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Message: 5500 - Contents - Hide Contents Date: Wed, 06 Nov 2002 07:55:19 Subject: Re: Tree zoom duals From: wallyesterpaulrus --- In tuning-math@y..., "monz" <monz@a...> wrote:>> >> What would be really cool is a 3D applet, which let you look at >> a 7-limit picture from various directions. We could have either >> ets as points, linear temperaments as lines, and commas as planes, >> or a dual picture with commas as points, linear temperaments >> again as lines, and ets as planes. > > >> this is all stuff that i've always intended from the beginning > to have available in my JustMusic software. > > JustMusic software, (c) 1999 by Joseph L. Monzo * [with cont.] (Wayb.)is that really true? do you have any of the ideas above referenced anywhere?
Message: 5501 - Contents - Hide Contents Date: Wed, 6 Nov 2002 00:14:59 Subject: Re: Tree zoom duals From: monz ----- Original Message ----- From: "wallyesterpaulrus" <wallyesterpaulrus@xxxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Tuesday, November 05, 2002 11:55 PM Subject: [tuning-math] Re: Tree zoom duals> --- In tuning-math@y..., "monz" <monz@a...> wrote: > >>>>>> What would be really cool is a 3D applet, which let you look at >>> a 7-limit picture from various directions. We could have either >>> ets as points, linear temperaments as lines, and commas as planes, >>> or a dual picture with commas as points, linear temperaments >>> again as lines, and ets as planes. >> >> >>>> this is all stuff that i've always intended from the beginning >> to have available in my JustMusic software. >> >> JustMusic software, (c) 1999 by Joseph L. Monzo * [with cont.] (Wayb.) >> is that really true? do you have any of the ideas above referenced > anywhere?what's on the webpage and in the archives of the Yahoo JustMusic group is all i've ever made public about it. the rest of my ideas are in folders and folders of notes and old BASIC code ... stuff that i haven't even looked at in over 5 years. -monz (*really* wishing that i could find that Visual C++ programmer!!)
Message: 5502 - Contents - Hide Contents Date: Wed, 06 Nov 2002 08:38:45 Subject: Re: Tree zoom duals From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:>> Yer welcome--of course now you need to zoom out rather than in if >> you want to fool with this. >> what do you mean?The high-octane ets crowd in towards the origin, so you zoom in to find them. The high-octane commas expand out to infinity, so you would need to zoom out to include them.
Message: 5503 - Contents - Hide Contents Date: Wed, 06 Nov 2002 09:35:51 Subject: Re: Tree zoom duals From: Carl Lumma>The high-octane ets crowd in towards the origin, so you zoom in >to find them. The high-octane commas expand out to infinity, so >you would need to zoom out to include them. Rad!In both cases, the unzoomed versions show me what I want to see! -Carl
Message: 5504 - Contents - Hide Contents Date: Thu, 07 Nov 2002 15:18:03 Subject: Re: from the realms of private correspondence From: gdsecor --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:> [Paul wrote...]>>>> ...with 5:3 v. 6:5, doesn't it seem wrong that adding span >>>> should be able to change the relative concordance (10:3 v. >>>> 6:5)? >>>>>> not to me. we're not just adding span, we're also adding >>> complexity. ask george secor what he thinks. >> Heya George... out there? Do you find 5:3 or 6:5 more concordant?I say 5:3 is more consonant; my experience has indicated that consonance of intervals between 1:1 and 2:1 can be ordered by the size of the product of the integers in their ratio -- up to a point: as long as the numbers are small enough that the identity of an interval isn't subject to confusion, such that it is more likely to be interpreted as a tempered approximation of another (more consonant) interval (which gets us into the study of harmonic entropy, but I digress).> Howabout 10:3 and 6:5?This one's a tougher call. The product is the same, but the span is so different that we're beginning to compare apples with oranges. Are we talking about comparing these with the lower tones being the same pitch, or the same upper tones, or an average of the two? You could even compare a 6:5 (with lower tone on middle C) with a 6:5 two octaves lower, and I think you would agree that the lower one is more muddy, i.e., less consonant, so range of pitch also enters into the picture. Now to answer your question, I think I would judge them to be equally consonant if the average pitch for each of the two intervals were the same. (Just my present opinion -- subject to change with persuading new evidence.) --George
Message: 5505 - Contents - Hide Contents
Date: Thu, 7 Nov 2002 14:49:51
Subject: About metrics
From: Pierre Lamothe
It seems there is confusion about metrics.
A metric measure
something. One have to reflect clearly what is supposed to be measured
and for that, to distinguish first the pertinent entities.
At microtonal level, i.e. independently of any interactional system,
one have to reflect, the most appropriately possible, what corresponds
to the perception.
It is well known there exist two irreducible manner to order a set
of pitch height intervals. One could call that
the ordinary or melodic order and the harmonic order.
There exist in this way two distinct distances to be measured, one
on each ordered structure. One could call that
the melodic distance and the harmonic distance.
At macrotonal level, i.e. when rules are added on how interact the
elements, one have to reflect the geometry of the created global
entity. In pure JI, one can simply use the Euclidean distance, once is
determined the quadric or hyperquadric defining the scalar product and
the orthogonality on that space, created by choice.
Microtonal level
There is a unique representation of a rational interval X in an
appropriate primal basis <pi>.
X = <pi> (xi) = (p0^x0) (p1^x1) ... (pN^xN)
I argue that the simplest appropriate distances between the intervals
X and Y are
Ordinary or Melodic Distance
| sum ( ( yi - xi ) log pi ) |
Tenney or Harmonic Distance
sum ( | yi - xi | log pi )
I remark also that the harmonic distance concept has no sense in the
real segment of pitch heights.
Macrotonal level
Suppose one choose to use heptatonic modes in prime 5-limit. One knows
that the simplest val is [7 11 16], which is simply
Round ( 7 [log 2 log 3 log 5] / log 2 )
Using X = (w,x,y) rather than (x0,x1,x2), that means each interval has
now a class or degree D(X) = 7w + 11x + 16y, or simply 4x + 2y (mod 7)
in the octave.That means also the set of unison vectors, i.e. all the
intervals having the degree 0 in the octave, is a sublattice of the <3
5> ZxZ lattice.
Is it sufficient to define a system ? No.
Defining a system implies equally the choice of the minimal unison
vectors around the unison which determines the fundamental domain of
that system. In a plane, like here, that implies the choice of the six
" nearest " unison vectors giving an hexagonal shape in which the
unison, at the center, is the unique interval having the degree 0.
Nearest here implies an undefined macrotonal metric. One have to
define a macrotonal distance permitting to say which among such
vectors are the nearest of the unison. There are different
possibilities. Each choice determine a distinct system.
Suppose one choose to get the " simplest ".
Can one only observe the skewness of possible hexagons to determine
that? No.
Simplest can only have a microtonal sense since the macrotonal metric
is not yet chosen. It's where the harmonic distance has to be used.
Comparing the harmonic norm (i.e. harmonic distance from unison) for
all intervals of each hexagons, it appears clearly here that the
simplest case corresponds to unison vector U = 81/80, V = 25/24, U+V
and the three inverses. What is well-known to correspond at the
Zarlino system with modes using 16/15, 10/9 and 9/8 as steps.
An Euclidean space is defined as a vector space with a scalar product,
which in turn is defined by the choice of a bilinear form
corresponding to a quadric or hyperquadric. The appropriate choice to
reflect the geometric properties of such tonal systems is to fit the
quadric on the chosen minimal unison vectors.
The ellipse
3 x^2 + 13 y^2 + 3 xy = 49
passes on the six chosen unison vectors. Using then the matrix M =
[(3, 3/2) (3/2, 13)], the scalar product
(X|Y) = X M Y
define an appropriate structure of Euclidean space with the
correlative Euclidean distance, so there exist an orthonormal basis in
which the ellipse is a circle of radius 7, and the distance of (x,y)
from unison, for instance, is defined as sqrt (x^2 + y^2).
Pierre
[This message contained attachments]
Message: 5506 - Contents - Hide Contents Date: Thu, 07 Nov 2002 20:46:49 Subject: Re: from the realms of private correspondence From: Carl Lumma>> >lso, still don't get how anything sensible could satisfy >> property 1, or how my suggestion is a cop-out. >>Ordinary Euclidean distance does.I must not understand property 1. If I have colinear points a b c, how is the sum of AB and BC >= AC? -Carl
Message: 5507 - Contents - Hide Contents Date: Thu, 07 Nov 2002 09:21:12 Subject: from the realms of private correspondence From: Carl Lumma [Paul wrote...]>>>> You're wondering how to define the taxicab distance between >>>> three points? The shortest path that connects the subgraph... >>>> wouldn't punish all the dyads, and a fully-connecting path >>>> would be equiv. to the sum of the dyadic Tenney HD... maybe >>>> the area of the enclosed polygon... for a chord like 4:5:25, >>>> this leads to the problem that you can only compare it to other >>>> 5-limit-only chords. Well, I give up. >>>>>> actually, i figured this out once. it's proportional to the >>> total edge length of the hyper-rectangle defined by the points. >>> but that doesn't make it a metric! >>>> 1. Can you give an example for 4:5:25? >>ok, in that case the hyper-rectangle collapses down to two >dimensions, but the total edge length is unaffected. so no real >problem there. the vertices of the hyper-rectangle are the >pitches comprising the (non-octave reduced) euler genus whose >factors are the notes in your chord.Come again? What points in the lattice does the rectangle intersect?>>> do you know what "metric" means? >>>> According to mathworld, it's a function that satisfies the >> following: >> >> 1. g(x,y) + g(y,z) >= g(x,z) >> triangle inequality >> >> 2. g(x,y) = g(y,x) >> symmetric >> >> 3. g(x,x) = 0 >> >> 4. g(x,y) = 0 implies x = y. >> >> 1. Makes no sense to me. does Tenney HD satisfy it for dyads? >>once again, carl -- tenney HD is a metric for *pitches*, not >dyads (notwithstanding your "cop-out").When did you ever say that? What does it mean? What meaning do pitches have in terms of concordance?>> 4. This would depend on the lack of chords sharing the same >> set of factors. Without it, mathworld says we have a >> "pseudometric". >>Tenney's HD is a metric, not a pseudometric. property 4 implies >that any pitch has a zero harmonic distance from itself -- >that's all.No, that's property 3. Property 4 says any time you see zero distance you measuring the distance from a pitch to itself. Also, still don't get how anything sensible could satisfy property 1, or how my suggestion is a cop-out.>>> ...with 5:3 v. 6:5, doesn't it seem wrong that adding span >>> should be able to change the relative concordance (10:3 v. >>> 6:5)? >>>> not to me. we're not just adding span, we're also adding >> complexity. ask george secor what he thinks.Heya George... out there? Do you find 5:3 or 6:5 more concordant? Howabout 10:3 and 6:5? -Carl
Message: 5508 - Contents - Hide Contents Date: Thu, 07 Nov 2002 20:51:08 Subject: Re: from the realms of private correspondence From: Carl Lumma>I say 5:3 is more consonant; my experience has indicated that >consonance of intervals between 1:1 and 2:1 can be ordered by >the size of the product of the integers in their ratioHow long have you been using this product limit, may I ask?>Howabout 10:3 and 6:5? > >This one's a tougher call. The product is the same, but the span >is so different that we're beginning to compare apples with >oranges.That's my interpretation of span -- it dilutes concordance without adding discordance. But according to the product limit, we've just added discordance to 5:3 by stretching it by an octave.>Are we talking about comparing these with the lower tones being >the same pitch, or the same upper tones, or an average of the two? >You could even compare a 6:5 (with lower tone on middle C) with a >6:5 two octaves lower, and I think you would agree that the lower >one is more muddy, i.e., less consonant, so range of pitch also >enters into the picture.You haven't been talking to Dave Keenan, have you? My feeling is that we want a measure that works when comparing dyads in the same pitch range, and that product limit always was such a measure.>Now to answer your question, I think I would judge them to be >equally consonant if the average pitch for each of the two >intervals were the same. Noted. Thanks! -Carl
Message: 5509 - Contents - Hide Contents Date: Thu, 07 Nov 2002 09:51:01 Subject: Re: from the realms of private correspondence From: Gene Ward Smith --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:> Also, still don't get how anything sensible could satisfy > property 1, or how my suggestion is a cop-out.Ordinary Euclidean distance does.
Message: 5511 - Contents - Hide Contents Date: Thu, 07 Nov 2002 21:44:21 Subject: Fwd: Re: from the realms of private correspondence From: Carl Lumma> It is my hypothesis that this is one of the complications of > dissonance that led to pitch grammars in which some intervals > are understood to be dissonant, even if they are not > especially rough, or lacking a strongly defined tone of > reference/precedence.Your hypothesis is that because psychoacoustic dissonance varies from listener to listener, music had to develop a grammar to make the perception of dissonance more... uniform? -Carl
Message: 5512 - Contents - Hide Contents Date: Thu, 07 Nov 2002 21:58:07 Subject: Re: from the realms of private correspondence From: gdsecor --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>> I say 5:3 is more consonant; my experience has indicated that >> consonance of intervals between 1:1 and 2:1 can be ordered by >> the size of the product of the integers in their ratio >> How long have you been using this product limit, may I ask?I don't understand how the term "limit" got into your question, or is this what others have called it? I've used the product of the integers for almost as long as I've been studying alternative tunings -- since the mid-1960s.>> Howabout 10:3 and 6:5? >> >> This one's a tougher call. The product is the same, but the span >> is so different that we're beginning to compare apples with >> oranges. >> That's my interpretation of span -- it dilutes concordance without > adding discordance. But according to the product limit, we've > just added discordance to 5:3 by stretching it by an octave.Depends on which direction you stretch it. Transpose the bottom tone downward and you'll add discordance, but transpose it upward and you won't add nearly as much -- but in neither case are you comparing two intervals in the same register. So transpose both tones by half an octave and you'll add discordance somewhere in between.>> Are we talking about comparing these with the lower tones being >> the same pitch, or the same upper tones, or an average of the two? >> You could even compare a 6:5 (with lower tone on middle C) with a >> 6:5 two octaves lower, and I think you would agree that the lower >> one is more muddy, i.e., less consonant, so range of pitch also >> enters into the picture. >> You haven't been talking to Dave Keenan, have you?Not about this. I picked this up from Herm Helmholtz, but have not had the opportunity to discuss it with him either. --George
Message: 5513 - Contents - Hide Contents Date: Thu, 07 Nov 2002 22:17:59 Subject: Re: from the realms of private correspondence From: Carl Lumma>I don't understand how the term "limit" got into your question, >or is this what others have called it?It comes from that it can be used as an alternative to odd limit. Interestingly, IIRC Paul showed odd-limit is as close to the product thing as we can get in an octave-equivalent measure. Is that right, Paul?>I've used the product of the integers for almost as long as I've >been studying alternative tunings -- since the mid-1960s.I got it from Denny Genovese, who was using it at least since the mid 80's, and maybe since the mid 70's. When did Tenney come up with his HD?>> That's my interpretation of span -- it dilutes concordance >> without adding discordance. But according to the product limit, >> we've just added discordance to 5:3 by stretching it by an >> octave. >>Depends on which direction you stretch it. Transpose the bottom >tone downward and you'll add discordance, but transpose it upward >and you won't add nearly as muchTransposing the bottom tone upward flips the ratio. Perhaps you meant the top tone upward? Listening just now (in two registers), I find 10:3 less concordant than 5:3, and 12:5 more concordant than 6:5. 6:5 suffers unfairly perhaps from too little span.>> You haven't been talking to Dave Keenan, have you? >>Not about this. I picked this up from Herm Helmholtz, but have not >had the opportunity to discuss it with him either. :) -Carl
Message: 5514 - Contents - Hide Contents Date: Thu, 07 Nov 2002 23:24:04 Subject: Re: from the realms of private correspondence From: Gene Ward Smith --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:> I must not understand property 1. If I have colinear > points a b c, how is the sum of AB and BC >= AC?AB + BC = AC The two parts of the line segment AC add up to the whole in length.
Message: 5515 - Contents - Hide Contents Date: Thu, 07 Nov 2002 23:28:23 Subject: Re: from the realms of private correspondence From: Gene Ward Smith --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:> I got it from Denny Genovese, who was using it at least since > the mid 80's, and maybe since the mid 70's. When did Tenney > come up with his HD?I think a lot of people used it. I used it, along with more complicated things aimed at scale creation. I don't know why Euler didn't use it, and have no idea who first did.
