Topic: Some 15-note scale/temperaments
2 scales
| File | Description | Notes | Period (ยข) |
|---|---|---|---|
| fifaug | Three circles of four (56/11)^(1/4) fifths with 11/7 as wolf | 15 | 1200.0 |
| porc15 | Pocupine[15] in 7-limit minimax tuning | 12 | 976.4 |
Thread (5 messages)
From: Gene Ward Smith (2003-12-29) Subject: 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.
From: Paul Erlich (2004-01-03)
Subject: Re: Some 15-note scale/temperaments
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 (2004-01-06)
Subject: Some 15-note scale/temperaments
>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
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From: wallyesterpaulrus (2004-01-06) Subject: Re: Some 15-note scale/temperaments --- 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
From: stephenszpak (2004-01-06)
Subject: Re: Some 15-note scale/temperaments
--- 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