Some 15-note scale/temperaments

These seem to be popular of late, so I'm giving two useful examples.

The first consists of three circles of (56/11)^(1/4) fifth, completed
by a "wolf" of exactly 11/7. Notable is how two different sizes of
major third show up in a 5/4-5/4-11/7 176/175-magical augmented triad.

! fifaug.scl
Three circles of four (56/11)^(1/4) fifths with 11/7 as wolf
15
!
95.623008
113.130973
208.753982
304.376991
400.000000
495.623008
513.130973
608.753982
704.376991
800.000000
895.623008
913.130973
1008.753982
1104.376991
1200.000000

Here are the 0-4-9 pattern triads:

[400.000000, 304.376992, 704.376992]
[400.000000, 382.492035, 782.492035]
[400.000000, 304.376991, 704.376991]
[400.000000, 304.376991, 704.376991]
[400.000000, 304.376991, 704.376991]
[400.000000, 304.376992, 704.376992]
[400.000000, 382.492035, 782.492035]
[400.000000, 304.376991, 704.376991]
[400.000000, 304.376991, 704.376991]
[400.000000, 304.376991, 704.376991]
[400.000000, 304.376992, 704.376992]
[400.000000, 382.492035, 782.492035]
[400.000000, 304.376991, 704.376991]
[400.000000, 304.376991, 704.376991]
[400.000000, 304.376991, 704.376991]

Here is Porcupine[15] in the 7-limit minimax tuning (which I picked
since it gives us something close to 11-limit rms tuning.)

! porc15.scl
Pocupine[15] in 7-limit minimax tuning
12
!
60.839199
162.737257
223.576457
325.474514
386.313714
488.211772
549.050971
650.949029
711.788228
813.686286
874.525486
976.423543
1037.262743
1139.160801
1200.000000

Here are the 0-4-9 triads with this tuning. I apologize to Jon for the
entirely unnessessary and useless precision.

[427.3725722703284, 325.4745144540656, 752.8470867243940]
[386.3137138648360, 325.4745144540656, 711.7882283189016]
[427.3725722703280, 325.4745144540656, 752.8470867243936]
[386.3137138648360, 325.4745144540654, 711.7882283189014]
[427.3725722703280, 325.4745144540660, 752.8470867243940]
[386.3137138648360, 325.4745144540656, 711.7882283189016]
[427.3725722703280, 284.4156560485732, 711.7882283189012]
[386.3137138648358, 325.4745144540658, 711.7882283189016]
[427.3725722703284, 284.4156560485732, 711.7882283189016]
[386.3137138648360, 325.4745144540656, 711.7882283189016]
[386.3137138648356, 325.4745144540660, 711.7882283189016]
[386.3137138648360, 325.4745144540656, 711.7882283189016]
[386.3137138648362, 325.4745144540656, 711.7882283189018]
[386.3137138648356, 325.4745144540656, 711.7882283189012]
[386.3137138648360, 325.4745144540656, 711.7882283189016]

The wolf fifths are less than a cent away from being 17/11's, which
may or may not inspire you. The sharp major thirds are, in this
version of porky, exact 32/25's, but porcupine eats 225/224 and this
is supposed to count as a 9/7 therefore.

Oh, and how could we have forgotten Blackwood[15]:

0
84.66
155.34
240.00
324.66
395.34
480.00
564.66
635.34
720.00
804.66
875.34
960.00
1044.66
1115.34
(1200.00)

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> These seem to be popular of late, so I'm giving two useful examples.
>
> The first consists of three circles of (56/11)^(1/4) fifth,
completed
> by a "wolf" of exactly 11/7. Notable is how two different sizes of
> major third show up in a 5/4-5/4-11/7 176/175-magical augmented
triad.
>
> ! fifaug.scl
> Three circles of four (56/11)^(1/4) fifths with 11/7 as wolf
> 15
> !
> 95.623008
> 113.130973
> 208.753982
> 304.376991
> 400.000000
> 495.623008
> 513.130973
> 608.753982
> 704.376991
> 800.000000
> 895.623008
> 913.130973
> 1008.753982
> 1104.376991
> 1200.000000
>
> Here are the 0-4-9 pattern triads:
>
> [400.000000, 304.376992, 704.376992]
> [400.000000, 382.492035, 782.492035]
> [400.000000, 304.376991, 704.376991]
> [400.000000, 304.376991, 704.376991]
> [400.000000, 304.376991, 704.376991]
> [400.000000, 304.376992, 704.376992]
> [400.000000, 382.492035, 782.492035]
> [400.000000, 304.376991, 704.376991]
> [400.000000, 304.376991, 704.376991]
> [400.000000, 304.376991, 704.376991]
> [400.000000, 304.376992, 704.376992]
> [400.000000, 382.492035, 782.492035]
> [400.000000, 304.376991, 704.376991]
> [400.000000, 304.376991, 704.376991]
> [400.000000, 304.376991, 704.376991]
>
> Here is Porcupine[15] in the 7-limit minimax tuning (which I picked
> since it gives us something close to 11-limit rms tuning.)
>
> ! porc15.scl
> Pocupine[15] in 7-limit minimax tuning
> 12
> !
> 60.839199
> 162.737257
> 223.576457
> 325.474514
> 386.313714
> 488.211772
> 549.050971
> 650.949029
> 711.788228
> 813.686286
> 874.525486
> 976.423543
> 1037.262743
> 1139.160801
> 1200.000000
>
> Here are the 0-4-9 triads with this tuning. I apologize to Jon for
the
> entirely unnessessary and useless precision.
>
> [427.3725722703284, 325.4745144540656, 752.8470867243940]
> [386.3137138648360, 325.4745144540656, 711.7882283189016]
> [427.3725722703280, 325.4745144540656, 752.8470867243936]
> [386.3137138648360, 325.4745144540654, 711.7882283189014]
> [427.3725722703280, 325.4745144540660, 752.8470867243940]
> [386.3137138648360, 325.4745144540656, 711.7882283189016]
> [427.3725722703280, 284.4156560485732, 711.7882283189012]
> [386.3137138648358, 325.4745144540658, 711.7882283189016]
> [427.3725722703284, 284.4156560485732, 711.7882283189016]
> [386.3137138648360, 325.4745144540656, 711.7882283189016]
> [386.3137138648356, 325.4745144540660, 711.7882283189016]
> [386.3137138648360, 325.4745144540656, 711.7882283189016]
> [386.3137138648362, 325.4745144540656, 711.7882283189018]
> [386.3137138648356, 325.4745144540656, 711.7882283189012]
> [386.3137138648360, 325.4745144540656, 711.7882283189016]
>
> The wolf fifths are less than a cent away from being 17/11's, which
> may or may not inspire you. The sharp major thirds are, in this
> version of porky, exact 32/25's, but porcupine eats 225/224 and this
> is supposed to count as a 9/7 therefore.

