weighted Bosanquet lattices

I don't know if anybody's ever thought about this, but
if Paul's got the right idea with weighted complexity, perhaps
keyboard layouts should be weighted too. The idea is that
physical distance on the keyboard from 1 to 2, 3, 5, etc.
should be proportional to log 2, 3, 5, etc. For every mapping
of every temperament, the variables would seem to be the
relative lengths of the two vectors mapping the keyboard and
the angle between them.

-Carl

Carl,

--- In tuning-math@yahoogroups.com, you wrote:
>
> I don't know if anybody's ever thought about this, but
> if Paul's got the right idea with weighted complexity, perhaps
> keyboard layouts should be weighted too. The idea is that
> physical distance on the keyboard from 1 to 2, 3, 5, etc.
> should be proportional to log 2, 3, 5, etc. For every mapping
> of every temperament, the variables would seem to be the
> relative lengths of the two vectors mapping the keyboard and
> the angle between them.

Draw me a picture? That would definitely make it easier to get my
head around this.

Yahya

This list is delaying/eating my posts virtually every time, so
I'm playing the 'slam the list with the post until it responds'
game, just for fun.

I know that my ISP has been blacklisted by outblaze a few
times recently... Spam really has made e-mail almost unuseable
for millions. Junk paper mail, though far more wasteful of
trees and such, doesn't seem cause so serious problems.

But I digress. Here's my post:

>Carl,
>
>--- In tuning-math@yahoogroups.com, you wrote:
>> I don't know if anybody's ever thought about this, but
>> if Paul's got the right idea with weighted complexity, perhaps
>> keyboard layouts should be weighted too. The idea is that
>> physical distance on the keyboard from 1 to 2, 3, 5, etc.
>> should be proportional to log 2, 3, 5, etc. For every mapping
>> of every temperament, the variables would seem to be the
>> relative lengths of the two vectors mapping the keyboard and
>> the angle between them.
>
>Draw me a picture? That would definitely make it easier to get my
>head around this.
>
>Yahya

Here's Gene's page

http://64.233.167.104/search?
q=cache:vJB9BLlKDYkJ:bahamas.eshockhost.com/~xenharmo/bosanquet.html+b
osanquet+lattices
(It doesn't look like Google caches images, and Gene's page is in
a perpetual state of downness, so that's the best I can do.)

but it may be easier to think about Bill Wesley's "array" vs.
the traditional piano keyboard.

http://www.thearraymbira.com/images/notes/5_theMBOX_Octaves%5B1%5D.gif

Octaves are very close together, and the left/right axis is
fifths. So there's no monotonic pitch scale here. But the
distance to a 5-limit interval is more proportional to its
complexity than on the piano.

There are two axes on a generalized keyboard, an angle between
them, and an aspect ratio between them. The idea is that for
a rank 2 temperament, find a map such that when the period and
generator are the two axes, some angle and aspect ratio makes
the distances come out proportional to the complexity of the
corresponding identities.

Dave's spreadsheet that Gene links to doesn't say what criteria
it uses to find its "ideal" layouts, but it apparently isn't
the one I'm describing here.

-Carl

>Carl,
>
>--- In tuning-math@yahoogroups.com, you wrote:
>> I don't know if anybody's ever thought about this, but
>> if Paul's got the right idea with weighted complexity, perhaps
>> keyboard layouts should be weighted too. The idea is that
>> physical distance on the keyboard from 1 to 2, 3, 5, etc.
>> should be proportional to log 2, 3, 5, etc. For every mapping
>> of every temperament, the variables would seem to be the
>> relative lengths of the two vectors mapping the keyboard and
>> the angle between them.
>
>Draw me a picture? That would definitely make it easier to get my
>head around this.
>
>Yahya

Here's Gene's page

http://64.233.167.104/search?q=cache:vJB9BLlKDYkJ:bahamas.eshockhost.com/~xenharmo/bosanquet.html+bosanquet+lattices
(It doesn't look like Google caches images, and Gene's page is in
a perpetual state of downness, so that's the best I can do.)

but it may be easier to think about Bill Wesley's "array" vs.
the traditional piano keyboard.

http://www.thearraymbira.com/images/notes/5_theMBOX_Octaves%5B1%5D.gif

Octaves are very close together, and the left/right axis is
fifths. So there's no monotonic pitch scale here. But the
distance to a 5-limit interval is more proportional to its
complexity than on the piano.

