Relative complexity and scale construction

I have a nifty new concept, "relative complexity", which should be useful for constructing scales in non-linear regular temperaments.

If w is the wedgie of a temperament, and q is an interval, then define the relative complexity of q with respect to w as the (geometric) complexity of w^q. If the kernel of w is generated by
<c1, ..., ck> then this is the complexity of c1^c2^...^ck^q. In the case where w is a linear temperament, the relative complexity of q is proportional to the number of generator steps for q. Hence, scale construction using relative complexity can be thought of as a generalization of scales constructed by contiguous generators, such as MOS.

For an example, consider the 126/125 "Starling" planar temperament. If we order equivalence classes of intervals 1 <= q < sqrt(2) by relative complexity, we get

1, 6/5, 4/3, 5/4, 7/5, 10/9, 21/20, 15/14, 8/7, 7/6, 16/15, 125/112, 80/63, ...

Here 6/5, for instance, represents the class {(6/5)*(125/125)^n} of intervals; I choose the representative of smallest height in each case.

Now we may use these intervals in the above order to construct scales; I give the JI version, then the version in 336-et (using the mapping h336+v3-v5 = [336, 532, 781, 943] which totally nails Starling.)

[1]
[0]
[336]

[1, 5/3]
[0, 249]
[249, 87]

[1, 6/5, 5/3]
[0, 87, 249]
[87, 162, 87]

[1, 6/5, 3/2, 5/3]
[0, 87, 196, 249]
[87, 109, 53, 87]

[1, 6/5, 4/3, 3/2, 5/3]
[0, 87, 140, 196, 249]
[87, 53, 56, 53, 87]

[1, 6/5, 4/3, 3/2, 8/5, 5/3]
[0, 87, 140, 196, 227, 249]
[87, 53, 56, 31, 22, 87]

[1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3]
[0, 87, 109, 140, 196, 227, 249]
[87, 22, 31, 56, 31, 22, 87]

[1, 6/5, 5/4, 4/3, 10/7, 3/2, 8/5, 5/3]
[0, 87, 109, 140, 174, 196, 227, 249]
[87, 22, 31, 34, 22, 31, 22, 87]

[1, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3]
[0, 87, 109, 140, 162, 174, 196, 227, 249]
[87, 22, 31, 22, 12, 22, 31, 22, 87]

[1, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 9/5]
[0, 87, 109, 140, 162, 174, 196, 227, 249, 283]
[87, 22, 31, 22, 12, 22, 31, 22, 34, 53]

[1, 10/9, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 9/5]
[0, 53, 87, 109, 140, 162, 174, 196, 227, 249, 283]
[53, 34, 22, 31, 22, 12, 22, 31, 22, 34, 53]

[1, 10/9, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 9/5, 40/21]
[0, 53, 87, 109, 140, 162, 174, 196, 227, 249, 283, 314]
[53, 34, 22, 31, 22, 12, 22, 31, 22, 34, 31, 22]

[1, 21/20, 10/9, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 9/5, 40/21]
[0, 22, 53, 87, 109, 140, 162, 174, 196, 227, 249, 283, 314]
[22, 31, 34, 22, 31, 22, 12, 22, 31, 22, 34, 31, 22]

[1, 21/20, 10/9, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 9/5, 28/15, 40/21]
[0, 22, 53, 87, 109, 140, 162, 174, 196, 227, 249, 283, 302, 314]
[22, 31, 34, 22, 31, 22, 12, 22, 31, 22, 34, 19, 12, 22]

[1, 21/20, 15/14, 10/9, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 9/5, 28/15, 40/21]
[0, 22, 34, 53, 87, 109, 140, 162, 174, 196, 227, 249, 283, 302, 314]
[22, 12, 19, 34, 22, 31, 22, 12, 22, 31, 22, 34, 19, 12, 22]

[1, 21/20, 15/14, 10/9, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 7/4, 9/5, 28/15, 40/21]
[0, 22, 34, 53, 87, 109, 140, 162, 174, 196, 227, 249, 271, 283, 302, 314]
[22, 12, 19, 34, 22, 31, 22, 12, 22, 31, 22, 22, 12, 19, 12, 22]

[1, 21/20, 15/14, 10/9, 8/7, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 7/4, 9/5, 28/15, 40/21]
[0, 22, 34, 53, 65, 87, 109, 140, 162, 174, 196, 227, 249, 271, 283, 302, 314]
[22, 12, 19, 12, 22, 22, 31, 22, 12, 22, 31, 22, 22, 12, 19, 12, 22]

[1, 21/20, 15/14, 10/9, 8/7, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4, 9/5, 28/15, 40/21]
[0, 22, 34, 53, 65, 87, 109, 140, 162, 174, 196, 227, 249, 261, 271, 283, 302, 314]
[22, 12, 19, 12, 22, 22, 31, 22, 12, 22, 31, 22, 12, 10, 12, 19, 12, 22]

[1, 21/20, 15/14, 10/9, 8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4, 9/5, 28/15, 40/21]
[0, 22, 34, 53, 65, 75, 87, 109, 140, 162, 174, 196, 227, 249, 261, 271, 283, 302, 314]
[22, 12, 19, 12, 10, 12, 22, 31, 22, 12, 22, 31, 22, 12, 10, 12, 19, 12, 22]

