Horagrams - beginner's guide?

Dear mathematically inclined comrades of whom I am envious

For some time now I've been meaning to get to grips with the mighty
horagrams because I think I could put them to good musical use. Now
they're beginning to p**s me off so much that I've decided I need some
help. I followed the earlier thread between David Finnamore and Paul
Erlich which shed some, but not a lot, of much needed light and I'm
grateful to these two for their greater understanding of things
mathematical. So I have two questions :-

1. Beginning with the index which looks like a mathematician's bad trip,
I can see some sort of tree/lambdoma affair with the fractions or ratios
extending downwards arrived at by adding numerators and denominators.
And I can also see (just) how Wilson arrives at the rows of calculations
below. These are some sort of mean or mid-point of two higher fractions.
What I don't see is why he does this and what basis in 'reality' do the
final numbers (to 12 decimal points) have?

2. Moving on to the pages with columns - calc.data, log2, pitch and
approx 8ve, I think I can see some connection between the numbers in
calc.data, such as 8 1 phi, read right to left as the arrow indicates,
but I can't see a consistent correlation. Presumably the log is arrived
at from the data, no idea how, and the other two columns follow. The
approx 8ve column I thought seemed reasonable but I can't see how 3/19
comes from 1.114727845 and 7/3 from 1.140386925. Then my ultimate
question is (with all due respect) - so what?

If I can clear this lot up then I might be able to sort myself out with
the horagrams themselves but don't hold your breath.

Any clues from anyone please and I'd be eternally grateful indeed.

BTW I intend using the horagrams eventually as a pitch, rhythm and
texture organisation device for some 22 tet guitar composition.
Horagrams would seem to have more substance than the bar codes I used to
use for objective compositional structures.

Kind Regards

--- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:
> Dear mathematically inclined comrades of whom I am envious
>
> For some time now I've been meaning to get to grips with the mighty
> horagrams because I think I could put them to good musical use. Now
> they're beginning to p**s me off so much that I've decided I need
some
> help. I followed the earlier thread between David Finnamore and Paul
> Erlich which shed some, but not a lot, of much needed light and I'm
> grateful to these two for their greater understanding of things
> mathematical.

You might want to look back in the archives at discussions involving
Jason Yust, Dave Keenan, Dan Stearns, Kraig Grady, and myself. Some
of it was in October of last year. Dan Stearns develops his own
version of the horagram stuff, unaware of Wilson's work, which is
then shown and explained to him.

> 1. Beginning with the index which looks like a mathematician's bad
trip,
> I can see some sort of tree/lambdoma affair with the fractions or
ratios
> extending downwards arrived at by adding numerators and
denominators.

Right. Each of those fractions ordinarily refers to a logarithmic
fraction of an octave, so in particular to a generator that is a
particular ET interval.

> And I can also see (just) how Wilson arrives at the rows of
calculations
> below. These are some sort of mean or mid-point of two higher
fractions.
> What I don't see is why he does this and what basis in 'reality' do
the
> final numbers (to 12 decimal points) have?

They are the limiting values you get if you continue a zig-zag
pattern down the tree indefinitely. If you use these values as
generators, all the MOSs in the horagram beyond a certain point have
their large step and small step in the Golden Ratio (1.618....) to
one another. Margo Schulter posted in detail about the derivation of
the "Golden mean" between two fractions that you're seeing Wilson use
here. Perhaps she can point you to an appropriate post by herself.

> 2. Moving on to the pages with columns - calc.data, log2, pitch and
> approx 8ve, I think I can see some connection between the numbers in
> calc.data, [snip]

You'll have to tell me what webpage you're looking at.

> Then my ultimate
> question is (with all due respect) - so what?

Wilson views the Golden Horagrams as Archetypes for MOS scales, as
they stay as far as possible away from ETs. The "specialness" of this
approach only shows up for any given infinite sequence of scales and
not for any one scale taken alone. Hence it's more a curiosity than a
musically important result, in my opinion. Besides Wilson seems to
have neglected the cases where the octave is an integer multiple
(other than 1) times the interval of repetition -- a major
shortcoming in my eyes.

I wrote,

> Margo Schulter posted in detail about the derivation of
> the "Golden mean" between two fractions that you're seeing Wilson
use
> here.

You can read this post, which was co-authored by Dave Keenan, here:

http://dkeenan.com/Music/NobleMediant.txt

It was posted to the tuning list on 17-Sep-2000 -- if you look there,
you'll find some responses I and others posted.