There are two axes on a generalized keyboard, an angle between
them, and an aspect ratio between them. The idea is that for
a rank 2 temperament, find a map such that when the period and
generator are the two axes, some angle and aspect ratio makes
the distances come out proportional to the complexity of the
corresponding identities.

Dave's spreadsheet that Gene links to doesn't say what criteria
it uses to find its "ideal" layouts, but it apparently isn't
the one I'm describing here.

-Carl

So can we deduce that posting from the web avoids the problem
for me?

-Carl

At 08:38 PM 8/20/2006, you wrote:
>This list is delaying/eating my posts virtually every time, so
>I'm playing the 'slam the list with the post until it responds'
>game, just for fun.
>
>I know that my ISP has been blacklisted by outblaze a few
>times recently... Spam really has made e-mail almost unuseable
>for millions. Junk paper mail, though far more wasteful of
>trees and such, doesn't seem cause so serious problems.
>
>But I digress. Here's my post:
>
>>Carl,
>>
>>--- In tuning-math@yahoogroups.com, you wrote:
>>> I don't know if anybody's ever thought about this, but
>>> if Paul's got the right idea with weighted complexity, perhaps
>>> keyboard layouts should be weighted too. The idea is that
>>> physical distance on the keyboard from 1 to 2, 3, 5, etc.
>>> should be proportional to log 2, 3, 5, etc. For every mapping
>>> of every temperament, the variables would seem to be the
>>> relative lengths of the two vectors mapping the keyboard and
>>> the angle between them.
>>
>>Draw me a picture? That would definitely make it easier to get my
>>head around this.
>>
>>Yahya
>
>Here's Gene's page
>
>http://64.233.167.104/search?
>q=cache:vJB9BLlKDYkJ:bahamas.eshockhost.com/~xenharmo/bosanquet.html+b
>osanquet+lattices
>(It doesn't look like Google caches images, and Gene's page is in
>a perpetual state of downness, so that's the best I can do.)
>
>but it may be easier to think about Bill Wesley's "array" vs.
>the traditional piano keyboard.
>
>http://www.thearraymbira.com/images/notes/5_theMBOX_Octaves%5B1%5D.gif
>
>Octaves are very close together, and the left/right axis is
>fifths. So there's no monotonic pitch scale here. But the
>distance to a 5-limit interval is more proportional to its
>complexity than on the piano.
>
>There are two axes on a generalized keyboard, an angle between
>them, and an aspect ratio between them. The idea is that for
>a rank 2 temperament, find a map such that when the period and
>generator are the two axes, some angle and aspect ratio makes
>the distances come out proportional to the complexity of the
>corresponding identities.
>
>Dave's spreadsheet that Gene links to doesn't say what criteria
>it uses to find its "ideal" layouts, but it apparently isn't
>the one I'm describing here.
>
>-Carl
>
>
>
>
>
>
>Yahoo! Groups Links
>
>
>
>

--- In tuning-math@yahoogroups.com, Carl Lumma wrote:
>
> So can we deduce that posting from the web avoids the problem
> for me?
>
> -Carl

If you hit it only twice, yes.

Yahya

Hi Carl,

--- In tuning-math@yahoogroups.com, you wrote:
> >--- In tuning-math@yahoogroups.com, you wrote:
> >> I don't know if anybody's ever thought about this, but
> >> if Paul's got the right idea with weighted complexity, perhaps
> >> keyboard layouts should be weighted too. The idea is that
> >> physical distance on the keyboard from 1 to 2, 3, 5, etc.
> >> should be proportional to log 2, 3, 5, etc. For every mapping
> >> of every temperament, the variables would seem to be the
> >> relative lengths of the two vectors mapping the keyboard and
> >> the angle between them.
> >
> >Draw me a picture? That would definitely make it easier to get
my head around this.
> >
> >Yahya
>
> Here's Gene's page
>
> http://64.233.167.104/search?
q=cache:vJB9BLlKDYkJ:bahamas.eshockhost.com/~xenharmo/bosanquet.html+
bosanquet+lattices
> (It doesn't look like Google caches images, and Gene's page is in
> a perpetual state of downness, so that's the best I can do.)