[1, 21/20, 15/14, 10/9, 8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4, 9/5, 28/15, 15/8, 40/21]
[0, 22, 34, 53, 65, 75, 87, 109, 140, 162, 174, 196, 227, 249, 261, 271, 283, 302, 305, 314]
[22, 12, 19, 12, 10, 12, 22, 31, 22, 12, 22, 31, 22, 12, 10, 12, 19, 3, 9, 22]

[1, 21/20, 16/15, 15/14, 10/9, 8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4, 9/5, 28/15, 15/8, 40/21]
[0, 22, 31, 34, 53, 65, 75, 87, 109, 140, 162, 174, 196, 227, 249, 261, 271, 283, 302, 305, 314]
[22, 9, 3, 19, 12, 10, 12, 22, 31, 22, 12, 22, 31, 22, 12, 10, 12, 19, 3, 9, 22]

[1, 21/20, 16/15, 15/14, 10/9, 8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4, 224/125, 9/5, 28/15, 15/8, 40/21]
[0, 22, 31, 34, 53, 65, 75, 87, 109, 140, 162, 174, 196, 227, 249, 261, 271, 280, 283, 302, 305, 314]
[22, 9, 3, 19, 12, 10, 12, 22, 31, 22, 12, 22, 31, 22, 12, 10, 9, 3, 19, 3, 9, 22]

[1, 21/20, 16/15, 15/14, 10/9, 125/112, 8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4, 224/125, 9/5, 28/15, 15/8, 40/21]
[0, 22, 31, 34, 53, 56, 65, 75, 87, 109, 140, 162, 174, 196, 227, 249, 261, 271, 280, 283, 302, 305, 314]
[22, 9, 3, 19, 3, 9, 10, 12, 22, 31, 22, 12, 22, 31, 22, 12, 10, 9, 3, 19, 3, 9, 22]

[1, 21/20, 16/15, 15/14, 10/9, 125/112, 8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 63/40, 8/5, 5/3, 12/7, 7/4, 224/125, 9/5, 28/15, 15/8, 40/21]
[0, 22, 31, 34, 53, 56, 65, 75, 87, 109, 140, 162, 174, 196, 218, 227, 249, 261, 271, 280, 283, 302, 305, 314]
[22, 9, 3, 19, 3, 9, 10, 12, 22, 31, 22, 12, 22, 22, 9, 22, 12, 10, 9, 3, 19, 3, 9, 22]

[1, 21/20, 16/15, 15/14, 10/9, 125/112, 8/7, 7/6, 6/5, 5/4, 80/63, 4/3, 7/5, 10/7, 3/2, 63/40, 8/5, 5/3, 12/7, 7/4, 224/125, 9/5, 28/15, 15/8, 40/21]
[0, 22, 31, 34, 53, 56, 65, 75, 87, 109, 118, 140, 162, 174, 196, 218, 227, 249, 261, 271, 280, 283, 302, 305, 314]
[22, 9, 3, 19, 3, 9, 10, 12, 22, 9, 22, 22, 12, 22, 22, 9, 22, 12, 10, 9, 3, 19, 3, 9, 22]

These scales vary in their degree of regularity, but Carl might find the 8 or 13 note scales acceptable, for example.

> [1, 5/3]
> [0, 249]
> [249, 87]
>
> [1, 6/5, 5/3]
> [0, 87, 249]
> [87, 162, 87]
>
> [1, 6/5, 3/2, 5/3]
> [0, 87, 196, 249]
> [87, 109, 53, 87]
>
> [1, 6/5, 4/3, 3/2, 5/3]
> [0, 87, 140, 196, 249]
> [87, 53, 56, 53, 87]
>
> [1, 6/5, 4/3, 3/2, 8/5, 5/3]
> [0, 87, 140, 196, 227, 249]
> [87, 53, 56, 31, 22, 87]
>
> [1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3]
> [0, 87, 109, 140, 196, 227, 249]
> [87, 22, 31, 56, 31, 22, 87]

Sorry to be dense, but where are the planar temperaments (the
JI versions, or the 336-et versions, or...)? How can there be
a 5-limit planar temp.?

> [1, 6/5, 5/4, 4/3, 10/7, 3/2, 8/5, 5/3]
> [0, 87, 109, 140, 174, 196, 227, 249]
> [87, 22, 31, 34, 22, 31, 22, 87]
//
> These scales vary in their degree of regularity, but Carl
> might find the 8 or 13 note scales acceptable, for example.

Thirteen is more notes than I'm looking for at the moment.
The 8-note scale seems reasonably well-approximated in 12-et...
And I think I prefer the 7-note version...

Anyway, I have no idea what you're up to here, other than
finding a way to extend our notion of complexity to non-linear
temperaments... of course I see why our current definition
cannot work, and I roughly get the idea behind geometric
complexity, but I'm afraid you've lost me with the relative
version.

