Some Porcupine theory - Porcupine in 22-tet is a miracle

Here's the SITREP so far on Porcupine[7].

Porcupine works a bit differently than meantone because of the two
MOS's being right next to each other in size. For starters, as the
chroma is c=L-s, you end up with a different chroma if you're working
with porcupine[7] or porcupine[8] - ~50 or ~100 cents, respectively.
Furthermore, the L-s for porcupine[7] happens to be the s for
porcupine[8], which confuses things.

This means that there are two intervals that you can "chromatically"
alter a note by; we will be making use of both of these. These
intervals are, relative to porcupine[7], what I called c_1 and c_2
before: c_1 = L-s = ~55 cents. c_2 = |s-c| = ~105 cents. So the 105
cent interval is the second-level chroma for porcupine[7], but the
first level chroma for porcupine[8].

The fact that there are two intervals means that there are two
different patterns to work out: the more "destructive" 100 cent
alteration, and the more "refined" 50 cent alteration. Because of the
size of these intervals I will refer to the 100 cent one as the chroma
and the 50 cent one as the diesis. We will work with porcupine[7] for
now. I will also stick with only diesic alterations for now.

First off, I'll define porcupine notation. I'll use the one that I've
seen both Herman Miller and Paul Erlich using, but with a few twists.
So if we assume the generator is going up, then we'll call the first
note "A" and go from there:

Porcupine[7]: A B C D E F G A = ssssssL
Porcupine[8]: A B C D E F G H A = LLLLLLLs

Some important porcupine[7] modes:
Major modes:
G A B C D E F G = Lssssss <-- Maybe this should be the "Major" mode.
F G A B C D E F = sLsssss <-- The whole tone between the G and A makes
this feel more "Suspended" than the above, perhaps in a "lydian" sense

Minor modes:
D E F G A B C D = sssLsss <-- The symmetric mode, kind of like Dorian.
Maybe this should be the "Minor" mode.
E F G A B C D E = ssLssss <-- Or maybe this should be the "Minor"
mode, because there's a 8:11:12 over the root and an 8:11:12 over the
minor third, making crazy interlocking 8:11:12 minor triads. Very
interesting.

The rest of the modes don't have a perfect fifth. But, really, you
need to get away from MOS's anyway. You've been using them too much,
haven't you? Yes, you have. So here's your chance to break free: you
can fix some of these flat fifth modes by raising the fifth, which
brings us into MODMOS territory.

We only want to look at MODMOS's that are proper today, and if you
remember the list of 22-tet proper scales there were a million of
them. How will we ever keep track of them all? Luckily, this turns out
to be extremely easy. If you're in 22-tet, we have the following
mathematical miracle on our side:

You can modify any one of the notes in this scale in either direction
by the L-s or ~50 cents alteration vector, and the scale will be both
proper and still fit within the 15-note chromatic MOS. In fact, you
can modify as many as you want, and whichever ones you want, and as
long as you're modifying them uniformly flat or sharp, the scale will
be proper. If you're modifying multiple mixtures of sharps and flats,
the scale will "usually" be proper with a few exceptions that
generally involve augmenting the large step or augmenting the major
third into a 9/7. But if you're really making use of that much
chromaticism then you're probably not thinking in terms of propriety
anyway.

AGAIN: EVERY SINGLE ONE OF THE 1-DIESIS ALTERATIONS OF PORCUPINE[7] IS
PROPER IN 22-TET AND FITS IN THE 15-NOTE MOS. NO NEED TO MEMORIZE
ANYTHING, IT JUST WORKS NO MATTER WHAT YOU DO. This is apparently a
miracle.

This also applies to generators that are smaller than or equal to that
of 22-tet; if it's smaller some of these scales will be strictly
proper, and if it's larger some of them will be improper.

How do we notate this? I've denoted alteration by the 55 cent "diesis"
sized interval by ^ and v, and alteration by the ~100 cent "chroma"
sized interval by # and b. So check it out:

Periodicity block view (assume octave-inverted generator)

A^ B^ C^ D^ E^ F^ G^ A B C D E F G Av Bv Cv Dv Ev Fv Gv

The first thing that should be evident from all of this is that Av and
what I called "H" earlier are the same thing. H is synonymous with Av,
and this is true of any porcupine tuning independent of the size of
the generator. Now let's look at the porcupine[8] block view:

Ab Bb Cb Db Eb Fb Gb Hb A B C D E F G H A# B# C# D# E# F# G# H#

Very simple. By learning these enharmonically equivalent names, we can
easily train ourselves to think in both porcupine[7] and porcupine[8].
Good stuff.

You could also use "Half sharp" for the diesis and "double sharp" for
the chroma, but in this case the metaphor doesn't work as well,
because A^^ only equals A# in 22-tet. Otherwise they're different
notes.

