X_System!

Was anyone aware of this?

http://lumma.org/tuning/MilneSetharesPlamondon--X_System.pdf

-Carl

>Was anyone aware of this?
>
> http://lumma.org/tuning/MilneSetharesPlamondon--X_System.pdf

This is pretty big. I even got a cite! This is wrong, though:

""MOS scales are maximally even (Clough and Douthett 1991) –
indeed MOS and maximal evenness are mathematically equivalent
descriptions (Breed 2005); and they are proper (Rothenberg 1969).""

Not all MOS are proper. And I possibly recall something about
maximal evenness not being what people thought it was. The
paper cites

http://x31eq.com/proof.html

but ME doesn't seem to be defined on this page. Wikipedia says
it was used "as part of the Ising model especially in physics to
model electron behaviour"

http://en.wikipedia.org/wiki/Maximal_evenness

Wow!

Ah yes, here's something from Clough (Tuning list circa 1999)

""I apprecaite Paul Erlich's clarification re maximal evenness (ME)
and distributional evenness (DE). I had not realized that what Paul
had called ME is equivalent to DE, as defined a few years ago (in
conference presentations) by Nora Engebretsen and me. For those
interested, our paper covering evenness and related features will
be coming out soon in _Music Theory Spectrum_:

J. Clough, J. Kochavi, and N. Engebretsen, "Scales, Sets, and
Interval Cycles: A Taxonomy.".

John Clough
SUNY at Buffalo""

So DE is 'at most two step sizes', but this still doesn't say
what ME is.

http://links.jstor.org/sici?sici=0195-6167(199921)21%3A1%3C74%3ASSAICA%3E2.0.CO%3B2-M

"Unfortunately, I do not have access to JSTOR from my location."
(What else is new?)

Tonalsoft has a blank page for DE. But for ME, it quotes
Scala's tips.par, which indicates that ME is DE plus the condition
that the steps differ in size by a single step of some ET.

Lame.

So MOS = DE (if non-octave periods are allowed) or
just = Myhill (if they're not).

-Carl

Carl Lumma wrote:
>>Was anyone aware of this?
>>
>>http://lumma.org/tuning/MilneSetharesPlamondon--X_System.pdf

I hadn't heard of it. If I remember I'll check it out next time I'm online. But for now ...

> This is pretty big. I even got a cite! This is wrong, though:
> > ""MOS scales are maximally even (Clough and Douthett 1991) �
> indeed MOS and maximal evenness are mathematically equivalent
> descriptions (Breed 2005); and they are proper (Rothenberg 1969).""

AIUI, MOS and ME are not the same. However it's always possible to tune a class of MOS scales such that you get a ME scale.

> Not all MOS are proper. And I possibly recall something about
> maximal evenness not being what people thought it was. The
> paper cites

I haven't read the original paper, so I think it's what Agmon though it was. He also thought that maximal evenness implies propriety.

> http://x31eq.com/proof.html
> > but ME doesn't seem to be defined on this page.

[I've broken netiquette and split up your paragraph. Sorry, but your paragraphs aren't good. I am an English teacher ;) ]

I give the definition as

p(n) = floor(na/b)

If that isn't maximal evenness, I'd like to know what it is, because it's a useful property.

> Wikipedia says
> it was used "as part of the Ising model especially in physics to
> model electron behaviour"

> http://en.wikipedia.org/wiki/Maximal_evenness
> > Wow!

Yes, this has been mentioned before. If you have things trying to get as far apart as they can with some kind of quantized space, it's the criteria for maximal evenness.

> Ah yes, here's something from Clough (Tuning list circa 1999)
> > ""I apprecaite Paul Erlich's clarification re maximal evenness (ME)
> and distributional evenness (DE). I had not realized that what Paul
> had called ME is equivalent to DE, as defined a few years ago (in
> conference presentations) by Nora Engebretsen and me. For those
> interested, our paper covering evenness and related features will
> be coming out soon in _Music Theory Spectrum_:

If that's true, then ME isn't what I thought it was.

> J. Clough, J. Kochavi, and N. Engebretsen, "Scales, Sets, and
> Interval Cycles: A Taxonomy.". > > John Clough
> SUNY at Buffalo""
> > So DE is 'at most two step sizes', but this still doesn't say
> what ME is.