Paul Erlich wrote:

> --- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:
> > Dear mathematically inclined comrades of whom I am envious
> >
> > For some time now I've been meaning to get to grips with the mighty
> > horagrams because I think I could put them to good musical use. Now
> > they're beginning to p**s me off so much that I've decided I need
> some
> > help. I followed the earlier thread between David Finnamore and Paul
> > Erlich which shed some, but not a lot, of much needed light and I'm
> > grateful to these two for their greater understanding of things
> > mathematical.
>
> You might want to look back in the archives at discussions involving
> Jason Yust, Dave Keenan, Dan Stearns, Kraig Grady, and myself. Some
> of it was in October of last year. Dan Stearns develops his own
> version of the horagram stuff, unaware of Wilson's work, which is
> then shown and explained to him.
>
> > 1. Beginning with the index which looks like a mathematician's bad
> trip,
> > I can see some sort of tree/lambdoma affair with the fractions or
> ratios
> > extending downwards arrived at by adding numerators and
> denominators.
>
> Right. Each of those fractions ordinarily refers to a logarithmic
> fraction of an octave, so in particular to a generator that is a
> particular ET interval.
>
> > And I can also see (just) how Wilson arrives at the rows of
> calculations
> > below. These are some sort of mean or mid-point of two higher
> fractions.
> > What I don't see is why he does this and what basis in 'reality' do
> the
> > final numbers (to 12 decimal points) have?
>
> They are the limiting values you get if you continue a zig-zag
> pattern down the tree indefinitely. If you use these values as
> generators, all the MOSs in the horagram beyond a certain point have
> their large step and small step in the Golden Ratio (1.618....) to
> one another. Margo Schulter posted in detail about the derivation of
> the "Golden mean" between two fractions that you're seeing Wilson use
> here. Perhaps she can point you to an appropriate post by herself.
>
> > 2. Moving on to the pages with columns - calc.data, log2, pitch and
> > approx 8ve, I think I can see some connection between the numbers in
> > calc.data, [snip]
>
> You'll have to tell me what webpage you're looking at.

http://www.anaphoria.com/hrgm01.html

> Then my ultimate

> > question is (with all due respect) - so what?
>
> Wilson views the Golden Horagrams as Archetypes for MOS scales, as
> they stay as far as possible away from ETs. The "specialness" of this
> approach only shows up for any given infinite sequence of scales and
> not for any one scale taken alone. Hence it's more a curiosity than a
> musically important result, in my opinion. Besides Wilson seems to
> have neglected the cases where the octave is an integer multiple
> (other than 1) times the interval of repetition -- a major
> shortcoming in my eyes.

Well, that's certainly clearer now, though I'm unsure as to the importance of MOS scales. I know what
they are but not why they are considered significant. I'll digest the above explanations and come up
with some more questions if that's OK.

Kind regards

Paul Erlich wrote:

> I wrote,
>
> > Margo Schulter posted in detail about the derivation of
> > the "Golden mean" between two fractions that you're seeing Wilson
> use
> > here.
>
> You can read this post, which was co-authored by Dave Keenan, here:
>
> http://dkeenan.com/Music/NobleMediant.txt
>
> It was posted to the tuning list on 17-Sep-2000 -- if you look there,
> you'll find some responses I and others posted.

Wonderful - I'm on the case now.

Regards

>
>

--- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:
>
>
> Paul Erlich wrote:
>
> >
> > You'll have to tell me what webpage you're looking at.
>
> http://www.anaphoria.com/hrgm01.html

That doesn't work for me. Are you sure you spelled everything right?

> Well, that's certainly clearer now, though I'm unsure as to the
importance of MOS scales. I know what
> they are but not why they are considered significant.

That's been discussed extensively. One major breakthrough was
my "Hypothesis" presented here and discussed more over at tuning-
math. My personal feeling is that omnitetrachordal variants of MOS
scales are melodically nicer than MOS scales themselves, though the
MOS scales tend to have more of the consonant chords. Of course,
the "Hypothesis" and the question of consonant chords are irrelevant
for the Horagrams, because consonance doesn't figure into them in any
way.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:

> > Well, that's certainly clearer now, though I'm unsure as to the
> importance of MOS scales. I know what
> > they are but not why they are considered significant.

I'd say the most significant thing about them is that sometimes they
are the result of iterating a single generator modulo an interval
such as 2 or sqrt(2). They are one way of getting scales interesting
both harmonically and melodically, though hardly the only way--no one
has commented as yet on my miracle-magic square, for another example,
but I at least think it has great promise, though it is not a MOS.

