ET's and unison vectors

You've probably seen or made charts of equal tunings sorted by the size of
their fifths and major thirds.

http://www.io.com/~hmiller/png/et-scales.png

What's interesting is that the tunings on this chart are lined up in a neat
interlocking set of rows, and that the slopes of the lines correspond to
5-limit unison vectors!

The row along the top consists of scales that share the unison vector
128/125 (0 -3). They all have 400 cent major thirds, with fifths ranging
from small to large: 21, 32, 12, 39, 27, 42, and 15-TET. The line continues
off the chart, with 9-TET to the left and 18-TET to the right.

Now look at the triangle with 12, 19, and 22-TET at its corners. Outside
this triangle is a handful of scales arranged in a pattern of interlocking
lines. Inside the triangle is a swarm of scales that fall mainly into the
"near-just" category, with all intervals within 5 cents of just; a few of
them have minor thirds slightly farther than 5 cents from just, and one
(101-TET) has a major third a bit more than 5 cents away from just.

The left edge of the triangle consists of meantone scales, with unison
vector 81/80 (4 -1). From top to bottom: 12, 67, 55, 98, 43, 117, 74, 105,
31, 81, 50, 69, 88, and 19-TET. Continuing onward, 45 and 26-TET lie along
the same line.

The right edge of the triangle consists of diaschismic scales, with unison
vector 2048/2025 (-4 -2). There's actually a pretty good number of scales
in this family: 12, 70, 58, 46, 126, 80, 114, 148, 34, 124, 90, 56, 78,
22-TET on the edge of the triangle, plus 54 and 32-TET if you continue the
line to the lower right.

The bottom edge of the triangle, from 19-TET to 22-TET, consists of scales
with slightly narrow major thirds, sharing a unison vector of 3125/3072 (-1
5). Five major thirds up equals a fifth plus an octave. The sequence goes:
19, 79, 60, 41, 104, 63, 85, 107, 22-TET. Also along the same line are
35-TET, to the left, and 25-TET, to the right.

The schismic temperaments, with unison vector 32805/32768 (8 1), are
collected on a line that goes straight through the middle of the swarm of
near-just tunings. There seem to be quite a few of these. 53-TET lies on a
major 3-way intersection near the center of the chart.

Scattered around the outside of the triangle are a number of small groups
of tunings in an interlocked pattern of lines; here are a few of the more
obvious ones.

(-4 -5) : (12) 73 61 49 37 (25)
(8 7) : (12) 95 83 71 59
(4 -4) : (12) 64 52 40 28
(3 4) : 28 47 19 48 29
(-2 7) : 26 29 32
(5 -3) : 37 59 22 51 29

Off the chart to the left is the tiny series 11, 9, 16, 23-TET, with the
unison vector 135/128 (3 1). Below the chart is the even smaller series 13,
10, 17-TET with the unison vector 25/24 (-1 2). 7-TET sits at the
intersection of these two lines, on a vertical line together with a few
other multiples of 7-TET, with 14-TET above the chart. 5-TET and 8-TET are
way off by themselves up and to the right of the chart. 6-TET is so far off
to the right that it might as well be on another planet.

--
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,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

--- In tuning-math@y..., Herman Miller <hmiller@I...> wrote:
> You've probably seen or made charts of equal tunings sorted by the
size of
> their fifths and major thirds.
>
> http://www.io.com/~hmiller/png/et-scales.png
>
> What's interesting is that the tunings on this chart are lined up
in a neat
> interlocking set of rows, and that the slopes of the lines
correspond to
> 5-limit unison vectors!

You're so cool, Herman! This chart looks almost exactly like the
plots I did of the triads within the early part of the harmonic
series . . . spooky.
>
> The bottom edge of the triangle, from 19-TET to 22-TET, consists of
scales
> with slightly narrow major thirds, sharing a unison vector of
3125/3072 (-1
> 5). Five major thirds up equals a fifth plus an octave.

Apparently, this is called the small diesis. This must me one of the
unison vectors of the 10-note chain-of-major-thirds scale that comes
up from time to time.

