Full thread (7 messages)
From: Gene Ward Smith (2004-08-10)
Subject: 7-limit 12-note epimorphic scales containing a hexany
If you break down the 7/6 steps of a hexany into either
(15/14)(21/20)(28/27) = 7/6 or (16/15)(21/20)(25/24) = 7/6, and if you
split 8/7 as (15/14)(16/15), you end up with 576 scales. Using the
characteristic polynomial we can quickly find the scale and inverse
which maximize 7-limit harmony; it turns out that the highest count of
consonant intervals and triads is reached by pris.scl and its inverted
form, which is related to how pris was found before.
If instead we use 9-limit consonances, we get another scale and its
inverse which maximizes 9-limit intervals and consonances which seems
to be new, and which like pris can be considered a relative of prism.
Here it is:
! hexy.scl
Maximized 9-limit harmony containing a hexany
12
!
21/20
9/8
7/6
5/4
4/3
7/5
3/2
8/5
5/3
7/4
28/15
2
From: Carl Lumma (2004-08-11)
Subject: Re: 7-limit 12-note epimorphic scales containing a hexany
>If you break down the 7/6 steps of a hexany into either
>(15/14)(21/20)(28/27) = 7/6 or (16/15)(21/20)(25/24) = 7/6, and
>if you split 8/7 as (15/14)(16/15), you end up with 576 scales.
>Using the characteristic polynomial we can quickly find the
>scale and inverse which maximize 7-limit harmony; it turns out
>that the highest count of consonant intervals and triads is
>reached by pris.scl and its inverted form, which is related to
>how pris was found before.
Is pris your version of prism, which swaps 3/2 for 112/75?
>If instead we use 9-limit consonances, we get another scale and
>its inverse which maximizes 9-limit intervals and consonances
>which seems to be new, and which like pris can be considered a
>relative of prism.
>Here it is:
>
>! hexy.scl
>Maximized 9-limit harmony containing a hexany
>12
>!
>21/20
>9/8
>7/6
>5/4
>4/3
>7/5
>3/2
>8/5
>5/3
>7/4
>28/15
>2
By the way, Scala's 'compare scale' feature is broken in
several ways, so I wouldn't trust it.
-Carl
From: Manuel Op de Coul (2004-08-11)
Subject: Re: [tuning-math] Re: 7-limit 12-note epimorphic scales containing a hexany
Carl wrote:
>By the way, Scala's 'compare scale' feature is broken in
>several ways, so I wouldn't trust it.
? I haven't seen a bug report.
Manuel
From: Carl Lumma (2004-08-11)
Subject: Re: 7-limit 12-note epimorphic scales containing a hexany
> Carl wrote:
> >By the way, Scala's 'compare scale' feature is broken in
> >several ways, so I wouldn't trust it.
>
> ? I haven't seen a bug report.
Gimme a couple of days.
-Carl
From: Manuel Op de Coul (2004-09-07)
Subject: Re: [tuning-math] Re: 7-limit 12-note epimorphic scales containing a hexany
Carl wrote:
>By the way, Scala's 'compare scale' feature is broken in
>several ways, so I wouldn't trust it.
Still haven't seen anything substantiating this allegation.
Manuel
From: Carl Lumma (2004-09-07)
Subject: Re: [tuning-math] Re: 7-limit 12-note epimorphic scales containing a hexany
>Carl wrote:
>
>>By the way, Scala's 'compare scale' feature is broken in
>>several ways, so I wouldn't trust it.
>
>Still haven't seen anything substantiating this allegation.
>
>Manuel
Thanks for reminding me. I'll get this to you this week.
-Carl
From: Carl Lumma (2004-09-07)
Subject: Re: 7-limit 12-note epimorphic scales containing a hexany
> >Carl wrote:
> >
> >>By the way, Scala's 'compare scale' feature is broken in
> >>several ways, so I wouldn't trust it.
> >
> >Still haven't seen anything substantiating this allegation.
> >
> >Manuel
>
> Thanks for reminding me. I'll get this to you this week.
The problem is, I forget what I was doing. Kurt saw it;
maybe he remembers.
-Carl