## Magrathean diacritics

cmloegcmluin
Posts: 721
Joined: Tue Feb 11, 2020 3:10 pm
Location: San Francisco, California, USA
Real Name: Douglas Blumeyer
Contact:

### Re: Magrathean diacritics

Dave Keenan wrote:
Thu May 28, 2020 5:56 pm
Do we get any useful 1 tina candidates by subtracting yellow commas for successive tinas?
For the 7-tina, each of your two favorites is a combination of two of our yellow commas: the 11:119n is our yellow 9-tina minus our yellow 2-tina, while the 7:425n is our yellow 5-tina plus our yellow 2-tina.

Adding our yellow 4- and 3- tinas gives a comma which we would reject for having a SoPF of 54.

tina	comma		limit	SoPF	error	cents	monzo			ratio
7.08	605/637n	13	54	0.08	0.995	[ 8 -5 1 -2 2 -1 ⟩	154880/154791


Is there something special we should be considering for the 7-tina, given its special position as half of 14 tinas?

For the 1-tina, our yellow 3-tina minus our yellow 2-tina gives the 65:833n, and our yellow 4-tina minus our yellow 3-tina gives the 196625n.

Our yellow 5 - 4 gives

tina	comma		limit	SoPF	error	cents	monzo			ratio
1.06	16807/75625n	11	77	0.06	0.149	[ -1 2 -4 5 -2 ⟩	151263/151250


Our yellow 6 - 5 gives

tina	comma		limit	SoPF	error	cents	monzo			ratio
0.79	1625/184877n	13	74	0.21	0.111	[ 10 -2 3 -5 -1 1 ⟩	1664000/1663893


Our yellow 9 - 8 gives a rubbish 16 for the abs 3 exp.

cmloegcmluin
Posts: 721
Joined: Tue Feb 11, 2020 3:10 pm
Location: San Francisco, California, USA
Real Name: Douglas Blumeyer
Contact:

### Re: Magrathean diacritics

Dave Keenan wrote:
Thu May 28, 2020 5:41 pm
Thanks for explaining re Ash's earlier commas.
cmloegcmluin wrote:
Thu May 28, 2020 3:54 pm
Re: the 7th, I kinda just like George's. But I feel terribly biased about it.
You were a bad boy putting all that stuff in the table about who first suggested each comma, and all that ALAGA George stuff.
I really just had those in there to help keep things straight. And also because a lot of my formulas were relative to rows that contained George in them. Hopefully no one got it in their heads that this was a competition!

Really I didn't mean I liked it because it was George's. I just meant I didn't feel I had an iota of objectivity to preferring it.
It should have been double-blind placebo-controlled. That means I poke both your eyes out and say "Here, take this".
I'm not blind until you poke both my eyes out. Wouldn't you have to poke them each out twice?!
I agree that any comma we might give for the dot, should map to zero tinas, using the 8539.00834edo mapping.
Did I say that? In any case, I agree too.

Dave Keenan
Posts: 1024
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

### Re: Magrathean diacritics

I really have to distance myself from Sagittal for a while, to get some other things finished.

You guys please sort it out between yourselves, what text we're going to have with each of the 1 thru 9 tina accents and the dots, in the Magrathean Sagittal section of the SMuFL specification, considering what we've done in previous sections.

Cml, perhaps you would post here what (I think) we already agreed to have in the Olympian section.

Don't assume you have to stick with my yellows.

cmloegcmluin
Posts: 721
Joined: Tue Feb 11, 2020 3:10 pm
Location: San Francisco, California, USA
Real Name: Douglas Blumeyer
Contact:

### Re: Magrathean diacritics

Dave Keenan wrote:
Fri May 29, 2020 12:49 pm
Cml, perhaps you would post here what (I think) we already agreed to have in the Olympian section.
It's already in the Sagittal-SMuFL-Map hosted on the site: http://sagittal.org/Sagittal-SMuFL-Map.pdf

Alright, @Ash9903b4 , we've got this from here.

A strategy for the 1-tina occurred to me tonight. So its primary comma will probably not often be the difference between the value of the symbol it notates and the symbol otherwise the same but without it -- probably even less so than the 1-mina represents its primary comma, as we've seen. Nonetheless, the 1-tina will get used a lot, strictly more than any other tina symbol (besides the 3 and 6 which are equivalent to the 1 and 2 mina in the Extreme precision level, as Dave noted earlier). So if we imagine the 1-tina does represent its primary comma as often as it can, then what would its effect on the SoPF<3 be on each of the current symbols in Sagittal?

