Tuning-Math Digests messages 5906 - 5930

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Message: 5906

Date: Thu, 09 Jan 2003 00:40:27

Subject: Re: Minimax generator

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> 
> Thanks. I've read Monz's summary of Woolhouse's book which I've
found very
> helpful. I was just wondering, though, why he doesn't take the
square root
> when calculating Root-Mean-Square. Is it unneccessary? 

If you only want to find an optimum generator it is unnecessary, in
fact you don't even need to take the mean, just minimise the sum of
the squares of the errors. But if you want to express the resulting
overall error in a perceptually meaningful fashion, e.g. for comparing
different temperaments, then you need to take the root of the mean of
the squares, so it has dimensions of log-of-frequency-ratio, and
therefore (usually) units of cents.

> I also see a little
> problem with the final calculation for rms for meantone. If
multiplication
> is commutative, you could take the results to ALSO mean (3/2)/(7/26)
> ^(81/80) which is obviously a wrong result. Clarification, anyone?
Thanks.

I find it easier to follow when expressed in the logarithmic frequency
domain (e.g. in cents) as 702.0c - 21.5c * 7/26


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Message: 5908

Date: Thu, 09 Jan 2003 00:51:44

Subject: Re: Poptimal generators

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> > By "this endeavour" I meant specifically the mathematical modelling 
> of
> > perceptual optimality of generators for musical temperaments, not
> > mathematic or statistics in general. I also only said "possibly". 
> I'm
> > happy to be corrected.
> 
> consider yourself corrected :)

So who, before you, has championed the use of mean-absolute error to
find optimum generators?

> nope. it´s just that an infinite number of tunings, a continuous 
> range of generators, can be considered poptimal for a given 
> temperament -- thus "absolutely and ideally perfect" cannot be taken 
> seriously, and was obviously poking fun of similar claims by people 
> like lucy.

Ah! Well the problem was that I didn't understand that a continuous
range was being referred to. Gene was quoting specific rational
fractions of an octave for the generators. And if I didn't understand
that, I suspect a lot of other people didn't either.

Hope you're having a great trip apart from the trains.


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Message: 5911

Date: Thu, 09 Jan 2003 21:34:32

Subject: Re: Nonoctave scales and linear temperaments

From: Carl Lumma

>>Why not optimize the generator size for the map, and let
>>it target the consonances?  Presumably because in some
>>tunings the errors for say 3 and 5 will cancel on consonances
>>like 5:3.
> 
>i'm not following you, or where you differ from what's
"standard" around here . . .

As I say, I don't know how much differing from what's
standard.  Calculations are seldom posted here at the
undergrad level.

As usual, I'm trying to figure things out by synthesizing
something and asking about it.  When I hit something that
works, I keep it.

>why don't you post a complete calculation for the meantone
>case, or if you wish, some other, more contrived case . . .

Map for 5-limit meantone...

      2  3  5
gen1  1  1 -2
gen2  0  1  4

Complexity for each identity...

2= 1
3= 2
5= 6

Let's weight by 1/base2log(i)...

2= 1.00
3= 1.26
5= 2.58

Now gen1 and gen2 are variables, and minimize...

error(2) + 1.26(error(3)) + 2.58(error(5))

I don't know how to do such a calculation, or even
if it's guarenteed to have a minimum.  It would
give us minimum-badness generators, not minimum
error gens.

The log2(i) weighting is only off the top of my head.
One could imagine no weighting.  One could imagine
weighting so steep we could find the optimal generators
for harmonic limit infinity.

If this does cause us to miss temperaments with good
composite consonances like 5:3, we can go back to
minimizing the error of all the intervals in the givin
limit, and keep the summed graham complexity.

-Carl


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Message: 5912

Date: Thu, 09 Jan 2003 02:52:21

Subject: Re: Nonoctave scales and linear temperaments

From: Carl Lumma

>you´re talking prime limit, which is fine for the mapping,
>as usual.

I'm talking 'put whatever you want in the map'.

> but for the optimization of the generator size, we need a
> list of consonances to target.

Why not optimize the generator size for the map, and let
it target the consonances?  Presumably because in some
tunings the errors for say 3 and 5 will cancel on consonances
like 5:3.

