Yahoo Tuning Groups Ultimate Backup tuning 22-equal day!

On 22 Feb 2002 13:54:45 -0800, paul@stretch-music.com wrote:

>hey, it's 2/22/2002.
>
>i hope everyone will play a little 22 today.
>
>let the diatonic scale be *spurlos versenkt* (to quote max meyer)!

Great idea!

http://www.io.com/~hmiller/midi/canon22.mid
http://www.io.com/~hmiller/midi/G22.MID
http://www.io.com/~hmiller/midi/marrgarrel-22.mid
http://www.io.com/~hmiller/midi/krocnard-22.mid
http://www.io.com/~hmiller/midi/porcupine-22.mid
http://www.io.com/~hmiller/midi/star.mid

--
see my music page ---> ---<http://www.io.com/~hmiller/music/index.html>--
hmiller (Herman Miller) "If all Printers were determin'd not to print any
@io.com email password: thing till they were sure it would offend no body,
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🔗Herman Miller <hmiller@IO.COM>

🔗Orphon Soul, Inc. <tuning@orphonsoul.com>

On 2/22/02 4:54 PM, "paul@stretch-music.com" <paul@stretch-music.com> wrote:

> hey, it's 2/22/2002.
>
> i hope everyone will play a little 22 today.
>
> let the diatonic scale be *spurlos versenkt* (to quote max meyer)!
>

YIKES. Only have 45 minutes left in the day. Playtime energies very
depleted as I've been up since 5am.

Tell you what though Paul. You asked me to post the convergence web for 22
so I'll do that for now sans math.

Of course there are an infinite number of them, but in the first few
permutations of 5th limit intervals run through the Brun algorithm, I wind
up referring to this one a lot, from the frequently occuring 22 34 46 53 etc
list:

D#
A E B F# C# *
Bb F C G D A E B F#
* Eb Bb F C G
Db

* - Rounding the half octave up from even temperaments produces a split when
you round back down so failing accuracy considerations, splits are more or
less a 50-50 situation. Here, for 11/22, either use the 64:45 Ab or the
45:32 G#, but don't use both. Meaning, in a just situation, don't change
the context midnote, like playing G up to the half octave and trying to slip
into A. This is part of the whole itch that leads to preconvergence webs.

Anyway. I saw something close to this once. Everything was the same except
the D# and Db were in the center row. This is more than legal within the
22\34\46 preconvergence. (which I used to call Chromatic MeanTone, if you
remember, to you this was more a function of just exploiting the
disappearing diaschisma)

Either way, this gave me an interesting interpolation idea, actually MOLDING
preconvergence grids into discrete webs. The logic you reveal could be as
concealed, as I was just reminded, like the card in your hand when you're
playing Stud Poker, or moreso, Battleship. Only YOU know what the map looks
like, you develop your own logic. Sort of a "custom raga"; 22
cross-reference not intended.

Marc

🔗clumma <carl@lumma.org>

> http://www.io.com/~hmiller/midi/krocnard-22.mid

What is this?

> http://www.io.com/~hmiller/midi/porcupine-22.mid

The version I downloaded has some strangeness -- Media Player
calculates the length at 4:something, but only plays anything
for the first :30 or so; Winamp claims it's a minute exactly,
but crashes :30 or so into it.

By the way, Herman, have you ever heard Wendy Carlos' Digital
Moonscapes?

>Of course, 11-et is every other note of 22, so this is also
>appropriate:
>
>http://www.io.com/~hmiller/midi/Unfinished-11et.mid

That's amazing! How'd you do that?

-Carl

🔗Herman Miller <hmiller@IO.COM>

On Sat, 23 Feb 2002 19:10:43 -0000, "clumma" <carl@lumma.org> wrote:

>> http://www.io.com/~hmiller/midi/krocnard-22.mid
>
>What is this?

A 22-retuned version of http://www.io.com/~hmiller/midi/krocnardsklestj.mid
("Krocnardsklestj", written in 1988, originally in 12).

>> http://www.io.com/~hmiller/midi/porcupine-22.mid
>
>The version I downloaded has some strangeness -- Media Player
>calculates the length at 4:something, but only plays anything
>for the first :30 or so; Winamp claims it's a minute exactly,
>but crashes :30 or so into it.

It's only 26 seconds long. I just put up a new version that I made by
copying and pasting from the old one into a new file, and it seems to work
fine now.

