## general methods for linear temperament notation

cmloegmcluin
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### general methods for linear temperament notation

[This topic has been broken out from where it originally came up, on the Porcupine discussion topic here.]

Thanks Herman. That's very helpful. I guess regular temperaments are not a concept I'm deeply familiar with! Not enough to see that right off the bat, anyway.

I think the only reason I caught myself in time, after first submitting my post without its third paragraph where I made that guess, is that somewhat recently I finally came to understand the definition of eigenmonzo. Despite taking some linear algebra in college, the "eigen-" prefix doesn't mean anything to me.

I think I had seen the table "Spectrum of Porcupine Tunings by Eigenmonzos" on the Xen Wiki page for porcupine before, but that's confusing because it gives a bunch of EDO degrees as examples of eigenmonzos, so that one might think eigenmonzo was just a synonym for "generator". That table's second column is labelled "neutral second" as if it's a different piece of information than the cents measurement of the first column. And several of the rows at the top and bottom seem to be outside the range of possibility for porcupine generators. If I'm wrong about any of this let me know. Otherwise maybe that page needs some attention! Actually now I can see that not all of them are cents measurements of the left column, like 5/4; I can't figure out what relationship that would have to 162.737¢. I have figured out that minimax is simply one of several ways of quantifying a certain "best" temperament.

I finally found on this page a statement which made sense to me: "Given a regular temperament tuning T, an eigenmonzo is a rational interval q such that T(q) = q; that is, T tunes q justly." Though even that I think could be rewritten in a lay way. I don't see the importance of the fractional monzo business I see often in the context of eigenmonzos. Anyway...

Re: those vals inside a monzo, that's not something I see everyday! I had a vague memory of seeing something like that recently. I think I found it, on the Vals and tuning space page of the Xen Wiki. Unfortunately there are just too many leaps of logic and assumptions there for me to make heads or tails of it. So I'm afraid the mapping [<1, 2, 3, 2], <0, -3, -5, 6]> doesn't convey much to me with my current level of understanding.

Re: the other way of specifying, by the commas, I've always felt confusing and surprised when I see tunings characterized by what commas are tempered out. Mostly because I wonder what's special about the list they give, because wouldn't there be infinite commas tempered out by any given temperament? How do you intuit which ones to care about? And in what sense would that information give you any feel or opinions about the temperament?

I guess this is spiraling out of control. Maybe a topic for experts in a temperament discussing how to notate it isn't the best place to ask entry-level questions about temperament in general. Please don't feel any obligation to reply in detail or anytime soon. I have a lot to learn.

cmloegmcluin
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### Re: general methods for linear temperament notation

Also, I suspect @Dave Keenan may eventually ask me to write a program which allows a regular (or is linear the preferred term?) temperament notation to be generated from some inputs, as a feature of a web-based Sagittal notation calculator...

cmloegmcluin
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### Re: general methods for linear temperament notation

I think that if we first had a pseudocode/recipe/flowchart/etc. for notating an arbitrary regular temperament, that could be really handy. Something like what we've got already for deriving symbols for new primes in the Prime Factor notation if necessary.

Learning Sagittal notation has helped me learn a number of microtonal concepts. I expect that learning how to notate regular temperaments in general would be a great way to frame them in terms of stuff I already understand.

Ah! I see that they are linear temperaments when they're rank-2 but one generator is a pure octave.

I just came across this chart from George that helped me understand minimax. This is how he came up with the secor:

(Edit: I used to say something about miracle temperament here, but I extracted it to its own topic here: viewtopic.php?p=4076#p4076)
Attachments
secor minimax diagram.jpg

herman.miller
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### Re: general methods for linear temperament notation

It looks like there's two different things in that page under the "eigenmonzo" column, and neither one of them is actually a monzo so it seems that there may be some confusion going on there. But the numbers that look like fractions (5/4, 3/2) appear to be intervals that are unchanged when the generator of a porcupine tuning is taken to be the number in the second column. If you use a 162.737 cent generator with a period of 1200.0 cents, then the tempered approximation of 5/4 is an exact 5/4, and with a 166.015 cent generator, the tuning of 4/3 is exactly just. Now 166.015 isn't a very good porcupine tuning, but it's at least somewhat consistent (at least as far as the 5-limit intervals). With a generator as small as 138.573, that's not even what I'd call consistent: the [2, 3> at 2815.719 cents is a better approximation of 5 than the [3, -5> (2907.135 cents, or 120.821 cents sharp).

I tried to understand what the https://en.xen.wiki/w/Fractional_monzo was trying to express with its description of what an eigenmonzo is, but that doesn't even look like a monzo either, so I'm confused. My understanding of a monzo is that it's a vector of exponents of prime numbers, like [-7, 3, 1> as a convenient notation for 2^-7 * 3^3 * 5^1 (or 135/128).

But you don't need any of that to define Sagittal notations for regular temperaments. All you need is the generator mapping, which shows how the primes can be approximated by the temperament.