DE should be about more than counting the step sizes. Otherwise the quartertone rast would count:

4 3 3 4 4 3 3

> http://links.jstor.org/sici?sici=0195-6167(199921)21%3A1%3C74%3ASSAICA%3E2.0.CO%3B2-M
> > "Unfortunately, I do not have access to JSTOR from my location."
> (What else is new?)
> > Tonalsoft has a blank page for DE. But for ME, it quotes
> Scala's tips.par, which indicates that ME is DE plus the condition
> that the steps differ in size by a single step of some ET.

That's conistent with what I think it is.

> Lame.
> > So MOS = DE (if non-octave periods are allowed) or
> just = Myhill (if they're not).

According to that Carey and Clampitt paper, these three are much the same thing. But it overlooks non-octave periods. I thought DE allowed them, Myhill didn't, and MOS weren't precisely enough defined. But I could be wrong.

Graham

>> Ah yes, here's something from Clough (Tuning list circa 1999)
>>
>> ""I apprecaite Paul Erlich's clarification re maximal evenness (ME)
>> and distributional evenness (DE). I had not realized that what Paul
>> had called ME is equivalent to DE, as defined a few years ago (in
>> conference presentations) by Nora Engebretsen and me. For those
>> interested, our paper covering evenness and related features will
>> be coming out soon in _Music Theory Spectrum_:
>
>If that's true, then ME isn't what I thought it was.
//
>> Tonalsoft has a blank page for DE. But for ME, it quotes
>> Scala's tips.par, which indicates that ME is DE plus the condition
>> that the steps differ in size by a single step of some ET.
>
>That's conistent with what I think it is.

IIRC Paul originally didn't know ME has the single-step
condition, but he may have later corrected his (22-tET) paper.
Hell, I half-remember correcting it myself.

Anyway, the Clough and Scala tips quotes above agree, so what
you think it is may not be constant.

>> So MOS = DE (if non-octave periods are allowed) or
>> just = Myhill (if they're not).
>
>According to that Carey and Clampitt paper, these three are much the
>same thing. But it overlooks non-octave periods. I thought DE allowed
>them, Myhill didn't, and MOS weren't precisely enough defined. But I
>could be wrong.

That's my understanding also.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> Was anyone aware of this?
>
> http://lumma.org/tuning/MilneSetharesPlamondon--X_System.pdf

No, but you, me, Graham and Paul are all cited. The Regular Tuning
Paradigm marches on!

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> "Unfortunately, I do not have access to JSTOR from my location."
> (What else is new?)

I can get JSTOR stuff; in fact, I have a bunch of pdf theory files
from there. What are you interested in?

At 05:46 PM 6/26/2006, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
>> "Unfortunately, I do not have access to JSTOR from my location."
>> (What else is new?)
>
>I can get JSTOR stuff; in fact, I have a bunch of pdf theory files
>from there. What are you interested in?

The link you snipped:

http://links.jstor.org/sici?sici=0195-6167(199921)21%3A1%3C74%3ASSAICA%3E2.0.CO%3B2-M

Which should be:

J. Clough, J. Kochavi, and N. Engebretsen, "Scales, Sets, and
Interval Cycles: A Taxonomy.".

-C.

Carl Lumma wrote:
> Was anyone aware of this?
> > http://lumma.org/tuning/MilneSetharesPlamondon--X_System.pdf
> > -Carl

Hmm.... Interesting. Wow, that schismatic keyboard layout is awkward! Just try to play the 25/24 semitone on that thing! I'd slide the octave over two steps at least to make it less of a stretch to reach the 5-limit intervals.

For porcupine, I think a horizontal mapping of the steps would work better. The keyboard could be split into horizontal ranges mapped to different octaves. I tried the mapping suggested in the article (on my crude generalized keyboard -- that is, a computer keybard) and it seemed awkward. I can play a 22-note chromatic porcupine scale with ease using my preferred mapping (using only three rows of the computer keyboard).

This page linked from the article has an interesting chart that appears to be a variation of Wilson's scale tree:

http://web.syr.edu/~rsholmes/music/xen/scale_mos.html

It's great to see that someone else out there is thinking about keyboard mappings.

>> Was anyone aware of this?
>>
>> http://lumma.org/tuning/MilneSetharesPlamondon--X_System.pdf
>>
>> -Carl
>
>Hmm.... Interesting. Wow, that schismatic keyboard layout is awkward!
>Just try to play the 25/24 semitone on that thing! I'd slide the octave
>over two steps at least to make it less of a stretch to reach the
>5-limit intervals.

I guess every layout is awkward for some things and catering to
others.