> (-4 -5) : (12) 73 61 49 37 (25)
> (8 7) : (12) 95 83 71 59
> (4 -4) : (12) 64 52 40 28
> (3 4) : 28 47 19 48 29
> (-2 7) : 26 29 32
> (5 -3) : 37 59 22 51 29
>
> Off the chart to the left is the tiny series 11, 9, 16, 23-TET,
with the
> unison vector 135/128 (3 1). Below the chart is the even smaller
series 13,
> 10, 17-TET with the unison vector 25/24 (-1 2). 7-TET sits at the
> intersection of these two lines, on a vertical line together with a
few
> other multiples of 7-TET, with 14-TET above the chart. 5-TET and 8-
TET are
> way off by themselves up and to the right of the chart. 6-TET is so
far off
> to the right that it might as well be on another planet.

Herman, you rock!

--- In tuning-math@y..., Herman Miller <hmiller@I...> wrote:
> You've probably seen or made charts of equal tunings sorted by the
size of
> their fifths and major thirds.
>
> http://www.io.com/~hmiller/png/et-scales.png

Hey that's neat.

Instead of starting endless threads in my own terms on this list, I figured
I'd just wait until something familiar showed up. It took a month, but now
I can finally answer one of Paul's questions.

On 6/25/01 10:02 PM, "Herman Miller" <hmiller@IO.COM> wrote:

> You've probably seen or made charts of equal tunings sorted by the size of
> their fifths and major thirds.
>
> http://www.io.com/~hmiller/png/et-scales.png

I used to refer to this exact chart quite a bit, but drawing it with no axes
or scale legend left it as sort of a vision hovering in mental space. I'd
also never plotted it on a large area. Graphic wise, going back 5-10 years
and doing a quick graph in HyperCard on the available monitors, with the
different color dots even, it's easier to visualize the inaccuracies now.
Having tick marks every 5 cents, I have to say, makes for a great quick
reference to the exact cent offsets. After spending so much more time
dealing with the percentage inaccuracies of intervals in ET's, this is a
very stable way to visualize it on the cent-scale. Thank you Herman.

> What's interesting is that the tunings on this chart are lined up in a neat
> interlocking set of rows, and that the slopes of the lines correspond to
> 5-limit unison vectors!

I'd never heard the term "unison vector" until recently. This is why I was
referring to these different coincidences in terms of their equivalences;
saying "4P5 = M3" instead of "a unison vector of 81:80" I would think in
playing, which I'm thinking guitar anyway, it would be easier in practice to
remember the equivalence rather than the ratio that disappears, no? But
then again, I haven't yet given reason to do otherwise. Stay tuned. You
might like this one.

One day I started zooming in on the middle of the graph and replotting it, I
think I wound up at something like 256x magnification - using the same
"size" dots, the concentration of temperament dots winds up being more and
more dense. Amazingly enough, though, if you magnify this graph more and
more, and run the plotting way into the thousands, say... the center of the
graph winds up looking like a perfectly formed SQUARE centered exactly on
the just point. I tried to come up with a proof for it; the closest I got
was the idea that the most any one temperament could be off from the middle
is slightly less than one-half the chromatic step, which means every other
point to infinity has to be within the square defined by the just point plus
or minus a half chromatic step. Also, as you approach infinity, the dot
density increases much more than the width decreases.

It's the coolest thing though, to set a loop from say 3000 to 12 and watch
the plotting in reverse. You see the square build and build, until
eventually it starts exploding into a whirlwind of triangles.

* * * * * * * * * * * * * * * *

Since it's a common enough vision that just intonation is "pure", I was
calling these temperament groups "strains" (as in virii) of meantone, again,
not realizing so many people associate meantone with the third series
mentioned here. Taking it literally, it seemed to me these should be
thought of as different classes of meantone since there was some "tone"
split into a "mean".

> The row along the top consists of scales that share the unison vector
> 128/125 (0 -3). They all have 400 cent major thirds, with fifths ranging
> from small to large: 21, 32, 12, 39, 27, 42, and 15-TET. The line continues
> off the chart, with 9-TET to the left and 18-TET to the right.

This I just called third-octave meantone, 3MT.

> The left edge of the triangle consists of meantone scales, with unison
> vector 81/80 (4 -1). From top to bottom: 12, 67, 55, 98, 43, 117, 74, 105,
> 31, 81, 50, 69, 88, and 19-TET. Continuing onward, 45 and 26-TET lie along
> the same line.

So this I just called Syntonic Meantone (SMT), or since it's four fifths to
a major third, SMT4... I'll explain why in a minute.

