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Message: 7201 Date: Mon, 04 Aug 2003 05:22:50 Subject: Re: Creating a Temperment /Comma From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> So, doesn't sound like you can break consistency in > >> regular temperaments!
> > > >how can you break something that isn't even defined?
> > You can't! Which is why I was said there isn't any > in linear temperaments. > > You mentioned something about infinite numbers of notes > always yielding better approximations.
that's why it isn't defined.
> What did you > mean by that?
consistency is only defined when each just interval has a best approximation in the tuning. with an infinite number of irrational notes, there is no best approximation, you can keep finding better and better ones.
> > -Carl
Message: 7202 Date: Mon, 04 Aug 2003 05:24:44 Subject: Re: review requested From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> I wrote this in a hurry, trying to explain the activity on this
list
> >> to a friend is as short a document as possible.
> > > >i just annotated it with a bunch of corrections, but then i hit the > >backspace key (which usually acts as "delete") and was sent back to > >the previous webpage. AAARRRRRGGGGHHHH!!
> > Sorry that happened. Thanks for trying! >
> >> I haven't read *The Forms of Tonality* since early 2001, but I
plan
> >> to do that again now. I'm sure it covers much of the same
ground.
> > > >not enough of it. you're getting closer to my whole philosophy on > >these things.
> > I don't think I've changed my position much.
your position? i mean you're getting closer than the forms of tonality alone.
Message: 7203 Date: Mon, 04 Aug 2003 05:32:43 Subject: Re: review requested From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >is the diatonic major second just or near-just?
> > Not sure where this is pointed.
you said something about the lattice being constructed out of the musical intervals used, or something . . .
> >> () We create a "pun" if we use the same name ("Ab") for both
> >> notes in such a pair.
> > > >i think "pun" refers to the same pitch being used for different > >musical functions.
> > It was my understanding that it was your assumption that two Ab > notes in a score refer to the same pitch, and thus, imply meantone > temperament for common practice music.
hmm . . . ok maybe i misread you or something . . . not sure . . .
> In which case my phrasing > above is ok. If we don't want to make that assumption I should > change it.
_the forms of tonality_ does not make that assumption. it is intended to appeal to the just intonation (network) crowd.
>
> >> () You can think of "simple" as giving more intervals > >> with fewer tones if the comma is tempered out.
> > > >??
> > Simple commas tend to define small blocks, so if all commas are > tempered out we get all the intervals with fewer notes.
but not more intervals.
> Even though > a linear temp. has infinitely many notes, there must be something > similar going on...
yes, more per pitch or whatnot.
> >> () As a matter of strange coincidence, the same math > >> is behind harmonic entropy!
> > > >behind or in front of?
> > ? > > Anyway, this clearly doesn't belong in the doc. But if you could > write a blurb on this for monz or someone to post, I think it'd > be interesting.
i'd love to, but what would i be writing a blurb on?
>
> >> * [with cont.] (Wayb.) > >> > >> Porcupine temperament * [with cont.] (Wayb.)
> > > >my piece "glassic" is even more directly based on this temperament, > >using its 7-note MOS for long stretches, and was just rebroadcast
on
> >wnyc!
> > Nice. Do you have a link?
it used to be on tuning-punks, chris bailey or someone may have rescued them . . .
> If I ever decide to publish this, it > will be as a web page with inline graphics, and I'll ask Herman > for a link to MPO. Or, I'm happy to provide links at lumma.org. >
> >>I haven't read *The Forms of Tonality* since early 2001, but I
plan
> >>to do that again now. I'm sure it covers much of the same ground.
> > > >not enough of it.
> > I assume you mean I'm not covering enough of your ground?
no, i mean the forms of tonality doesn't cover enough of what you're covering here. and hoped to make a tuning-math collaborative paper on, but there were too many disagreements to get started.
> My goal > is to make a document much shorter than TFOT. Actually, maybe I > haven't even do so. TFOT was pretty short IIRC! > > -Carl
carl, please help us see beyond our differences and produce a document which will be beautiful and matter to the future of music . . .
Message: 7204 Date: Mon, 04 Aug 2003 05:34:40 Subject: Re: review requested From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> Heya monz, >
> >>>> () You can think of "simple" as giving more > >>>> intervals with fewer tones if the comma is > >>>> tempered out.
> >>> > >>>??