-Carl

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

> > [1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3]
> > [0, 87, 109, 140, 196, 227, 249]
> > [87, 22, 31, 56, 31, 22, 87]
>
> Sorry to be dense, but where are the planar temperaments (the
> JI versions, or the 336-et versions, or...)? How can there be
> a 5-limit planar temp.?

The 5-limit intervals are merely representing intervals of the temperament; the tuning in 336-et is given on the next line.

> Thirteen is more notes than I'm looking for at the moment.

Twelve, then. :)

> The 8-note scale seems reasonably well-approximated in 12-et...

Reasonable for whom and what purpose? I wouldnt' use it.

> Anyway, I have no idea what you're up to here, other than
> finding a way to extend our notion of complexity to non-linear
> temperaments...

The idea is that this gives the classes of the planar temperament in terms of a lattice, where distance is measured in a way connected to the approximation(s) of the temperament, which it already is or can be in a linear temperament or JI. This fills in the gap between them.

> of course I see why our current definition
> cannot work, and I roughly get the idea behind geometric
> complexity, but I'm afraid you've lost me with the relative
> version.

I defined a notion of distance on ocatave equivalence classes of
p-limit JI. I then used this to define a notion of distance on wedge products of classes, getting geometric complexity. Measuring the complexity after wedging in the interval gives us *relative* complexity for that interval with respect to the temperament in question. In the case of linear temperaments, this reduces to number of generator steps, and in the case of JI, to my original weighted Euclidean distance measure.

>>>[1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3]
>>>[0, 87, 109, 140, 196, 227, 249]
>>>[87, 22, 31, 56, 31, 22, 87]
>>
>>Sorry to be dense, but where are the planar temperaments (the
>>JI versions, or the 336-et versions, or...)? How can there be
>>a 5-limit planar temp.?
>
>The 5-limit intervals are merely representing intervals of the
>temperament; the tuning in 336-et is given on the next line.

Ok.

>>Thirteen is more notes than I'm looking for at the moment.
>
>Twelve, then. :)

Ten's my limit.

>>Anyway, I have no idea what you're up to here, other than
>>finding a way to extend our notion of complexity to non-linear
>>temperaments...
>
>The idea is that this gives the classes of the planar temperament
>in terms of a lattice, where distance is measured in a way
>connected to the approximation(s) of the temperament, which it
>already is or can be in a linear temperament or JI. This fills
>in the gap between them.

Graham complexity measures the complexity of a temperament based
on the minimum number of *notes* of the temperament you'd need
to get *all* of a target set of intervals. This aspect is what
attracts me to Graham complexity, and it seems it *is* extensible
to planar and higher temperaments. . .

In the case linear temperaments, the it is equivalent to measuring
the taxicab distance of the target intervals on a lattice 'defined
by the temperament' (the chain of generators). Is it accurate to
say that you intend to extend this second aspect of Graham
complexity to planar temperaments, and not the first? If so, then
I only need to understand how you build the lattice, and how you
measure distance on it...

>I defined a notion of distance on ocatave equivalence classes of
>p-limit JI. I then used this to define a notion of distance on
>wedge products of classes, getting geometric complexity. Measuring
>the complexity after wedging in the interval gives us *relative*
>complexity for that interval with respect to the temperament in
>question. In the case of linear temperaments, this reduces to
>number of generator steps, and in the case of JI, to my original
>weighted Euclidean distance measure.

...sounds like you measure distance with a fancy Euclidean metric,
which I'm happy to accept as such. However, I'd like to have a
picture of how the lattice you're measuring the distance on looks.
Perhaps the lattice you just posted will help, but it doesn't
look 'special' to me (or to monz, apparently) yet...

-Carl

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

> In the case linear temperaments, the it is equivalent to measuring
> the taxicab distance of the target intervals on a lattice 'defined
> by the temperament' (the chain of generators). Is it accurate to
> say that you intend to extend this second aspect of Graham
> complexity to planar temperaments, and not the first? If so, then
> I only need to understand how you build the lattice, and how you
> measure distance on it...

Perhaps I should choose a better name than "relative complexity"--what about "relative distance" using the "relative metric"? In any case, for linear temperaments, you can consider it taxicab if you like, but it is all occurring along a single line and I am considering that to be Euclidean distance.

> ...sounds like you measure distance with a fancy Euclidean metric,
> which I'm happy to accept as such. However, I'd like to have a
> picture of how the lattice you're measuring the distance on looks.

The lattice is *defined* by the distance. If you look at the plots of
what I have up so far (126/125, 225/224, 1728/1715) it should be obvious how closely the lattice is connected with the planar temperament, so I'm not quite sure what the problem is.

> Perhaps the lattice you just posted will help, but it doesn't
> look 'special' to me (or to monz, apparently) yet...

Why not? Have you tried relating the lattice to the temperaments which correspond to the lattice?

>>In the case linear temperaments, the it is equivalent to measuring
>>the taxicab distance of the target intervals on a lattice 'defined
>>by the temperament' (the chain of generators). Is it accurate to
>>say that you intend to extend this second aspect of Graham
>>complexity to planar temperaments, and not the first? If so, then
>>I only need to understand how you build the lattice, and how you
>>measure distance on it...
>
>Perhaps I should choose a better name than "relative complexity"--
>what about "relative distance" using the "relative metric"? In any
>case, for linear temperaments, you can consider it taxicab if you
>like, but it is all occurring along a single line and I am
>considering that to be Euclidean distance.

