[Newspoetry] tuning lattices and mandalas (fwd)
david moses fruchter
dmf23 at neuron.net
Thu Jun 22 17:53:21 CDT 2000
i apologize for posting such a long non-newspoem, but i thought the
following might be interesting to those on this list who are into "new
music". i can imagine making an argument that it is in fact a newspoem,
but i do not make that argument.
note: you'll have to use a fixed-width or monospace font (such as
courier) for the ASCII diagrams below to appear correctly.
love,
david
ONPH (obligatory newspoetry haiku):
my state kills tonight
i am a part of what kills
i kill part of me
---------- Forwarded message ----------
Date: Thu, 22 Jun 2000 15:30:24 -0700 (PDT)
From: pr0k r!ndz <spigot at neuron.net>
To: leri at daft.com
Subject: tuning lattices and mandalas
well i've talked about tuning lattices in the past, and more recently
on #leri pasting some weird ascii art relating to them. someone
asked for if i could explain them, so i wrote this. i thought it
might be of interest to someone on leri at . it's kinda long, but
a lot of it is ascii art, so it's not quite as long as a line-count
would imply...
the basic question is, what on earth is *this* about:
E
/|\
/ | \
D#-------,A#-------,E#
\`. /,'/|\`.\ ,'/
\ `C'-/-|-\-`G' /
\/|\/.,Dx,\/|\/
/\|/,'Bbb`.\|/\
/ ,F#-------`C# \
/,' \`.\|/,'/ `.\
Ab-----\-`Eb-/-----`Bb
\ | /
\|/
A
so... starting with basics -- you know that a stable sound pitch is
when the air is vibrating at a repeating frequency, right? so we
could have a pitch at say 400 hertz (waves per second). if we want
to create new pitches that might sound in tune to this starting pitch,
we could create them by multiplying or dividing 400 hertz in various ways.
so we could multiply 400 hertz by 2 and get a new pitch of 800 hertz.
if you listen to these both at the same time, they'll sound very much
in tune (they are octaves). we could multiply by 2 twice,
400 * 2 * 2 = 1600, more octaves. or, we could multiply 400 by 3 and
get 1200 hertz. listening to these two pitches together will also
sound in tune, in a different way. we can ignore multplying by
4 since that's the same as multiplying by 2 twice. multiplying
400 hertz by 5 makes 2000 hertz, and this also sounds in tune
(actually 2000 hertz is a pretty high note, but i'll get to that in a
minute). multiplying by 6 "doesn't count" because it's just like
multiplying by 3s and 2s. 7 counts, and leads to new pitches, etc.
so what we're doing is taking a starting vibration frequency and
creating new vibration frequencies by multiplying the frequency by
various numbers. it turns out that prime numbers are what is important --
any number you can get by multiplying by 9, for example, you could also
get by multiplying by a couple 3s. so each prime number "opens up" a new
"dimension" of possible numbers you can create from the starting number.
make sense? ok...
if we just multiply 400 by various prime numbers, we're just going to
end up with larger and larger numbers and that's kinda limited.
so we can multiple and divide to get more numbers -- and to bring
the pitches back down to a reasonable level. human hearing goes
up to around 20,000 hertz, but in general, people enjoy harmony
around the range of 200 - 2000 hertz or so.
so now we can do things like, start at 400 hertz, multiple by 3
to get 1200 hertz, then divide by 2 to get 600 hertz. or we could
start at 400, multiple by 5 to get 2000, then divide by 3 to get
666.666 hertz, etc. we're just making new vibrations. the possible
combinations are endless -- you can of course keep multiplying and
dividing over and over with whatever numbers you want. there's an
infinite number of possibilities. but it turns out the simplier your
process, the more 'in tune' the notes will sound, generally speaking.
anyway, that's the "physics of sound" background info. next i gotta
describe the notation that gets used.
all this multiplying and dividing can be written in the form of fractions.
your starting pitch we can call "1", or, 1/1 (or, 1:1). if you
multiply 1 by 3, you get "3", of course, or 3/1. if you multiply
1 by 3, then divide that by 2, you get 3/2, and so on. writing
all this multiplication and division out as a single ratio makes
it much easier to read. it can also describe how the actual
wavelengths of vibrating air relate to each other. take, say,
the pitch 1/1 and the pitch 3/1 (400 hertz and 1200 hertz, say).
for every vibration in the 1/1 pitch, there are 3 in the 3/1 pitch,
right? now take the pitches 1/1 and 3/2. in this case, for every
2 vibrations of the 1/1 pitch, there are 3 in the 3/2 pitch.