>By the way, Herman, have you ever heard Wendy Carlos' Digital
>Moonscapes?

Yes, in fact I listened to it just recently after I heard the light cycle
music from the new Tron DVD (an early version of the music that became Io
from Digital Moonscapes).

>>Of course, 11-et is every other note of 22, so this is also
>>appropriate:
>>
>>http://www.io.com/~hmiller/midi/Unfinished-11et.mid
>
>That's amazing! How'd you do that?

Lots of trial and error. I found the chord progression a while back, but I
didn't have a melody to go with it. Real-time improvisation never worked
out quite right, since 12-TET habits sound really bad in 11-et (much more
so than in something like 18-et). I'd occasionally come up with something I
liked, but I couldn't repeat. It finally started coming together when I got
the idea to play one line at a time on the 11-tuned keyboard, then use
Midiconv to put the pitch bends in for that line and play it on the sound
card while I used the intermediate results to plan and practice the next
line on the keyboard. A final step was switching the timbres to find out
what sounded the best -- the melody was originally a flute, then I tried a
muted trumpet, but the flute didn't stand out enough from the background
and the muted trumpet just sounded fake. So it's a clarinet now.

🔗jpehrson2 <jpehrson@rcn.com>

🔗clumma <carl@lumma.org>

>It's only 26 seconds long. I just put up a new version that I made
>by copying and pasting from the old one into a new file, and it
>seems to work fine now.

Does here.

>>By the way, Herman, have you ever heard Wendy Carlos' Digital
>>Moonscapes?
>
>Yes, in fact I listened to it just recently after I heard the light
>cycle music from the new Tron DVD (an early version of the music
>that became Io from Digital Moonscapes).

Moonscapes is my favorite Carlos, and one of my favorite albums of
any. Without actually transcribing the progressions, Eden has
something in common with this section of porcupine, no?

>For fun, try the 12-TET version!

Sounds so wrong! I don't think I've ever heard 12 sound this wrong!
Amazing! Everyone on the list should check this out!

How does the Eden progression come out okay in 12? A few extra
fifths?

>A 22-retuned version of http://www.io.com/~hmiller/midi/
>krocnardsklestj.mid ("Krocnardsklestj", written in 1988,
>originally in 12).

Cool name. Which language? BTW, I remember seeing that you link to
the Lojban site. Are you active in the Lojban community (you may
reply off-list, or to metatuning)?

>>>http://www.io.com/~hmiller/midi/Unfinished-11et.mid
>>
>>That's amazing! How'd you do that?
/.../
>It finally started coming together when I got the idea to play one
>line at a time on the 11-tuned keyboard, then use Midiconv to put
>the pitch bends in for that line and play it on the sound card
>while I used the intermediate results to plan and practice the next
>line on the keyboard. A final step was switching the timbres to find
>out what sounded the best -- the melody was originally a flute, then
>I tried a muted trumpet, but the flute didn't stand out enough from
>the background and the muted trumpet just sounded fake. So it's a
>clarinet now.

Thanks for sharing your method! There are so many ways to work in
the Midi-relaying studio, it's always cool to hear 'em.

-Carl

🔗Herman Miller <hmiller@IO.COM>

On Sun, 24 Feb 2002 18:54:34 -0000, "clumma" <carl@lumma.org> wrote:

>Moonscapes is my favorite Carlos, and one of my favorite albums of
>any. Without actually transcribing the progressions, Eden has
>something in common with this section of porcupine, no?
>
>How does the Eden progression come out okay in 12? A few extra
>fifths?

Let's see; I think it goes something like this:

F major - Bb major - G major - C major - A major - D minor - G major - C
major - F major

C#
D A E B
F C G D A
Bb F C

So while the Porcupine progression goes up by major thirds and down by
fifths, the Eden progression goes down by minor thirds and fifths. It's a
more elaborate version of the comma pump.

The Porcupine progression, in the new Porcupine notation I've been working
out, goes like this:

E major - G# minor - C major - E# major - A major - C# major - F# major -
(B minor) - B major - E major

D# A# Ex
B F C# Gx
D A E# B#
C G# D#
E B

(The major difference between porcupine notation and diatonic notation is
that the interval C-E is a _minor_ third, which means that A-E# and C-G#
are perfect fifths.)

