Wilson Horagrams

All,

My initial attempts to tame the 1,3,5,7 hexany didn't get far. In
the process of researching at Anaphoria to find more info about
hexanies, dekanies, eikosanies, and so forth, I stumbled across
Wilson's "golden horagrams of the scale tree." I can see that they
contain lambdomic patterns of some kind. They appear to be vertical
slices of the tree bent around in a circle. Does anyone on the list
know what they are for? I'm working on the assumption - no more
than a guess, really - that the circle represents 2:1 or possibly
any interval of equivalence. I've generated some tunings by turning
the angles directly into cents - simply multiplying the number of
degrees by 3 1/3 (1200 cents/360 degrees).

Whether that's what Wilson intended, I have no idea, but it yields
some amazing scales. Unlike nearly all JI tunings I've played with,
most of these have a cohesiveness that is readily perceivable.
There's a beauty to them that feels eerily natural to me, often kind
of gooey or watery, if you know what I mean. Change one tone in a
scale by only a few cents - even if it brings a given interval
closer to Just - or accidentally invoke a tone from the next ring
out, and it stands out like a sore thumb. This even in the context
of already highly xenharmonic intervals. I'm kind of stunned.
Never experienced anything like it.

Nevertheless, one thing that makes me wonder whether I'm using them
as intended (besides the fact that there's nothing in the way of a
key or legend!) is that the horagrams are derived from the scale
tree, which in turn appears to be an alternative representation of
(sections of?) the Lambdoma, which is, in all other uses I've seen
of it, strictly rational. So one might expect the horagrams to
represent rational tunings, too, although the "slices" that become
horagrams look like they're taken from approx. phi times the length
of each "branch" of the tree.

This all assumes that the horagrams are supposed to represent
tunings, also only a guess in the dark. They might represent ways
of categorizing scales by some means that doesn't taking specific
tunings into account. Or ...?

In any event, the tunings generated by "my" method sound promising
to me. If anyone is interested, I'll post them on a web page.
There's probably too much material to publish on the List.

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

--- In tuning@egroups.com, "David J. Finnamore" <daeron@b...> wrote:

> I'm working on the assumption - no more
> than a guess, really - that the circle represents 2:1 or possibly
> any interval of equivalence. I've generated some tunings by turning
> the angles directly into cents - simply multiplying the number of
> degrees by 3 1/3 (1200 cents/360 degrees).

Correct!
>
> Whether that's what Wilson intended, I have no idea, but it yields
> some amazing scales. Unlike nearly all JI tunings I've played with,
> most of these have a cohesiveness that is readily perceivable.
> There's a beauty to them that feels eerily natural to me, often kind
> of gooey or watery, if you know what I mean. Change one tone in a
> scale by only a few cents - even if it brings a given interval
> closer to Just - or accidentally invoke a tone from the next ring
> out, and it stands out like a sore thumb. This even in the context
> of already highly xenharmonic intervals. I'm kind of stunned.
> Never experienced anything like it.

Which scales have you played with? There are lots of scales here --
one of the horagrams corresponds to the Kornerup meantones, while
another is Keenan Pepper's "neo-gothic" tuning, and other is the
2^phi tuning Jason Yust and I were discussing heavily this
summer . . . since then, the horagram scales have come up quite a bit
in discussions involving Margo Schulter, Dave Keenan, Dan Stearns,
and Kraig Grady.

> Nevertheless, one thing that makes me wonder whether I'm using them
> as intended (besides the fact that there's nothing in the way of a
> key or legend!) is that the horagrams are derived from the scale
> tree, which in turn appears to be an alternative representation of
> (sections of?) the Lambdoma, which is, in all other uses I've seen
> of it, strictly rational. So one might expect the horagrams to
> represent rational tunings, too, although the "slices" that become
> horagrams look like they're taken from approx. phi times the length
> of each "branch" of the tree.

You're on the right track -- each of the generators (1 per horagram)
is a noble number, meaning it's your position on the tree after
you've alternately zigged and zagged an infinite number of
consecutive times.

> Does anyone on the list
> know what they are for

As you've probably surmised, each horagram shows a number of
different scales built from repetitions of a single generator, given
in each case by a noble (Wilson would say "golden") fraction of an
octave. Each ring is a different MOS scale, and besides a few "extra"
MOSs at the beginning of the series (in the centermost rings), the
pattern always ends up as a Fibonacci-like sequence; e.g., 5, 7, 12,
19, 31, 50, . . . for the Kornerup horagram
http://www.anaphoria.com/hrgm33.html) or the Fibonacci sequence
itself (as in http://www.anaphoria.com/hrgm28.html).