> The right edge of the triangle consists of diaschismic scales, with unison
> vector 2048/2025 (-4 -2). There's actually a pretty good number of scales
> in this family: 12, 70, 58, 46, 126, 80, 114, 148, 34, 124, 90, 56, 78,
> 22-TET on the edge of the triangle, plus 54 and 32-TET if you continue the
> line to the lower right.

Paul, this is what I was calling Chromatic Meantone or CMT... the even ET's
where two minor seconds equal a major second. You said it's "simply
temperaments in which the diaschisma disappears". I couldn't figure out how
to explain what I was trying to communicate to you about the slight
differences in the series(pl.) until now. I'll explain in a minute.

> The bottom edge of the triangle, from 19-TET to 22-TET, consists of scales
> with slightly narrow major thirds, sharing a unison vector of 3125/3072 (-1
> 5). Five major thirds up equals a fifth plus an octave. The sequence goes:
> 19, 79, 60, 41, 104, 63, 85, 107, 22-TET. Also along the same line are
> 35-TET, to the left, and 25-TET, to the right.

I always liked this "line" on the chart because it doesn't pass through 12.
I don't really write up or down more than two or three major thirds in a
piece, so this series has a nice cushioning effect for me; that you can take
a note and two major thirds up or down, and walk that stack of thirds up and
down the cycles of fourths and fifths. Running out of ideas, since I saw it
stretching up to a sharp sixth from a note, I called this Augmented
Meantone.

> The schismic temperaments, with unison vector 32805/32768 (8 1), are
> collected on a line that goes straight through the middle of the swarm of
> near-just tunings. There seem to be quite a few of these. 53-TET lies on a
> major 3-way intersection near the center of the chart.

Oh you mean 12, 17, 29, 41, 53, 65, 77, 89, 94, 101, 118, 135, 142, 147,
171, 183, 200, 207, 219, 224, 248, 253, 272, 277, 289, 301, 313, 330, 378,
383, 395, 407, 419, 431, 436, 443, 460, 484, 489, 501, 508, 525, 537, 542,
561, 566, 607, 614, 619, 631, 643, 655, 667, 679, 696, 720, 725, 737, 749,
761, 773, 778, 785, 797, 802, 843, 850, 879, 891, 908, 932, 956, 961, 973,
985, 997, 1009, 1014, 1021, 1033, 1038, 1067, 1079, 1086, 1091, 1103, 1115,
1127, 1139, 1144, 1151, 1168, 1192, 1209, 1233, 1245, 1250, 1269, 1274,
1303, 1315, 1322, 1327, 1339, 1351, 1363, 1375, 1387, 1404, 1421, 1428,
1433, 1457, 1469, 1481, 1486, 1493, 1510, 1551, 1558, 1563, 1587, 1599,
1616, 1640, 1657, 1664, 1669, 1681, 1693, 1705, 1717, 1722, 1729, 1746,
1775, 1787, 1794, 1799, 1811, 1823, 1828, 1835, 1847, 1852, 1876, 1900,
1905, 1917, 1941, 1953, 1958, 1982, 2011, 2030, 2047, 2059, 2064, 2071,
2083, 2112, 2129, 2136, 2141, 2153, 2165, 2177, 2189, 2194, 2218, 2259,
2271, 2295, 2324, 2348, 2353, 2365, 2377, 2389, 2401, 2430, 2454, 2471,
2483, 2495, 2536, 2589... and 2707?

Hell EVERYONE knows THAAAT... :-P

I liked this name, since an A-Db was the 5:4 I called this Enharmonic
Meantone.

> Scattered around the outside of the triangle are a number of small groups
> of tunings in an interlocked pattern of lines; here are a few of the more
> obvious ones.
>
> (4 -4) : (12) 64 52 40 28

This I called quarter octave, 4MT.

> (-2 7) : 26 29 32

Yup. This series actually sounds pretty interesting if you use the flat
major thirds.

* * * * * * * * * * * * * * * *

What "other" equivalences...?
All of these examples exclude those multiples of temperaments whose thirds
and fifths are also multiples. (e.g. 24, 106...)

Take the paradox of four fifths equalling a just major third.
You would see this as "a unison vector of 81:80".

Now take four fifths equalling a just major sixth.
You would see this as "a unison vector of 81:80".

Yes?