So there are definitely similarities in the two progressions, but the Eden
progression wouldn't work in 15 any more than the Porcupine progression
works in 12.

🔗clumma <carl@lumma.org>

>>How does the Eden progression come out okay in 12? A few extra
>>fifths?
>
>Let's see; I think it goes something like this:
>
>F major - Bb major - G major - C major - A major - D minor -
>G major - C major - F major

Right, except it's A minor, not major.

>So while the Porcupine progression goes up by major thirds and
>down by fifths, the Eden progression goes down by minor thirds
>and fifths. It's a more elaborate version of the comma pump.

Ah, right. Manuel/Paul- does this example fit in to your recent
discussion over the derrivation of the syntonic comma?

>The Porcupine progression, in the new Porcupine notation I've
>been working out, goes like this:
>
> E major - G# minor - C major - E# major - A major - C# major -
> F# major - (B minor) - B major - E major
>
> D# A# Ex
> B F C# Gx
> D A E# B#
> C G# D#
> E B
>
>(The major difference between porcupine notation and diatonic
>notation is that the interval C-E is a _minor_ third, which
>means that A-E# and C-G# are perfect fifths.)

Shorthand: (E, G#min, C, E#, A, C#, F#, B, E).

What about this:

E#
/ \ / \ / / / .
C#--G#--D#--A#^-E#^-
/ \ / \ / \ / \ / \ ,
A---E---B---F#^-C#^-G#^
/ \ / \ / \ / \ / \ /
F---C---G---D---A^--E^

So in tunings with positive fifths, the # raises two steps
and the ^ raises 1, and with normal fifths the # works as
normal and you ignore the ^. The progression is then:

15 (E, G#min, C#, E#=E^, A^, C#^, F#^, B, E)

12 (E, G#min, C#, E#=F, Bb, D, G, C, F), or
(E, G#min, C#, E#, A#, Cx, Fx, B#, E#), right?

Does this make sense?

-Carl

🔗manuel.op.de.coul@eon-benelux.com

🔗Carl Lumma <carl@lumma.org>

🔗Herman Miller <hmiller@IO.COM>

On Mon, 25 Feb 2002 08:14:10 -0000, "clumma" <carl@lumma.org> wrote:

>What about this:
>
> E#
> / \ / \ / / / .
> C#--G#--D#--A#^-E#^-
> / \ / \ / \ / \ / \ ,
> A---E---B---F#^-C#^-G#^
> / \ / \ / \ / \ / \ /
>F---C---G---D---A^--E^
>
>So in tunings with positive fifths, the # raises two steps
>and the ^ raises 1, and with normal fifths the # works as
>normal and you ignore the ^. The progression is then:
>
>15 (E, G#min, C#, E#=E^, A^, C#^, F#^, B, E)

Right, except it actually starts on Eb. (C-E is a minor third in porcupine
notation, so porcupine E corresponds to traditional Eb.)

>12 (E, G#min, C#, E#=F, Bb, D, G, C, F), or
> (E, G#min, C#, E#, A#, Cx, Fx, B#, E#), right?
>
>Does this make sense?

Yes, that's one way of looking at it. I've used similar notations except
with the D a fifth below A rather than the one above G. But if you're going
to start notating commas, I think it's almost easier to use the Pythagorean
fifths and write Eb, G-, C-, E--=Eb+, Ab+, C, F, Bb, Eb.

. --- G#
. / \
. -- E---B
. / \ / \
. - C---G---D---A---E
. \ / \ / \ / \ / \
. start-> Eb--Bb--F---C---G
. \ / \ / \ / \
. + Db--Ab--Eb--Bb

Porcupine notation is to the porcupine tunings what Graham Breed's decimal
notation is for the MIRACLE temperament. The notes A-G are equally spaced,
based on the generator of the porcupine scale (2 steps of 15-ET, 3 of
22-ET, or 5 of 37-ET). Two steps up is a minor third, three steps a perfect
fourth, and five steps a minor sixth. A step above G is notated as Ab, and
the cycle keeps going, adding more and more flats. Similarly, a step below
A is G#, and more sharps are added as you go down the scale.