As I say, I'm fine by that. But you didn't answer if your method
keeps the 'minimum-notes' aspect of Graham complexity.

>>...sounds like you measure distance with a fancy Euclidean metric,
>>which I'm happy to accept as such. However, I'd like to have a
>>picture of how the lattice you're measuring the distance on looks.
>
>The lattice is *defined* by the distance. If you look at the plots
>of what I have up so far (126/125, 225/224, 1728/1715) it should
>be obvious how closely the lattice is connected with the planar
>temperament, so I'm not quite sure what the problem is.

Unfortunately, it isn't obvious to me.

>>Perhaps the lattice you just posted will help, but it doesn't
>>look 'special' to me (or to monz, apparently) yet...
>
>Why not? Have you tried relating the lattice to the temperaments
>which correspond to the lattice?

What are the temperaments? I'm looking at "126/125", but I'm not
sure what that means. It means you've tempered out one out of
three unison vectors in a 7-limit block? What are the other two?

>You geometers might want to look at it geometrically--take the 3D
>lattice of 7-limit JI classes, rotate it so that things separated
>by the comma in question are stacked--in other words, the comma is
>perpendicular to your "eye"--and then project down onto a plane,
>making the comma vanish. Since we now have a number of 7-limit
>classes coinciding, we pick the one of smallest tenney height
>within the ocatave to label the point.

This gets you a block, but not a temperament.

-Carl

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

> As I say, I'm fine by that. But you didn't answer if your method
> keeps the 'minimum-notes' aspect of Graham complexity.

I don't know what you are asking for--it's two dimensional.

> >You geometers might want to look at it geometrically--take the 3D
> >lattice of 7-limit JI classes, rotate it so that things separated
> >by the comma in question are stacked--in other words, the comma is
> >perpendicular to your "eye"--and then project down onto a plane,
> >making the comma vanish. Since we now have a number of 7-limit
> >classes coinciding, we pick the one of smallest tenney height
> >within the ocatave to label the point.
>
> This gets you a block, but not a temperament.

It does no such thing. We are just projecting a 3D lattice down to a 2D lattice; no blocks appear.

>>As I say, I'm fine by that. But you didn't answer if your method
>>keeps the 'minimum-notes' aspect of Graham complexity.
>
>I don't know what you are asking for--it's two dimensional.

Graham complexity tells me the minimum number of notes of the
temperament I need to play all the identities in question.
Does relative complexity?

>>>You geometers might want to look at it geometrically--take the 3D
>>>lattice of 7-limit JI classes, rotate it so that things separated
>>>by the comma in question are stacked--in other words, the comma is
>>>perpendicular to your "eye"--and then project down onto a plane,
>>>making the comma vanish. Since we now have a number of 7-limit
>>>classes coinciding, we pick the one of smallest tenney height
>>>within the ocatave to label the point.
>>
>>This gets you a block, but not a temperament.
>
>It does no such thing. We are just projecting a 3D lattice down
>to a 2D lattice; no blocks appear.

By hiding commatic versions behind the pitches visible on the
plane, you're applying unison vector(s), but not tempering.
Perhaps you don't have enough uvs to close a block, but you're
certainly on you're way. No?

-Carl

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

> Graham complexity tells me the minimum number of notes of the
> temperament I need to play all the identities in question.
> Does relative complexity?

What you want would be geometric complexity, and it doesn't. However a measure of the convex hull of the identities would certainly be possible.

>> By hiding commatic versions behind the pitches visible on the
> plane, you're applying unison vector(s), but not tempering.

I'm showing relationships between tempered classes; it is analogous to the 5-limit JI lattice.

> Perhaps you don't have enough uvs to close a block, but you're
> certainly on you're way. No?

If you think 5-limit JI is on the way to being a block.

>>Graham complexity tells me the minimum number of notes of the
>>temperament I need to play all the identities in question.
>>Does relative complexity?
>
>What you want would be geometric complexity,

Why do I want that?

>and it doesn't.

You mean "and relative complexity doesn't"?

>However a measure of the convex hull of the identities would
>certainly be possible.

Sounds like a good idea.

>>By hiding commatic versions behind the pitches visible on the
>>plane, you're applying unison vector(s), but not tempering.
>
>I'm showing relationships between tempered classes; it is
>analogous to the 5-limit JI lattice.
>
>>Perhaps you don't have enough uvs to close a block, but you're
>>certainly on you're way. No?
>
>If you think 5-limit JI is on the way to being a block.

Maybe Paul can shed some light on this, when he's feeling
better.