> >> > >>Simple commas tend to define small blocks, so if all commas are > >>tempered out we get all the intervals with fewer notes. Even > >>though a linear temp. has infinitely many notes, there must be > >>something similar going on...
> > > >Carl, it's simply a matter of differing dimensions. > > > >i think i must be misunderstanding this discussion,
> > I'm not sure which part of the above quote you're referring to. > The last part there is a aggressive abstraction of Paul's > complexity heuristic, which may not be accurate.
well then i didn't understand it either!
Message: 7205 Date: Mon, 04 Aug 2003 05:36:34 Subject: Re: Creating a Temperment /Comma From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >consistency is only defined when each just interval has a > >best approximation in the tuning. with an infinite number > >of irrational notes, there is no best approximation, you > >can keep finding better and better ones.
> > But you have to respect the map, according to Gene.
if that's so, no equal temperament can be inconsistent either.
> So > we can still define something like best approx. of n/p > must be best approx. n - best approx p.
how?? either you respect the map, or you use the best approximation . . . (?)
Message: 7206 Date: Mon, 04 Aug 2003 05:38:23 Subject: Re: review requested From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >>>not enough of it. you're getting closer to my whole philosophy on > >>>these things.
> >> > >> I don't think I've changed my position much.
> > > >your position? i mean you're getting closer than the forms of > >tonality alone.
> > Oh, thanks. I thought you meant I needed to get closer to it. > Then maybe it's worth getting everybody's Seal of Approval on > the present doc and publishing it on the web, with inline > graphics. I'll need a link to glassic.
maybe john starrett can rescue it. and maybe ara and i can record a better version of it. we're only living together for another month . . .
> If you haven't changed > the file since the mp3.com days, I have it, and can provide a > link for you.
cool!
Message: 7207 Date: Mon, 04 Aug 2003 05:59:20 Subject: Re: Creating a Temperment /Comma From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >consistency is only defined when each just interval has a > >best approximation in the tuning. with an infinite number > >of irrational notes, there is no best approximation, you > >can keep finding better and better ones.
> > But you have to respect the map, according to Gene.
You either do or don't respect the map with one generator; in the same way, you can respect or not respect the map with two. I'm always on the side of respect.
Message: 7208 Date: Mon, 04 Aug 2003 10:17:37 Subject: Re: review requested From: Graham Breed Paul Erlich wrote:
> no, i mean the forms of tonality doesn't cover enough of what you're > covering here. and hoped to make a tuning-math collaborative paper on, > but there were too many disagreements to get started.
I got started with this document:
How to find linear temperaments * [with cont.] (Wayb.)
which covers the parts I'm directly interested in. Now I have more time
on my hands, I could write up the method from unison vectors as well.
But not this week, because there's good weather forecast.
I don't know about the philosophy behind unison vectors, because that's
not really my thing. But I suggest the idea of Constant Structure is
important:
Definitions of tuning terms: constant structur... * [with cont.] (Wayb.)
A full set of unison vectors define a constant structure. If you temper
out all commas, all intervals of a particular kind are equally well
approximated. If you temper none of them out, some intervals are
perfect and some are wolves. With a linear temperament, some are
"official" approximations, and others wolves, and so on.
Is the caveat about unison vector sizes necessary? I think a
periodicity block will always obey the mapping, but the notes might not
be in ascending order!
Graham
Message: 7209 Date: Mon, 04 Aug 2003 02:30:10 Subject: Re: review requested From: Carl Lumma
>Is the caveat about unison vector sizes necessary?
Which caveat? The one about the uvs being smaller than the smallest 2nd? This is necessary if you want a constant structure. Or, at least, that was the consensus back in 2000. -Carl
Message: 7210 Date: Mon, 04 Aug 2003 10:39:39 Subject: Re: review requested From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> I got started with this document: > > How to find linear temperaments * [with cont.] (Wayb.)
I hope to have something complimentary soon. First I need to discuss wedge products, and I've just added a lot about that. Maybe it will make sense to someone.