. G---D---A E#--B# Fx--Cx
. \ / \ \ / \
. Eb--Bb F---C G#--D#--A# Ex--Bx
. / \ \ / \ \ / \
. Gb--Db--Ab E---B F#--C# Gx--Dx--Ax
. \ / \ \ / \ \ / \
. Ebb-Bbb Fb--Cb G---D---A E#--B# Fx--Cx
. \ / \ \ / \ \ / \
. Gbb-Dbb-Abb Eb--Bb F---C G#--D#--A#
. \ / \ \ / \ \ / \
. Fbb-Cbb Gb--Db--Ab E---B F#--C#
.
. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
. A # b B # b C # b D # b E # b F # b G # x b A
. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Also, I'm not sure if this is generally true for the porcupine scales, but
in the ones I'm mainly interested in (15, 22, and 37), six steps up is a
good 7/4 approximation. (I imagine this is like the A# approximation in
meantone -- something that works for many of the porcupine scales, but
probably not all of them.)

🔗dkeenanuqnetau <d.keenan@uq.net.au>

Sorry I'm just dropping in on this thread without having read the
foregoing.

I take Porcupine temperament to be an approximately 163c generator
with primes 3,5,7,11 mapping to -3,-5,6,-4 generators, (11-limit sans
9's). I'd notate the 7 note MOS as

A Bv C^ D Ev F^ G

v and ^ represent both the syntonic comma and the undecimal diesis and
correspond to 1 step of either 15 or 22-tET, and 2-steps of 37-tET.
Feel free to substitute your favourite comma symbols, everyone else
does :-)

Here's a 22-note chain of it with a JI-approximation slide rule under
it.

G#v A^ B C#v D^ E F#v G^ A Bv C^ D Ev F^ G Av Bb^ C Dv Eb^ F Gv
-5--11-3--------1----------------7

Note that this can be viewed as 3 interleaved chains of fourths. One
chain with v, one with ^ and one plain, so it's clear how to extend it
indefinitely. However for chains longer than 22 and indeed for 37-tET
itself, one can introduce the 13-comma symbols, say {}, (even though
13s are poorly approximated) so that, for example, G#^ is enharmonic
to A{.

Thanks for this, guys. It solved my problem of how to notate 37-tET.
And provided some insights into how the common JI/ET notation system
can also work for linear temperaments.

Here's 37-tET
D D}v D^ Eb^ E{ Ev E{^ E F F}v F^ F} F#v Gv G{^ G G}v G^ G} A{ Av A{^
A A}v A^ Bb^ B{ Bv B{^ B C C{^ C^ C} C#v Dv D{^ D

🔗Carl Lumma <carl@lumma.org>

>>Ah, right. Manuel/Paul- does this example fit in to your recent
>>discussion over the derivation of the syntonic comma?
>
>No, I see no connection. There's only one recognisable fifth and
>one recognisable third in 22-tET.

In 22-tET? ?

I wasn't following the discussion, but Paul wrote:

>the syntonic comma is usually the difference between
>three 4:3s (minus an octave) and a 6:5

Herman Miller wrote:

>So while the Porcupine progression goes up by major thirds
>and down by fifths, the Eden progression goes down by minor
>thirds and fifths. It's a more elaborate version of the comma
>pump.

Herman Miller - Mizarian Porcupine Overture

+M3 -P5 +M3 -P5 +M3 -P5 -P5 -P5 = +250:243
Eb Gm C E A C# F# B E

Bb--Bb C B C#---C#---C# D# E
G---G----G G# A G# A# B----B
Eb D E----E----E E# F#---F# G#

Wendy Carlos - Eden

-P5 -m3 -P5 -m3 -P5 -P5 -P5 -P5 = -6561:6400, (81:80)^2
F Bb G C A Dm G C F

A Bb B C C# D----D E F
F---F G----G A----A B C----C
C D----D E----E F G----G A

So isn't the Eden progression a way of getting the syntonic
comma with minor thirds and fourths, as Paul described, which
won't be the same as the [4 1] way in ets like 64 and 105
(as Paul mentioned)?

-Carl

🔗manuel.op.de.coul@eon-benelux.com

🔗genewardsmith <genewardsmith@juno.com>

--- In tuning@y..., Herman Miller <hmiller@I...> wrote:

> Also, I'm not sure if this is generally true for the porcupine scales, but
> in the ones I'm mainly interested in (15, 22, and 37), six steps up is a
> good 7/4 approximation. (I imagine this is like the A# approximation in
> meantone -- something that works for many of the porcupine scales, but
> probably not all of them.)