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
> >>Graham complexity tells me the minimum number of notes of the
> >>temperament I need to play all the identities in question.
> >>Does relative complexity?
> >
> >What you want would be geometric complexity,
>
> Why do I want that?
>
> >and it doesn't.
>
> You mean "and relative complexity doesn't"?
>
> >However a measure of the convex hull of the identities would
> >certainly be possible.
>
> Sounds like a good idea.
>
> >>By hiding commatic versions behind the pitches visible on the
> >>plane, you're applying unison vector(s), but not tempering.
> >
> >I'm showing relationships between tempered classes; it is
> >analogous to the 5-limit JI lattice.
> >
> >>Perhaps you don't have enough uvs to close a block, but you're
> >>certainly on you're way. No?
> >
> >If you think 5-limit JI is on the way to being a block.
>
> Maybe Paul can shed some light on this, when he's feeling
> better.
>
> -Carl

i can't.

now answer the question, carl!

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

> Graham complexity tells me the minimum number of notes of the
> temperament I need to play all the identities in question.
> Does relative complexity?

Graham complexity along both generators, and thier product, might be what we need. We could attempt to minimize the product.

>>Graham complexity tells me the minimum number of notes of the
>>temperament I need to play all the identities in question.
>>Does relative complexity?
>
>Graham complexity along both generators, and thier product,
>might be what we need. We could attempt to minimize the product.

Sounds right. I wonder how this follows relative and/or
geometric complexity...

I should point out that I *don't know* the difference between
this notes measure, and geometric complexity measures -- I'm
asking. What is it exactly that we're trying to measure about
temperaments, from a music-theoretical POV?

-Carl

>>>>Perhaps you don't have enough uvs to close a block, but you're
>>>>certainly on you're way. No?
>>>
>>>If you think 5-limit JI is on the way to being a block.
>>
>>Maybe Paul can shed some light on this, when he's feeling
>>better.
>>
>>-Carl
>
>i can't.
>
>now answer the question, carl!

What question?

-Carl

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

> Sounds right. I wonder how this follows relative and/or
> geometric complexity...

It's not clear to me how to really get a Graham generalization yet. A simple function computing numbers of complete p-limit chords in a parallelogram would be good.

I looked at 11-limit chords leaving off 9 for the "Wonder" temperament I just uploaded a graph for;
any of the pairs of generators <5, 15/14>, <5, 7/6>, <5, 4/3>
<15/14, 4/3>, <15/14, 12/7> will work, and give the minimal value of 9 for the product. So, for instance, the block based on generators of
<33/32, 3087/3025> works, giving a Marvel-tempered block of 30 notes which should have (3-2)*(10-2) = 8 of these major chords. This doesn't seem very satisfactory; better would be something which takes a Fokker block and gives a quick answer, which I'll think about.

Gene Ward Smith wrote something which Mozilla has decided to delete along with the start of my message. So I'll have to write that bit again and then look for another program that can authenticate with an SMTP server.

Okay, it was something about the general rule for parallelograms. I decided to simplify that by only looking at rectangles. The complexity of a planar temperament is the pair (i, j) where i and j are the numbers of each generator needed for a complete chord. You can then describe a rectangular scale as (I, J) where the number of notes in the scale is I*J.

The general rule for the number of complete otonal chords in the scale is (I-i)*(J-j).

Example: 5-limit JI (planar scale not temperament) with generators of a fifth and major third. Take the rectangle

A E B F#
F C G D

The complexity is (1, 1). The size of the scale is (2, 4). That gives us (2-1)*(4-1)=3 major chords. They are F, C and G.

Example: 4-limit JI with generators of a chromatic semitone and minor third. Take the rectangle

B# D# F# A
B D F Ab

The complexity is (1, 2) because you need a chromatic semitone and two minor thirds to make up a fifth. The size of the scale is (2, 4) again. So there are (2-1)*(4-2)=2 major chords. They are B and D.

The problem here is that the same tuning gives different complexities for different choices of generators. That's presumably what Gene was thinking of when he suggested parallelograms rather than rectangles. I can write the first scale using the second mapping as

F#
B D
E G
A C
F

Which is a parallelogram. So different choices of generator will always give one? You can say that the real complexity is the smallest one you get by choosing different pairs of generators. That pushes the problem a step back -- how do you find the simplest generators? Well, you probably want to do that anyway if you're working with planar temperaments.

To have one number to compare planar temperaments, multiplying the two components of the complexity is the simplest option, as Gene suggested somwhere else. But we can be a bit cleverer.

Take a temperament with complexity (i, j) with i<j. The simplest scales involving complete chords are with (i+1, j+n) notes, where n>=1. The number of (otonal) complete chords is (i+1-i)*(j+n-j)=n. The number of notes is (i+1)*(j+n) = ij + j + ni + n = j(i+1) + n(i+1). As n gets large, you can ignore j and make i the complexity. But if you're using that many notes you can probably get an equivalent approximation with fewer notes by using a linear temperament.

And that brings us to the really difficult problem. It's already there in comparing equal with linear temperaments. The logical way of extending my complexity measure to equal temperaments gives them all a complexity of zero. So there's no reason why you should ever prefer a linear temperament over an equal temperament.

Hmm.

Okay, say you're using meantone with 5-limit harmony. The minimax optimum is quarter comma. For most of us, that's equivalent to 31-equal. So if you're using more than 31 notes, you can simplify the scale with no real disadvantage by making it equally tempered.