Message: 7211 Date: Sat, 09 Aug 2003 01:41:19 Subject: tctmo! From: Carl Lumma Here's the doc... * [with cont.] (Wayb.) ...Herman, you're welcome to link to the Mizarian Porcupine Overture at * [with cont.] (Wayb.). Or if you have a url for it elsewhere, I can use that instead. Paul, Glassic is at * [with cont.] (Wayb.). -Carl
Message: 7212 Date: Sun, 10 Aug 2003 04:47:08 Subject: Bosanquet keyboards and linear temperaments From: Gene Ward Smith Here is a url for a picture of a Bosanquet keyboard: bosanquet * [with cont.] (Wayb.) I will assume this is laid out in a flat array, and that the hexagons are actually regular hexagons. I will also assume it is a meantone keyboard, rather than specifically a 31-et keyboard. It then has the following properties: (1) There is a "sharp" axis, c, c sharp, etc., where if q is a 5- limit rational number, then h5(q) (where h5=[5, 8, 12] is the 5-limit standard val for the 5-et) counts the number of steps along this axis of the "sharp" coordinate. This axis is inclined at an angle of 76.102 degrees upward. (2) 120 degrees away there is a "flat" axis, c, d flat, e double flat and so forth, inclined at an angle of -43.898 degrees downward. Given q, the number of steps along the flat axis for the flat coordinate is h7(q), where h7 = [7, 11, 16] is the 5-limit standard val for the 7- et. (3) Because of the inclination of the h5 and h7 axis, octaves are exactly along a horizontal line. (4) The keys are divided into white and black colors. If [a, b] are the coordinates for a key, with "a" being the h5 coordinate and "b" being the h7 coordinates, then if a+b mod 12 is 0, 2, 4, 5, 7, 9, or 11 the key is colored white; if it is 1, 3, 6, 8, or 10 mod 12, it is colored black. This is a 7-note MOS (the diatonic scale) in 7+5=12 equal terms. The values of for the inclination of the h5 and h7 axes follow from the assumption that the keys are regular hexagons. This means the axes are 120 degrees apart, and so we can use the inner product H where H([a1, a2], [b1, b2]) = a1b1 + a2b2 - (a1b2+b1a2)/2 to determine angles and distances. Afterwards, however, we are free to apply an affine transformation if we prefer another shape of hexagon. We can now generalize this for any linear temperament. Let us instead take miracle, and h10 and h11 in the place of h5 and h7. The h10 axis is now inclined 64.715 degrees upward, and h11 is inclined -55.285 degrees downward. To get the key coloring, we take a version of the 11-note miracle MOS in 21-equal; and reduce it to a set of eleven numbers mod 21. If a+b mod 21 is a member of this set, we color white; if not, we color black.
Message: 7214 Date: Sun, 10 Aug 2003 04:49:32 Subject: Re: tctmo! From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> Here's the doc... > > * [with cont.] (Wayb.)
That should say some of the thinking, not all of it. Of course we could get ambitious.
Message: 7215 Date: Sun, 10 Aug 2003 20:03:08 Subject: Re: tctmo! From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >Is the argument about signal-processing capabilities basic to > >the thinking on tuning-math?
> > You can hardly get more basic. Why are we concerned with error?
To keep things in tune.
> >Moreover, you have not discussed octave equivalence and you need > >to at this point.
> > Octaves are just part of the fixed set, part of JI.
Mostly, we've been looking at octave-reduced consonance sets.
> The original title of this document was "Scales and Temperaments". > Its original intent was to explain what I consider to be one of the > more important realizations of our work -- that the difference > between scales and temperaments is largely semantic.
It sometimes is (ciculating temperaments) but not always (regular temperaments.)
> >Moreover, you have not defined your terms. Do the chosen elements > >generate a group, and is this "stacking"? Where and how does the > >lattice turn up?
> > The intended audience doesn't know or care what a group is. The > lattice generating business could probably benefit from an example > and/or diagram.
They'd better, since you need groups to define lattices. A lattice is a free abelian group of finite rank with additional structure, the additional structure being an embedding into a normed real vector space. It is better not to add additional structure unless you have a use for it; that merely confuses things.
> >This should be telling us that one generator gives us an infinite > >chain, which can be embedded as the integers into the real numbers, > >two elements give us a free group of rank two, embeddable in R2 as > >a lattice in various ways, and so forth.
> > Gene, this is a 'for complete idiots' type document. I wonder how > many here even know what the heck this means.
I don't think it is a for complete idiots subject.
> >you need to put an inner product/quadratic form/metric/norm
(however
> >you want to define it) onto your space to make the 5-limit lattice
be
> >A2 and the 7-limit lattice be A3 = D3. The "and so on" gets you
into
> >new problems. Is any of this really basic?
