!the scale tree in two dimensions & errors of 5-limit consonances from JI in ETs

monz, kraig, joseph, carl, marc, bill s., everyone:

something about all these graphs we've been looking at ( including
the first one at
/tuning/files/dict/eqtemp.htm ) that seems
to have escaped everyone's attention is that

they are two-dimensional scale trees!

this is brought out spectacularly in this new graph:

***********************************************************
/tuning/files/perlich/tree.gif
***********************************************************

which differs from previous variations, especially last week's
miller4.gif, only in that the smaller numbers are printed in bigger
fonts, and vice versa!

along any straight line in the graph that passes through several ETs
(and there are many!), we recognize an instance of the "traditional",
wilson-style scale tree for a particular linear mapping or set of
parent MOSs.

for example, the "meantone", or "5+7", tree (i've omitted the scales
where none of the best approximations to 5-limit JI consonances are
implied by meantone, but could just as well have kept them in)

.....................................................................5
...................................................................//.
................................................................././..
.............................................................../../...
............................................................./.../....
7........................................................./...../.....
|\\...................................................../....../......
||\.\................................................./......./.......
|||...\............................................./......../........
|.\\....\........................................./........./.........
\.|.\.....\...................................../........../..........
|||..\......\................................./.........../...........
||.\.|........\............................./............/............
||.|..\.........\........................./............./.............
||.|...\..........\...................../............../..............
|\..\..|............\................./.............../...............
|.|.|...\.............\............./................/................
|.|.|....\..............\........./................./.................
\.|..\....\...............\...../................../..................
.||..|....|.................\./.................../...................
.|\..|.....\................12.................../....................
.|.|..\.....\.............../|\................./.....................
.|.|..|.....|..............//|.\.............../......................
.|.|..|......\............/.||..\............./.......................
.|.|...\......\........../..||...\.........../........................
.|.\...|.......\......../...||....\........./.........................
.|..|..|.......|......./.../||.....\......./..........................
.|..|...\.......\...../....|||......\...../...........................
.|..|...|........\.../.....|||.......\.../............................
.|..|...|........|../...../.||........\./.............................
.|..\....\........\/......|./|........17..............................
.\...|...|........19......||.|........................................
..|..|...|.......//\......||.|........................................
..|..|....\...../.|||..../.|.|........................................
..|..|....|.../...|||....|.|.|........................................
..|..\....|../....|||....|.|.|........................................
..|...|....26.....||\.../..|.|........................................
..|...|..../|..../.|.|..|..|.|........................................
..|...|....||....|.|.|..|../.|........................................
..|...|.../.|....|.|.|..|.|../........................................
..|...\../..\....|.|..\/..|.|.........................................
..|....||....|...|.\..31..|.|.........................................
..|....33....|../...|./|..|.|.........................................
..|.../......|..|...|.||..|.|.........................................
..\../.......\..|...|.|\..|.|.........................................
...40.........|.|...|.|.|.|.|.........................................
..............|.|...|.|.\./.|.........................................
..............|/....|.|.43..|.........................................
..............45....\/...\..|.........................................
....................50....\./.........................................
..........................55..........................................

is clearly visible along one line in the graph. the "pelogic",
or "7+9", tree is visible as well, as are any others you should care
to name.

as a consequence:

pick an ET inside the graph (inconsistent ones may appear in more
than one place; pick one spot and stick to it). locate any line of
collinear ETs that it lies on. starting from the ET you picked, move
slowly in one direction along the line until you find a number with a
bigger font. do the same in the other direction to find a second
number. add up the two numbers ("tree roots") you found. the result
should equal the ET you picked! (the big curve on top is the bottom
of a humungous "3")

this is more than just a neat coincidence. the two numbers you found
tell you the cardinalities of prominent (in 5-limit terms) parent
MOSs within the scale, both based on a single generator and
complementing one another. even

for example, focus on 19.

looking at the meantone line
( if you don't know what i mean by that, look again at the first
graph at /tuning/files/dict/eqtemp.htm ),
19 can be seen as having "tree roots" of 12 and 7. this reflects the
familiar "yasserian" fact that a 19-equal keyboard can be profitably
arranged with 7 white keys (forming a diatonic scale generated by the
perfect fifth -- the generator of meantone) and 12 black keys
(forming an unequal chromatic scale, also generated by the same
meantone perfect fifth). each of these numbers, 12 and 7, as well as
any even bigger-font numbers along the same line with no still-bigger
font numbers intervening (in this case 5), signify a fifth-
generated "grandparent" MOS subset of 19-equal. if there is a bigger-
font number intervening, as is the case for 17 (12 intervenes), the
MOS is a "cousin".