So, if you're that conveniently simple composer who's only interested in the approximation to JI and the number of notes in the scale, the only reason for using 5-limit meantone is if you can get by with fewer than 31 notes. Then it's either simpler or appreciably more accurate than any equal temperament. When you hit 34 notes, you can get better harmony with an equal temperament, although it behaves differently, so I'm not sure if that's relevant. It is sort of because the most efficient equal temperaments up to this point are meantones.

It looks to me that the best way of comparing the complexity of equal and linear temperaments is stating the number of notes in the smallest equal temperament that does an equivalent job. So 31 is a special number for 5-limit meantone because the error in 31-equal is about the same as the lowest possible error in meantone. You could always objectify this -- say they're equivalent when the optimum error is at least some proportion of the equal tempered error. You can also say that, for a given target error, a certain equal temperament will do the job and there's no point in looking at linear temperaments that require more notes.

And in practical terms, if you have some idea of how many notes you can deal with and how accurate you want the tuning, you can then compare the number of chords that linear or equal temperaments will give you.

Although I can't see a way of listing equal and linear temperaments together in a general and satisfactory way, you could place them on a graph of accuracy against number of notes. At any point, you show the temperament (of whatever kind) that gives the most complete chords. In fact, if anybody knows how to draw such graphs, I'd quite like to see a few.

Adding planar temperaments doesn't really change much. The biggest problem is deciding what the "typical" scale with a given number of notes is. But you can always go for the simplest one, as I show above. So you can then add planar temperaments to that graph of size against accuracy. They'll appear for high accuracies with a large but not absurdly large number of notes. You can also say the number of notes beyond which a planar temperament is equivalent to a linear temperament that's consistent with it.

Then you can look at spatial temperaments, and good luck to you ;)

Graham

Graham wrote:

> Okay, say you're using meantone with 5-limit harmony. The minimax
> optimum is quarter comma. For most of us, that's equivalent to
> 31-equal. So if you're using more than 31 notes, you can simplify
the
> scale with no real disadvantage by making it equally tempered.
>
> So, if you're that conveniently simple composer who's only
interested in
> the approximation to JI and the number of notes in the scale, the
only
> reason for using 5-limit meantone is if you can get by with fewer
than
> 31 notes. Then it's either simpler or appreciably more accurate
than
> any equal temperament. When you hit 34 notes, you can get better
> harmony with an equal temperament, although it behaves differently,
so
> I'm not sure if that's relevant. It is sort of because the most
> efficient equal temperaments up to this point are meantones.
>
> It looks to me that the best way of comparing the complexity of
equal
> and linear temperaments is stating the number of notes in the
smallest
> equal temperament that does an equivalent job. So 31 is a special
> number for 5-limit meantone because the error in 31-equal is about
the
> same as the lowest possible error in meantone. You could always
> objectify this -- say they're equivalent when the optimum error is
at
> least some proportion of the equal tempered error. You can also
say
> that, for a given target error, a certain equal temperament will do
the
> job and there's no point in looking at linear temperaments that
require
> more notes.

Hi Graham!

I'd say you overestimate the utility of 31-equal as a 5-limit
meantone system. If you measure the 5-limit errors of meantone equal
temperaments as a proportion of the scale division it's actually 19-
equal that comes out as the best. But 31-equal is the winner in the 7-
limit.

Kalle

Kalle Aho wrote:

>I'd say you overestimate the utility of 31-equal as a 5-limit >meantone system. If you measure the 5-limit errors of meantone equal >temperaments as a proportion of the scale division it's actually 19-
>equal that comes out as the best. But 31-equal is the winner in the 7-
>limit.
>
I was talking about the absolute error, not this "utility" measure. 31-equal gets 5-limit harmony to within 5.96 cents. Quarter comma meantone improves on that slightly with 5.38 cents. That means the best possible worst error with meantone is only 90% of the worst error with 31-equal. 19-equal has a worst error of 7.37 cents, and you can lose 27% of that by tuning to quarter comma meantone.

Somewhere between the two, you can say you don't care and 31-equal and quarter comma meantone are really the same. If the last 10% (or 0.6 cents) is still important, you can set a narrower margin. But at some point it'll be reached.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:

> So you can then add planar temperaments to that graph of size against
> accuracy. They'll appear for high accuracies with a large but not
> absurdly large number of notes. You can also say the number of notes
> beyond which a planar temperament is equivalent to a linear temperament
> that's consistent with it.

Why a large number of notes? The Pauline tempered Duodene only has 12, and I was wrong about then all being covered by tetrads. You can cut the corners off, and get the following scale:

! pship.scl
!
Pauline (225/224) tempered 10 note scale
10
!
268.145800
384.152140
500.158479
583.835182
699.841521
768.304279
884.310618
967.987322
1083.993661
2/1

This has 4 major and 4 minor triads, 2 major and 2 minor tetrads,
2 supermajor and 2 subminor triads, one each of what I call supermajor and subminor tetrads (1-9/7-3/2-9/5 and 1-7/6-3/2-5/3), and
a 1-7/6-7/5-5/3 dimininished 7th chord. I wouldn't call this a lot of notes, but the tempering is very much in evidence.