Porcupine really has only one logical way of extending itself to 7 or 11 limit, so I think those should as porcupine also. In them 15, 22, 37 and 59 (which you might add to your porcupine list) work in the same way. I'll spare you the wedgies, but the Minkowski-Tenney reduced bases are [64/63, 250/243] for the 7-limit and [55/54, 64/63, 100/99] for the 11-limit. Porky comes in with a fairly low badness score (under 400) for the 5, 7 and 11-limits, but the lowest was 11-limit, incidentally.

🔗manuel.op.de.coul@eon-benelux.com

🔗dkeenanuqnetau <d.keenan@uq.net.au>

🔗manuel.op.de.coul@eon-benelux.com

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., "clumma" <carl@l...> wrote:
> >>How does the Eden progression come out okay in 12? A few extra
> >>fifths?
> >
> >Let's see; I think it goes something like this:
> >
> >F major - Bb major - G major - C major - A major - D minor -
> >G major - C major - F major
>
> Right, except it's A minor, not major.
>
> >So while the Porcupine progression goes up by major thirds and
> >down by fifths, the Eden progression goes down by minor thirds
> >and fifths. It's a more elaborate version of the comma pump.
>
> Ah, right. Manuel/Paul- does this example fit in to your recent
> discussion over the derrivation of the syntonic comma?

you can narrow down the comma pump to what happens between the two C
major chords . . . there it is clearly the difference between a minor
third and three fourths.

🔗paulerlich <paul@stretch-music.com>

🔗Carl Lumma <carl@lumma.org>

🔗dkeenanuqnetau <d.keenan@uq.net.au>

🔗genewardsmith <genewardsmith@juno.com>

🔗Orphon Soul, Inc. <tuning@orphonsoul.com>

🔗Herman Miller <hmiller@IO.COM>

On Tue, 26 Feb 2002 08:03:21 -0000, "dkeenanuqnetau" <d.keenan@uq.net.au>
wrote:

>Sorry I'm just dropping in on this thread without having read the
>foregoing.
>
>I take Porcupine temperament to be an approximately 163c generator
>with primes 3,5,7,11 mapping to -3,-5,6,-4 generators, (11-limit sans
>9's). I'd notate the 7 note MOS as
>
>A Bv C^ D Ev F^ G
>
>v and ^ represent both the syntonic comma and the undecimal diesis and

That would be 22:21, I presume? So if 81:80 and 22:21 are the same interval
in this temperament, that means that 1760:1701 is also a unison vector?

>correspond to 1 step of either 15 or 22-tET, and 2-steps of 37-tET.
>Feel free to substitute your favourite comma symbols, everyone else
>does :-)

It reminds me of Easley Blackwood's 15-TET notation, which makes sense in
theory, but isn't all that easy to read. I'll have to try a few things in
different notations to see how easy it is to follow. Of course, the nice
thing about this notation is that it makes it easy to take a 15, 22, or
37-TET sequence and "transpose" it to another temperament.

>Here's a 22-note chain of it with a JI-approximation slide rule under
>it.
>
>G#v A^ B C#v D^ E F#v G^ A Bv C^ D Ev F^ G Av Bb^ C Dv Eb^ F Gv
>-5--11-3--------1----------------7
>
>Note that this can be viewed as 3 interleaved chains of fourths. One
>chain with v, one with ^ and one plain, so it's clear how to extend it
>indefinitely. However for chains longer than 22 and indeed for 37-tET
>itself, one can introduce the 13-comma symbols, say {}, (even though
>13s are poorly approximated) so that, for example, G#^ is enharmonic
>to A{.

13/8 is only off by 2.7 cents in 37-ET, and so is 13/11; 13/7 is even
better (1.4 cents flat). 13/10 is nearly perfect at 0.2 cents flat. The one
that's way off is 13/9, which makes sense because 9/8 is so far off to
begin with.