-Carl


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Message: 5913

Date: Thu, 09 Jan 2003 22:22:39

Subject: Re: Poptimal generators

From: Carl Lumma

>>So who, before you, has championed the use of mean-absolute
>>error to find optimum generators?
> 
>i'm not championing them, but i've seen them suggested several
>times.  i think carl lumma was one of those people.

I don't remember it, but I won't deny it.  Lately I've been
behind RMS, though I'm interested in the possiblity of a
universal thinger such as the heuristic or the poptimal stuff.

>you must have skipped over most of gene's original message. why
>do you think we're having this discussion about absolute error
>criteria in the first place? it's because gene suggested using
>the entire range of p-norms, with p from 2 to infinity, and i
>suggested moving the lower limit down to 1.

How does this sort of thing compare with the heuristic approach?

-Carl


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Message: 5914

Date: Thu, 09 Jan 2003 14:29:05

Subject: Re: A common notation for JI and ETs

From: David C Keenan

At 01:16 AM 9/01/2003 +0000, Dave Keenan <d.keenan@xx.xxx.xx> wrote:
>--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
><gdsecor@y...> wrote:
>--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...>
>wrote:
> > In case anyone has already looked at the .bmp for my latest
>suggestion
> > regarding symbolising the 5'-comma (5-schisma) up and down:
> >
> > It was riddled with vertical alignment errors so I've had another
>go at it.
> >
> > See Yahoo groups: /tuning- *
>math/files/Dave/5Schismas.bmp
>
> > -- Dave Keenan
> > Brisbane, Australia
>
>I've looked at it and thought about it for a couple of days.
>
>Good points:
>
>1) The +-5' symbols can easily be used in conjunction with any
>existing symbol.
>2) They clearly indicate (by vertical position) the line or space of
>the note being modified.
>3) There is also a helpful indication (a difference in vertical
>position in addition to slope) of whether the 5' is plus or minus.
>
>Comment on point 3:  The slope is most meaningful if the new symbols
>are placed to the left, as in the upper staff (which you also favor).

Yes. I'm happy to eliminate right-hand accents from consideration.

>Problems:
>
>1) The +-5' symbols are detached from the others, so are too easy to
>overlook (particularly if this is the only thing modifying a natural
>note).

I see this as a good point. It really doesn't matter much if a performer 
misses a 2 cent modification. On a flexible-pitch instrument many would not 
be able to do anything about them anyway. They just aren't that accurate. 
On a fixed pitch instrument there will presumably not be two notes 
available that are only 2 cents apart.

It would be much worse if the performer couldn't interpret the symbol at 
all (or quickly enough) because the conjoined 5' modification made it look 
unfamiliar.

>2) Since they are detached from the others, we technically have two
>new modifying symbols used together, so the double-symbol version of
>the notation might now become a triple-symbol version -- something to
>think about.

Yes, technically 3 symbols, but in reality no different to adding an accent 
to a roman character. Because they are so close together and because the 
accent is small relative to the character, it is perceived as a single 
character.

We can even refer to these small slanting lines as acute and grave.

>Since looking at this I also tried something else, which I have added
>to this file (on the third staff):
>
>Yahoo groups: /tuning- *
>math/files/secor/notation/Schisma.gif
>
>Note:  If you don't see 4 staves in the figure, then click on the
>refresh button on your browser to ensure that you're looking at the
>latest version of the file.
>
>I tried small arrowheads to indicate the 5' down and up symbols.  In
>the 3rd staff I attached them to the point of an existing sagittal
>symbol; for the up-arrow I removed the pixel at the end of the shaft
>to clarify the symbol.  The big advantage here is that we would avoid
>having detached symbol elements.

Yes. But unfortunately they make it look like you're modifying a note 
aligned with the place between the 5' arrowhead and the rest of the flags.

>In the 4th staff (up to the first double bar) I placed the arrows to
>the left of existing sagittal symbols, but they could just as easily
>be placed to the right, or on either side, depending on where they
>would look or fit best.
>
>Wherever you put them, I think that these small arrowheads are easier
>to see than those tiny slanted lines,

Based on making symbols proportional to their size in cents relative to 
strict Pythagorean, the 5' symbol should only have about 6 pixels because 
the 19 comma flag has 10 and corresponds to 3.4 cents. The small arrowheads 
(or circumflex and caron) contain 8 pixels.

>and they give a better
>indication of direction of alteration.