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
> >>>>Perhaps you don't have enough uvs to close a block, but you're
> >>>>certainly on you're way. No?
> >>>
> >>>If you think 5-limit JI is on the way to being a block.
> >>
> >>Maybe Paul can shed some light on this, when he's feeling
> >>better.
> >>
> >>-Carl
> >
> >i can't.
> >
> >now answer the question, carl!
>
> What question?

do you think 5-limit JI is on the way to being a block?

> > >
> > >now answer the question, carl!
> >
> > What question?
>
> do you think 5-limit JI is on the way to being a block?

No.

The lattices Gene are posting contain pitches
outside of 5-limit JI.

-C.

Me:
>> So you can then add planar temperaments to that graph of size against >>accuracy. They'll appear for high accuracies with a large but not >>absurdly large number of notes. You can also say the number of notes >>beyond which a planar temperament is equivalent to a linear temperament >>that's consistent with it.

Gene:
> Why a large number of notes? The Pauline tempered Duodene only has 12, and I was wrong about then all being covered by tetrads. You can cut the corners off, and get the following scale:

Oh, I was assuming you'd need a lot of notes to get an accurate temperament that's still simpler than JI.

> ! pship.scl
> !
> Pauline (225/224) tempered 10 note scale

Why is this still being called the Pauline temperament? I showed that it isn't the temperament Pauline gave.

> This has 4 major and 4 minor triads, 2 major and 2 minor tetrads,
> 2 supermajor and 2 subminor triads, one each of what I call supermajor and subminor tetrads (1-9/7-3/2-9/5 and 1-7/6-3/2-5/3), and
> a 1-7/6-7/5-5/3 dimininished 7th chord. I wouldn't call this a lot of notes, but the tempering is very much in evidence.

It isn't that many notes, but it isn't that accurate either. It's simple as planar temperaments go. I make it equivalent to h12&h60 beyond 33 notes. Or 34 notes if you lop the corners off. 36 in practice as h12&h60 goes up in steps of 12 notes.

Graham

I wrote:
> It isn't that many notes, but it isn't that accurate either. It's > simple as planar temperaments go. I make it equivalent to h12&h60 > beyond 33 notes. Or 34 notes if you lop the corners off. 36 in > practice as h12&h60 goes up in steps of 12 notes.

Well, that's wrong because my simplification of planar temperaments is wrong. It looks like you get more chords per note if you have equal numbers of steps in each generator direction, instead of holding one at its smallest value and expanding the scale in only one direction.

Here's a 10 note scale with 2 utonal and 2 otonal 7-limit tetrads when you temper out 225:224

D# A# E#
E B F# C#
C G D

Expanding in one direction, you get a 13 note scale with 3 of each tetrad

D# A# E# B#
E B F# C# G#
C G D A

And the next in the pattern is a 16 note scale with 4 of each tetrad

G# D# A# E# B#
A E B F# C# G#
F C G D A

But there's also this 14 note scale that has 4 of each tetrad

D# A# E#
E B F# C#
C G D A
Ab Eb Bb

So it can be done with 14 notes when the theory predicts 16 (and the even cruder theory 18)

18 notes can give 6 of each tetrad

G# D# A# E#
A E B F# C#
F C G D A
Db Ab Eb Bb

This is the point where we get extra tetrads with meantone. Then there's a 23 note scale with 9 of each tetrad

B# Fx Cx Gx
C# G# D# A# E#
A E B F# C#
F C G D A
Db Ab Eb Bb

By my original formula, it would take 30 notes to get this many tetrads.

So, the complexity of a planar temperament is (i, j) and the scale is (I, J). The number of notes is then I*J. The number of complete chords of each type is (I-i)*(J-j). This still slightly overestimates the number of notes, because some notes in the rectangle won't be used in any chords, but that's only a minor correction.

I'm going to set I-i = J-j = n. Then the number of chords is n**2 (n squared) and the number of notes is ij + n(i+j) + n**2.

I've worked out that the number of notes where the number of chords in an equal and planar temperament are the same is

((c-ij)/(i+j))**2 + c

where c is the complexity of the linear temperament and (i, j) is the complexity of the planar temperament. Using that formula, the 225:224 planar temperament doesn't become equivalent to the h12&h60 linear temperament until they both have 49 notes. So the planar temperament holds it's own for larger scales than I said before.

Ennealimmal (which is much more accurate) becomes simpler around 60 notes.

It also takes more notes before the planar temperament becomes simpler than JI. This scale has 12 notes and 3 of each tetrad, a feat that requires 13 notes for a 225:224 tempered scale. You'll really need to view source and turn off word wrapping for this to come out right.

C#
/ \
B#------Fx------Cx
\ / / \ \ /
\ A-/---\-E /
\ / \ /
/ D#------A#\
/ \ \ / / \
F-----\-C-/-----G
\ /
F#

You can also get 4 of each tetrad from 14 notes by filling in the two 5-limit triads in the middle (is that a stellated hexany?) So the planar temperament rules where you have more than 14 up to around 49 or 60 notes -- a large, but not absurdly large number of notes to me.

