## developing a notational comma popularity metric

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### Re: developing a notational comma popularity metric

Okay. Feel free to share your thoughts as you figure out what to use as our error metric. I may not be in front of a computer tomorrow but I might have some time to ponder things.

Dave Keenan
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### Re: developing a notational comma popularity metric

The best I have been able to do, by adding to the LPE metric either u * abs(err) or u * 2^abs(err), is to match one more comma. Up from 101 matches to 102 matches. Not very inspiring.

"Err" here is the number of minas the comma is away from the nearest whole mina (1/233rd of an apotome). This won't work properly for those few split minas. But it's good enough to show that including error doesn't help much in nailing down any particular set of parameter values. Still, it gives us some badness measures to try. And maybe a wide range of the badnesses that work for the extreme level, give the same comma for each tina. A man can dream.

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### Re: developing a notational comma popularity metric

There’s an argument for error being infinity at +/- 1 ina, even if the inside radius is super tiny, i.e. it’s nowhere near infinity until it’s really really close to the edge.

But I can also see it being reasonable to still admit commas right up to the exact edge from either side, i.e. error never reaches infinity, just something really big.

Dave Keenan
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### Re: developing a notational comma popularity metric

[Response to the above. Originally posted Sun Nov 01, 2020 5:05 pm (Australian Eastern Standard Time):]

Independent of any badness metric, we're implementing a sharp cutoff at ± the half ina (± the quarter ina in the case of the half-tina dot).

The fact that adding u × AERR only gives one more extreme match, may be because George and I may not have actually given any weight to the mina error when assigning the commas to the extreme level.

[Continued in the Magrathean diacritics thread]

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### Re: developing a notational comma popularity metric

While brewing some coffee just now I clicked on this video by the YouTube channel 12Tone:

Don't be fooled by his name "12Tone"; this is now the 5th or so video of his I've seen where he addresses music theory outside of 12-EDO.

So has anyone ever heard of Euler's gradus-suavitalis, or "degree of agreeableness" before? He's using it to measure consonance, or what he called "softness" apparently. That's certainly different from popularity, but not completely different. His formula goes like this:

$$E(n) = 1 + \sum_{k=1}^{r} a_k(p_k - 1)$$

Where p is the prime and a is the prime exponent.

His formula ignores direction, i.e. 20/1 is the same as 5/4. And in his formula, 2's and 3's affect the grade.

So for example, [ -4 4 -1⟩ would be 4(2-1) + 4(3-1) + 1(5-1) = 4(1) + 4(2) + 1(4) = 4 + 8 + 4 = 16.

The monzo terms get geared down one multiplicative order from exponentiation to multiplication, which should seem familiar: for most of the stages of N2D3P9's development, we were gearing down one multiplicative order by taking the logarithm of our ratios. In the end we recognized our metric's constants as expressible as simple low whole numbers in terms of our chosen logarithmic base of 2 and thus we could simplify our formula without altering the rankings it produced.

In the video, golf is used as an example of a system where points are bad, which amuses me because golf was the only example I could come up with for an inverted score a couple weeks ago over email with Dave. Perhaps the problem was with the word "score"; maybe "grade" is what we were looking for all along. Grade "A" is the best, even though A is the first/lowest letter.

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### Re: developing a notational comma popularity metric

cmloegcmluin wrote: Fri Nov 13, 2020 7:16 am His formula goes like this:

$$E(n) = 1 + \sum_{k=1}^{r} a_k(p_k - 1)$$
Note that it would probably be ideal to get rid of that $$1 +$$; it was only really put there so that 1/1 wouldn't be zeroth-grade, and plainly 0 is an acceptable output now that it's just another number.
Grade "A" is the best, even though A is the first/lowest letter.
More like "'First-class' is the highest priority, even though 1 is the first/lowest number". ...Whatever.
Last edited by volleo6144 on Tue Nov 24, 2020 3:03 am, edited 1 time in total.
A random guy who sometimes doodles about Sagittal and microtonal music in general in his free time.

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### Re: developing a notational comma popularity metric

volleo6144 wrote: Fri Nov 13, 2020 7:45 am Note that it would probably be ideal to get rid of that $1 +$; it was only really put there so that 1/1 wouldn't be zeroth-grade, and plainly 0 is an acceptable output now that it's just another number.
Agreed.
More like "'First-class' is the highest priority, even though 1 is the first/lowest number". ...Whatever.
Yeah, that works too. Unfortunately we're already using "class" in this domain for set membership (2,3-free classes), so it's off-limits.

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### Re: developing a notational comma popularity metric

Just got recommended another 12tone video, and this one he touches upon Euler's gradus suavitalis, as well as another function which confusingly measures "roughness" of pitches (not to be confused with mathematical roughness as in the 5-rough numbers our metrics grade), where roughness is another variant on consonance or simplicity:

This roughness function was developed by Helmholtz (I assume the same "H" as in "EHEJIPN"). Unlike the gradus function in the other 12tone video, it is not broken down specifically in the video. It is talked about in this paper linked from the video page, but I can't find a specific definition of the function. Just this:
Helmholtz defined the roughness of an interval between tones p and q on the basis of the sum of beat intensities I_n + I_m associated with the n^{th} harmonic of p and the m^{th} harmonic of q (Helmholtz 1954).
There's more information in there if you want to try to figure it out, though. The 1954 work by Helmholtz it refers to is "On the Sensations of Tone", a book whose title I'm certainly familiar with, but I've never read it because Kyle Gann warned me it was "unreadable".

Dave Keenan
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### Re: developing a notational comma popularity metric

Yes. I'm familiar with Euler's gradus suavitatis. I included it among a number of similar measures in a spreadsheet/chart I published a long time ago: http://dkeenan.com/Music/HarmonicComplexity.zip

And I believe Helmholtz' roughness is closely related to Sethares' dissonance, in this spreadsheet: http://dkeenan.com/Music/SetharesDissonance.zip

This roughness function was developed by Helmholtz (I assume the same "H" as in "EHEJIPN").

Yes. And the second E is for Alexander Ellis who "translated" Sensations of Tone into English. The scare quotes around "translated" are not because he was unfaithful to the original (far from it). It is because he annotated and appended to the translation so copiously and so well, that the final result is really a book by Helmholtz and Ellis, of which Ellis wrote 60%!
The 1954 work by Helmholtz it refers to is "On the Sensations of Tone", a book whose title I'm certainly familiar with, but I've never read it because Kyle Gann warned me it was "unreadable".
I am mystified by the 1954 publication date. Helmholtz died in 1894 and Ellis in 1890. Here are the publication dates in Wikipedia.
https://en.wikipedia.org/wiki/Sensations_of_Tone

I guess the 1954 edition is just the most recent reprinting of the 1885 edition. I agree it's probably not a book to read from cover to cover, but I'm not sure why Kyle says it's "unreadble". In any case, it's pretty much the founding work of "our field".

And if you go looking in it for support for the symbols used in EHEJIPN, you will find none. But you will find support for the semantic principle it has in common with Sagittal (but not Johnston), of notating relative to a chain of Pythagorean fifths, with comma alterations for primes above 3.

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