>For porcupine, I think a horizontal mapping of the steps would work
>better. The keyboard could be split into horizontal ranges mapped to
>different octaves. I tried the mapping suggested in the article (on my
>crude generalized keyboard -- that is, a computer keybard) and it seemed
>awkward. I can play a 22-note chromatic porcupine scale with ease using
>my preferred mapping (using only three rows of the computer keyboard).

Paul E. was Matlabing some hexagonal layouts of all kinds of
temperaments a while back ... I wish that would have continuued.

>This page linked from the article has an interesting chart that appears
>to be a variation of Wilson's scale tree:
>
>http://web.syr.edu/~rsholmes/music/xen/scale_mos.html
>
>It's great to see that someone else out there is thinking about keyboard
>mappings.

Yep.

-Carl

I should say that this paper hasn't been published. It's a
work-in-progress, and the authors are looking for feedback.

-Carl

Hi Carl,

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >Was anyone aware of this?
> >
> > http://lumma.org/tuning/MilneSetharesPlamondon--X_System.pdf
>
> <snip>
>
> Tonalsoft has a blank page for DE. But for ME, it quotes
> Scala's tips.par, which indicates that ME is DE plus the
> condition that the steps differ in size by a single step
> of some ET.
>
> Lame.

Hey, i've been busy!

If you (or anyone else) has contributions or corrections
for the Tonalsoft Encyclopedia, *please* email them to me
or post them here.

As i said before, i'm a little disappointed that folks are
tending to put new stuff about tuning into Wikipedia instead
of at Tonalsoft first. Wikipedia should be a distillation
of something that's written about more in-depth at Tonalsoft
(or wherever).

Even tho i've done probably 95% of the work that's in there,
The Tonalsoft Encyclopedia has always been thought of by me
as a group effort by the whole tuning community.

-monz
http://tonalsoft.com
Tonescape microtonal music software

Carl Lumma wrote:
>>> Was anyone aware of this?
>>>
>>> http://lumma.org/tuning/MilneSetharesPlamondon--X_System.pdf
>>>
>>> -Carl
>> Hmm.... Interesting. Wow, that schismatic keyboard layout is awkward! >> Just try to play the 25/24 semitone on that thing! I'd slide the octave >> over two steps at least to make it less of a stretch to reach the >> 5-limit intervals.
> > I guess every layout is awkward for some things and catering to
> others.

Right. There are also advantages to mapping schismatic the same as meantone. And as the Thummer keyboard appears to have keys more closely spaced than a computer keyboard, the intervals are probably less awkward. But at first glance, Wilson's Bosanquet layout seems like it would be easier for this particular temperament.

One of my interests in playing with keyboard layouts is to figure out which melodic and harmonic patterns would be most natural for the keyboard layouts of Zireen musical instruments. Different regions developed their own keyboard layouts based on square or hexagonal arrays of buttons on a reed organ type instrument. So the layouts I'm interested in aren't just limited to the ones that would work out well on a Thummer or a MicroZone, but anything that could be of use in guiding my musical exploration (the possibilities are so vast that I need a few landmarks!).

But that paper looks like it might have a lot of useful information if I can find the time to figure out what it's saying. Trial and error can go so far, but I can see that a better understanding of the mathematics would be helpful.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> Carl Lumma wrote:
> > Was anyone aware of this?
> >
> > http://lumma.org/tuning/MilneSetharesPlamondon--X_System.pdf
> >
> > -Carl
>
> <snip>
>
>
> It's great to see that someone else out there is thinking
> about keyboard mappings.
>

Jim Plamondon is the guy who invented the Jammer (also known
as the Thummer)

http://www.thummer.com/

-monz
http://tonalsoft.com
Tonescape microtonal music software

>If you (or anyone else) has contributions or corrections
>for the Tonalsoft Encyclopedia, *please* email them to me
>or post them here.
>
>As i said before, i'm a little disappointed that folks are
>tending to put new stuff about tuning into Wikipedia instead
>of at Tonalsoft first. Wikipedia should be a distillation
>of something that's written about more in-depth at Tonalsoft
>(or wherever).
>
>Even tho i've done probably 95% of the work that's in there,
>The Tonalsoft Encyclopedia has always been thought of by me
>as a group effort by the whole tuning community.