1. OT, on giving URLs:
TinyURL gives this:
http://tinyurl.com/l9mkb

for the cached page address, and this:
http://tinyurl.com/pto4y

for the actual page. These things are easier to post and use ..

2. I can almost follow Gene's page, and associated links, tho the
algebra does involve some terminology I haven't studied in detail.
It's good to see that Gene uses more examples than is customary in
this kind of theory, and they do make the reading a little easier.
In particular, I found Dave Keenan's spreadsheet for creating
keyboard layouts, linked by Gene, quite helpful.

> but it may be easier to think about Bill Wesley's "array" vs.
> the traditional piano keyboard.
>
> http://www.thearraymbira.com/images/notes/5_theMBOX_Octaves%5B1%
5D.gif
>
> Octaves are very close together, and the left/right axis is
> fifths. So there's no monotonic pitch scale here. But the
> distance to a 5-limit interval is more proportional to its
> complexity than on the piano.
>
> There are two axes on a generalized keyboard, an angle between
> them, and an aspect ratio between them. The idea is that for
> a rank 2 temperament, find a map such that when the period and
> generator are the two axes, some angle and aspect ratio makes
> the distances come out proportional to the complexity of the
> corresponding identities.

That last sentence is a succinct summary of purpose - thanks!

> Dave's spreadsheet that Gene links to doesn't say what criteria
> it uses to find its "ideal" layouts, but it apparently isn't
> the one I'm describing here.

Perhaps we could ask Dave to explain his criteria?

Best,
Yahya

>1. OT, on giving URLs:
>TinyURL gives this:
>http://tinyurl.com/l9mkb
>
>for the cached page address, and this:
>http://tinyurl.com/pto4y
>
>for the actual page.

Actual page was down at the time. And I'd wager I've been using
tinyurl (and makeashorterlink) before anyone you know. But sometimes,
I just don't have the time.

>> Dave's spreadsheet that Gene links to doesn't say what criteria
>> it uses to find its "ideal" layouts, but it apparently isn't
>> the one I'm describing here.
>
>Perhaps we could ask Dave to explain his criteria?

I searched the archives for keyboardmapper, and it's finally looking
like the new Yahoo groups search is better than regexing my mail
files btw, but didn't find any explanation of the spreadsheet.
In fact, I found something from Dave hinting that he's never
explained it. I did find something that says he requires an
ascending pitch axis, though.

-Carl

> >> Dave's spreadsheet that Gene links to doesn't say what criteria
> >> it uses to find its "ideal" layouts, but it apparently isn't
> >> the one I'm describing here.
> >
> >Perhaps we could ask Dave to explain his criteria?
>
> I searched the archives for keyboardmapper, and it's finally looking
> like the new Yahoo groups search is better than regexing my mail
> files btw, but didn't find any explanation of the spreadsheet.
> In fact, I found something from Dave hinting that he's never
> explained it. I did find something that says he requires an
> ascending pitch axis, though.

Yes. That's pretty much all there is to it.

http://dkeenan.com/Music/KeyboardMapper.xls

I assume that pitch increases uniformly from left to right on the
keyboard (bottom to top in the spreadsheet chart). The period axis is
the same as the pitch axis.

The other dimension is then used to ensure that chains of generators
are in straight lines, which would seem to be the definition of a
linear-temperament-specific keyboard.

The other constraints are provided by the user:
Generator size
Period size
No of notes per period
Aspect ratio

However, when i suggest that one can sometimes obtain a more regular
hexagonal (or square) layout by slightly varying the generator from
that actually used for the tuning, then I am allowing that pitch need
not increase strictly uniformly from left to right but may only do so
approximately.

You can see how it works more easily if you change the period to say
200 cents and the generator to around 100 cents and choose only 3
notes per period. Then play around with the generator-size slider.

An interesting question about these keyboard layouts is whether there
is, in some sense, a finite number of useful regular hexagonal and
square layouts and what the generators (and aspect ratios) are for
these, and how they relate to the "good" linear temperaments.