> That's been discussed extensively. One major breakthrough was
> my "Hypothesis" presented here and discussed more over at tuning-
> math.

I hope this won't annoy you, but I'm not clear why it is a
breakthrough. It depends on tempering a block, which it is not
necessary to do.

My personal feeling is that omnitetrachordal variants of MOS
> scales are melodically nicer than MOS scales themselves, though the
> MOS scales tend to have more of the consonant chords.

Time for me to think about omnitetrachordality, perhaps.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:
>
> > > Well, that's certainly clearer now, though I'm unsure as to the
> > importance of MOS scales. I know what
> > > they are but not why they are considered significant.
>
> I'd say the most significant thing about them is that sometimes

Why not always?

> they
> are the result of iterating a single generator modulo an interval
> such as 2 or sqrt(2).

In the case of Horagrams, it's always 2 (although you can certainly
interpret them in terms of a different interval of repetition, such
as 2^(1/2), 2^(1/29), or 3.

> They are one way of getting scales interesting
> both harmonically and melodically,

Melodically interesting because there are only two step sizes, and no
more than two sizes of any generic interval.

> though hardly the only way--no one
> has commented as yet on my miracle-magic square, for another
example,
> but I at least think it has great promise, though it is not a MOS.

It looks alright, though not sure what's so "special" about it.

> > That's been discussed extensively. One major breakthrough was
> > my "Hypothesis" presented here and discussed more over at tuning-
> > math.
>
> I hope this won't annoy you, but I'm not clear why it is a
> breakthrough. It depends on tempering a block, which it is not
> necessary to do.

Well, if you don't temper the block, you get a CS scale, which is
often so MOS-like that Kraig Grady will refer to it as MOS. Tempering
smooths over the "wolves", and then happens to result in an MOS. If
it weren't for that, I'd hardly have any interest in MOS at all.
That's why it's a "breakthrough" in the context of understanding the
importance of MOS for me (also I think it's a new development).

If you're going to build a new instrument with fixed pitches, or
create a keyboard tuning, or refret a guitar, wouldn't you agree that
eliminating "wolves" through slight tempering is quite valuable? Can
you see any powerful reason _not_ to do so, aside from some new-age
mystical mumbo-jumbo about the sacredness of rational numbers?

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> > though hardly the only way--no one
> > has commented as yet on my miracle-magic square, for another
> example,
> > but I at least think it has great promise, though it is not a MOS.

> It looks alright, though not sure what's so "special" about it.

It's compares reasonably well to MOS of about that size in terms of
amount of harmony (for instance, the 9-note Orwell I'm writing in at
the moment), and since it is planar it can be tuned even more
precisely. It shows up as a part of MOS scales in a natural way, as I
was showing with Blackjack and some magic and orwell scales, and
which would also be true of schismic if you went far enough, and it
also is naturally included in the larger three-step scales one gets
by continuing the Osmium construction, such as a nifty 19-note scale.
It strikes me therefore as a reasonably significant formation.

> Well, if you don't temper the block, you get a CS scale, which is
> often so MOS-like that Kraig Grady will refer to it as MOS.
Tempering
> smooths over the "wolves", and then happens to result in an MOS.

But why introduce a block at all--why not simply proceed as I've been
doing with my articles with names like "41" or "46"?

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > > though hardly the only way--no one
> > > has commented as yet on my miracle-magic square, for another
> > example,
> > > but I at least think it has great promise, though it is not a
MOS.
>
> > It looks alright, though not sure what's so "special" about it.
>
> It's compares reasonably well to MOS of about that size in terms of
> amount of harmony (for instance, the 9-note Orwell I'm writing in
at
> the moment),

Well, that's not an ideal example :)

> and since it is planar it can be tuned even more
> precisely. It shows up as a part of MOS scales in a natural way, as
I
> was showing with Blackjack and some magic and orwell scales, and
> which would also be true of schismic if you went far enough, and it
> also is naturally included in the larger three-step scales one gets
> by continuing the Osmium construction, such as a nifty 19-note
scale.
> It strikes me therefore as a reasonably significant formation.

Is it true that the only 7-limit unison vector it swallows is
224:225? I'd like to draw a lattice.

> > Well, if you don't temper the block, you get a CS scale, which is
> > often so MOS-like that Kraig Grady will refer to it as MOS.
> Tempering
> > smooths over the "wolves", and then happens to result in an MOS.
>
> But why introduce a block at all--why not simply proceed as I've
been
> doing with my articles with names like "41" or "46"?