> > Certainly not worded like that. If you can furnish a for-dummies > version, I'm happy to add it. Then again, I suggested you add it to > your own website weeks ago.
I've been adding more to my website; you should check it out. Lattices are a good idea for another addition, along with a host of other things.
> I think stacking is quite basic, and familiar to musicians and > tuning list readers.
I've never heard of it, so I can't see how it is integral to thinking here.
> Groups aren't more basic to tuning list readers. I'd love to start > with groups, if I knew what they had to do with music.
The p-limit intervals form a group. Equal temperaments are groups. Linear temperaments are groups. Groups and group homomorphisms are everywhere on this list. They are absolutely basic.
> > 5.1-- The complexity of a comma may be measured by the > > distance on the lattice between the pitches that generate it. > > Thus, simpler commas tend to delimit smaller blocks. > > > >This makes no sense to me.
> > I'm not sure how I could possibly rephrase this.
What do you mean by "the pitches that generate it"?
Message: 7216
Date: Sun, 10 Aug 2003 05:46:13
Subject: Re: tctmo!
From: Gene Ward Smith
Outline
1-- Assume...
1.1-- Music involves repetition. Often, instead of exact
repetition, some things change while the rest stay the same.
1.1.1-- A theme played in a different mode keeps
generic
intervals (3rds, etc.) the same while pitches change.
1.1.1.1-- This is, in fact, only true for
Rothenberg-proper scales, such as the familiar
diatonic scale in 12-tone equal temperament.
1.1.2-- A theme played in a different key keeps
absolute
intervals the same while pitches change.
1.1.2.1-- As Rothenberg efficiency, the more
we
must rely on rules such as those traditionally
enforced in Western tonal music to make key
changes recognizable.
This is scale theory and should go under the heading of scales,
rather than as an introduction to everything.
1.2-- Possible intervals between notes are to be taken from
some
fixed set of just or near-just intervals, in order to best
exploit the signal-processing capabilities of the human
hearing
system to deliver information to the listener.
Is the argument about signal-processing capabilities basic to the
thinking on tuning-math? There are various reasons one could put
forward for limiting your intervals to a fixed set. Moreover, you
have not discussed octave equivalence and you need to at this point.
2-- So, to build a scale, we take our chosen just intervals and
*stack*
them. This generates a lattice.
This drags scales into an area where they are not required. Moreover,
you have not defined your terms. Do the chosen elements generate a
group, and is this "stacking"? Where and how does the
lattice turn up?
2.1-- In the 3-limit we get a chain. In the 5-limit, we get a
planar lattice. 7-limit, we can use the face-centered cubic
lattice. And so on.
You are assuming octave equivalence without having mentioned it. This
should be telling us that one generator gives us an infinite chain,
which can be embedded as the integers into the
real numbers, two elements give us a free group of rank two,
embeddable in R2 as a lattice in various ways, and so forth. You
haven't explained where the fcc 7-limit lattice arises from;
you need to put an inner product/quadratic form/metric/norm (however
you want to define it) onto your space to make the 5-limit lattice be
A2 and the 7-limit lattice be A3 = D3. The "and so on" gets you into
new problems. Is any of this really basic?
3-- Eventually, we will run into pitches that are very close to
pitcheswe already have. The small intervals between such pairs of
pitches are called commas.
This makes it sound as if the lattices needed to be introduced to do
this, which is not the case. Groups are more basic, why not start
there?
3.1-- We create a "pun" if we use the same name ("Ab") for
both
notes in such a pair.
3.2-- We create a "comma pump" by writing a chord progression
whose starting and ending "tonic" involve a "pun". Every time
the chord progression is repeated, our pitch standard moves by
the comma involved. Or...
We can create...
4-- We can temper the comma(s) out! Doing so collapses the lattice.
If
we temper enough commas out, we arrive at a finite "block" -- a
section
of the lattice delimited by the commas.
Once again, I suggest you start from groups, not lattices.
5-- It seems that most scales which have been popular around the world
and throughout history correspond fairly well to temperaments where
very simple commas have been tempered out.
5.1-- The complexity of a comma may be measured by the
distance
on the lattice between the pitches that generate it. Thus,
simpler commas tend to delimit smaller blocks.