So here is the list of every single diesic alteration MODMOS (!!!
means that I think it's particularly interesting):

Alterations down:
Gv A B C D E F Gv - I put this in mainly for completeness. Different
modes of this scale are usually better than this one.
G Av B C D E F G - Just another mode of porcupine major, the
"suspended" mode (needs better name)
G A Bv C D E F G - (!!!) G porcupine major with a 6/5 instead of 5/4.
This is the same as the "interlocking 8:11:12" mode I said above, but
with a 9/8 over the root instead of 10/9, which strengthens it
tonally.
G A B Cv D E F G - (!!!) G porcupine major with a 4/3 instead of an
11/8, sounds "Ionian"
G A B C Dv E F G - G porcupine major with a flatted fifth; why you'd
use this I have no idea
G A B C D Ev F G - (!!!) G porcupine major with the 5/3 replaced with
8/5; sounds "Radiohead"-ish
G A B C D E Fv G - (!!!!!) G porcupine major with the 11/6 replaced
with a 7/4. Sounds super-otonal and resonant. This one is a winner

Alterations up:
G^ A B C D E F G^ - Just another mode of porcupine major
G A^ B C D E F G - Augmented second, not sure what the point is
G A B^ C D E F G - G porcupine major with a 9/7 instead of 5/4, use if
you hate your listeners
G A B C^ D E F G - (!!) G porcupine major with a 45/32 instead of an
11/8; having the 5/4 over the 9/8 makes this sound kind of "Lydian"
G A B C D^ E F G - (!) G porcupine major with a raised fifth; an
"augmented" mode which might be nice for some dissonance
G A B C D E^ F G - (!!) G porcupine major with the 5/3 raised to
become 12/7; makes the whole thing sound "brighter" and more "open"
G A B C D E F^ G - (!!!!!) G porcupine major with the 11/6 replaced
with a 15/8. Sounds "major" in the diatonic sense. Another winner

Each one of these 14 scales has 7 modes, so you might think you'll
need to spend forever learning them. The trick is to just think in
terms of intervals - do you want an 11/8 in this scale? Or maybe a
4/3? Or perhaps you want 45/32? Do you want 11/6, or 15/8, or 7/4? So
just remember the rule, again: start with the porcupine[7] mode and
just alter things individually, and whatever alteration you make is a
proper and valid MODMOS. No need thinking about it any more than that.