That's true. To some degree acute and grave reproduce the problem of the 
Bosanquet comma slash that your arrow shaft solved. But I think this is 
greatly mitigated by the up and down displacement and because they relate 
to the arrow shaft on the symbol that they are modifying.

I now agree that they should only be placed to the left so that the grave 
symbol retains its linguistic meaning of low or falling pitch and acute - 
high or rising pitch.

Full arrowheads already have a sagittal association with the prime 11 
whereas the slanted lines preserve the association with 5.

Code Charts (PDF Version) * is useful to check for clashes with existing 
music symbols. I didn't find any except the "marcato" symbol (downward 
arrowhead) which appears below (not beside) the notehead.

I just tried adding very short shafts to the acute and grave to make their 
direction clearer but this makes them look totally like separate symbols 
rather than accents, in fact it makes them look like 5-comma (not 5'-comma) 
symbols intended for grace notes.

>While I was writing this I got a couple of other ideas that use 5'
>flags, so I quickly added them on the fourth staff.  I lowered the
>short straight -5' flag to the same vertical position that we seem to
>be agreeing on to see how that would look and made 3 symbols that way.

I agree with including a bare shaft when a 5' accent mark would otherwise 
occur on its own.

>Then, after the next double bar, I used the small arrowheads as right
>flags and tried some symbols that way.  (The 5:7 comma is also there
>for comparison.)  After I looked at them for a little while, I
>decided to move the 5' flags one pixel to the right, so that they are
>almost, but not quite touching the rest of the sagittal symbol (to
>avoid confusion with a concave right flag).  I think that this last
>group is my preference in that:
>
>1) The 5' flags are clear and logical;
>2) The 5' symbol elements aren't off by themselves, therefore don't
>get overlooked;
>3) Their vertical positions are well placed;
>4) They aren't larger than concave flags.

Well, I'd go along with kerning the acute nearer to (the left of) the 
symbol being modified, when that symbol has a left flag (as in the 
pythagorean comma symbol), but I'd still prefer that the 5' symbols were 
defined as separate symbols in the font, for what are, I hope, obvious 
reasons, and I'd still prefer that the unkerned distance was two pixels 
(such as in the diaschisma symbol).

Pythag comma   '/|
Diaschisma     `/|

-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 5919

Date: Fri, 10 Jan 2003 22:26:04

Subject: Re: Notating Kleismic

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
<gdsecor@y...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith 
> <genewardsmith@j...>" <genewardsmith@j...> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" 
> <d.keenan@u...> wrote:
> > > However, if you _were_ using kleismic for 7-limit with the least
> > > complex 7's, you would probably be dissatisfied with a 53-ET based
> > > notation.
> > 
> > Why? Neither one is using the best value of the "7" of the et in 
> question, and in both cases the 7-limit intervals are much more out 
> of tune than the 5-limit intervals. It is cheesy no matter how you 
> notate it, and I don't see why it is any *more* dissapointing for 53 
> than for 72.

You're right. I hadn't realised that 72-ET's 7 wasn't kleismic's least
complex 7.

> Since 7 is +22 and 11 is -21 in the series of ~5:6's in 53, 72, and 
> 125, these are the normal positions for the kleismic temperament.  So 
> we should be using symbols of a different nature to notate a lot of 
> the tones closer to the origin, which would seem to call for the 5^2 
> symbol, //|.  The notation for 125 uses not only this, but also both 
> the 7-comma and 11-diesis symbols if the tuning is extended far 
> enough to take in ratios of 7 and 11, so 125 seems like a logical 
> choice to me.

This agrees better with the ET-independent notation I gave too. 
So
/   //   ///
would become
/|  //|  (|)


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Message: 5920

Date: Fri, 10 Jan 2003 23:12:46

Subject: Re: Notating Pajara

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> This is the system with wedgie [2, -4, -4, 2, 12, -11] which we used
to call Paultone. It has [1/2, 5/56] as poptimal in both the 7-limit
and the 9-limit, and my recommendation is that the 56-et be used to
notate it. The alternative is 22, but with all due respect for Paul's
favorite division, 56 is in much better tune. As a way of tuning a
22-tone MOS and playing Decatonic, it's something Paul might try if he
hasn't already.
>

I believe 56-ET would notate a chain of 22 just the same as 22-ET
would, since the only symbol required is the 5-comma symbol /|. But
beyond that, I suspect that the best 7 of 56-ET may not be the 7 of
pajara/paultone.


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