This makes no sense to me.
5.1.1-- The size of the denominator of a comma
provides
a good "heuristic" for its complexity (Paul Erlich).
5.2-- The complexity of a temperament may be defined as
notes/intervals.
Again, this makes no sense to me.
5.2.1-- The complexity of a temperament is closely
related to the complexity of the commas which define
it.
Not true, I'm afraid. The commas can be arbitarily horrible so long
as there is more than one of them.
Message: 7217 Date: Sun, 10 Aug 2003 22:55:16 Subject: Re: tctmo! From: Graham Breed Gene Ward Smith wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: > >
>>>Is the argument about signal-processing capabilities basic to >>>the thinking on tuning-math?
>> >>You can hardly get more basic. Why are we concerned with error?
> > To keep things in tune.
There's only a brief mention of signal processing. It's a definition of "in tune".
> Mostly, we've been looking at octave-reduced consonance sets.
Octave equivalence should certainly be mentioned in there.
> They'd better, since you need groups to define lattices. A lattice is > a free abelian group of finite rank with additional structure, the > additional structure being an embedding into a normed real vector > space. It is better not to add additional structure unless you have a > use for it; that merely confuses things.
No. A lattice, in this context is a "network of points which show the simple translation vectors on which a structure is based." (H. M. Rosenberg, "The Solid State", third edition pp2-3)
> I don't think it is a for complete idiots subject.
Yes, we should at least target fairly intelligent idiots. But there's no reason to assume knowledge of group theory.
> The p-limit intervals form a group. Equal temperaments are groups. > Linear temperaments are groups. Groups and group homomorphisms are > everywhere on this list. They are absolutely basic.
"A scale is a group"?
>>> 5.1-- The complexity of a comma may be measured by the >>> distance on the lattice between the pitches that generate it. >>> Thus, simpler commas tend to delimit smaller blocks. >>> >>>This makes no sense to me.
>> >>I'm not sure how I could possibly rephrase this.
> > What do you mean by "the pitches that generate it"?
A comma is an interval between two pitches.
Graham
Message: 7218 Date: Sun, 10 Aug 2003 04:50:35 Subject: Re: tctmo! From: Carl Lumma
>This is scale theory and should go under the heading of scales, >rather than as an introduction to everything.
They are merely examples of repetition where some things remain fixed and others change. I need this to explain stacking.
>Is the argument about signal-processing capabilities basic to >the thinking on tuning-math?
You can hardly get more basic. Why are we concerned with error? Error from what? What separates what we do here from all the other kooked-out music theory out there? Acoustics.
>There are various reasons one could put forward for limiting >your intervals to a fixed set.
Such as? And not any fixed set, specifically just intonation.
>Moreover, you have not discussed octave equivalence and you need >to at this point.
Octaves are just part of the fixed set, part of JI.
>2-- So, to build a scale, we take our chosen just intervals and >*stack* them. This generates a lattice. > >This drags scales into an area where they are not required.
The original title of this document was "Scales and Temperaments". Its original intent was to explain what I consider to be one of the more important realizations of our work -- that the difference between scales and temperaments is largely semantic.
>Moreover, you have not defined your terms. Do the chosen elements >generate a group, and is this "stacking"? Where and how does the >lattice turn up?
The intended audience doesn't know or care what a group is. The lattice generating business could probably benefit from an example and/or diagram.
> 2.1-- In the 3-limit we get a chain. In the 5-limit, we get a > planar lattice. 7-limit, we can use the face-centered cubic > lattice. And so on. > >You are assuming octave equivalence without having mentioned it.
You can show the octaves and just bump everything up. Maybe I should say something about that.
>This should be telling us that one generator gives us an infinite >chain, which can be embedded as the integers into the real numbers, >two elements give us a free group of rank two, embeddable in R2 as >a lattice in various ways, and so forth.
Gene, this is a 'for complete idiots' type document. I wonder how many here even know what the heck this means.
>you need to put an inner product/quadratic form/metric/norm (however >you want to define it) onto your space to make the 5-limit lattice be >A2 and the 7-limit lattice be A3 = D3. The "and so on" gets you into >new problems. Is any of this really basic?
Certainly not worded like that. If you can furnish a for-dummies version, I'm happy to add it. Then again, I suggested you add it to your own website weeks ago. I think stacking is quite basic, and familiar to musicians and tuning list readers.
