> From: paulerlich <paul@stretch-music.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Sunday, December 23, 2001 3:47 PM

> Subject: [tuning-math] Re: a different example (was: coordinates from

unison-vectors)

>

>

> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:

>

> > Therefore, my conclusion is that 7/25-comma meantone, 50-EDO,

> > and the (6 -14),(-4 1) periodicity-block are all intimately

> > related, and essentially identical.

>

> It would seem that you're totally ignoring the issue of how the

> intervals are _tuned_, whether they're wolves or not, and focusing on

> a rather superficial kind of similarity that results from sticking to

> a 2-d JI lattice, and not using a cylinder to view 7/25-comma

> meantone, and a torus to view 50-tET. In the latter cases, your

> association of fixed JI pitches to each note in the tuning system is

> at work, and Dave Keenan and I have been fighting to explain for

> years now, it's the _intervals_ that matter in these lattices, not

> the _pitches_. Think of the simple example of "where do you put the

> 1/1". C? D? The key of the piece? The key of the section? The current

> chord? What if you have a chord like C-A-G-E-D?

>

> As to the meantones in this range, I don't think your lattices

> illuminate their similarity to any greater degree than is already

> obvious from, say, the cents values of the fifths. The numeric data

> on ratios you're pointing to in these studies has, I argue, no

> musical or acoustical relevance to the perception of these meantones.

Paul, I recognize that the *ambiguity* of the rational implications of

temperaments has become an integral part of Eurocentric musical theory

and practice. It's hard for me to visualize what happens on a cylinder

or torus since I'm dealing with planar graphs.

But if I'm sufficiently interested in pursuing this line of reasoning,

then why not just go ahead and make the calculations, graphs, spreadsheets,

etc., and then eventually if others who are interested in my approach

do some experiments to test their validity, perhaps some may be found. :)

-monz

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> It's hard for me to visualize what happens on a cylinder

> or torus since I'm dealing with planar graphs.

Well then, the Hall article I'm sending you tomorrow may help.

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> It's hard for me to visualize what happens on a

cylinder

> or torus since I'm dealing with planar graphs.

Here is 31-tET mapped onto the surface of a toroid as a 5-limit

lattice. If you print out the lattice below (in a monospaced font),

cut out the rectangle (cutting a half character width or height inside

the lines), loop and tape it first side to side and then top to

bottom, and you'll have it. Unfortunately you have to flatten it after

the first looping to get it to loop in the other dimension, unless you

printed it on rubber.

[If you're viewing this from Yahoo's web interface, you will need to

choose Message Index then Expand Messages to see it correctly

formatted.]

-------------------------------

| Gx |

| Cx |

| Fx |

| B# |

| E# |

| A# |

| D# |

| G# |

| C# |

| F# |

| B |

| E |

| A |

|D |

| G |

| C |

| F |

|b B|

| Eb |

| Ab |

| Db |

| Gb|

| Cb |

| Fb |

| BB |

| EB |

| AB |

| DB |

| GB |

| Ax |

| Ex |

-------------------------------

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> > It's hard for me to visualize what happens on a

> cylinder

> > or torus since I'm dealing with planar graphs.

>

> Here is 31-tET mapped onto the surface of a toroid as a 5-limit

> lattice. If you print out the lattice below (in a monospaced font),

> cut out the rectangle (cutting a half character width or height

inside

> the lines), loop and tape it first side to side and then top to

> bottom, and you'll have it. Unfortunately you have to flatten it

after

> the first looping to get it to loop in the other dimension, unless

you

> printed it on rubber.

I'd suggest just taping the right edge to the left edge, as the

resulting cylinder represents the 31 central tones of _any_ meantone,

not just 31-tET.

> From: dkeenanuqnetau <d.keenan@uq.net.au>

> To: <tuning-math@yahoogroups.com>

> Sent: Sunday, December 23, 2001 4:47 PM

> Subject: [tuning-math] Re: a different example

>

> Here is 31-tET mapped onto the surface of a toroid as a 5-limit

> lattice. If you print out the lattice below (in a monospaced font),

> cut out the rectangle (cutting a half character width or height inside

> the lines), loop and tape it first side to side and then top to

> bottom, and you'll have it.

Thanks, Dave! Actually, I seem to recall that you posted

something like this once before.

-monz

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Hi Gene and Paul,

I'm finally getting around to replying to the "different example"

periodicity-blocks we created a couple of days ago. Gene, you

must have been sleepy, because there are three errors in your

post; I'll correct them in the quote.