The current sum of SoPF<3 for all Sagittal symbols is 3414 (if you're curious, the maximum SoPF<3 is 47, for everyone's favorite 47-limit comma, the 1/47S (don't worry, it's closely followed by 46)). For each of the 1-tina candidates, I took the sum across all of them with the tina added as well as with it subtracted. Here are the results:

121:1225n
subtracting it → 8578

65:833n
subtracting it → 8677

7:221n
subtracting it → 7727

196625n
subtracting it → 9102

5:10241n
subtracting it → 8751

By this measure, the 7:221n is the clear victor, having the least impact on the total SoPF>3.

Re: the 7-tina, I vote for 7:425n:
- made out of the 9, which is the 6 and the 3 which are already in Sagittal
- lowest SoPF>3
- most occurrences in Extreme Precision
- low abs 3 exp
- superparticular

Dave Keenan
Posts: 1024
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

### Re: Magrathean diacritics

I've turned off email notifications, but the above snuck thru beforehand. I'm not addicted. I can stop any time I want.

Brilliant idea, looking at its effect on existing primary commas. But why not also look at its effect on 3-exponent, which relates to how many sharps or flats will be needed to notate it? And why not apply the same test to other numbers of minas? Or at least the difficult ones. Like 7.

But error must still be considered.

cmloegcmluin
Posts: 721
Joined: Tue Feb 11, 2020 3:10 pm
Location: San Francisco, California, USA
Real Name: Douglas Blumeyer
Contact:

### Re: Magrathean diacritics

Dave Keenan wrote:
Fri May 29, 2020 4:39 pm
I've turned off email notifications, but the above snuck thru beforehand. I'm not addicted. I can stop any time I want.
C'mon Dave... don't you wanna be cool? Party with us
Brilliant idea, looking at its effect on existing primary commas.
Thanks! At first I didn't love it, because I was afraid it would amplify whatever distortions, however slight, in Sagittal's attempt to best represent harmonic space. But after a good night's sleep, I think it's rather the opposite if anything — sorta smoothing the rough edges.
But why not also look at its effect on 3-exponent, which relates to how many sharps or flats will be needed to notate it? And why not apply the same test to other numbers of minas? Or at least the difficult ones. Like 7.
I should do the same sort of check for the absolute 3 exponent too, indeed. I'll just add a couple more columns to the table when I get a chance, so we can see how it affects all the mina choices.

Ash9903b4
Posts: 16
Joined: Sun Mar 15, 2020 11:27 am

### Re: Magrathean diacritics

cmloegcmluin wrote:
Fri May 29, 2020 3:56 pm
Alright, @Ash9903b4 , we've got this from here.
I think I’ll have to postpone this for real life reasons as well, but I might work on it independently a little and post my findings in a few weeks.

cmloegcmluin
Posts: 721
Joined: Tue Feb 11, 2020 3:10 pm
Location: San Francisco, California, USA
Real Name: Douglas Blumeyer
Contact:

### Re: Magrathean diacritics

Still planning to do those two new columns for "total SoPF>3" and "total abs3exp" soon.

In the meantime, I performed a regression analysis on our thresholds for various of these metrics across the existing precision levels, to validate where we've set them.

We set our threshold for SoPF>3 at 51. But if you look at the maximum SoPF>3 for each precision level and consider it proportionally to its EDA size ((21,18), (47,30) (58,32), (233,46), (809,??)) you get a logarithmic fit:
11.3854*ln(0.263201*EDA)

Which suggests that perhaps we should consider commas with SoPF>3 up to 61.

For abs3exp threshold we started with 12 and Dave has asked to try extending to 13. Well, the maximum abs3exp is currently 14, and the progression by precision level goes 8, 11, 14, 14. For intervals as important as the very small ones that will recur throughout the notation we should aim for much lower though; if we only consider the JI intervals corresponding to the single steps of 21-EDA, 47-EDA, 58-EDA, and 233-EDA, then their abs3exps are 6, 3, 8, 2, for which there is no good line of fit, but they are overall quite a bit lower. And I also don't really see this metric as naturally growing with each more precise level (as makes sense for the SoPF>3) so much as leveling out at 14. But a logarithmic fit for ((21,8), (47,11), (58,14), (233,14)) would give closer to 15. So... I'll give up to 15 a shot.