Either way is equivalent, of course -- with the block approach you're
emphasizing Joe Monzo's finity concept, in that you're plopping
yourself down in the just lattice, expanding to include more and more
pitches connected by consonant intervals, and stopping just before
you add more pitches "equivalent" to ones you already have.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., genewardsmith@j... wrote:

> > It's compares reasonably well to MOS of about that size in terms
of
> > amount of harmony (for instance, the 9-note Orwell I'm writing in
> at
> > the moment),
>
> Well, that's not an ideal example :)

Why do you say that?

> > It strikes me therefore as a reasonably significant formation.

> Is it true that the only 7-limit unison vector it swallows is
> 224:225? I'd like to draw a lattice.

There are four primes up to the 7-limit, which is therefore 4D, and a
planar temperament is (naturally!) 3D, so the kernel is 1D, and
225/224 generates it. You can add 385/384 for the 11-limit.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@y..., genewardsmith@j... wrote:
>
> > > It's compares reasonably well to MOS of about that size in
terms
> of
> > > amount of harmony (for instance, the 9-note Orwell I'm writing
in
> > at
> > > the moment),
> >
> > Well, that's not an ideal example :)
>
> Why do you say that?

Well, I'd say that the 9-note Orwell's harmonic resources are a bit
spotty, at least by my usual standards.

> > > It strikes me therefore as a reasonably significant formation.
>
> > Is it true that the only 7-limit unison vector it swallows is
> > 224:225? I'd like to draw a lattice.
>
> There are four primes up to the 7-limit, which is therefore 4D, and
a
> planar temperament is (naturally!) 3D, so the kernel is 1D, and
> 225/224 generates it. You can add 385/384 for the 11-limit.

So is this right for the 7-limit?:

4/3·······1/1······
/|\ / `. ,'
/ | \ / 12/7
5/4·······-S \ /
/ \`. /,'/ `.\ /
/ \ S··/·····8/5
7/6 / \ | /
,' `. / \|/
4/3·······1/1·······3/2
/|\ / `. ,'
/ | \ / 12/7
5/4·······-S \ /
/ \`. /,'/ `.\ /
/ \ S··/·····8/5
7/6 / \ | /
,' `. / \|/
······1/1·······3/2

Looks OK, though I'm not blown away by it . . . looks closely related
to the Lumma-Fokker scale . . .

What 11-limit chords does it have?

In general, though, I think what you're saying is basically that if
you take a block, and leave two, rather than only one, of the unison
vectors untempered, you get a good non-MOS scale. This makes perfect
sense to me, and I think is what Dave Keenan was talking about in his
goodbye post to the list, when he brought up a 19-tone subset of
MIRACLE (note that Dave has posted since then but emphasizes that he
isn't back).

genewardsmith@juno.com wrote:

> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:
>
> > > Well, that's certainly clearer now, though I'm unsure as to the
> > importance of MOS scales. I know what
> > > they are but not why they are considered significant.
>
> I'd say the most significant thing about them is that sometimes they
> are the result of iterating a single generator modulo an interval
> such as 2 or sqrt(2). They are one way of getting scales interesting
> both harmonically and melodically, though hardly the only way--no one
> has commented as yet on my miracle-magic square, for another example,
> but I at least think it has great promise, though it is not a MOS.

I've saved your miracle-magic square post and I look forward to finding time to look more closely.

Kind Regards

>

Paul Erlich wrote:

> --- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:
> >
> >
> > Paul Erlich wrote:
> >
> > >
> > > You'll have to tell me what webpage you're looking at.
> >
> > http://www.anaphoria.com/hrgm01.html
>
> That doesn't work for me. Are you sure you spelled everything right?

Ah - maybe it's dead because I seem to remember something about all the Wilson archives going to
PDF. I don't have the URL for those I'm afraid.

>
>
> > Well, that's certainly clearer now, though I'm unsure as to the
> importance of MOS scales. I know what
> > they are but not why they are considered significant.
>
> That's been discussed extensively. One major breakthrough was
> my "Hypothesis" presented here and discussed more over at tuning-
> math. My personal feeling is that omnitetrachordal variants of MOS
> scales are melodically nicer than MOS scales themselves, though the
> MOS scales tend to have more of the consonant chords. Of course,
> the "Hypothesis" and the question of consonant chords are irrelevant
> for the Horagrams, because consonance doesn't figure into them in any
> way.

Interesting. I'm not quite up to the level for the math list. Once I understand what you've
explained I'll try to make sense of the horagrams themselves.

Best Wishes