Topic: RE: [tuning-math] Digest Number 1238

2 scales

Thread (10 messages)

From: Yahya Abdal-Aziz (2005-03-22)
Subject: RE: [tuning-math] Digest Number 1238

Hi Gene,

1.  Have you devised a naming scheme for the holey
scales yet?

---

2.  We can decompose the ball 
[1, 7/6, 5/4, 11/8, 3/2, 5/3, 7/4, 11/6]
into
[1, 7/6, 5/4, 11/8, 3/2, 5/3, 7/4, 11/6] =
[1, 7/6, 5/3, 11/6] U [5/4, 11/8, 3/2, 7/4]

This means that, musically, we could characterise it
as a tetrad together with its transposition by a fifth, 
with the last two notes octave reduced.  This would
seem to offer many compositional uses.

---

3.  In integers, the ratios multiply out to:
{24, 28, 40, 44} U {36, 42, 30, 33}
= {24, 28, 30, 33, 36, 40, 42, 44}
with steps of size 4, 2, 3, 3, 4, 2, and 2
(48ths of a string length? quarter-tones?)

== C D Eb F- Gb Ab Bbb Cbb  (F- is F half flat)
which is clearly not diatonic according to any of the
recent "definitions" :-)

Regards,
Yahya


-----Original Message-----
________________________________________________________________________
   Date: Mon, 21 Mar 2005 08:39:33 -0000
   From: "Gene Ward Smith" <[email protected]>
Subject: Re: 11-limit holey scales 1


--- In [email protected], "Gene Ward Smith" <gwsmith@s...>
wrote:

> Shell 1 radius^2 = 8/5
> {|1 0 0 0>, |0 1 0 0>, |-1 0 1 0>, |-1 0 0 1>,
> |0 0 1 0>, |0 0 0 0>, |-1 1 0 0>, |0 0 0 1>}
> 
> 8 note ball
> [1, 7/6, 5/4, 11/8, 3/2, 5/3, 7/4, 11/6]

This can be characterized as [3,5,7,11] union 3*[3,5,7,11].
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From: Yahya Abdal-Aziz (2005-03-22)
Subject: RE: [tuning-math] Digest Number 1238

Gene,

Isn't that a 12-note scale?

Regards,
Yahya


-----Original Message-----
________________________________________________________________________
   Date: Sun, 20 Mar 2005 23:42:07 -0000
   From: "Gene Ward Smith" <[email protected]>
Subject: 11-limit holey scales 2


This time the hole is around [0, 1/2, 1/2, 1/2].

Shell 1 radius^2 = 2
g1 := {5, 7, 11, 15, 21, 33, 35, 55, 77, 77/3, 35/3, 55/3}

Ball scale 8 notes
[1, 35/33, 7/6, 5/4, 14/11, 15/11, 3/2, 35/22, 5/3, 7/4, 20/11, 21/11]

...
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From: Gene Ward Smith (2005-03-22)
Subject: Re: Digest Number 1238

--- In [email protected], "Yahya Abdal-Aziz" <yahya@m...> wrote:
> 
> 
> Hi Gene,
> 
> 1.  Have you devised a naming scheme for the holey
> scales yet?

No, but I was thinking the 12-note holes I described could be called
"bihexanies" since they are the union of two hexanies. The 12-note
hole is two chords a fifth apart, which is a fairly obvious way to
make a scale but it's more interesting if the scale you get is a hole.

> 3.  In integers, the ratios multiply out to:
> {24, 28, 40, 44} U {36, 42, 30, 33}
> = {24, 28, 30, 33, 36, 40, 42, 44}
> with steps of size 4, 2, 3, 3, 4, 2, and 2
> (48ths of a string length? quarter-tones?)

It's 11-limit JI, so any good 11-limit system works, and in particular
the twelfth-tones of 72-equal would be fine.

> == C D Eb F- Gb Ab Bbb Cbb  (F- is F half flat)
> which is clearly not diatonic according to any of the
> recent "definitions" :-)

That discussion hasn't made much sense to me.

From: Gene Ward Smith (2005-03-22)
Subject: Re: Digest Number 1238

--- In [email protected], "Yahya Abdal-Aziz" <yahya@m...> wrote:
> 
> Gene,
> 
> Isn't that a 12-note scale?

Yes, it's a "bihexany", and I think an interesting scale for people to
investigate who want a 12-note, 11-limit JI scale which is decidedly
and not just tentitively 11-limit. Something of the same idea as the
eikosany, but more manageable. An interesting feature it has is that
540/539, rank 3 tempering would make a lot of sense for it. That
merges 90/77 with 7/6 and 77/60 with 9/7. 9801/9800 could also go, and
that suggests tempering via octoid, the 152&224 system, with TM basis
{540/549, 1375/1372, 4000/3993}.