-- Dave

>> I searched the archives for keyboardmapper, and it's finally looking
>> like the new Yahoo groups search is better than regexing my mail
>> files btw, but didn't find any explanation of the spreadsheet.
>> In fact, I found something from Dave hinting that he's never
>> explained it. I did find something that says he requires an
>> ascending pitch axis, though.
>
>Yes. That's pretty much all there is to it.
>
>http://dkeenan.com/Music/KeyboardMapper.xls
>
>I assume that pitch increases uniformly from left to right on the
>keyboard (bottom to top in the spreadsheet chart). The period axis is
>the same as the pitch axis.
>
>The other dimension is then used to ensure that chains of generators
>are in straight lines, which would seem to be the definition of a
>linear-temperament-specific keyboard.
>
>The other constraints are provided by the user:
> Generator size
> Period size
> No of notes per period
> Aspect ratio

Thanks for explaining. In this thread, I was wondering why
keyboard instruments tend to have an ascending pitch axis, whereas
guitars (for example) do not. Bill Wesley has proposed, built,
and recorded music for keyboards that are laid out by octaves
and fifths, like a guitar. It seems like a good idea. After all,
overplaying "runs" is a famous blunder in classical, romantic, and
jazz music. Jonathan Glasier also told me about a trick the
San Diego microtonalists (sometimes including Wesley and monz,
I think) use to force novel improvisations -- they randomly assign
pitches of a new scale to a halberstadt MIDI keyboard. The quality
of the music Jonathan gave me on CD seems to vouch for the method.
So maybe the thing to minimize is the 'Graham complexity' distance
over the keyboard.

>An interesting question about these keyboard layouts is whether there
>is, in some sense, a finite number of useful regular hexagonal and
>square layouts and what the generators (and aspect ratios) are for
>these, and how they relate to the "good" linear temperaments.

Indeed.

-Carl

Hi dave

These links are not accessible , any problem?

<<<<<Here are some spreadsheets with charts for predicting the dissonance of intervals.
SetharesDissonance.zip <http://dkeenan.com/Music/SetharesDissonance.zip> 636kB
HarmonicComplexity.zip <http://dkeenan.com/Music/HarmonicComplexity.zip> 155kB>>>>>

Shaahin Mohaajeri

Tombak Player & Researcher , Microtonal Composer

My web site <http://240edo.tripod.com/>

My farsi page in Harmonytalk <http://www.harmonytalk.com/id/908>

My tombak musics in Rhythmweb <http://www.rhythmweb.com/gdg>

My article in DrumDojo <http://www.drumdojo.com/world/persia/tonbak_acoustics.htm>

________________________________

From: tuning-math@yahoogroups.com [mailto:tuning-math@yahoogroups.com] On Behalf Of Dave Keenan
Sent: Monday, October 23, 2006 4:15 AM
To: tuning-math@yahoogroups.com
Subject: [tuning-math] Re: weighted Bosanquet lattices

> >> Dave's spreadsheet that Gene links to doesn't say what criteria
> >> it uses to find its "ideal" layouts, but it apparently isn't
> >> the one I'm describing here.
> >
> >Perhaps we could ask Dave to explain his criteria?
>
> I searched the archives for keyboardmapper, and it's finally looking
> like the new Yahoo groups search is better than regexing my mail
> files btw, but didn't find any explanation of the spreadsheet.
> In fact, I found something from Dave hinting that he's never
> explained it. I did find something that says he requires an
> ascending pitch axis, though.

Yes. That's pretty much all there is to it.

http://dkeenan.com/Music/KeyboardMapper.xls <http://dkeenan.com/Music/KeyboardMapper.xls>

I assume that pitch increases uniformly from left to right on the
keyboard (bottom to top in the spreadsheet chart). The period axis is
the same as the pitch axis.

The other dimension is then used to ensure that chains of generators
are in straight lines, which would seem to be the definition of a
linear-temperament-specific keyboard.

The other constraints are provided by the user:
Generator size
Period size
No of notes per period
Aspect ratio

However, when i suggest that one can sometimes obtain a more regular
hexagonal (or square) layout by slightly varying the generator from
that actually used for the tuning, then I am allowing that pitch need
not increase strictly uniformly from left to right but may only do so
approximately.

You can see how it works more easily if you change the period to say
200 cents and the generator to around 100 cents and choose only 3
notes per period. Then play around with the generator-size slider.