The Tonalsoft encyclopedia's a great place for a lot of the
things that aren't appropriate on Wikipedia. But Wikipedia
has the following advantages:

() More people read it.
() It rallies people from outside this community and
allows them to contribute.
() It has powerful typesetting and revision-tracking
features.
() The content is open and reusable, as opposed to
copyrighted by a private, for-profit corporation.

-Carl

>>>> Was anyone aware of this?
>>>>
>>>> http://lumma.org/tuning/MilneSetharesPlamondon--X_System.pdf
>>>>
>>>> -Carl
>>> Hmm.... Interesting. Wow, that schismatic keyboard layout is awkward!
>>> Just try to play the 25/24 semitone on that thing! I'd slide the octave
>>> over two steps at least to make it less of a stretch to reach the
>>> 5-limit intervals.
>>
>> I guess every layout is awkward for some things and catering to
>> others.
>
>Right. There are also advantages to mapping schismatic the same as
>meantone. And as the Thummer keyboard appears to have keys more closely
>spaced than a computer keyboard, the intervals are probably less
>awkward. But at first glance, Wilson's Bosanquet layout seems like it
>would be easier for this particular temperament.

There's something to be said for putting fifths nearby rather
than 2nds -- Bill Wesley convinced me of this. And the Thummer
layout is based on a performance-proven concertina
keyboard ("Hayden system"). And if you think of guitars as
layouts...

>But that paper looks like it might have a lot of useful information if I
>can find the time to figure out what it's saying. Trial and error can go
>so far, but I can see that a better understanding of the mathematics
>would be helpful.

"The generating set chosen by the algorithm in Section 2.3.1"

Hey, this looks interesting, but they don't give examples and
I don't have time to work through it right now. Gene, do you
have a feeling how this compares to Hermite?

It doesn't look like they've introduced the term "T-curve"
by the time they start using it on page 43 ... it looks like
they're talking about the number of pitch classes you get on
a fixed section of the keyboard as the generator of a rank 2
temperament is swept over an octave.

-Carl

Carl Lumma wrote:
> I should say that this paper hasn't been published. It's a
> work-in-progress, and the authors are looking for feedback.

They'll have to sort out the MOS/ME section then. Especially if they want to cite me in support of it :P

It's generally a very promising bit of research. If they can get this software out and working with the Thummer, it'd make some advanced concepts concrete and graspable. Imagine an affordable generalized keyboard with a knob to change the temperament and another knob to temper the timbre!

Anyway, to the criticisms:

Their concept of primary consonances looks like a poor stand-in for odd limits. Why not include 9 in the 11-limit? There are other places where they look at familiar things in an unfamiliar way, but I haven't digested them yet.

The opening sentence/paragraph of 5.1 looks wrong. (Page 44 of the file, 40 by the page numbering.) I don't know what they meant to say but it must be close to the opposite of what they do say. Either that or I don't understand what they are saying, so they could make it clearer.

Graham

Carl, if you're in touch with the authors of that paper (I don't know if Sethares is reading here) could you pass on that there's a mistake in section 2.4.4: "perfect fifths, major thirds, minor sixths" should read "perfect fifths, major thirds, major sixths" ("perfect fifths, major thirds, minor thirds" would also be logical in this sentence, but it's sixths that are intended).

Also, I find this definition weird:

Definition: A scale is unalterably improper if it is improper, as defined above, but can be made proper by the addition or removal of a single note to that scale.

To me "unalterably improper" should mean it's not possible to alter the scale to make it proper - but their definition is quite the opposite. I like the notion, but surely there's a better name.

Thanks for posting the article --Jon

>Carl, if you're in touch with the authors of that paper (I don't know if
>Sethares is reading here) could you pass on that there's a mistake in
>section 2.4.4: "perfect fifths, major thirds, minor sixths" should read
>"perfect fifths, major thirds, major sixths" ("perfect fifths,
>major thirds, minor thirds" would also be logical in this sentence, but
>it's sixths that are intended).
>
>Also, I find this definition weird:
>
>Definition: A scale is unalterably improper if it is improper, as defined
>above, but can be made proper by the addition or removal of a single note
>to that scale.
>
>To me "unalterably improper" should mean it's not possible to alter the
>scale to make it proper - but their definition is quite the opposite. I
>like the notion, but surely there's a better name.
>
>Thanks for posting the article --Jon

I'm collecting errata for Bill. I'll add this.

-Carl

>The opening sentence/paragraph of 5.1 looks wrong. (Page 44 of the
>file, 40 by the page numbering.) I don't know what they meant to say
>but it must be close to the opposite of what they do say. Either that
>or I don't understand what they are saying, so they could make it clearer.

