666667
500.000000
700.000000
733.333333
885.714286
966.666667
1119.047619
1200.000000
! MmsMmsMmsMm
! propsep.scl
proper septicyclic 1029/1024-tempered scale in 252 et
11
!
38.095238
152.380952
304.761905
385.714286
538.095238
619.047619
700.000000
852.380952
933.333333
1085.714286
1200.000000
! MmMsMmMsmms
From: Carl Lumma (2006-07-21)
Subject: Re: [tuning-math] CS vs Proper
>Here they both are in 252-et tuning:
>
>! septicyc.scl
The name says it all. Just kidding. :)
-Carl
From: Gene Ward Smith (2006-07-22)
Subject: Re: CS vs Proper
--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
> One is constant structure but not proper, and the other is proper but
> not constant structure.
Constant structure does not seem to be as stringent a criterion; we have
466 of these which are CS but only one which is proper. Since strictly
proper ==> CS if I'm using the right definiton for CS (that is, it
does for the definition I'm using) that's not a surprise.
The definition for CS I'm using, in case anyone wants to object, is that
i != j ==> Class(i) intersect Class(j) = {}; which is to say, the
intervals in distinct interval classes are distinct.
From: Kraig Grady (2006-07-22)
Subject: Re: [tuning-math] Re: CS vs Proper
i assume you did it this way to allow overlapping. the only thing i can
think might violate the spirit of this if one had a scale or series of
tones where none is repeated.
So i don't know how you put in that at least some intervals are
repeated somehow, or if it really necessary until a real example comes
up. possible a class implied a family of more than one
Gene Ward Smith wrote:
>
> --- In tuning-math@yahoogroups.com
> <mailto:tuning-math%40yahoogroups.com>, "Gene Ward Smith"
> <genewardsmith@...> wrote:
>
> > One is constant structure but not proper, and the other is proper but
> > not constant structure.
>
> Constant structure does not seem to be as stringent a criterion; we have
> 466 of these which are CS but only one which is proper. Since strictly
> proper ==> CS if I'm using the right definiton for CS (that is, it
> does for the definition I'm using) that's not a surprise.
>
> The definition for CS I'm using, in case anyone wants to object, is that
> i != j ==> Class(i) intersect Class(j) = {}; which is to say, the
> intervals in distinct interval classes are distinct.
>
>
--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles