Yahoo Tuning Groups Ultimate Backup tuning-math from the realms of private correspondence

[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

🔗Gene Ward Smith <genewardsmith@juno.com>

🔗gdsecor <gdsecor@yahoo.com>

--- 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

🔗Carl Lumma <clumma@yahoo.com>

>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?

>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

🔗gdsecor <gdsecor@yahoo.com>

--- 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

🔗Carl Lumma <clumma@yahoo.com>

>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 much

Transposing 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

🔗Gene Ward Smith <genewardsmith@juno.com>

🔗Carl Lumma <clumma@yahoo.com>

🔗Gene Ward Smith <genewardsmith@juno.com>

🔗Carl Lumma <clumma@yahoo.com>

>>Yeah, I must be reading the notation here...
>>
>>http://mathworld.wolfram.com/Metric.html
>
>It says the distance from A to B, plus the distance from B to C
>(two legs of a triangle) is at least as big as the distance from
>A to C (the third leg.)

Mint!

Of course. Sorry.

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.

-Carl

🔗Gene Ward Smith <genewardsmith@juno.com>

--- 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||

🔗gdsecor <gdsecor@yahoo.com>

--- 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:

/tuning-math/message/4945

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

🔗Carl Lumma <clumma@yahoo.com>

>>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?

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

🔗Carl Lumma <clumma@yahoo.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:
> [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?

its vertices are

1 4

5 20

25 100

125 500

and it also intersects 5 (again), 20 (again), 25 (again), 100
(again), 2, and 10.

> >>>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?

pitches are what you start with when you want to calculate
concordance. the fact that you can transpose them (together) without
affecting the concordance is one of the essential features in making
this a metric at all.

> >>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.

true enough!

> Also, still don't get how anything sensible could satisfy
> property 1,

of course any sensible metric satisfies property 1. euclidean,
taxicab, you name it!

> or how my suggestion is a cop-out.

because it deprives "metric" of all meaning.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:

> 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.

if you're in the same pitch range, you can use product, you can use
denominator, you can use sum, you can use numerator, and they'll all
give you the same ranking. it's when comparing dyads in *different*
pitch ranges that these diverge. product is at least preferable to
the other options, in that case, because it leaves the curve with no
overall trend -- it's asymptotically flat.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- 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.

probably it was benedetti. in the renaissance. he had very
interesting reasons for choosing it -- margo has posted on this on
the tuning list . . .

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:
> >>Yeah, I must be reading the notation here...
> >>
> >>http://mathworld.wolfram.com/Metric.html
> >
> >It says the distance from A to B, plus the distance from B to C
> >(two legs of a triangle) is at least as big as the distance from
> >A to C (the third leg.)
>
> Mint!
>
> Of course. Sorry.
>
> 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.

what do you mean? of course tenney HD meets property 4, as i
demonstrated in my e-mail which was your jumping-off point for all
this . . .

🔗Gene Ward Smith <genewardsmith@juno.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- 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)|

no, it's

|| 2^z 3^a 5^b 7^c . . . || = |z*log(2)| + |a*log(3)| + |b*log(5)| +
|c*log(7)| . . . which ends up being the log of (numerator times
denominator)

>
> or something like that, it is a vector space norm and automatically
induces a metric by
>
> d(X, Y) = ||X - Y||

where "-" is really an arithmetic division, not an arithmetic
subtraction . . .

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:
> > 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.
>
> I also don't see how Tenney HD meets property 4... take
> dyads 33:26 and 39:22 (Paul's example).
>
> -Carl

just because two pairs of points have the same distance from one
another, or (more directly) just because two different points have
the same distance from the origin, doesn't mean that they have to be
the same point, or anything like that. property 4 only says that if
the distance between two points is *zero*, they have to be the same
point.

🔗Carl Lumma <clumma@yahoo.com>

>>>>>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?
>
>its vertices are
>
>1 4
>
>5 20
>
>25 100
>
>125 500
>
>and it also intersects 5 (again), 20 (again), 25 (again), 100
>(again), 2, and 10.

My chord is:

25
|
5
|
1

Here are your verticies on the lattice:

125 - x - 500
| | |
25 - x - 100
| | |
5 - x - 20
| | |
1 - x - 4

Obviously, you intend some of these to be on
extra dimensions. Why? How did you figure
out that the perimeter of these structures
would be a consistent taxicab distance for three
points?

>>>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?
>
>pitches are what you start with when you want to calculate
>concordance. the fact that you can transpose them (together)
>without affecting the concordance is one of the essential
>features in making this a metric at all.

I don't understand how a pitch can have concordance.

-Carl

🔗Gene Ward Smith <genewardsmith@juno.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:
> >>>>>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?
> >
> >its vertices are
> >
> >1 4
> >
> >5 20
> >
> >25 100
> >
> >125 500
> >
> >and it also intersects 5 (again), 20 (again), 25 (again), 100
> >(again), 2, and 10.
>
> My chord is:
>
> 25
> |
> 5
> |
> 1

you said 4:5:25!

> Here are your verticies on the lattice:
>
> 125 - x - 500
> | | |
> 25 - x - 100
> | | |
> 5 - x - 20
> | | |
> 1 - x - 4

i don't know if you read my diagram right. it was meant to represent
the eight vertices of (in concept) a cube.