looking at the kleismic line, we enter the realm of dave keenan's
chain-of-minor-thirds scales. 19's parents here are 4- and 15-note
chain-of-minor-thirds MOSs. 11 is a "grandparent" MOS. these
genealogical terms should make sense if you construct scale trees
like the one i illustrated above -- though there's an obscene amount
of "incest" here -- note that in the klesmic tree, 19's grandma(11)
and pa(4) gave birth to 19's own ma(15), and, as you can see in the
meantone tree above, 19's grandpa(5) and ma(7) gave birth to 19's own
pa(12).

the 5-limit magic line has 19 begat by 16 and 3, with one of the
three instances of the number 13 showing up as a granparent. these
scales are chains of *major* thirds.

these facts may be rather obscure to someone who hasn't worked with
19-equal in a 5-limit context very much . . . so how about we look at
12.

anyone familiar with the development of western music through the
20th century should appreciate the significance of 12 being:

5+7 along the meantone (fifth-generated) line -- the pentatonic
scale, plus a disjoint diatonic scale, give the 12-tone totality;

8+4 along the diminished/octatonic (minor-third-generated) line --
the famous octatonic scale, plus a disjoint diminished seventh chord,
give the 12-tone totality;

3+9 along the augmented/diesic (major-third-generated) line -- the
augmented triad, plus a disjoint tcherepnin scale, give the 12-tone
totality.

so far, all the MOS systems we've seen are generated by one of the 5-
limit consonant intervals, and indeed each of their defining lines in
the graph line up rather well with the blue axis for the
corresponding consonant interval quite well. but this is not always
the case. for example, the line indicating the vanishing 16875:16384
(which monz has in his table, but its name -- negri -- is missing, as
is the corresponding line), which is yet another that passes through
19 (and in this case the relevant parents are 10 and 9), corresponds
to a generator that is between 1/9 and 1/10 of an octave. perhaps
more familiar to list members is a line that passes from 10, down
(and slightly rightward) through 51, 41, (72 if you can make it out,)
31, 52, 21, and a barely-visible 11 peeking out from the bottom of
the graph -- this is the 5-limit incarnation of miracle, or
ampersand, and all these numbers give MOSs of miracle generators,
which are of course similar to "semitones" in size.

well, this should be enough to chew on for a while -- i welcome
questions, comments, corrections, and discussions, of course. i
happen to think that

/tuning/files/perlich/tree.gif

might strike some of you as one of the coolest charts you've seen
around here, and with the addition of some labeled red "family tree"
lines a la monz, this one chart might allow one to easily see
anything anyone would ever need to know about the 5-limit properties
of ETs!

wow, paul, this is AWESOME!!!!

can i put it into my "equal-temperaments" definition?
or should it go somewhere else?

-monz

----- Original Message -----
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: Wednesday, October 16, 2002 9:51 AM
Subject: [tuning] !the scale tree in two dimensions & errors of 5-limit
consonances from JI in ETs

> monz, kraig, joseph, carl, marc, bill s., everyone:
>
> something about all these graphs we've been looking at
> ( including the first one at
> /tuning/files/dict/eqtemp.htm )
> that seems to have escaped everyone's attention is that
>
> they are two-dimensional scale trees!
>
> this is brought out spectacularly in this new graph:
>
> ***********************************************************
> /tuning/files/perlich/tree.gif
> ***********************************************************
>
> <snip>

--- In tuning@y..., "monz" <monz@a...> wrote:
> wow, paul, this is AWESOME!!!!
>
> can i put it into my "equal-temperaments" definition?

you betcha!!!!!!! and more, zoomed-in ones to come, per your offlist
order. we can always add some red lines later.

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>

/tuning/topicId_39667.html#39667

>
> well, this should be enough to chew on for a while -- i welcome
> questions, comments, corrections, and discussions, of course. i
> happen to think that
>
> /tuning/files/perlich/tree.gif
>
> might strike some of you as one of the coolest charts you've seen
> around here, and with the addition of some labeled red "family
tree"
> lines a la monz, this one chart might allow one to easily see
> anything anyone would ever need to know about the 5-limit
properties
> of ETs!

