A trick for Tim Reeves: porcupine temperament

Alright Tim, consider the following. Let's start with harmonics 8-16,
which using the notation you've described before, lays out as follows:

8:9:10:11:12:13:14:15:16 - A:B:C:D:E:F:G:H:A

You've described this scale as a sort of maximally "natural" scale
that basic mathematics reveals. Now, as you're also aware, the usual
7-note diatonic scale is not justly intoned, and hence diverges from
this in a way that you might perhaps consequently call "less natural."
However, this scale is musically useful enough that billions of people
have been entranced by it throughout the course of history, despite
its divergence from the harmonic series. So one good approach to
finding new possibilities is to hedonistically say "to hell with
naturalness" and find other scales that likewise warp and diverge from
the harmonic series in a useful and musically entrancing way.

Let's start with that A-B-C-D-E pentachord, which is in a perfect JI
8:9:10:11:12 ratio. There's an awful lot of intervals in this
pentachord that are close in size to one another, so one way to start
is to deliberately detune things so various intervals in this
pentachord end up being the same size. The usual way to do this is to
make the A-B-C equal in size by flattening the B (or sharpening the C)
a bit, until the 9/8 and 10/9 become equal - this is the familiar
"meantone" temperament that we're all used to. If you do some tricks
and reflect this equal A-B-C pattern along the scale in an interesting
way, you can arrive at a sort of maximally symmetric scale, which ends
up being the diatonic scale that we're all familiar with. Part of the
point of this list is to get away from that, so a good way to go is to
equate other intervals and reflect them instead.

Let's break with tradition and NOT make 9/8 and 10/9 equal. Instead,
we're going to make 10/9, 11/10, and 12/11 are all equal, so that A-B
is a big interval, and B-C-D-E are all small intervals that are some
kind of average of 10/9, 11/10, and 12/11. We'll distribute the
tempering error evenly throughout this process, and if you work the
math out that means we sharpen the B a bit, flatten the C slightly,
flatten the D, and sharpen the E slightly as well, until B-C-D-E are
all separated by a constant interval of about 163 cents, and the A-B
is about 224 cents. So the 5-note scale ends up being, in cents - 0.0,
223.51496, 386.26247, 549.00997, 711.75748. If you mash it, it still
sounds like 8:9:10:11:12, but warbles a little bit.

Note that that B-E is a perfect fourth (which is now slightly flat
because of all the tempering). Well, there's also a perfect fourth
between that highest note - the E - and the octave, right? So we're
going to hedonistically throw caution to the wind, run all over the
12:13:14:15:16, and just double the pattern of steps. If we do this,
we arrive at another one of these maximally symmetric scales, which
comprises part of what is called "porcupine" temperament:

! /Users/mike/Desktop/porcupine.scl
!
POTE porcupine[7] MOS
7
!
223.51496
386.26247
549.00997
711.75748
874.50499
1037.25249
2/1

The latter half is completely divergent from 12:13:14:15:16, but this
to my ears adds harmonic variety and sounds awesome anyway.

If you find that you like this scale, there are two other temperaments
right behind it called "negri" and "mohajira," both of which are also
awesome.

-Mike

Mike,

That was a helpful description of Procupine. Thanks.

Regards,
Jake

On 7/11/11, Mike Battaglia <battaglia01@...> wrote:
> Alright Tim, consider the following. Let's start with harmonics 8-16,
> which using the notation you've described before, lays out as follows:
>
> 8:9:10:11:12:13:14:15:16 - A:B:C:D:E:F:G:H:A
>
> You've described this scale as a sort of maximally "natural" scale
> that basic mathematics reveals. Now, as you're also aware, the usual
> 7-note diatonic scale is not justly intoned, and hence diverges from
> this in a way that you might perhaps consequently call "less natural."
> However, this scale is musically useful enough that billions of people
> have been entranced by it throughout the course of history, despite
> its divergence from the harmonic series. So one good approach to
> finding new possibilities is to hedonistically say "to hell with
> naturalness" and find other scales that likewise warp and diverge from
> the harmonic series in a useful and musically entrancing way.
>
> Let's start with that A-B-C-D-E pentachord, which is in a perfect JI
> 8:9:10:11:12 ratio. There's an awful lot of intervals in this
> pentachord that are close in size to one another, so one way to start
> is to deliberately detune things so various intervals in this
> pentachord end up being the same size. The usual way to do this is to
> make the A-B-C equal in size by flattening the B (or sharpening the C)
> a bit, until the 9/8 and 10/9 become equal - this is the familiar
> "meantone" temperament that we're all used to. If you do some tricks
> and reflect this equal A-B-C pattern along the scale in an interesting
> way, you can arrive at a sort of maximally symmetric scale, which ends
> up being the diatonic scale that we're all familiar with. Part of the
> point of this list is to get away from that, so a good way to go is to
> equate other intervals and reflect them instead.
>
> Let's break with tradition and NOT make 9/8 and 10/9 equal. Instead,
> we're going to make 10/9, 11/10, and 12/11 are all equal, so that A-B
> is a big interval, and B-C-D-E are all small intervals that are some
> kind of average of 10/9, 11/10, and 12/11. We'll distribute the
> tempering error evenly throughout this process, and if you work the
> math out that means we sharpen the B a bit, flatten the C slightly,
> flatten the D, and sharpen the E slightly as well, until B-C-D-E are
> all separated by a constant interval of about 163 cents, and the A-B
> is about 224 cents. So the 5-note scale ends up being, in cents - 0.0,
> 223.51496, 386.26247, 549.00997, 711.75748. If you mash it, it still
> sounds like 8:9:10:11:12, but warbles a little bit.
>
> Note that that B-E is a perfect fourth (which is now slightly flat
> because of all the tempering). Well, there's also a perfect fourth
> between that highest note - the E - and the octave, right? So we're
> going to hedonistically throw caution to the wind, run all over the
> 12:13:14:15:16, and just double the pattern of steps. If we do this,
> we arrive at another one of these maximally symmetric scales, which
> comprises part of what is called "porcupine" temperament:
>
> ! /Users/mike/Desktop/porcupine.scl
> !
> POTE porcupine[7] MOS
> 7
> !
> 223.51496
> 386.26247
> 549.00997
> 711.75748
> 874.50499
> 1037.25249
> 2/1
>
> The latter half is completely divergent from 12:13:14:15:16, but this
> to my ears adds harmonic variety and sounds awesome anyway.
>
> If you find that you like this scale, there are two other temperaments
> right behind it called "negri" and "mohajira," both of which are also
> awesome.
>
> -Mike
>
>
> ------------------------------------
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Hi Mike
Thanks for the suggestion. I had never heard of the porcupine tuning but it does look interesting, especially in your approach,  I like your hedonistic attitude towards all of this , it makes doing all this kind of fun instead of boring and tedious.   can't wait to hear it and try it out
Tim
.--- On Mon, 7/11/11, Jake Freivald <jdfreivald@...> wrote:

From: Jake Freivald <jdfreivald@...>
Subject: Re: [tuning] A trick for Tim Reeves: porcupine temperament
To: tuning@yahoogroups.com
Date: Monday, July 11, 2011, 5:03 AM

Mike,

That was a helpful description of Procupine. Thanks.

Regards,
Jake

On 7/11/11, Mike Battaglia <battaglia01@gmail.com> wrote:
> Alright Tim, consider the following. Let's start with harmonics 8-16,
> which using the notation you've described before, lays out as follows:
>
> 8:9:10:11:12:13:14:15:16 - A:B:C:D:E:F:G:H:A
>
> You've described this scale as a sort of maximally "natural" scale
> that basic mathematics reveals. Now, as you're also aware, the usual
> 7-note diatonic scale is not justly intoned, and hence diverges from
> this in a way that you might perhaps consequently call "less natural."
> However, this scale is musically useful enough that billions of people
> have been entranced by it throughout the course of history, despite
> its divergence from the harmonic series. So one good approach to
> finding new possibilities is to hedonistically say "to hell with
> naturalness" and find other scales that likewise warp and diverge from
> the harmonic series in a useful and musically entrancing way.
>
> Let's start with that A-B-C-D-E pentachord, which is in a perfect JI
> 8:9:10:11:12 ratio. There's an awful lot of intervals in this
> pentachord that are close in size to one another, so one way to start
> is to deliberately detune things so various intervals in this
> pentachord end up being the same size. The usual way to do this is to
> make the A-B-C equal in size by flattening the B (or sharpening the C)
> a bit, until the 9/8 and 10/9 become equal - this is the familiar
> "meantone" temperament that we're all used to. If you do some tricks
> and reflect this equal A-B-C pattern along the scale in an interesting
> way, you can arrive at a sort of maximally symmetric scale, which ends
> up being the diatonic scale that we're all familiar with. Part of the
> point of this list is to get away from that, so a good way to go is to
> equate other intervals and reflect them instead.
>
> Let's break with tradition and NOT make 9/8 and 10/9 equal. Instead,
> we're going to make 10/9, 11/10, and 12/11 are all equal, so that A-B
> is a big interval, and B-C-D-E are all small intervals that are some
> kind of average of 10/9, 11/10, and 12/11. We'll distribute the
> tempering error evenly throughout this process, and if you work the
> math out that means we sharpen the B a bit, flatten the C slightly,
> flatten the D, and sharpen the E slightly as well, until B-C-D-E are
> all separated by a constant interval of about 163 cents, and the A-B
> is about 224 cents. So the 5-note scale ends up being, in cents - 0.0,
> 223.51496, 386.26247, 549.00997, 711.75748. If you mash it, it still
> sounds like 8:9:10:11:12, but warbles a little bit.
>
> Note that that B-E is a perfect fourth (which is now slightly flat
> because of all the tempering). Well, there's also a perfect fourth
> between that highest note - the E - and the octave, right? So we're
> going to hedonistically throw caution to the wind, run all over the
> 12:13:14:15:16, and just double the pattern of steps. If we do this,
> we arrive at another one of these maximally symmetric scales, which
> comprises part of what is called "porcupine" temperament:
>
> ! /Users/mike/Desktop/porcupine.scl
> !
> POTE porcupine[7] MOS
>  7
> !
>  223.51496
>  386.26247
>  549.00997
>  711.75748
>  874.50499
>  1037.25249
>  2/1
>
> The latter half is completely divergent from 12:13:14:15:16, but this
> to my ears adds harmonic variety and sounds awesome anyway.
>
> If you find that you like this scale, there are two other temperaments
> right behind it called "negri" and "mohajira," both of which are also
> awesome.
>
> -Mike
>
>
> ------------------------------------
>
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> of these addresses (from the address at which you receive the list):
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