> Obviously, you intend some of these to be on
> extra dimensions. Why?

they're not on extra dimensions, but the edges overlap, since it's
squashed down from

> How did you figure
> out that the perimeter of these structures
> would be a consistent taxicab distance for three
> points?

it's easy. there are 12 edges. the three representing each of the
pitches' distances from 1/1 (when they are expressed as simply as
possible as harmonics thereof) are each present four times. so you
can divide through by four, and you simply have log(a) + log(b) + log
(c), which equals log(a*b*c). get it?

> >>>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?
> >
> >pitches are what you start with when you want to calculate
> >concordance. the fact that you can transpose them (together)
> >without affecting the concordance is one of the essential
> >features in making this a metric at all.
>
> I don't understand how a pitch can have concordance.

it doesn't! that's why we have a concordance *metric*!

🔗Carl Lumma <clumma@yahoo.com>

>>I don't understand how a pitch can have concordance.
>
>They have it with respect to "1".

Dyad. So what you have is a function that assigns a value
to pitches, and then you subtract them. The function assigns
zero to 1/1 but you're still subtracting. I have a function
that assigns a value to triads, and I can subtract them.
What's the difference?

>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.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

>>>its vertices are
>>>
>>>1 4
>>>
>>>5 20
>>>
>>>25 100
>>>
>>>125 500
>>>
>>>and it also intersects 5 (again), 20 (again), 25 (again), 100
>>>(again), 2, and 10.
>>
>> My chord is:
>>
>> 25
>> |
>> 5
>> |
>> 1
>
> you said 4:5:25!

That's true, but I didn't mean it. :)

>> Here are your verticies on the lattice:
>>
>> 125 - x - 500
>> | | |
>> 25 - x - 100
>> | | |
>> 5 - x - 20
>> | | |
>> 1 - x - 4
>
>i don't know if you read my diagram right. it was meant to
>represent the eight vertices of (in concept) a cube.

Ah, it was a diagram!

>> How did you figure
>> out that the perimeter of these structures
>> would be a consistent taxicab distance for three
>> points?
>
> it's easy. there are 12 edges. the three representing
> each of the pitches' distances from 1/1 (when they are
> expressed as simply as possible as harmonics thereof)
> are each present four times.

> so you can divide through by four, and you simply have
> log(a) + log(b) + log (c), which equals log(a*b*c). get it?

Oh, dear, I certainly don't... You're in favor of my
suggestion after all, just that you don't consider it
a metric?

>>I don't understand how a pitch can have concordance.
>
>it doesn't! that's why we have a concordance *metric*!

I understand what you're getting at now on this point, but I
still would call both Tenney HD and my suggestion pseudometrics
based on the notation at mathworld.

-Carl

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:

> > so you can divide through by four, and you simply have
> > log(a) + log(b) + log (c), which equals log(a*b*c). get it?
>
> Oh, dear, I certainly don't... You're in favor of my
> suggestion after all, just that you don't consider it
> a metric?

right. actually, i remember my suggestion better now. it's the total
length of the edges of the smallest hyperrectangle that can enclose
the points (a direct generalization of the tenney metric for dyads).
and this ends up being proportional to the *LCM*, rather than to the
product. so 4:5:6 and 10:12:15 actually end up having the same
generalized tenney complexity, since they both have LCM of 60. this
accords better with the lattice itself being symmetric along
utonal/otonal lines. i don't see how one could ever hope to embody
favoritism for otonal over utonal in a lattice, as much as i believe
in such favoritism myself.

> >>I don't understand how a pitch can have concordance.
> >
> >it doesn't! that's why we have a concordance *metric*!
>
> I understand what you're getting at now on this point, but I
> still would call both Tenney HD and my suggestion pseudometrics
> based on the notation at mathworld.

Tenney HD is most certainly a metric, not a pseudometric!

🔗Gene Ward Smith <genewardsmith@juno.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- 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 . . .

🔗Gene Ward Smith <genewardsmith@juno.com>

--- 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:

http://www.wikipedia.org/wiki/Normed_vector_space

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.

🔗Gene Ward Smith <genewardsmith@juno.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Carl Lumma <clumma@yahoo.com>

>>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:

I did. Unfortunately, most of it is straight over my head.
Why this is so is a matter of some interest to me... I can't
tell if it's really hard, just some notational hurdle, or both.

>http://www.wikipedia.org/wiki/Normed_vector_space

That's better, thanks.

>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.

You should fix the page... it's a Wiki, after all. Just click
"Edit this page" near the top.

-Carl

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:
> >>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:
>
> I did. Unfortunately, most of it is straight over my head.
> Why this is so is a matter of some interest to me... I can't
> tell if it's really hard, just some notational hurdle, or both.

if you sit down and verify for yourself all the properties of "metric"
for euclidean distance and taxicab distance, i think you'll begin to
have a better idea of what a metric is, in general.

note that my geometric generalization of tenney's HD function so
that it amounts to the LCM of the N integers when the N-ad is
expressed in lowest harmonic terms (the LCM of great
importance to measures like Euler's and, more recently,
Marion's) allows us to forget that it's a metric altogether . . . and
to forget about taxicab, since the construction of the rectangle for
a dyad already contains the definition of taxicab inside it. of
course, i have no idea what such a construction is called, but
who cares?