Looks so.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> The Tonalsoft encyclopedia's a great place for a lot of the
> things that aren't appropriate on Wikipedia. But Wikipedia
> has the following advantages:
>
> () More people read it.
> () It rallies people from outside this community and
> allows them to contribute.
> () It has powerful typesetting and revision-tracking
> features.
> () The content is open and reusable, as opposed to
> copyrighted by a private, for-profit corporation.

It's also much easier to write a Wikipedia article.

Carl Lumma wrote:
> I should say that this paper hasn't been published. It's a
> work-in-progress, and the authors are looking for feedback.
> > -Carl

Who should I send comments to? I noticed an error in the description of porcupine temperament in section 2.1; the generator is 1/3 of a perfect fourth (not a fifth). (The notation in Figure 8 is correct.)

Interesting stuff. The term "disruptive comma" could be useful -- has this come up before?

"Any comma that can be invoked by a functional progression within a particular musical system is termed a _disruptive comma_."

These are the sort of progressions that we've referred to as "comma pumps", but I don't recall coming across a term for the particular commas found in these progressions.

>> I should say that this paper hasn't been published. It's a
>> work-in-progress, and the authors are looking for feedback.
>>
>> -Carl
>
>Who should I send comments to? I noticed an error in the description of
>porcupine temperament in section 2.1; the generator is 1/3 of a perfect
>fourth (not a fifth). (The notation in Figure 8 is correct.)

The paper is just a rough draft. I've probably inundated Bill
by now, by you could send them to him.

>Interesting stuff. The term "disruptive comma" could be useful -- has
>this come up before?
>
>"Any comma that can be invoked by a functional progression within a
>particular musical system is termed a _disruptive comma_."
>
>These are the sort of progressions that we've referred to as "comma
>pumps", but I don't recall coming across a term for the particular
>commas found in these progressions.

Not sure what they mean by this.

-Carl

Carl Lumma wrote:

>> Right. There are also advantages to mapping schismatic the same as >> meantone. And as the Thummer keyboard appears to have keys more closely >> spaced than a computer keyboard, the intervals are probably less >> awkward. But at first glance, Wilson's Bosanquet layout seems like it >> would be easier for this particular temperament.
> > There's something to be said for putting fifths nearby rather
> than 2nds -- Bill Wesley convinced me of this. And the Thummer
> layout is based on a performance-proven concertina
> keyboard ("Hayden system"). And if you think of guitars as
> layouts...

I don't care so much that the 2nds are nearby, but it's a bit of a stretch for the 3rds. The Thummer layout seems about perfect for meantone, but the schismatic layout requires some awkward skips for a diatonic scale. Although it does seem quite usable for the left hand, so perhaps mirror-inverting the right hand would help....

At 07:27 PM 6/28/2006, you wrote:
>Carl Lumma wrote:
>
>>> Right. There are also advantages to mapping schismatic the same as
>>> meantone. And as the Thummer keyboard appears to have keys more closely
>>> spaced than a computer keyboard, the intervals are probably less
>>> awkward. But at first glance, Wilson's Bosanquet layout seems like it
>>> would be easier for this particular temperament.
>>
>> There's something to be said for putting fifths nearby rather
>> than 2nds -- Bill Wesley convinced me of this. And the Thummer
>> layout is based on a performance-proven concertina
>> keyboard ("Hayden system"). And if you think of guitars as
>> layouts...
>
>I don't care so much that the 2nds are nearby, but it's a bit of a
>stretch for the 3rds. The Thummer layout seems about perfect for
>meantone, but the schismatic layout requires some awkward skips for a
>diatonic scale. Although it does seem quite usable for the left hand, so
>perhaps mirror-inverting the right hand would help....

I guess the devil would say not to play the diatonic scale
sequentially. "Runs" are, after all, the most overused musical
devices ever, and one wonders what blame the halberstadt deserves
in this.

-Carl

Carl Lumma wrote:

> I guess the devil would say not to play the diatonic scale
> sequentially. "Runs" are, after all, the most overused musical
> devices ever, and one wonders what blame the halberstadt deserves
> in this.

Well, I don't want to focus too much on this one keyboard mapping, but basically the things that end up being easy are the 3-limit intervals (at the expense of the major third and major sixth). You could do some interesting things with tritones and such, but aren't the 3rds and 6ths one of the big attractions of schismatic?