For prime limit, @Ash9903b4 stuck with 19 and I went up to 37. A logarithmic fit to ((21,17), (47,23), (58,23), (233,37*)) says that 47** should be the limit with 809-EDA. Of course we are extremely unlikely to find useful n-tinas with SoPF>3 ≤ 61 but limit 47. But we could try.

For error, it only really makes sense to consider the errors for the primary commas for the single steps of each EDA. For 21-EDA, e.g., it's the . which at 5.758¢ is 1.063 steps of 21-EDA (where 1 step of 21-EDA is 5.414¢) and thus has error of 0.063. The errors for the 4 established precision levels are thus 0.063, 0.396, 0.003, and 0.153, which don't really have much of a pattern, but if you ask Wolfram for a linear fit it'll tell you the next one should be about 0.125, which is half the threshold we've been using so far. All our yellow commas so far are within that (well, except the 5-tina, which is on the cusp at 0.13). So perhaps we should be stricter about the error.

By the way, I realized that at least I have not being calculating tina error quite correctly. I've simply been taking the absolute value of the difference between the tina column and the whole tina, so that e.g. the 7:221n which is 0.83 has an error of 0.17. But actually I should have been taking the error of the absolute ratio, or as Dave and I call it the undirected value of the ratio between the comma and one step of 809-EDA. So I'll get that right in my next chart.

In any case, I'm approaching the results of this analysis not in terms of where we should move a yes/no cutoff, but more like how we should weight these factors in a forthcoming consolidated tina badness metric.

*Assuming we do eliminate the 1/47S from the Extreme precision level.
**But we may see the 1/47S come right back in the Insane precision level, then!

cmloegcmluin
Posts: 721
Joined: Tue Feb 11, 2020 3:10 pm
Location: San Francisco, California, USA
Real Name: Douglas Blumeyer
Contact:

### Re: Magrathean diacritics

Update: I think I'll just take the complexity metric empirically used in the development of Sagittal which was unearthed over here: viewtopic.php?p=1659#p1659 [Edit: I will have to incorporate the tina error into it somehow, though]

Dave Keenan
Posts: 1024
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

### Re: Magrathean diacritics

If we come up with an improvement over SoPF>3, as a measure of the complexity of the set of 2,3-equivalent ratios that can be notated by a symbol for a given comma, as requested here: viewtopic.php?p=1676#p1676, then we would presumably also want to substitute it for SoPF>3 in any badness measure we might use in this thread. This is a specific kind of complexity measure — one that we have designed to be strongly (inversely) correlated with popularity.

A "badness" measure is typically a combination of a complexity measure and an error measure. But we have two kinds of complexity measure in this case. The other kind is the complexity of the resulting notation, in the sense of how many sharps or flats may need to be used to cancel out (or mostly cancel out) the 3-exponent of the comma. Or on a finer scale, how many fifths the resulting sharped or flatted letter name is from 1/1 along the chain of fifths. The Revo (pure) notation can provide at most two sharps or flats, which corresponds to a change of 14 in the absolute value of the 3-exponent. And it has the additional limitation that positive comma alterations cannot be applied to a double-sharp and negative cannot be applied to a double-flat.

On waking this morning, I had some thoughts about how to munge both the raw 3-exponent and the raw tina error, to make them compatible with SoPF>3 (and hopefully any future modified-SoPF>3), in the sense that it would then make sense to add them (with some weighting factors) to the SoPF>3, to obtain an overall badness measure.

Let's take tina-error first, as it seems the more straightforward of the two.

There should be zero penalty for a zero error, and the penalty for an error of 0.5-tina should be infinite. This suggests munged_error(error) = 1/(0.5 - error) - 1/0.5.

This gives munged_error(0.25) = 2. We would then have to decide what decrease in SoPF>3 we would be willing to trade for an increase in error from 0 to 0.25 tinas. Maybe we'd be willing to trade that for the removal of a prime 11 or two prime 5's, in which case we'd multiply the munged_error by 11/2 = 5.5 or 10/2 = 5.0 before adding it to the SoPF>3.

With the 3-exponent, there should be zero penalty for zero 3-exponent and infinite penalty for a 3-exponent of 14 (or maybe 15). So munged_3_exp(3_exp) = 1/(14 - 3_exp) - 1/14. Then we have to think about what decrease in SoPF>3 we'd be willing to trade for a single sharp (3-exp = 7), where munged_3_exp(7) = 1/14. Maybe we'd be willing to trade that too, for the removal of a prime 11 or two prime 5's, in which case we'd multiply the munged_error by 11×14 = 154 or 10×14 = 140 before adding it to the SoPF>3.