so just by seeing the ratio 3/2, we know it's mathematical
relationship to 1/1. the same thing works for say, 17/11 --
for every 11 vibrations in 1/1 note, there are 17 in the 17/11 note.
anyway, so that's the ratio notation...
now, since we're dealing only with fractions, we can eliminate the idea
of "division" and only do multiplication. if you want to multiply by 3,
you multiply by 3/1. if you want to divide by 3, you multiply by 1/3.
it all works out the same.
oh, one final and potentially weird thing all this -- it's traditional
in music to think of octaves as being "the same" note. you can play a C
on a piano, and then play the C an octave up and say, well they are
both Cs -- they are the same "class" of note, just one is twice as high
as the other. mathematically, octaves involve multiplying or dividing
by 2. so in our ratio notation, you can *always* multiple your ratio
by 2/1 or 1/2 and it will be class of note. tuning theory cares about
what class the note is and not how high or low it is. a C is a C is a C.
so even though 3/1 is twice as high as 3/2, for the purposes of tuning
theory, they are the same. 3/1 = 3/2, and 5/4 = 20/4, etc. that might
seem weird, but it actually makes things simpler -- like this --
to keep things as clear as possible, ratios are always written
so that they are between 1/1 and 2/1. since multiplying and
dividing by 2 "doesn't matter", we can cram all our notes into
a single octave. this is called "octave reduction" and it
makes things much simpler once you get over the weirdness of it.
so *now*, if we start with 1/1 at 400 hertz and make a new note
by multiplying it by 3, we get 3/1 or 1200 hertz. 3/1 is more than
2/1, so we gotta apply octave reduction to it, resulting ina 3/2 or
600 hertz. practically speaking, this means when you create a new
note/ratio, you just multiply it by 2/1 or 1/2 until the result in
between 1/1 and 2/1.
oh, also to keep things simple, ratios are always reduced
to their simplest factors -- in other words, 20/10 is the
same as 2/1, so we write it as 2/1. and 35/25 is the same as
7/5, so that's how it's written.
ok, now we can talk about lattices and mandalas!
lattices are just a visual way to see the relationships between
these ratio-pitches. the idea is, since each new prime number
you use opens up a new dimension, you graph your pitches using
a new axis for each prime number. the prime "2" doesn't count
because of octave reduction.
so if we're taking our primes in order from low to high, our
first dimension is the prime "3". if we are only dealing with
multiplying by 3/1 or 1/3, we've got a 1-dimensional situation.
on a straight line, we could put 1/1 in the middle, and say that
each step *left* is an additional multiplication by 1/3, while
each step *right* is an additional multiplication by 3/1, resulting
in this:
<-- 1/81 -- 1/27 -- 1/9 -- 1/3 -- 1/1 -- 3/1 -- 9/1 -- 27/1 -- 81/1 -->
or, in hertz:
4.9 -- 14.8 -- 44.4 -- 133.3 -- 400 -- 1,200 -- 3,600 -- 10,800 -- 32,400
right? of course we can't even *hear* most of these pitches, so
we do the octave reduction and get everything between 1/1 and 2/1
by multiplying everything by 1/2 or 2/1 until we get:
128/81 -- 32/27 -- 16/9 -- 4/3 -- 1/1 -- 3/2 -- 9/8 -- 27/16 -- 81/64
that make sense? all those ratios are now between 1/1 and 2/1.
in hertz these pitches are:
632.1 -- 474.1 -- 711.1 -- 533.3 -- 400 -- 600 -- 450 -- 675 -- 506.25
now all the pitches are between 400 and 800 hertz and we can hear
them nicely. these are pitches all made with only the prime "3",
and so such a set of notes is called a "3-limit" set. it's also
called "pythagorean". note that the line extends infinitely in both
directions.
now if we also start to use the prime "5", we need another dimension.
naturally, we add a vertical dimension and say that a step *up* is
to multiply by 5/1, and a step *down* is to multiply by 1/5.
working out the math, and doing octave reduction, we can make a
2-dimensional lattice like this:
40/27---10/9-----5/3-----5/4----15/8----45/32--135/128
| | | | | | |
| | | | | | |
32/27---16/9-----4/3-----1/1-----3/2-----9/8----27/16
| | | | | | |
| | | | | | |
256/135--64/45---16/15----8/5-----6/5----9/5-----27/20
this lattice extends infinitely in all four directions,
i've only drawn the very center. you can figure the hertz
out for yourself if you want, it's easy! just multiply 400
by the ratio in question. many of the pitches in this lattice are
very close to the notes on a piano (piano note ratios, and all equal
tempered instruments, are *irrational*, but i'm not going to get into
that now, the lattice pitches are, for the most part, very *close*).