[<1, 2, 3, 2], <0, -3, -5, 6]> porcupine
[<1, 1, 3, 3], <0, 6, -7, -2]> miracle

With that, you can take any Sagittal represented as a monzo (e.g. [-4, 4, -1> for 81/80) and do a matrix multiplication in a spreadsheet to see which interval of the temperament this symbol represents. For porcupine it's [+1, -7> or up one period and down 7 generators; for miracle it's [-3, +31>. The tricky part is deciding which Sagittals to use. Previously I've proposed one notation for miracle temperament using these accidentals:

[-4, 4, -1> (-3, +31)
[7, -4, 0, 1, -1> (+4, -41)
[6, -2, 0, -1> (+1, -10)
[-5, 1, 0, 0, 1> (-2, +21)
[-3, -1, 2> (+2, -20)
[-7, 3, 1> (-1, +11)
[-11, 7> (-4, +42)

A sequence of miracle generators starting from D up to A would look like this:
D E E F F G G A

cmloegmcluin
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### Re: general methods for linear temperament notation

herman.miller wrote: Sat Feb 20, 2021 12:06 pm It looks like there's two different things in that page under the "eigenmonzo" column, and neither one of them is actually a monzo so it seems that there may be some confusion going on there. But the numbers that look like fractions (5/4, 3/2) appear to be intervals that are unchanged when the generator of a porcupine tuning is taken to be the number in the second column.
Haha, okay, so I'm not crazy. Yeah, I'm fine with people using a quotient where a monzo is expected... they're both rational at least.
If you use a 162.737 cent generator with a period of 1200.0 cents, then the tempered approximation of 5/4 is an exact 5/4,
I see my mistake. I was looking for the familiar 386.314 value and couldn't find it. It didn't occur to me until now to also search for the inverse, i.e. subtracted from 1200 = 813.686 value, which occurs after only 5 g's.

I get now that the eigenmonzo doesn't have to be the generator, if the generator is a fraction of it.

Thanks for this! If we take the conversation re: miracle further, perhaps I should start a new topic for it.

Dave Keenan
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### Re: general methods for linear temperament notation

Douglas, It would be good if you would split out the non porcupine stuff to a new thread. I'll leave it up to you as to whether it's a thread about Miracle notation or about how to generate linear or rank-2 notations in general (or both).

cmloegmcluin
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### Re: general methods for linear temperament notation

Thanks for the nudge, Dave. I decided to split out only a topic for general linear temperament notation, because to extract the parts specific to Miracle would have required doing some surgery on Herman's writing

If you have enough dedicated stuff to say about Miracle, of course, you are free to start another new topic and just quote our work here or otherwise link back to things here.

(Update: the miracle discussion continued here so I went ahead and broke it out to its own thread here: viewtopic.php?f=6&t=531)

cmloegmcluin
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### Re: general methods for linear temperament notation

Also, for the record, there are some thoughts about general linear temperament notation at least in the first post of the mavila temperament topic here.

Dave Keenan
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### Re: general methods for linear temperament notation

cmloegcmluin wrote: Sat Feb 20, 2021 7:50 am Also, I suspect @Dave Keenan may eventually ask me to write a program which allows a regular (or is linear the preferred term?) temperament notation to be generated from some inputs, as a feature of a web-based Sagittal notation calculator...
No. Like EDO notations., it's still more art than science, weighing too many competing priorities., some of which we don't even know about because they haven't arisen in any linear temperament we've looked at so far (which isn't many).

Dave Keenan
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### Re: general methods for linear temperament notation

cmloegcmluin wrote: Sat Feb 20, 2021 7:50 am Also, I suspect @Dave Keenan may eventually ask me to write a program which allows a regular (or is linear the preferred term?) temperament notation to be generated from some inputs, as a feature of a web-based Sagittal notation calculator...
Actually. It should be a simple extension to your existing code base, to at least do the grunt work of determining how many periods and generators each Olympian symbol's default comma maps to, for a given temperament. A user might copy and paste the temperament's mapping from Graham Breed's Temperament Finder.

It's a standard linear-algebra operation of multiplying a matrix by a column vector to get another column vector. The matrix is the mapping. The first column vector is the prime-exponent vector (monzo) for the comma. The resulting column vector is the generator-count vector (with the first generator being the period).

I note that when we write 81/80 as [ -4 4 -1 ⟩
that's just a lazy way of writing the column vector:

$$\begin{bmatrix} -4\\ 4\\ -1 \end{bmatrix}$$

And so this (Meantone) Mapping:
[ ⟨  1  2  4 ]
⟨  0 -1 -4 ] ⟩

is just the matrix:

$$\begin{bmatrix} 1 & 2 & 4\\ 0 & -1 & -4 \end{bmatrix}$$

That tells us that prime 2's approximation consists of 1 period and 0 generators. i.e. the period is an approximate octave, prime 3's approxcimation consists of 2 periods and -1 generators, i.e the generator is an approximate 4/3, and prime 5's approximation consists of 4 periods and -4 generators.

To find out how many periods and generators 81/80 corresponds to in meantone, just compute the following product:

$$\begin{bmatrix} 1 & 2 & 4\\ 0 & -1 & -4 \end{bmatrix} \begin{bmatrix} -4\\ 4\\ -1 \end{bmatrix}$$

That's just:

$$\begin{bmatrix} 1×-4 + 2×4 + 4×-1 \\ 0×-4 + -1×4 + -4×-1 \end{bmatrix}=\begin{bmatrix} 0\\ 0 \end{bmatrix}$$

= [0 0⟩

That's not a monzo. It's a generator-count vector. So we see that the 5-comma is zero periods and zero generators in meantone and so and are of no use in notating meantone.

I note that Herman and I both use plus signs "+" for positive generator counts, which serves as a clue that they are not monzos. Herman further distinguished his generator count vectors above (but not everywhere) by using parentheses instead of square and angle brackets.

Graham Breed has a tutorial here:
http://x31eq.com/matrices.htm
and if that isn't basic enough, he has a matrix math tutorial here:
http://x31eq.com/matritut.htm