Message: 5517 - Contents - Hide Contents Date: Fri, 08 Nov 2002 10:43:08 Subject: Re: from the realms of private correspondence From: Gene Ward Smith --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:> In which case, I'd think property 4 is the only one my cop-out > doesn't meet, and Tenney HD also doesn't meet it. If this > really isn't true, I'm hoping someone will refute it.What's Tenney HD? If you mean || 3^a 5^b 7^c || = |a/log(3)| + |b/log(5)| + |c/log(7)| or something like that, it is a vector space norm and automatically induces a metric by d(X, Y) = ||X - Y||
Message: 5519 - Contents - Hide Contents Date: Fri, 08 Nov 2002 16:43:59 Subject: Re: from the realms of private correspondence From: gdsecor --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>>> That's my interpretation of span -- it dilutes concordance >>> without adding discordance. But according to the product limit, >>> we've just added discordance to 5:3 by stretching it by an >>> octave. >> [GS:]>> Depends on which direction you stretch it. Transpose the bottom >> tone downward and you'll add discordance, but transpose it upward >> and you won't add nearly as much > [CL:]> Transposing the bottom tone upward flips the ratio. Perhaps you > meant the top tone upward? Yes. > Listening just now (in two registers), I find 10:3 less concordant > than 5:3, and 12:5 more concordant than 6:5. 6:5 suffers unfairly > perhaps from too little span.I don't think that I would agree, but perhaps the timbres that we're hearing these in might have something to do with it: Yahoo groups: /tuning-math/message/4945 * [with cont.] I haven't had any desire to delve too deeply into the all of the variables that would be considered in a scientific treatment of the subject (too many other things to do) and am happy to leave that to others to pursue. I gave my opinion only because you asked for it. :) --George
Message: 5520 - Contents - Hide Contents Date: Fri, 08 Nov 2002 23:56:25 Subject: Re: from the realms of private correspondence From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote: >>> probably it was benedetti. in the renaissance. he had very >> interesting reasons for choosing it -- margo has posted on this on >> the tuning list . . . >> Wow--it goes back a ways. I was screwing my thinking cap on, and >decided I got it from a book--which probably means indirectly from >Tenney. In any case I quickly decided I liked p+q better than pq as >a consonance measure anyway.p+q is the mann measure. but pq ranks things just like p+q within any narrow range, and pq as a limit leaves the harmonic entropy curve with no overall trend, while p+q ends up with some trend. i like no overall trend because we can then say that "span" is a truly independent component (independent from harmonic entropy) of our overall assessment. but this discussion really belongs on the harmonic entropy list . . .
Message: 5521 - Contents - Hide Contents Date: Fri, 08 Nov 2002 20:05:54 Subject: Re: from the realms of private correspondence From: Carl Lumma> I haven't had any desire to delve too deeply into the all of the > variables that would be considered in a scientific treatment of > the subject (too many other things to do) and am happy to leave > that to others to pursue. I gave my opinion only because you > asked for it. :)For sure. Thanks again. -Carl
Message: 5522 - Contents - Hide Contents Date: Fri, 08 Nov 2002 23:58:25 Subject: Fwd: Re: from the realms of private correspondence From: wallyesterpaulrus --- In tuning-math@y..., <Josh@o...> wrote:> For what it's worth, the Danner thing I mentioned before > deals mostly with trichords. > > MUSIC PERCEPTION also has some articles on subjective > trichord similarity, which might also help gauge > "normal" interpretation of comparative dissonance. > Also some good work on scale construction as resolution > of inharmonic partials in Indian music, and in piano tuning > (!). > Have you guys seen this journal?i've seen a little bit. what's this inharmonicity article? i might try to check it out if i can, if only to poke holes in it . . .
Message: 5523 - Contents - Hide Contents Date: Fri, 08 Nov 2002 20:15:13 Subject: Re: from the realms of private correspondence From: Carl Lumma>> >n which case, I'd think property 4 is the only one my cop-out >> doesn't meet, and Tenney HD also doesn't meet it. If this >> really isn't true, I'm hoping someone will refute it. >>What's Tenney HD?Tenney Harmonic Distance. Note that it is only defined for dyads. I attempted to extend it to triads. Paul claims that in so doing, I removed its metric status.>If you mean > >|| 3^a 5^b 7^c || = |a/log(3)| + |b/log(5)| + |c/log(7)| > >or something like that,That's right, except that its |a * log(3)|... etc, and it includes 2's. So it is equivalent to just log(a*b) for a dyad a:b in lowest terms. I generalized it to log(a*b*c) for a triad a:b:c.>it is a vector space norm and automatically induces a metric >by d(X, Y) = ||X - Y||Not sure I follow that... -Carl
Message: 5524 - Contents - Hide Contents Date: Fri, 08 Nov 2002 23:58:31 Subject: Re: from the realms of private correspondence From: Gene Ward Smith --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>> Because of the group structure, this automatically gives you >> concordance for dyads, which is why thinking aobut it in terms >> of norms rather than metrics makes the most sense. >> If you know what norms are and how to work with them. I'm > still struggling with metrics. But do tell. Maybe Paul > will follow.Did you see my mathworld citation? Here is another: Normed vector space - Wikipedia, the free ency... * [with cont.] (Wayb.) There is an error on this page--the field need not be either C or R, but can be any local field of characteristic 0. In particular, it can be the rational numbers.
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