> From: monz <joemonz@yahoo.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Sunday, December 23, 2001 3:31 PM

> Subject: [tuning-math] a different example (was: coordinates from

unison-vectors)

>

>

> ... So anyway, I put in the [(3,5) unison-vector] matrix:

>

> ( 6 -14)

> (-4 1)

>

> and could see that the resulting periodicity-block had a strong

> correlation (in the sense of my meantone-JI implied lattices)

> with the neighborhood of 2/7- to 3/11-comma meantone.

>

> The determinant is 50, so this agrees with my observation.

> From: genewardsmith <genewardsmith@juno.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Sunday, December 23, 2001 8:04 PM

> Subject: [tuning-math] Re: a different example

>

>

> ... First I put the 2 back into the above commas, and get

>

> q1 = 2^23 3^6 5^-15 and q2 = 80/81.

Er... there's a typo in that last exponent: it should be

q1 = 2^23 3^6 5^-14.

> ... I have one et, h50, such that h50(q1) = h50(q2) = 0.

> Now I search for something where h(q1)=0 and h(q2)=1,

> obtaining h19, h69, -h31 and -h81. The simplest of these

> is h19, and I choose it.

>

> Next, I look for something such that h(q1)=1 and h(q2)=0,

> and I get h16, -h34, -h84; I choose h16.

>

> Now I form a 3x3 matrix from these, and invert it:

>

> [ 50 19 16]

> [ 79 30 25]^(-1) =

> [116 44 37]

>

> [-10 -1 5]

> [ 23 6 -14]

> [ 4 -4 1]

>

> The rows of the inverted matrix correspond to the commas

> q0 = 2^-10 3^-1 5 = 3125/3072 (small diesis),

Typo: that should be q0 = 2^-10 3^-1 5^5

> q1 = 2^23 3^6 5^14, and q2=80/80.

Typo: according to the signs on the integers in the bottom

row of the last matrix, q2 = 80/81. If that's not right,

then it should be q2 = [-4 4 -1] = 81/80.

>

> Now I calculate the scale; the nth step is

>

> scale[n] = q0^n * q1^round(19n/50) * q2^round(16n/50),

>

> where "round" rounds to the nearest integer. It doesn't matter

> which paticular hn I selected when I do this, or where I start

> and end; though my definition of "nearest integer" does matter.

>

> I got in this way:

>

> 1, 3125/3072, 128/125, 25/24, 82944/78125, 16/15, 625/576,

> 3456/3125, 10/9, 15625/13824, 144/125, 125/108, 18432/15625,

> 6/5, 625/512, 768/625, 5/4, 15625/12288, 32/25, 125/96,

> 20736/15625, 4/3, 3125/2304, 864/625, 25/18, 110592/78125,

> 36/25, 625/432, 4608/3125, 3/2, 15625/10368, 192/125, 25/16,

> 24576/15625, 8/5, 625/384, 1024/625, 5/3, 15625/9216, 216/125,

> 125/72, 27648/15625, 9/5, 3125/1728, 1152/625, 15/8, 78125/41472,

> 48/25, 125/64, 6144/3125

>

> How does this compare with your results?

> From: paulerlich <paul@stretch-music.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Sunday, December 23, 2001 8:33 PM

> Subject: [tuning-math] Re: a different example (was: coordinates from

unison-vectors)

>

>

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

>

> With 50 notes, some arbitrary decision has to be made -- no note can

> be exactly in the center, since 50 is an even number. But you should

> be getting the following block or its reflection through the origin:

>

>

> p5's M3's

> ---- -----

>

> 3 -7

> 4 -7

> 1 -6

> 2 -6

> 3 -6

> 4 -6

> 1 -5

> 2 -5

> 3 -5

> 0 -4

> 1 -4

> 2 -4

> 3 -4

> 0 -3

> 1 -3

> 2 -3

> 3 -3

> 0 -2

> 1 -2

> 2 -2

> -1 -1

> 0 -1

> 1 -1

> 2 -1

> -1 0

> 0 0

> 1 0

> -2 1

> -1 1

> 0 1

> 1 1

> -2 2

> -1 2

> 0 2

> -3 3

> -2 3

> -1 3

> 0 3

> -3 4

> -2 4

> -1 4

> 0 4

> -3 5

> -2 5

> -1 5

> -4 6

> -3 6

> -2 6

> -1 6

> -4 7

Sorry about the long quote, but I wanted both of these sets of

data to be together here, because after going thru all the trouble

of prime-factoring Gene's set, I see that it's identical to Paul's.

Now, how does this compare with my results? Well... the shape

and size of the periodicity-block is exactly the same; the only

difference is that my block is not quasi-centered on 1/1, as

Gene/Paul's is and as I wanted mine to be.

I'd appreciate it if you guys could fix my pseudo-code so that

I can use my own spreadsheet method and still get the correct results.

Again, my spreadsheet is at

/tuning-math/files/monz/matrix%20math.xls

Thanks.

-monz

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