"4P5 = M3"
The temperaments where:
two octaves below four of the closest note to a 3:2
equals the closest note to a 5:4
...are:

12, 19, 26, 31, 43, 45, 50, 55, 67, 69, 74, 81, 88, 98, 105 and 117.

"3P5 = M6"
The temperaments where:
one octave below two of the closest note to a 3:2
equals the closest note to a 5:3
...are:

12, 19, 26, 31, 33, 40, 43, 45, 50, 55, 64, 69, 74, 81 and 88.

Okay one more, "5P5 = M7" (SMT5)
The temperaments where:
two octaves below five of the closest note to a 3:2
equals the closest note to a 15:8
...are:

12, 19, 31, 43, 50, 55, 67, 79, 81, 91, 98, 105, 117, 122, 141 and 153.

For a while, I thought the program was screwed up until I realized, if you
change the equivalence, the "unison vector" stays the same, but you get a
different series of temperaments. Unison vector. That's about 10 words
less than what I used to call it.

This is hard enough to visualize at first.

The case of the "diaschismic" differences in series didn't occur to me until
a couple years late, as the strains are harder to imagine.

"4P4 = 2M3"
You would see this at first as "a unison vector of 2048:2025 (-4 -2)".

The temperaments where:
one octave below four of the closest note to a 4:3
equals two of the closest note to a 5:4
...are:

12, 22, 32, 34, 46, 54, 56, 58, 70, 78, 80, 90, 114, 124, 126 and 148,
and thank you again, Herman, for bringing this facet forward.
(scarily enough you can also include *10*...)

"6P5 = 2M6"
You would see this at first as "a unison vector of 2048:2025 (-4 -2)".

The temperaments where:
three octaves below six of the closest note to a 3:2
equals two of the closest note to a 5:3
...are:

12, 22, 34, 46, 56, 58, 70, 78, 80, 90, 104, 114, 124, 126, 148, 150, 172,
206 and 218.

At first glance, or not graphing high enough, you might not notice the
absence of 54 or the presence of 104; if not, the difference doesn't show up
until after 148. This is a *lot* more subtle than the syntonic version!!!

So. You were absolutely right. The associated "unison vector" in each of
these cases *is* an interval that will "disappear". But instead of six of
one and half a dozen of the other, you can still make them disappear by
using different "equivalences".

As I said...

"8P4 = M3" (EMT8)
...a unison vector of 32805/32768 (8 1).

The temperaments where:
three octaves below eight of the closest note to a 4:3
equals the closest note to a 5:4
are: :::deep breath:::

12, 17, 29, 41, 53, 65, 77, 89, 94, 101, 118, 135, 142, 147, 171, 183, 200,
207, 219, 224, 248, 253, 272, 277, 289, 301, 313, 330, 378, 383, 395, 407,
419, 431, 436, 443, 460, 484, 489, 501, 508, 525, 537, 542, 561, 566, 607,
614, 619, 631, 643, 655, 667, 679, 696, 720, 725, 737, 749, 761, 773, 778,
785, 797, 802, 843, 850, 879, 891, 908, 932, 956, 961, 973, 985, 997, 1009,
1014, 1021, 1033, 1038, 1067, 1079, 1086, 1091, 1103, 1115, 1127, 1139,
1144, 1151, 1168, 1192, 1209, 1233, 1245, 1250, 1269, 1274, 1303, 1315,
1322, 1327, 1339, 1351, 1363, 1375, 1387, 1404, 1421, 1428, 1433, 1457,
1469, 1481, 1486, 1493, 1510, 1551, 1558, 1563, 1587, 1599, 1616, 1640,
1657, 1664, 1669, 1681, 1693, 1705, 1717, 1722, 1729, 1746, 1775, 1787,
1794, 1799, 1811, 1823, 1828, 1835, 1847, 1852, 1876, 1900, 1905, 1917,
1941, 1953, 1958, 1982, 2011, 2030, 2047, 2059, 2064, 2071, 2083, 2112,
2129, 2136, 2141, 2153, 2165, 2177, 2189, 2194, 2218, 2259, 2271, 2295,
2324, 2348, 2353, 2365, 2377, 2389, 2401, 2430, 2454, 2471, 2483, 2495,
2536, 2589... and 2707.
... and ...

"9P4 = M6" (EMT9)
...another unison vector of 32805/32768 (8 1).