Next up will be the #/b alterations.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Here's the SITREP so far on Porcupine[7].
>
> Porcupine works a bit differently than meantone because of the two
> MOS's being right next to each other in size. For starters, as the
> chroma is c=L-s, you end up with a different chroma if you're working
> with porcupine[7] or porcupine[8] - ~50 or ~100 cents, respectively.
> Furthermore, the L-s for porcupine[7] happens to be the s for
> porcupine[8], which confuses things.
>
> This means that there are two intervals that you can "chromatically"
> alter a note by; we will be making use of both of these. These
> intervals are, relative to porcupine[7], what I called c_1 and c_2
> before: c_1 = L-s = ~55 cents. c_2 = |s-c| = ~105 cents. So the 105
> cent interval is the second-level chroma for porcupine[7], but the
> first level chroma for porcupine[8].
>
> The fact that there are two intervals means that there are two
> different patterns to work out: the more "destructive" 100 cent
> alteration, and the more "refined" 50 cent alteration. Because of the
> size of these intervals I will refer to the 100 cent one as the chroma
> and the 50 cent one as the diesis. We will work with porcupine[7] for
> now. I will also stick with only diesic alterations for now.
>
> First off, I'll define porcupine notation. I'll use the one that I've
> seen both Herman Miller and Paul Erlich using, but with a few twists.
> So if we assume the generator is going up, then we'll call the first
> note "A" and go from there:
>
> Porcupine[7]: A B C D E F G A = ssssssL
> Porcupine[8]: A B C D E F G H A = LLLLLLLs
>
> Some important porcupine[7] modes:
> Major modes:
> G A B C D E F G = Lssssss <-- Maybe this should be the "Major" mode.
> F G A B C D E F = sLsssss <-- The whole tone between the G and A makes
> this feel more "Suspended" than the above, perhaps in a "lydian" sense
>
> Minor modes:
> D E F G A B C D = sssLsss <-- The symmetric mode, kind of like Dorian.
> Maybe this should be the "Minor" mode.
> E F G A B C D E = ssLssss <-- Or maybe this should be the "Minor"
> mode, because there's a 8:11:12 over the root and an 8:11:12 over the
> minor third, making crazy interlocking 8:11:12 minor triads. Very
> interesting.
>
> The rest of the modes don't have a perfect fifth. But, really, you
> need to get away from MOS's anyway. You've been using them too much,
> haven't you? Yes, you have. So here's your chance to break free: you
> can fix some of these flat fifth modes by raising the fifth, which
> brings us into MODMOS territory.
>
> We only want to look at MODMOS's that are proper today, and if you
> remember the list of 22-tet proper scales there were a million of
> them. How will we ever keep track of them all? Luckily, this turns out
> to be extremely easy. If you're in 22-tet, we have the following
> mathematical miracle on our side:
>
> You can modify any one of the notes in this scale in either direction
> by the L-s or ~50 cents alteration vector, and the scale will be both
> proper and still fit within the 15-note chromatic MOS. In fact, you
> can modify as many as you want, and whichever ones you want, and as
> long as you're modifying them uniformly flat or sharp, the scale will
> be proper. If you're modifying multiple mixtures of sharps and flats,
> the scale will "usually" be proper with a few exceptions that
> generally involve augmenting the large step or augmenting the major
> third into a 9/7. But if you're really making use of that much
> chromaticism then you're probably not thinking in terms of propriety
> anyway.
>
> AGAIN: EVERY SINGLE ONE OF THE 1-DIESIS ALTERATIONS OF PORCUPINE[7] IS
> PROPER IN 22-TET AND FITS IN THE 15-NOTE MOS. NO NEED TO MEMORIZE
> ANYTHING, IT JUST WORKS NO MATTER WHAT YOU DO. This is apparently a
> miracle.
>
> This also applies to generators that are smaller than or equal to that
> of 22-tet; if it's smaller some of these scales will be strictly
> proper, and if it's larger some of them will be improper.
>
> How do we notate this? I've denoted alteration by the 55 cent "diesis"
> sized interval by ^ and v, and alteration by the ~100 cent "chroma"
> sized interval by # and b. So check it out:
>
> Periodicity block view (assume octave-inverted generator)
>
> A^ B^ C^ D^ E^ F^ G^ A B C D E F G Av Bv Cv Dv Ev Fv Gv
>
> The first thing that should be evident from all of this is that Av and
> what I called "H" earlier are the same thing. H is synonymous with Av,
> and this is true of any porcupine tuning independent of the size of
> the generator. Now let's look at the porcupine[8] block view:
>
> Ab Bb Cb Db Eb Fb Gb Hb A B C D E F G H A# B# C# D# E# F# G# H#
>
> Very simple. By learning these enharmonically equivalent names, we can
> easily train ourselves to think in both porcupine[7] and porcupine[8].
> Good stuff.
>
> You could also use "Half sharp" for the diesis and "double sharp" for
> the chroma, but in this case the metaphor doesn't work as well,
> because A^^ only equals A# in 22-tet. Otherwise they're different
> notes.
>
> So here is the list of every single diesic alteration MODMOS (!!!
> means that I think it's particularly interesting):
>
> Alterations down:
> Gv A B C D E F Gv - I put this in mainly for completeness. Different
> modes of this scale are usually better than this one.
> G Av B C D E F G - Just another mode of porcupine major, the
> "suspended" mode (needs better name)
> G A Bv C D E F G - (!!!) G porcupine major with a 6/5 instead of 5/4.
> This is the same as the "interlocking 8:11:12" mode I said above, but
> with a 9/8 over the root instead of 10/9, which strengthens it
> tonally.
> G A B Cv D E F G - (!!!) G porcupine major with a 4/3 instead of an
> 11/8, sounds "Ionian"
> G A B C Dv E F G - G porcupine major with a flatted fifth; why you'd
> use this I have no idea
> G A B C D Ev F G - (!!!) G porcupine major with the 5/3 replaced with
> 8/5; sounds "Radiohead"-ish
> G A B C D E Fv G - (!!!!!) G porcupine major with the 11/6 replaced
> with a 7/4. Sounds super-otonal and resonant. This one is a winner
>
> Alterations up:
> G^ A B C D E F G^ - Just another mode of porcupine major
> G A^ B C D E F G - Augmented second, not sure what the point is
> G A B^ C D E F G - G porcupine major with a 9/7 instead of 5/4, use if
> you hate your listeners
> G A B C^ D E F G - (!!) G porcupine major with a 45/32 instead of an
> 11/8; having the 5/4 over the 9/8 makes this sound kind of "Lydian"
> G A B C D^ E F G - (!) G porcupine major with a raised fifth; an
> "augmented" mode which might be nice for some dissonance
> G A B C D E^ F G - (!!) G porcupine major with the 5/3 raised to
> become 12/7; makes the whole thing sound "brighter" and more "open"
> G A B C D E F^ G - (!!!!!) G porcupine major with the 11/6 replaced
> with a 15/8. Sounds "major" in the diatonic sense. Another winner
>
> Each one of these 14 scales has 7 modes, so you might think you'll
> need to spend forever learning them. The trick is to just think in
> terms of intervals - do you want an 11/8 in this scale? Or maybe a
> 4/3? Or perhaps you want 45/32? Do you want 11/6, or 15/8, or 7/4? So
> just remember the rule, again: start with the porcupine[7] mode and
> just alter things individually, and whatever alteration you make is a
> proper and valid MODMOS. No need thinking about it any more than that.
>
> Next up will be the #/b alterations.
>
> -Mike
>
This is interesting, have you perchance looked at Erlich's paper
"Tuning, Tonality and Twenty-two Tone Temperament" (On main page links). I can't remember if he covers porcupine much there. I see
you mentioned him so perhaps you have read it....

I also have a paper on 22-tET, (on Affine(22), Z-relations etc. but
I will admit it is better fit for yahoo musicalsettheory than yahoo tuning-math. If you are interested in musical settheory discussions, I started that group back in November).

TT&TTTT paper has all sorts of cool horograms and also a new keyboard layout idea for 22t-ET.

- Paul