>3-- Eventually, we will run into pitches that are very close to >pitches we already have. The small intervals between such pairs of >pitches are called commas. > >This makes it sound as if the lattices needed to be introduced to do >this, which is not the case. Groups are more basic, why not start >there?
Groups aren't more basic to tuning list readers. I'd love to start with groups, if I knew what they had to do with music.
> 5.1-- The complexity of a comma may be measured by the > distance on the lattice between the pitches that generate it. > Thus, simpler commas tend to delimit smaller blocks. > >This makes no sense to me.
I'm not sure how I could possibly rephrase this.
> 5.1.1-- The size of the denominator of a comma > provides a good "heuristic" for its complexity > > 5.2-- The complexity of a temperament may be defined as > notes/intervals. > >Again, this makes no sense to me.
5.2 is a definition. Notes one has over consonant intervals formed between them. 5.1.1 is Paul's complexity heuristic. The size of the denominator of a pitch approximates the taxicab distance from the origin to it.
> 5.2.1-- The complexity of a temperament is closely > related to the complexity of the commas which define > it. > >Not true, I'm afraid. The commas can be arbitarily horrible so long >as there is more than one of them.
Eh? Again, this is the heuristic. It only works for one comma. But it should be applicable in the multiple-comma, with the addition of a concept called "straightness". -Carl
Message: 7219 Date: Sun, 10 Aug 2003 23:04:50 Subject: Re: tctmo! From: Graham Breed Carl Lumma wrote:
> You can hardly get more basic. Why are we concerned with error? > Error from what? What separates what we do here from all the > other kooked-out music theory out there? Acoustics.
I prefer to make my assumptions clear and leave acoustics out of it. It's then up to the reader whether or not they agree with it.
>>There are various reasons one could put forward for limiting >>your intervals to a fixed set.
> > Such as? And not any fixed set, specifically just intonation.
No, the way I work it can be any fixed set of pitches. That's why I
don't like these heuristics which tie it all to ratios for no good reason.
"Straightness" is another thing I never understood.
Graham
Message: 7220 Date: Sun, 10 Aug 2003 23:13:58 Subject: Re: Bosanquet keyboards and linear temperaments From: Graham Breed Gene Ward Smith wrote:
> I will assume this is laid out in a flat array, and that the hexagons > are actually regular hexagons. I will also assume it is a meantone > keyboard, rather than specifically a 31-et keyboard. It then has the > following properties:
Bosanquet didn't use hexagons. Erv Wilson did, and you can see his original papers at WILSON ARCHIVES * [with cont.] (Wayb.) It's generally a keyboard for scales generated by an octave and fifths. Bosanquet used 53 notes with a schismic mapping (the tuning may have been JI, I forget).
> We can now generalize this for any linear temperament...
Erv did that a long time ago.
Graham
Message: 7221 Date: Sun, 10 Aug 2003 23:43:04 Subject: Re: Bosanquet keyboards and linear temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Gene Ward Smith wrote: > Bosanquet didn't use hexagons. Erv Wilson did, and you can see his > original papers at > > WILSON ARCHIVES * [with cont.] (Wayb.)
Do you think Wilson lattices would be a better name than Bosanquet lattices for these?
> > We can now generalize this for any linear temperament...
> > Erv did that a long time ago.
I'll see if I can figure out what Erv is saying (not always easy for me, since he says it in pictures) and if I agree. Do you have a more exact reference?
Message: 7222 Date: Sun, 10 Aug 2003 21:24:18 Subject: Re: tctmo! From: Carl Lumma
>Schenker's motto, appearing on the title page of Der Freie Satz, was: > >Semper idem sed non eodem modo. > >"Always the same, but not in the same way."
I just learned what Shenkerian analysis was back in January. Totally cool.
>I like your outline Carl: very "Tractatus Logico Philosophicus".
Yeah, the numbering scheme. It was originally just bullets, but I figured this would allow people to reference a section without having to copy-and-paste it. -Carl
Message: 7223 Date: Sun, 10 Aug 2003 21:57:06 Subject: Re: tctmo! From: Carl Lumma
>> You can hardly get more basic. Why are we concerned with error? >> Error from what? What separates what we do here from all the >> other kooked-out music theory out there? Acoustics.