***This chart is absolutely "trippy" Paul! I'm still puzzling
through it...

Joseph

--- In tuning@y..., "Joseph Pehrson" <jpehrson@r...> wrote:

/tuning/topicId_39667.html#39688

> --- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
>
> /tuning/topicId_39667.html#39667
>
> >
> > well, this should be enough to chew on for a while -- i welcome
> > questions, comments, corrections, and discussions, of course. i
> > happen to think that
> >
> > /tuning/files/perlich/tree.gif
> >
> > might strike some of you as one of the coolest charts you've seen
> > around here, and with the addition of some labeled red "family
> tree"
> > lines a la monz, this one chart might allow one to easily see
> > anything anyone would ever need to know about the 5-limit
> properties
> > of ETs!
>
> ***This chart is absolutely "trippy" Paul! I'm still puzzling
> through it...
>
> Joseph

***I guess it *would* make sense that the larger the ET the more it
would come close to all the axes, which is certainly the way it looks
in this great new chart...

JP

--- In tuning@y..., "Joseph Pehrson" <jpehrson@r...> wrote:

>
> ***I guess it *would* make sense that the larger the ET the more it
> would come close to all the axes, which is certainly the way it
looks
> in this great new chart...

sometimes . . . but note that, for example, 12 is closer to
the 'crosshairs' than, say, any of the three instances of 33 -- so
33, even though it has way more notes than 12, cannot match 12's 5-
limit accuracy, no matter *how* you use 33!

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>

/tuning/topicId_39667.html#39694

wrote:
> --- In tuning@y..., "Joseph Pehrson" <jpehrson@r...> wrote:
>
> >
> > ***I guess it *would* make sense that the larger the ET the more
it
> > would come close to all the axes, which is certainly the way it
> looks
> > in this great new chart...
>
> sometimes . . . but note that, for example, 12 is closer to
> the 'crosshairs' than, say, any of the three instances of 33 -- so
> 33, even though it has way more notes than 12, cannot match 12's 5-
> limit accuracy, no matter *how* you use 33!

***Oh sure! That makes sense. That's why in your "famous" accuracy
chart (from TTTT) the accuracies don't go straight up in a 45 degree
angle, but "zig zag" around a bit, I think...

JP

hey there,

in case you (monz) or anyone were wondering what happened to 24, 30,
36, 38, and 44-tone equal temperaments, i made a better graph where
they're now visible:

/tuning/files/perlich/zoomer.gif

i used different colors so that the bigger numbers they overlay are
still legible.

also, 5-limit consistent equal temperaments are now in shades of blue
and violet, inconsistent ones in shades of orange and red.

every equal temperament up to 51 (except 1, 2, ans 6) is now clearly
visible in at least one place on this graph, and those from 52 to 99
are there too, just "scrunched".

i have a full set of zoom-ins (and a zoom-out) created with this
method, if these are what you'd like to include on your webpage.

--- In tuning@y..., "monz" <monz@a...> wrote:
> wow, paul, this is AWESOME!!!!
>
> can i put it into my "equal-temperaments" definition?
> or should it go somewhere else?
>
>
> -monz
>
>
>
> ----- Original Message -----
> From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
> To: <tuning@y...>
> Sent: Wednesday, October 16, 2002 9:51 AM
> Subject: [tuning] !the scale tree in two dimensions & errors of 5-
limit
> consonances from JI in ETs
>
>
> > monz, kraig, joseph, carl, marc, bill s., everyone:
> >
> > something about all these graphs we've been looking at
> > ( including the first one at
> > /tuning/files/dict/eqtemp.htm )
> > that seems to have escaped everyone's attention is that
> >
> > they are two-dimensional scale trees!
> >
> > this is brought out spectacularly in this new graph:
> >
> > ***********************************************************
> > /tuning/files/perlich/tree.gif
> > ***********************************************************
> >
> > <snip>

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>

/tuning/topicId_39667.html#39755

wrote:
> hey there,
>
> in case you (monz) or anyone were wondering what happened to 24,
30,
> 36, 38, and 44-tone equal temperaments, i made a better graph where
> they're now visible:
>
> /tuning/files/perlich/zoomer.gif
>
> i used different colors so that the bigger numbers they overlay are
> still legible.
>

****This is quite nice, and comes out quite nicely on my ink jet...

J. Pehrson