Although MOS scales are certainly melodically favorable in certain
ways, the particular advantage of a noble generator as opposed to any
other generator seems to only show itself when the whole infinite
series, or sizable part thereof, of MOS scales is examined. For any
_particular_ MOS scale, there seems to be no possible reason why one
would distinguish a noble generator from a non-noble one. Not being
negative; just suggesting that you might want to look at other, non-
noble MOS scales and you might find other things you like . . .

This is good news.

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:

> Although MOS scales are certainly melodically favorable in certain
> ways, the particular advantage of a noble generator as opposed to
any
> other generator seems to only show itself when the whole infinite
> series, or sizable part thereof, of MOS scales is examined. For any
> _particular_ MOS scale, there seems to be no possible reason why
one
> would distinguish a noble generator from a non-noble one. Not being
> negative; just suggesting that you might want to look at other, non-
> noble MOS scales and you might find other things you like . . .

So what I'm hearing might be due to MOS rather than to noble
generators, or a combination thereof. I'll try to get a firmer grasp
on the concept of MOS next then. Was the "particular advantage of a
noble generator" discussed earlier this year (my reading of the list
was spotty for most of the summer)? By "when the whole infinite
series, or sizable part thereof, of MOS scales is examined," do you
mean that seems to show up in theory (when looking at a set of
scales) but not in practice (when playing a particular scale)?

David Finnamore

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:

> Which scales have you played with?

On any horagram I've limited myself to rings with 12 or fewer steps.
This is partly because it allowed me to explore a large number of
them more quickly, viz., without the extra work of setting up
and finding fingerings for tunings that repeat at higher numbers of
steps. It's also because of the 7+/-2 rule. Even with 12 scale
steps perception of the order starts to blur.

For the first one I chose horagram #50: 7phi+2 / 17phi+5, because it
offers the familiar 5->7->12 pattern. It just melted me immediately.
This pans out as:

Cents Key 12 EDO deviation
0.00 C 44
59.73 C# 4
119.46 D -37
275.84 D# 20
335.57 E -20
491.95 F 36
551.68 F# -4
611.41 G -45
767.78 G# 12
827.52 A -28
983.89 A# 28
1043.62 B -12

More later, the boss is a-callin'.

> one of the horagrams corresponds to the Kornerup meantones, while
> another is Keenan Pepper's "neo-gothic" tuning, and other is the
> 2^phi tuning Jason Yust and I were discussing heavily this
> summer . . . since then, the horagram scales have come up quite a
bit
> in discussions involving Margo Schulter, Dave Keenan, Dan Stearns,
> and Kraig Grady.

See, I always get in trouble for not paying attention in class!
Actually, I do remember parts of these discussions. Many of the
digests containing them are filed in my "Further Research" folder. I
don't remember Wilson's horagrams being mentioned in connection with
them.

David Finnamore

--- In tuning@egroups.com, "David Finnamore" <daeron@b...> wrote:

> Was the "particular advantage of a
> noble generator" discussed earlier this year (my reading of the
list
> was spotty for most of the summer)? By "when the whole infinite
> series, or sizable part thereof, of MOS scales is examined," do you
> mean that seems to show up in theory (when looking at a set of
> scales) but not in practice (when playing a particular scale)?

Basically. A particular scale in a noble series of MOSs has the
property that the ratio between the large and small melodic steps
(measured logarithmically, e.g., in cents) is phi. Nothing special
about that in itself. However, each successive scale in the series
splits each of the large steps of the previous scale into two parts --
one part is equal to the small step of the previous scale, and the
other is equal to phi-1 = 1/phi times the small step of the previous
scale. Thus the ratio between the two step sizes of the new scale is
1:(phi-1) = 1:(1/phi) = phi. This magic works because phi is the
solution to the equation
x-1 = 1/x.
Anyway, any non-noble, non-ET generator will eventually produce an
MOS scale where the ratio of the step sizes is more uneven than phi.
However, one may have to look at very high in the series of MOSs for
a particular generator to find this uneven scale. So the advantage of
using a noble generator seems only clear if you extend the series of
MOSs very high.

--- In tuning@egroups.com, "David Finnamore" <daeron@b...> wrote:
>
> For the first one I chose horagram #50: 7phi+2 / 17phi+5, because
it
> offers the familiar 5->7->12 pattern. It just melted me
immediately.