>Thanks for this, guys. It solved my problem of how to notate 37-tET.
>And provided some insights into how the common JI/ET notation system
>can also work for linear temperaments.
>
>Here's 37-tET
>D D}v D^ Eb^ E{ Ev E{^ E F F}v F^ F} F#v Gv G{^ G G}v G^ G} A{ Av A{^
>A A}v A^ Bb^ B{ Bv B{^ B C C{^ C^ C} C#v Dv D{^ D

And here's the extended porcupine notation for 37-TET (using the semisharp
and semiflat symbols for single degrees of the scale):

D Dz D# Eb Et E Ez E# Fb Ft F Fz F# Gb Gt G Gz G# Abb Gx Ab At A Az A# Bb
Bt B Bz B# Cb Ct C Cz C# Db Dt D

(You could do almost the whole scale with double sharps and double flats,
but steps 16 and 21 would require triple sharps/flats.)

🔗Herman Miller <hmiller@IO.COM>

On Tue, 26 Feb 2002 21:28:14 -0000, "paulerlich" <paul@stretch-music.com>
wrote:

>--- In tuning@y..., Herman Miller <hmiller@I...> wrote:
>
>> Right, except it actually starts on Eb. (C-E is a minor third in
>porcupine
>> notation, so porcupine E corresponds to traditional Eb.)
>
>my piece 'glassic' uses this sort of 'notation' -- actually it isn't
>notated, but we mapped the basic 'heptatonic porcupine scale' to the
>white notes of the keyboard so that c-e becomes a minor third. i
>really like the 8:9:10:11:12 chord g-a-b-c-d.

Interesting. You know, that's always been one of my favorite of your 22-ET
pieces from the first time I heard it. And I've always wanted to try
something in 37-TET, but I couldn't figure out what to do with it. This
looks like a pretty useful temperament.

So we could call the 7-note scale built from this generator the "glassic
scale", and the 15-note scale the "porcupine scale". What's the origin of
the word "glassic"?

🔗Orphon Soul, Inc. <tuning@orphonsoul.com>

On 2/27/02 12:15 AM, "Herman Miller" <hmiller@IO.COM> wrote:

> Interesting. You know, that's always been one of my favorite of your 22-ET
> pieces from the first time I heard it. And I've always wanted to try
> something in 37-TET, but I couldn't figure out what to do with it. This
> looks like a pretty useful temperament.

Well you got a good 5th 7th and 11th harmonic. They have an interesting way
of folding over each other.

Depending on how you layer intervals,
you can use either fourth and fifth.

The 22/37 fifth isn't as sharp as 3/5.
The 21/37 fifth is a little flatter than 4/7 but since 9/37 is the
pythagorean minor third from 111, 0 - 9 - 21 isn't such a bad chord. The
two dyads seem to take care of each other. Whereas the 0 - 21 dyad itself
crumbles a bit.

Also a little use of the 3/37 generator normalizes the batch a bit. It's
close to a 12 semitone (3/36) but the 3/37 series passes through the 5:4 and
7:4, as well as the aforementioned 9/37 minor third.

If that helps any. I go back a long way with 37 but I haven't really been
interested in it again until about a year or so ago.

From my experience with guitar-fretboard-timbres, I miss 37 a lot. It has a
very strong sound. I need to make a new one soon.

Marc

🔗paulerlich <paul@stretch-music.com>

🔗Orphon Soul, Inc. <tuning@orphonsoul.com>

On 2/27/02 12:07 PM, "paulerlich" <paul@stretch-music.com> wrote:

> --- In tuning@y..., "Orphon Soul, Inc." <tuning@o...> wrote:
>
>> Also a little use of the 3/37 generator normalizes the batch a bit.
>
> hmm . . . other generators seem to come to mind first . . . besides
> porcupine, you also have that 6/37 generator giving you great 19-tone
> scales . . .
>

I should have gone with my instinct and looked up generator before I tried
using it in a sentence. I just looked it up in the Monzaurus and it doesn¹t
look like what I thought it meant.

I was trying to describe a scale based on a chromatic interval of 3/37,
which was sort of ambiguous as to whether I meant the 12 scale in 37, or a
pseudo-12 scale that repeats just short of the octave, which doesn¹t matter
because I meant it either way and I imagined a generator was one of those
two things.

Which, as I was saying, in the context, I was using, passes through the 5:4,
the 4:3, the 7:4 and a few other points of interest. Which, as I said,
makes for a break from the stranger geometry gotten from playing interval
origami with the large 3:2 and such.

Oh and also there's a very strong separation in ownership between the 2/37
and 3/37 minor seconds. In a 3/37 dyad the root is very submissive whereas
the 2/37 it's very absorbant.

22-equal day!

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