I may still be wrong on the details, but I think I'm getting closer

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:

> > ! pship.scl
> > !
> > Pauline (225/224) tempered 10 note scale
>
> Why is this still being called the Pauline temperament? I showed that
> it isn't the temperament Pauline gave.

Sorry, I missed that. This is a very important temperament, and needs a name.

> > This has 4 major and 4 minor triads, 2 major and 2 minor tetrads,
> > 2 supermajor and 2 subminor triads, one each of what I call supermajor and subminor tetrads (1-9/7-3/2-9/5 and 1-7/6-3/2-5/3), and
> > a 1-7/6-7/5-5/3 dimininished 7th chord. I wouldn't call this a lot of notes, but the tempering is very much in evidence.

> It isn't that many notes, but it isn't that accurate either.

The 225/224 planar is pretty accurate; you can use 72-et or for more accuracy even 228-et. Why do you say it isn't accurate?

72-et: [-1.955001, -2.980380, -2.159239]

228-et: [-1.955001, -2.103187, -.404853]

It's
> simple as planar temperaments go. I make it equivalent to h12&h60
> beyond 33 notes.

I don't have a name listed for this temperament yet, but Duodecimal ought to do. It is equivalent in the sense that if you use 72-et or
228-et as above, they are Duodecimal compatible; you can describe it as 225/224 plus the Pythagorean comma, though its actual TM reduced basis is <225/224, 250047/250000>. If you use the mapping
[[12,19,28,34],[0,0,-1,-2]] for it, you get generators of the Pythagorean minor second (256/243) and the 81/80~126/25 comma; this not neccessarily the best setup for 225/224 alone, it seems to me.

You can also add commas to get, for instance, Catakleismic or Miracle, so this isn't uniquely attached to 225/224, though the connection is close. If we used

[331, 524, 768, 929] [-2.257116, -2.023683, -.850075]

or

[631, 999, 1464, 1771] [-2.113480, -2.161574, -.838584]

as mappings, on the grounds they are near the rms optimum, of course the Pythagorean comma would not be a factor.

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

> I don't have a name listed for this temperament yet, but Duodecimal ought to do. It is equivalent in the sense that if you use 72-et or
> 228-et as above, they are Duodecimal compatible; you can describe it as 225/224 plus the Pythagorean comma, though its actual TM reduced basis is <225/224, 250047/250000>. If you use the mapping
> [[12,19,28,34],[0,0,-1,-2]] for it, you get generators of the Pythagorean minor second (256/243) and the 81/80~126/25 comma; this not neccessarily the best setup for 225/224 alone, it seems to me.

A closely related situation happens with the 126/125 temperament, which is covered by 108 and 120, and done to perfection by
[336, 532, 781, 943]. We can tune it using the linear temperament
of 126/125 and the Pythagorean comma, for which a mapping is
[[12,19,28,34], [0,0,-1,-3]], with generators a Pythagorean minor second and a 81/80~225/224 comma.

Gene Ward Smith wrote:

> The 225/224 planar is pretty accurate; you can use 72-et or for more accuracy even 228-et. Why do you say it isn't accurate?

I was comparing it with other 7-limit planar temperaments, not equal temperaments. It's only a marginal improvement on Miracle, whereas some planar temperaments can do a lot better (such as those Schismic-with-explicit-schisma notations).

But this isn't important. You can call it "pretty accurate" if you like and say that planar temperaments occupy the "pretty accurate" zone.

> I don't have a name listed for this temperament yet, but Duodecimal ought to do.

Why does it need a name? I didn't think anybody was writing in it. I've a feeling it came up on Usenet earlier in the year.

> It is equivalent in the sense that if you use 72-et or
> 228-et as above, they are Duodecimal compatible; you can describe it as 225/224 plus the Pythagorean comma, though its actual TM reduced basis is <225/224, 250047/250000>. If you use the mapping
> [[12,19,28,34],[0,0,-1,-2]] for it, you get generators of the Pythagorean minor second (256/243) and the 81/80~126/25 comma; this not neccessarily the best setup for 225/224 alone, it seems to me.

It's equivalent in the sense that it tempers out 225:224, and gets within 98% of the accuracy of 225:224.

> You can also add commas to get, for instance, Catakleismic or Miracle, so this isn't uniquely attached to 225/224, though the connection is close.

225:224 planar's worst error is 79.4% of Miracle's. You can call that equivalent if you like. In which case the equivalence occurs beyond 18 notes. Catakleismic is more complex and less accurate, so what's the point? h31&h63 gets to within nearly 85%, and has a complexity of 23. But as h12&h60's so much more accurate, and only slightly more complex (24) I went with that.

Graham

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
> > > >
> > > >now answer the question, carl!
> > >
> > > What question?
> >
> > do you think 5-limit JI is on the way to being a block?
>
> No.
>
> The lattices Gene are posting contain pitches
> outside of 5-limit JI.
>
> -C.

but they're planar arrangements of pitches, with like vectors
corresponding to like intervals. so there's no essential
difference . . .

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

> but they're planar arrangements of pitches, with like vectors
> corresponding to like intervals. so there's no essential
> difference . . .

You could even make zoomers and dualzoomers for 'em.