From: Gene Ward Smith (2005-03-22)
Subject: Re: Digest Number 1238

--- In [email protected], "Gene Ward Smith" <gwsmith@s...>
wrote:
> 
> --- In [email protected], "Yahya Abdal-Aziz" <yahya@m...>
wrote:
> > 
> > Gene,
> > 
> > Isn't that a 12-note scale?
> 
> Yes, it's a "bihexany", and I think an interesting scale for people to
> investigate who want a 12-note, 11-limit JI scale which is decidedly
> and not just tentitively 11-limit.

Just to amuse y'all, here is a short Schoenberg piece retuned to the
bihexany; of course, it makes a big difference what key we use! This is C.

<url:http://groups.yahoo.com/group/tuning-math/files/Gene/schoe19a-bihexany.mid
>

From: Carl Lumma (2005-03-23)
Subject: Re: [tuning-math] Re: Digest Number 1238

>> Gene,
>> 
>> Isn't that a 12-note scale?
>
>Yes, it's a "bihexany", and I think an interesting scale for people to
>investigate who want a 12-note, 11-limit JI scale which is decidedly
>and not just tentitively 11-limit. Something of the same idea as the
>eikosany, but more manageable. An interesting feature it has is that
>540/539, rank 3 tempering would make a lot of sense for it. That
>merges 90/77 with 7/6 and 77/60 with 9/7. 9801/9800 could also go, and
>that suggests tempering via octoid, the 152&224 system, with TM basis
>{540/549, 1375/1372, 4000/3993}.

Can you give a scala file for such an octoid tempering?

-Carl

From: Gene Ward Smith (2005-03-23)
Subject: Re: Digest Number 1238

--- In [email protected], Carl Lumma <ekin@l...> wrote:

> Can you give a scala file for such an octoid tempering?

Here it is in the 600 division, which is very close to the minimax.
224 is even closer to rms, but since this comes out so evenly I
thought I'd use it:

! bihexany-octoid.scl
Octoid tempering of bihexany, 600-equal
12
!
102.000000
268.000000
386.000000
418.000000
536.000000
702.000000
804.000000
884.000000
970.000000
1034.000000
1120.000000
1200.000000

From: Carl Lumma (2005-03-24)
Subject: Re: [tuning-math] Re: Digest Number 1238

>> Can you give a scala file for such an octoid tempering?
>
>Here it is in the 600 division, which is very close to the minimax.
>224 is even closer to rms, but since this comes out so evenly I
>thought I'd use it:

Whaddabout TOP?

>! bihexany-octoid.scl
>Octoid tempering of bihexany, 600-equal
>12
>!
>102.000000
>268.000000
>386.000000
>418.000000
>536.000000
>702.000000
>804.000000
>884.000000
>970.000000
>1034.000000
>1120.000000
>1200.000000

Thanks!

-Carl

From: Gene Ward Smith (2005-03-24)
Subject: Re: Digest Number 1238

--- In [email protected], Carl Lumma <ekin@l...> wrote:
> >> Can you give a scala file for such an octoid tempering?
> >
> >Here it is in the 600 division, which is very close to the minimax.
> >224 is even closer to rms, but since this comes out so evenly I
> >thought I'd use it:
> 
> Whaddabout TOP?

! bihex-top.scl
Bihexany in octoid TOP tuning
12
!
101.318325
267.590529
385.147324
417.624264
535.181059
701.453263
802.771588
883.963938
969.043793
1033.997672
1119.077527
1200.269877

! bihex540.scl
Bihexany in 540/539 tempering
12
!
101.621102
267.824229
386.256798
417.215888
535.648458
701.851584
803.472686
884.228421
969.675813
1033.620081
1119.067472
1199.823207

From: Carl Lumma (2005-03-24)
Subject: Re: [tuning-math] Re: Digest Number 1238

Danke.

-C.

>> Whaddabout TOP?
>
>! bihex-top.scl
>Bihexany in octoid TOP tuning
>12
>!
>101.318325
>267.590529
>385.147324
>417.624264
>535.181059
>701.453263
>802.771588
>883.963938
>969.043793
>1033.997672
>1119.077527
>1200.269877
>
>! bihex540.scl
>Bihexany in 540/539 tempering
>12
>!
>101.621102
>267.824229
>386.256798
>417.215888
>535.648458
>701.851584
>803.472686
>884.228421
>969.675813
>1033.620081
>1119.067472
>1199.823207