if you start with C as 1/1, the 12 notes of a piano are,
more or less:
C 1/1
C# 16/15
D 9/8
Eb 6/5
E 5/4
F 4/3
F# 45/32
G 3/2
Ab 8/5
A 27/16
Bb 16/9
B 15/8
note all these ratios are in our lattice already. so just by
multiplying and dividing by 3 and 5 a few times, we've already
made all 12 notes used on modern instruments, and then some!
(except that modern tuning is irrational, but that's a whole
other matter...).
now tuning and harmony theorists these days are pretty out there.
they are exploring these kinds of lattices for new and interesting
things. one way to look for surprises is to add another prime
number: 7. this adds a third dimension to the lattice, making
it hard to draw on paper or in ascii! but there are some tricks
to make it work, sorta. this is where the lattices start to
look funky.
what they do is leave the horizontal axis as the "prime 3". the vertical
axis ("prime 5") they skew 60 degrees. so in ascii, it's like this:
"3-axis" "5 axis"
/
--------- /
/
/
this results in a skewed lattice:
10/9----5/3----5/4---15/8---45/32
/ / / / /
/ / / / /
16/9----4/3----1/1----3/2----9/8
/ / / / /
/ / / / /
64/45--16/15---8/5----6/5----9/5
just to make things more complicated, people usually draw in more
lines, resulting in a triangular lattice:
10/9------5/3------5/4-----15/8-----45/32
/ \ / \ / \ / \ /
/ \ / \ / \ / \ /
/ \ / \ / \ / \ /
/ \/ \/ \/ \/
16/9------4/3------1/1------3/2------9/8
/ \ / \ / \ / \ /
/ \ / \ / \ / \ /
/ \ / \ / \ / \ /
/ \/ \/ \/ \/
64/45----16/15-----8/5------6/5------9/5
this is just the same as the square lattice i drew above, except
the verticals are tilted to the right, and diagonals have been added.
now in *this* lattice, we can define what moving in various
directions means:
move left: multiply by 3/1
move right: multiply by 1/3
move up-right: multiply by 5/1
move down-left: multiply by 1/5
move up-left: multiply by 5/3
move down-right: multiply by 3/5
in other words, the new diagonals we've added correspond to *two* steps.
going from 1/1 to 5/3 really means, in the simplest terms: one step
*up* on the five-axis + one step *left* on the three-axis, or,
1/1 * 5/1 * 1/3 = 5/3. so these new diagonals aren't really of the same
type as the other lines, but they can be useful to draw in to show
relationships, so long as you keep in mind their composite nature.
but we still haven't added the third-dimension of the prime 7.
ok, here goes. technically, the 3rd dimension sticks up out of,
and down into, the screen. can't draw that. but on this skewed
lattice we've drawn, we can give the illusion of a 3rd dimension
like this:
5/4
/|\
/ | \
/ 7/4 \
/,' `.\
1/1'-------3/2
the 7/4 in the middle is thought of as sticking up out of the page,
making a tetrahedron. this gives us *three* new directions to move
in (each of the three lines connecting the 7/4). the *basic* step
is moving in the direction from 1/1 to 7/4 ("northeast"). this is just
multiplying by 7/1 (octave reduction makes it 7/4). the other two new
directions are composite. anyway, we can describe all the possible
steps in this lattice as multiplying your starting ratio by another ratio:
5/1 5/3
/ \
1/3 ------ 3/1 / \
/ \
1/5 3/5
, 7/1 5/7 7/3 .
,' | `.