>From: "Stephen Szpak" <stephen_szpak@hotmail.com>
>To: tuning@yahoogroups.com
>CC: szpakmusic@hotmail.com
>Subject: Fwd: Re: understanding 15 EDO
>Date: Wed, 31 Dec 2003 21:07:12 -0500
>
>--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>--- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
>wrote:
>>
>> (If anyone wants to comment to this that's fine. Please try to be
>as
>>simple as possible.)
>>
STEPHEN SZPAK WRITES:::

This "limit" stuff is way hard. Paul Erlich comments below that the Kleismic-15
scale is 5-limit. How can it be 5-limit with the inclusion of the
11/8 at 565.84 cents???

>===========================================================
>You can view it as altering 15-equal, or perhaps better is to view 15-
>equal as the alteration of such scales. Gene gave you a couple of
>great examples. Another important one, at least for 5-limit, would be
>Kleismic-15:
>
> 0
> 68.319
> 180.44
> 248.76
> 317.08
> 385.4
> 497.52
> 565.84
> 634.16
> 702.48
> 814.6
> 882.92
> 951.24
> 1019.6
> 1131.7
>
>Lots of quite pure major and minor triads here.
stephen_szpak@hotmail.com

_________________________________________________________________
Take advantage of our limited-time introductory offer for dial-up Internet access. http://join.msn.com/?page=dept/dialup

--- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
wrote:
>
>
>
> >From: "Stephen Szpak" <stephen_szpak@h...>
> >To: tuning@yahoogroups.com
> >CC: szpakmusic@h...
> >Subject: Fwd: Re: understanding 15 EDO
> >Date: Wed, 31 Dec 2003 21:07:12 -0500
> >
> >--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> >--- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
> >wrote:
> >>
> >> (If anyone wants to comment to this that's fine. Please try to
be
> >as
> >>simple as possible.)
> >>
> STEPHEN SZPAK WRITES:::
>
> This "limit" stuff is way hard. Paul Erlich comments below
that the
> Kleismic-15
> scale is 5-limit. How can it be 5-limit with the
inclusion of the
> 11/8 at 565.84 cents???

I mean the precise tuning is only optimized with 5-limit harmony,
i.e., triads, in mind. Any approximations to higher-limit intervals
are only there by accident -- much like what happens with diminished
fourths approximating 5:4s in Pythagorean, and augmented sixths
approximating 7:4s in meantone. However, there are probably
temperaments very similar to this one (one of which might also be
called "Kleismic") where the corresponding pitch *does* intentionally
approximate 11/8 -- the approximation will almost certainly be better
(this one's 15 cents off), and all the intervals will change slightly
(since the generator will be slightly different). Perhaps Gene can
furnish a few examples.

Thanks for hangin',
Paul

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
> wrote:
> >
> >
> >
> > >From: "Stephen Szpak" <stephen_szpak@h...>
> > >To: tuning@yahoogroups.com
> > >CC: szpakmusic@h...
> > >Subject: Fwd: Re: understanding 15 EDO
> > >Date: Wed, 31 Dec 2003 21:07:12 -0500
> > >
> > >--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> > >--- In tuning@yahoogroups.com, "Stephen Szpak"
<stephen_szpak@h...>
> > >wrote:
> > >>
> > >> (If anyone wants to comment to this that's fine. Please try
to
> be
> > >as
> > >>simple as possible.)
> > >>
> > STEPHEN SZPAK WRITES:::
> >
> > This "limit" stuff is way hard. Paul Erlich comments below
> that the
> > Kleismic-15
> > scale is 5-limit. How can it be 5-limit with the
> inclusion of the
> > 11/8 at 565.84 cents???
>
> I mean the precise tuning is only optimized with 5-limit harmony,
> i.e., triads, in mind. Any approximations to higher-limit intervals
> are only there by accident -- much like what happens with
diminished
> fourths approximating 5:4s in Pythagorean, and augmented sixths
> approximating 7:4s in meantone. However, there are probably
> temperaments very similar to this one (one of which might also be
> called "Kleismic") where the corresponding pitch *does*
intentionally
> approximate 11/8 -- the approximation will almost certainly be
better
> (this one's 15 cents off), and all the intervals will change
slightly
> (since the generator will be slightly different). Perhaps Gene can
> furnish a few examples.
>
> Thanks for hangin',
> Paul

STEPHEN SZPAK WRITES:::::::::

I don't know why I thought the 565 was 4 cents from 551!
I think I understand the rest. Thanks.
Stephen Szpak