> >I prefer to make my assumptions clear and leave acoustics out of it. >It's then up to the reader whether or not they agree with it.
Everything we do starts from the fundamental assumption of JI. Or so it seems to me.
>>>There are various reasons one could put forward for limiting >>>your intervals to a fixed set.
>> >> Such as? And not any fixed set, specifically just intonation.
> >No, the way I work it can be any fixed set of pitches. That's why I >don't like these heuristics which tie it all to ratios for no good >reason.
Za?
>"Straightness" is another thing I never understood.
It has to do with the angle between the commas. If A and B are commas that vanish, and a and b are their lattice points, then the interval C between a and b also vanishes. The thing is, A and B could both be simple, but if the angle between them is wide, C could still be complex. So you have to account for this in a heuristic. -Carl
Message: 7224 Date: Sun, 10 Aug 2003 22:31:52 Subject: Re: tctmo! From: Carl Lumma
>> >Is the argument about signal-processing capabilities basic to >> >the thinking on tuning-math?
>> >> You can hardly get more basic. Why are we concerned with error?
> >To keep things in tune.
And why do we care about that?
>> >Moreover, you have not discussed octave equivalence and you need >> >to at this point.
>> >> Octaves are just part of the fixed set, part of JI.
> >Mostly, we've been looking at octave-reduced consonance sets.
Octaves are special, but there's no need to reflect this in a theory of temperaments, other than weighting the error and/or complexity functions. They're just like any other consonance to be mapped.
>> The original title of this document was "Scales and Temperaments". >> Its original intent was to explain what I consider to be one of the >> more important realizations of our work -- that the difference >> between scales and temperaments is largely semantic.
> >It sometimes is (ciculating temperaments) but not always (regular >temperaments.)
My plan is to paint circulating temperaments as small perturbations of regular temperaments, not significant in the big-picture sense.
>> The intended audience doesn't know or care what a group is. The >> lattice generating business could probably benefit from an example >> and/or diagram.
> >They'd better, since you need groups to define lattices. A lattice is >a free abelian group of finite rank with additional structure, the >additional structure being an embedding into a normed real vector >space. It is better not to add additional structure unless you have a >use for it; that merely confuses things.
It's jargon that's confusing. If one can draw a lattice it should be sufficient to understand this, without a rigorous definition.
>> >This should be telling us that one generator gives us an infinite >> >chain, which can be embedded as the integers into the real numbers, >> >two elements give us a free group of rank two, embeddable in R2 as >> >a lattice in various ways, and so forth.
>> >> Gene, this is a 'for complete idiots' type document. I wonder how >> many here even know what the heck this means.
> >I don't think it is a for complete idiots subject.
If you can't communicate with musicians and composers, what's the point? You don't have to communicate everything to them. You can work at another level. But at the end of the day, you should be able to explain the big picture.
>> I think stacking is quite basic, and familiar to musicians and >> tuning list readers.
> >I've never heard of it, so I can't see how it is integral to >thinking here.
Most musicians are familiar with structures like the circle of fifths.
>> Groups aren't more basic to tuning list readers. I'd love to >> start with groups, if I knew what they had to do with music.
> >The p-limit intervals form a group. Equal temperaments are groups. >Linear temperaments are groups. Groups and group homomorphisms are >everywhere on this list. They are absolutely basic.
I hope you can show this on your web site. I think I understand the mathworld definitions for group and homomorphism, but I'm not sure of the alternatives. That is, basic ratio arithmetic, which most folks on the tuning list already know, seems to fail without a lot of this stuff. I'm not aware of all the alternatives and their consequences, but many times when I do a text substitution on the jargon in one of your math things it winds up sounding totally obvious. Certainly, it doesn't end up sounding like something that would lead to the results you get. Either I misunderstand the definitions, you're leaving something out, or I just don't see the deep consequences of the ensemble of definitions.
>> > 5.1-- The complexity of a comma may be measured by the >> > distance on the lattice between the pitches that generate it. >> > Thus, simpler commas tend to delimit smaller blocks. >> > >> >This makes no sense to me.
>> >> I'm not sure how I could possibly rephrase this.
> >What do you mean by "the pitches that generate it"?
You're right, this is bad. "Define" is maybe better than "generate". Another way to say it might be "distance on the lattice between the comma and the origin". Currently taking suggestions for even better ways to say it. No jargon allowed. -Carl
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