David, the generator of this scale is just 1 cent different from the
22-tET fourth (or fifth). I have this 12-out-of-22 MOS as one of my
three 12-out-of-22 keyboard mappings on my synth. I mapped the chain
of fifths to the notes Eb-Bb-F. . .B-C#-G#, leaving the interval from
Eb to G# as exactly 600 cents. It has many wonderful properties, such
a host of consonant 4:6:7:9 chords, and a normally fingered "Bb
minor" scale sounding (and acting) like a JI _major_ scale!

David Finnamore wrote,

>For the first one I chose horagram #50: 7phi+2 / 17phi+5, because it
>offers the familiar 5->7->12 pattern. It just melted me immediately.
> This pans out as:

>Cents Key 12 EDO deviation
>0.00 C 44
>59.73 C# 4
>119.46 D -37
>275.84 D# 20
>335.57 E -20
>491.95 F 36
>551.68 F# -4
>611.41 G -45
>767.78 G# 12
>827.52 A -28
>983.89 A# 28
>1043.62 B -12

Hmm . . . looks like you've accidentally got the right note names too,
except that your "C" is actually a B# and your "F" is actually a E#. So with
this mapping, you can get a near-"JI major scale" by fingering an A _minor_
scale, and a near-"JI minor scale" by fingering a Bb _major_ scale. You've
got Greek enharmonic scales at B#-C#-D-E#-F#-G-A and B#-C#-D-E#-F#-G-A#. The
following "keys" (traditionally fingered) will feature three 6:7:9 minor
triads and three 1/9:1/7:1/6 major triads: D major, A major, E major, B
major, F# major, and C# major, and their relative minors . . .

P.S. I'd advocate something closer to 22-tET because the "JI minor thirds"
here are almost 20 cents sharp, while in 22-tET they're only 11 cents sharp
. . .

I wrote:
> > For the first one I chose horagram #50: 7phi+2 / 17phi+5, because
> > it offers the familiar 5->7->12 pattern.

Paul H. Erlich wrote:
> David, the generator of this scale is just 1 cent different from the
> 22-tET fourth (or fifth). I have this 12-out-of-22 MOS as one of my
> three 12-out-of-22 keyboard mappings on my synth. I mapped the chain
> of fifths to the notes Eb-Bb-F. . .B-C#-G#, leaving the interval from
> Eb to G# as exactly 600 cents. It has many wonderful properties, such
> a host of consonant 4:6:7:9 chords, and a normally fingered "Bb
> minor" scale sounding (and acting) like a JI _major_ scale!

How fortuitous! Of course, it's 13.5 cents different by the time
you get around the circle. I wonder whether all noble generator MOS
scales might have near approximations in EDOs (well, you know, EDOs
smaller than 8539)? Wilson offers rational approximations to the
generating intervals. All roads lead to Rome. But the road you
take influences your view of Rome once you get there.

> Hmm . . . looks like you've accidentally got the right note names too,

No, those are just the note names in the O5R/W tuning table. I see
what you've done, though, using the meantone naming convention.
That makes sense.

If you make little melodies by playing the "1," then add the "2,"
then the "3," then the "4 and 5" and stay pentatonic for a bit, then
the "6 and 7" and stay heptatonic before adding the other tones, you
can really notice the organizational principle. Well, I'd say it
just jumps out at you. This doesn't happen with JI tunings (unless
they're MOS, maybe?). In fact, play the heptatonic and then try
replacing either the "4" or "5" with an adjacent tone from the 12
ring and hear the system fall apart. That's what amazed me. I
suppose it works in 12 EDO, too, I just never thought to look at it
that way. (Uh-oh, duck, here come the rotten vegetables.)

> P.S. I'd advocate something closer to 22-tET because the "JI minor thirds"
> here are almost 20 cents sharp, while in 22-tET they're only 11 cents sharp

I'll try that. But I don't view them as JI minor thirds, and
they're nice in their own right.

Here are some other MOS/horagram/phi/noble scales that impressed me
immediately. I'll refer to them by their generator in cents.
273.85 and 280.61 have more-or-less equal pentatonics but
interesting 9-tone scales that "make sense." They don't feel like
you tried to shoehorn a couple more notes into a diatonic scale, as
9-toners often do.