1/7 ' | ` 3/7
7/5
SO! that's all there is. :) now you should be able to understand this:
5/4
/|\
/ | \
7/6-------7/4------21/16
\`. /,'/|\`.\ ,'/
\ 1/1-/-|-\-3/2 /
\/|\/49/40\/|\/
/\|/,12/7`.\|/\
/ 7/5------21/20\
/,' \`.\|/,'/ `.\
8/5-----\-6/5-/-----9/5
\ | /
\|/
42/25
this is called (i think), a "14-tone stellated hexany" or "The Mandala".
it took me a while to wrap my brain about the 3rd dimension here, but
i think i got it --
note the two big triangles. the upright one, with vertices 5/4,
9/5 and 8/5 is, let's say, flat on the surface of the computer screen.
this triangle is just a part of the simple 2-d lattice we made before:
10/9------5/3------5/4-----15/8-----45/32
/ \ / \ / \ / \ /
/ \ / \ / \ / \ /
/ \ / \ / \ / \ /
/ \/ \/ \/ \/
16/9------4/3------1/1------3/2------9/8
/ \ / \ / \ / \ /
/ \ / \ / \ / \ /
/ \ / \ / \ / \ /
/ \/ \/ \/ \/
64/45----16/15-----8/5------6/5------9/5
note the 1/1, 3/2, and 6/5 in the stellated lattice are just more
points from the simple 2-d plane. now look at the note 7/5 in the
stellated lattice. it's the top vertex of a tetrahedron sticking
up from the triangle around it. same with 7/4 and 21/20. so while
the upright big triangle is on the surface on the screen, the
upside-down big triangle is floating one step above the screen.
that means that the central rectangle (1/1, 3/2, 7/5, 21/20) is
not flat but skewed -- its upper edge is on the screen and its
lower edge is floating above it. the final weirdness is the two
pitches in the middle. 12/7 is the apex of a tetrahedron that
is sticking *down* into the computer screen, while 49/40 is the
apex of a tetrahedron sticking up *two* steps from the screen.
you can tell this because each step *above* the screen represents
multiplication by 7/1. "49" is a number you can only get to by
multiplying by 7 *twice*. so 49/40 is the top of the tetrahedron
with the base 7/4, 7/5, 21/20 (the small, upright, central triangle).
just the same, each step *down* into the screen represents multiplication
by 1/7. "12/7" therefore is *into* the screen relative to 1/1.
so 12/7 is the apex of an inverted tetrahedron with the base
1/1, 3/2, 6/5 (the small, upside-down, central triangle).
i have no idea how music made with these pitches would sound.
i also don't understand very well how people construct these
kind of things. i believe a common technique is to try to
find as small a set of notes as you can that yields as large
a number as possible of "consonant" intervals and chords between
all members of the set to each other.
and now, for your amusement!, here's the names of the ratios in this
stellated lattice, in order of pitch, and the number of half-steps
from 1/1 on a piano they represent, approximate note-names in
equal temperment, and technical name:
ratio #-of-half-steps approx-note-name technical name
1/1 0.00 "C" unison
21/20 0.84 "C#" minor semitone
7/6 2.66 "Eb" septimal minor third
6/5 3.15 "Eb" minor third
49/40 3.51 "E" neutral third
5/4 3.86 "E" major third
21/16 4.70 "F" narrow fourth
7/5 5.82 "F#" septimal tritone
3/2 7.01 "G" perfect fifth
8/5 8.13 "Ab" minor sixth
42/25 8.98 "A" quasi-tempered major sixth
12/7 9.33 "A" septimal major sixth
7/4 9.68 "Bb" harmonic seventh
9/5 10.17 "Bb" just minor seventh
and *finally*! i got this mandala lattice in a post on the
tuning list in which the author (Paul H. Erlich, i think) wrote
that with these 14 notes you get 36 consonant intervals.
however, he later pointed out that you can do better with
the so-called "7-limit tonality diamond", which also has
36 consonant intervals, but also 8 consonant 4-note chords,
and only 13 notes. what a strange discipline. i can't even
begin to imagine how one calculates the number of consonant
intervals and chords. and if you think all this was tricky,
just imagine what would happen if you added the dimensions of
the primes 11, 13, or 17, as some people do!
anyway --
"14-note Stellated Hexany" "7-limit Tonality Diamond"
E E---------B
/|\ /|\`. ,'/|\
/ | \ / | \ `Db / | \
D#-------,A#-------,E# / ,A#-------,E# \
\`. /,'/|\`.\ ,'/ /,' \`.\|/.'/ `.\
\ `C'-/-|-\-`G' / C'-----\-`G'-/-----`D
\/|\/.,Dx,\/|\/ \`. /,\/|\/.\ ,'/
/\|/,'Bbb`.\|/\ \ Bbb-/\|/\--Fb /
/ ,F#-------`C# \ \ | / ,C# \ | /
/,' \`.\|/,'/ `.\ \|/,' `.\|/
Ab-----\-`Eb-/-----`Bb Eb-------`Bb
\ | /
\|/
A
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