An interesting question about these keyboard layouts is whether there
is, in some sense, a finite number of useful regular hexagonal and
square layouts and what the generators (and aspect ratios) are for
these, and how they relate to the "good" linear temperaments.

-- Dave

--- In tuning-math@yahoogroups.com, "Mohajeri Shahin" <shahinm@...> wrote:
>
> Hi dave
>
>
>
> These links are not accessible , any problem?
>
>
>
> <<<<<Here are some spreadsheets with charts for predicting the
dissonance of intervals.
> SetharesDissonance.zip
<http://dkeenan.com/Music/SetharesDissonance.zip>
636kB
> HarmonicComplexity.zip
<http://dkeenan.com/Music/HarmonicComplexity.zip>
155kB>>>>>

Sorry Shahin,

I took those files away because I needed the space for Sagittal stuff.
I'd be happy to email them if you're really interested.
I think we pretty much agreed that Harmonic Entropy is the best model
so far and Product Complexity is the best of the ones you can
calculate in your head.

-- Dave

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
> Thanks for explaining. In this thread, I was wondering why
> keyboard instruments tend to have an ascending pitch axis, whereas
> guitars (for example) do not.

The only reason there is no pitch axis on a standard-tuned guitar is
because it isn't regular. When the open strings are all the same
interval apart then there _is_ a pitch axis. It just isn't parallel
to, or at right angles to, anything else.

> Bill Wesley has proposed, built,
> and recorded music for keyboards that are laid out by octaves
> and fifths, like a guitar. It seems like a good idea. After all,
> overplaying "runs" is a famous blunder in classical, romantic, and
> jazz music. Jonathan Glasier also told me about a trick the
> San Diego microtonalists (sometimes including Wesley and monz,
> I think) use to force novel improvisations -- they randomly assign
> pitches of a new scale to a halberstadt MIDI keyboard. The quality
> of the music Jonathan gave me on CD seems to vouch for the method.
> So maybe the thing to minimize is the 'Graham complexity' distance
> over the keyboard.

Or the log(p) weighted complexity Paul uses in 'A Middel Path'. So
you're saying, let the pitch axis end up where it may, just minimise
the stretch for the most consonant chords.

Sure. Sounds like a good approach. Sorry I don't have time to
spreadsheet it.

-- Dave

>Or the log(p) weighted complexity Paul uses in 'A Middel Path'. So
>you're saying, let the pitch axis end up where it may, just minimise
>the stretch for the most consonant chords.

Will there always be a pitch axis?

>Sure. Sounds like a good approach. Sorry I don't have time to
>spreadsheet it.

I'm sorry I don't, and I'm even more sorry you don't. :)

-Carl

--- In tuning-math@yahoogroups.com, "Mohajeri Shahin" <shahinm@...> wrote:
>
> Hi dave
>
> These links are not accessible , any problem?
>
> <<<<<Here are some spreadsheets with charts for predicting the
dissonance of intervals.
> SetharesDissonance.zip
<http://dkeenan.com/Music/SetharesDissonance.zip>
636kB
> HarmonicComplexity.zip
<http://dkeenan.com/Music/HarmonicComplexity.zip>
155kB>>>>>

I've put them up in the file area of the tuning_files yahoo group.
Please let me know if there is any problem with these URLs.

</tuning-math/files/Keenan/SetharesDissonance.zip>

</tuning-math/files/Keenan/HarmonicComplexity.zip>

-- Dave

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >Or the log(p) weighted complexity Paul uses in 'A Middel Path'. So
> >you're saying, let the pitch axis end up where it may, just minimise
> >the stretch for the most consonant chords.
>
> Will there always be a pitch axis?

It may help to look at Figure 2 of Paul Erlich's 'A Middle Path'.
(Sorry about the typo above, Paul).

There you will see octaves increasing upward and twelfths (1:3) to the
right. If you take these generator axes as continuous rather than
discrete then you can see that the diagonal lines join points of equal
pitch (Easier to see if you relate it back to figure 1 with the note
names). It doesn't matter what size the generators are, or what angle
their axes make. Because the generator axes are uniform, so are the
isopitch contours and therefore there is a pitch axis perpendicular to
the contours.

-- Dave