273.85
cents key deviation from 12 EDO
0.00 C -7
169.25 D -38
273.85 D# -33
443.10 E 36
547.70 F 41
716.95 G 10
821.55 G# 15
990.80 A# -16
1095.40 B -12

280.61
cents key deviation from 12 EDO
0.00 C -22
203.05 D -19
280.61 D# -41
483.66 E 62
561.22 F# -61
764.27 G 42
841.83 G# 20
1044.88 A# 23
1122.44 B 0

317.17 has a "Hungarian" 7-tone scale that evokes lively, magical
images for me.

cents key deviation from 12 EDO
0.00 C 5
68.68 C# -26
137.36 D -58
317.17 D# 22
385.85 E -9
454.53 F -40
634.34 F# 39
703.02 G 8
771.70 G# -23
951.51 A 57
1020.19 A# 25
1200.00 0 0

I did not like 331.67. For those of you wishing, as someone said
recently, to emancipate the dissonances, check this one out. The
exception that proves the rule.

cents key deviation from 12 EDO
0.00 C -4
126.69 C# 23
253.37 D 49
331.67 D# 28
458.36 F -46
585.05 F# -19
663.34 G -41
790.03 G# -14
916.72 A 13
995.02 A# -9
1121.70 B 18
1200.00 0 0

350.90 has an intriguing 7-tone scale with neutral tones. Chic.

cents key deviation from 12 EDO
0.00 C 1
56.27 C# -43
203.58 D 5
350.90 D# 52
407.16 E 8
554.48 F# -45
701.79 G 3
758.06 G# -41
905.37 A 6
1052.69 A# 54

Once again, "key" means the key that the pitch fell on in my synth's
tuning table. For cycles of fewer than 12 tones, I usually choose
to map symmetrically where possible. Some choices are arbitrary.
Suggestions welcome, as always.

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

David Finnamore wrote,

>How fortuitous! Of course, it's 13.5 cents different by the time
>you get around the circle. I wonder whether all noble generator MOS
>scales might have near approximations in EDOs

Of course they do (just pick the number of notes in one of the outer rings)
-- but for a sufficiently fine degree of approximation, noble fractions (of
the octave, i.e., logarithmic) are harder to approximate by rational
fractions (of the octave, i.e., logarithmic) than any other fractions.
(This, of course, works if you remove "of the octave, i.e., logarithmic" too
. . .)

>Wilson offers rational approximations to the
>generating intervals.

Hmm . . . there's no special rule about how well linear rationals
approximate logarithmic nobles . . .

Glad you found so many of the horagram scales useful. I'll have to play with
them at some point. The 350-cent generator has been discussed a lot, e.g. on
Graham Breed's website.

--- In tuning@egroups.com, "David J. Finnamore" <daeron@b...> wrote:

> ... I wonder whether all noble generator MOS scales might have
> near approximations in EDOs (well, you know, EDOs smaller than
> 8539)? Wilson offers rational approximations to the generating
> intervals. All roads lead to Rome. But the road you take
> influences your view of Rome once you get there.

It really struck me how appropriately this metaphor characterizes
one of the main things I'm trying to accomplish with my lattice
diagrams. The important thing for me is to find a totally
consistent way of accurately graphing the measurements of any
and all kinds of tunings.

Paul, this is exactly why I wanted to know how to factor the
degrees of meantone scales - it simply allows me to plug those
tunings into my lattice formula and 'put them on the map'. ;-)

(Most appropriate for me, a confirmed 'map geek'...)

-monz
http://www.ixpres.com/interval/monzo/homepage.html
'All roads lead to n^0'

Monz wrote,

>It really struck me how appropriately this metaphor characterizes
>one of the main things I'm trying to accomplish with my lattice
>diagrams. The important thing for me is to find a totally
>consistent way of accurately graphing the measurements of any
>and all kinds of tunings.

>Paul, this is exactly why I wanted to know how to factor the
>degrees of meantone scales - it simply allows me to plug those
>tunings into my lattice formula and 'put them on the map'. ;-)

But you don't have a unique factorization in general! For LucyTuning, for
example, you will have no way of deciding where to put the notes, since
there are an infinite number of ways of "factoring" the intervals -- all
involving irrational exponents of the primes.

I'd suggest a different way of latticing meantones and other temperaments.
Use a lattice where the line segments don't quite meet! This would be an
immediate way to see how far the temperament's intervals are from JI
intervals. And yet the basic "shape" of a chord or scale would look about
the same no matter what temperament you're using, which is certainly not the
case the way you're mapping meantones now.