The temperaments where:
three octaves below nine of the closest note to a 4:3
equals the closest note to a 5:3
are:

12, 29, 41, 53, 65, 77, 89, 94, 118, 135, 142, 147, 171, 183, 200, 207, 224,
248, 253, 277, 289, 301, 313, 330, 378, 383, 395, 407, 419, 431, 436, 460,
484, 489, 501, 525, 537, 542, 566, 595, 607, 619, 631, 643, 655, 667, 696,
713, 720, 725, 737, 749, 761, 773, 778, 785, 802, 843, 879, 884, 891, 908,
932, 949, 956, 961, 973, 985, 997, 1009, 1014, 1038, 1055, 1067, 1079, 1091,
1103, 1115, 1127, 1144, 1168, 1173, 1209, 1233, 1245, 1250, 1274, 1291,
1303, 1315, 1327, 1339, 1351, 1356, 1404, 1409, 1421, 1433, 1457, 1462,
1469, 1486, 1510, 1527, 1551, 1563, 1587, 1592, 1616, 1633, 1640, 1645,
1657, 1669, 1681, 1693, 1705, 1722, 1746, 1751, 1763, 1775, 1787, 1799,
1804, 1811, 1828, 1852, 1869, 1876, 1905, 1917, 1934, 1958, 1982, 1987,
1999, 2011, 2040, 2047, 2064, 2093, 2105, 2117, 2129, 2141, 2153, 2165,
2170, 2194, 2218, 2235, 2259, 2271, 2276, 2324, 2329, 2341, 2353, 2365,
2377, 2382, 2389, 2430, 2447, 2454, 2459, 2471, 2483, 2495, 2512, 2536,
2553, 2577, 2589, 2618, 2642, 2671, 2683, 2695, 2707, 2748, 2789, 2801,
2813, 2854, 2960... and 3013.

In both cases, 407 seems to capture the musical gist of the paradox quite
well.

Even though my terminology is occasionally makeshift, I do hope this article
somewhat amplifies the subtle distinction I've found, in the method of
accumulation of harmono-intervallic coincedences, that I was previously
trying to convey.

In occasional clarity,
Marc Jones

--- In tuning-math@y..., "Orphon Soul, Inc." <tuning@o...> wrote:
> Instead of starting endless threads in my own terms on this list, I
figured
> I'd just wait until something familiar showed up. It took a month,
but now
> I can finally answer one of Paul's questions.

OK, I'm looking forward to it (which one)?
>
> Taking it literally, it seemed to me these should be
> thought of as different classes of meantone since there was
some "tone"
> split into a "mean".

Huh? Let's take the scales were the diesis (128:125) vanishes.
What "tone" is being split into a "mean" there?

> Take the paradox of four fifths equalling a just major third.
> You would see this as "a unison vector of 81:80".
>
> Now take four fifths equalling a just major sixth.
> You would see this as "a unison vector of 81:80".
>
> Yes?

I don't think so. Did you mean three fifths equallying a just major
sixth? Then yes.
>
>
>
> "4P5 = M3"
> The temperaments where:
> two octaves below four of the closest note to a 3:2
> equals the closest note to a 5:4
> ...are:
>
> 12, 19, 26, 31, 43, 45, 50, 55, 67, 69, 74, 81, 88, 98, 105 and 117.
>
>
>
> "3P5 = M6"
> The temperaments where:
> one octave below two

You mean three?

of the closest note to a 3:2
> equals the closest note to a 5:3
> ...are:
>
> 12, 19, 26, 31, 33, 40, 43, 45, 50, 55, 64, 69, 74, 81 and 88.

Aha -- for me these two series would come out identical, because I
would exclude the 5-limit inconsistent ones from both -- such as 33.
In general, though, a given unison vector or set of unison vectors
vanishing implies a temperament, not an ET, unless the number of
unison vectors vanishing is equal to the number of dimensions. For
example, in Herman's chart, each point lies at the intersection of
two lines (actually more, but two is enough for an unambiguous
location), corresponding to the two unison vectors that vanish for
the ET represented by that point. And you can calculate the ET from
the absolute value of the determinant of the matrix of unison
vectors. The determinant is defined as (a*d - b*c) for a matrix

[a b]
[c d]

For example, 12-tET lies at the intersection of the line representing
the syntonic comma (4 -1) and the schisma (8 1). So the matrix is

[4 -1]
[8 1]

and the determinant is (4*1 - 8*(-1)) = 4+8 = 12. Voila! Try some
others . . .

As for the "paradox" or "subtle distinction" you bring up, I don't
put too much weight on it, since inconsistent ETs can be mapped to
the consonant intervals in more than one way, and it's really the
_mapping_, not the ET itself, that determines which unison vectors
are involved.

----- Original Message -----
From: D.Stearns <STEARNS@CAPECOD.NET>
To: <tuning-math@yahoogroups.com>
Sent: Tuesday, June 26, 2001 5:27 PM
Subject: Re: [tuning-math] Re: ET's, unison vectors (and other equivalences)

> Hi Paul and everyone,
>
> You can also use the 2d lattice as a basic model for plotting
> coordinates other that 3 and 5.
>
> Earlier I gave the [4,3] 7-tone, neutral third scale as an example
> with the unison vectors 52/49

= 2^2 * 7^-2 * 13^1 = [ 2 0 0 -2 0 1]

> and 28672/28561.

= 2^12 * 7^1 * 13^-4 = [ 12 0 0 1 0 -4]

> This would be an example of plugging 13 and 7 into
> a 2D lattice space while retaining the diatonic matrix.

As cab be seen at a glance in either of the prime-factor notations.

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

> ----- Original Message -----
> From: D.Stearns <STEARNS@CAPECOD.NET>
> To: <tuning-math@yahoogroups.com>
> Sent: Wednesday, June 27, 2001 12:28 AM
> Subject: Re: [tuning-math] Re: ET's, unison vectors (and other
equivalences)
>
>
> Hi Joe,
>
> In this case I don't think "2^2 * 7^-2 * 13^1", or, "[ 2 0 0 -2 0
> 1]" is any easier to swallow than the plain ol' ratio of 52/49.

Yup... you got me there. I just added the vector notation for that
one to be consistent... the reason I really started translating your
unison-vector ratios into prime-factor vector notation was because
those other ratios had *huge* integers and I couldn't tell what
was going on until I factored them.

>
> Also seeing as how this example is explicitly 2D, and just the
> diatonic periodicity block in some colorful new party clothes, I think
> the most direct shorthand would probably be,
>
> | -1 2 |
> | 4 -1 |

Right again, Dan. This notation really *is* simply a shorthand
for what I wrote as:

[ 0 0 2 0 -1]
[ 0 0 -1 0 4]

(Here, I've inverted the signs as you have, and ignore
powers of 2 as I usually would but didn't before; I've also
decided that I like the square brackets better than the bar
because there's less chance of confusion with "1".)

I'm curious tho... why did you put the 13-column first and
the 7-column second in your matrix?

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

/tuning-math/message/417

>
> ----- Original Message -----
> From: D.Stearns <STEARNS@C...>
> To: <tuning-math@y...>
> Sent: Tuesday, June 26, 2001 5:27 PM
> Subject: Re: [tuning-math] Re: ET's, unison vectors (and other
equivalences)
>
>
> > Hi Paul and everyone,
> >
> > You can also use the 2d lattice as a basic model for plotting
> > coordinates other that 3 and 5.
> >
> > Earlier I gave the [4,3] 7-tone, neutral third scale as an example
> > with the unison vectors 52/49
>
> = 2^2 * 7^-2 * 13^1 = [ 2 0 0 -2 0 1]
>
>
> > and 28672/28561.
>
> = 2^12 * 7^1 * 13^-4 = [ 12 0 0 1 0 -4]
>
>
> > This would be an example of plugging 13 and 7 into
> > a 2D lattice space while retaining the diatonic matrix.
>
> As cab be seen at a glance in either of the prime-factor notations.
>

I can see a cab at a glance, but I don't get this vector notation...

Could you please gently run it down again, or would that be for the
*arithmetic* list??

Thanks!

_________ ______ _____
Joseph Pehrson

--- In tuning-math@y...,
"D.Stearns" <STEARNS@C...> wrote:
> Hi Paul and everyone,
>
> You can also use the 2d lattice as a basic model for plotting
> coordinates other that 3 and 5.

We've done that before, for
example Margo and my
discussions (about 22-tET, among
other things) in the (3,7) plane.