## Magrathean diacritics

Dave Keenan
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### Re: Magrathean diacritics

Your coin-counting idea at least tells us that we don't need to go beyond 9.5 tinas additive. And if we want to allow users to make the choice between additive and subtractive in the range from 7.5 to 9.5 then we need to go up to 9.5. For your example that needs no tinas, but up to 4 minas, I'm happy to just use two 2 t symbols.

Thanks for giving the fractional tinas for the commas. They led to the thought that the dot up and dot down don't have to be ±¹⁄₂ t. They could be ±¹⁄₃ t. Is there an accurate EDO that's approximately 3 × 8539edo? That would also mean that the maximum error allowed for the various commas would be ±¹⁄₆ t.

And if we impose a requirement that n>7 tina commas must be 5-schisma complements of n<7 tina commas, then the fact that there are 13.90 tinas in the 5-schisma, not 14, implies even tighter constraints on the errors.

Those below 7 tinas can have an error of -0.167 t but only +0.067 t. And vice versa for those above 7 tinas.

It seems your list is still not complete, because when I subtract your 9.21 tina comma from the 5-schisma, I obtain the following 4.69 tina comma that isn't in your list.

tinas  comma        monzo                                     interval             cents      limit  pop
13.90  5s           [-15   8   1   0   0   0   0   0 ⟩         32805/32768         1.953721     5      5
9.21  41503n       [ -9  -4   0   3   2   0   0   0 ⟩         41503/41472         1.293601    11     43
4.69  ?????n       [ -6  12   1  -3  -2   0   0   0 ⟩         ?????/?????         ?.??????    11     48

But these do not make an acceptable pair for 5 and 9 tinas, because they are more than a sixth of a tina away from 5 and 9.

The obvious 9 tina comma is:

8.93  539n         [ 17  -5   0  -2  -1   0   0   0 ⟩        131072/130977        1.255240    11     25

It's annoying that its 5-schisma complement (a candidate for 5 tinas) has a 3-exponent of 13:

4.97  ???n         [-32  13   1   2   1   0   0   0 ⟩        ??????/??????        ?.??????    11     30

There is a clear winner for 8 tinas. It has the best popularity ranking by far, and it is the 5-schisma complement of the 6 tina (2 mina) comma.

7.98  13:77n       [-20  11   0   1   1  -1   0   0 ⟩      13640319/13631488      1.121197    13     31

The idea occurs, to obtain the 2 tina comma as the difference between the 8 and 6 tina commas, but no, it's complete rubbish, with a 3-exponent of 14 and a popularity rank of 67.

So what about the 5 tina as the difference between the 8 and 3 tinas? Perhaps unsurprisingly, it gives us the same comma we found for 5 tinas as 14 - 9 tinas. i.e. the one with a 3-exponent of 13 (popularity 30). Maybe we should use it anyway.

It seems, at first, that there are two potential complementary 4 tina/10 tina pairs:

3.84  17:245n      [-12  10  -1  -2   0   0   1   0 ⟩       1003833/1003520       0.539891    17     36
10.06  17:1225n     [ -3  -2   2   2   0   0  -1   0 ⟩          1225/1224          1.413829    17     41
and
4.07  7:3025n      [ -4  -3   2  -1   2   0   0   0 ⟩          3025/3024          0.572403    11     39
9.83  7:605n       [-11  11  -1   1  -2   0   0   0 ⟩       1240029/1239040       1.381317    11     34

But that 9.83 tinas is actually 9.830 or 9.829 and so is more than 1/6th of a tina away from 10 tinas. So it has to be the 17-limit pair.

Unfortunately the 5 tina comma obtained as 9 tinas minus 4 tinas has a 3-exponent of -15 (popularity 33).

Ash9903b4
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### Re: Magrathean diacritics

@Dave Keenan It looks like a lot of posts were deleted here?

Dave Keenan
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### Re: Magrathean diacritics

Ash9903b4 wrote:
Tue May 26, 2020 1:10 pm
@Dave Keenan It looks like a lot of posts were deleted here?
Yes, @Ash9903b4 . I'm so sorry. Please see viewtopic.php?f=15&t=473. But you will see that cmloegcmluin and I have reconstructed it all below, from our browser caches.

cmloegcmluin
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### Re: Magrathean diacritics

[Originally posted Mon May 25, 2020 3:51 pm East Australian time]

This is a very interesting discussion and I have a lot of other thoughts -- and I am happy to write my own script to check against @Ash9903b4 's results -- but before I head off to bed tonight, I just want to throw this question out there: in determining commas for the tinas, is it not also very important to consider what these commas produce when combined with the existing commas in Sagittal? This question would seem to become increasingly pressing with each new precision level. Are there alternate approaches we should complimenting our current strategies with, for example, seeing which of the most popular commas Sagittal does not yet represent, and biasing toward tina values that would exactly combine with existing primary commas to produce them?

Dave Keenan
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### Re: Magrathean diacritics

[Originally posted May 26, 2020 12:38 am]

I'm afraid I have zero interest in assigning symbols and commas to all 405 tinas in the half-apotome. Doing the 117 minas is difficult enough, and who uses them?

I'd much rather have you doing the notation calculator and educational materials.

Don't let the perfect be the enemy of the good.

All we need are some commas for the SMuFL documentation, to correspond to the tina symbols on a bare shaft. We just want to come up with some that we are unlikely to be embarrassed about later. But if we are embarrassed about them later, we can just change them. No one will have used them.

Dave Keenan
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### Re: Magrathean diacritics

[Post by cmloegcmluin Tue May 26, 2020 12:50 am East Australian time]

Yes at one point yesterday I came across this quote:
Dave Keenan wrote:
Thu Mar 19, 2020 12:29 pm
Now that I think of it, this seems far more useful than futzing around with Magrathean.
I shan't bother square-bracketing the pronoun, since pretty much anything is more useful than futzing with Magrathean.

Dave Keenan
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### Re: Magrathean diacritics

[Post by cmloegcmluin Tue May 26, 2020 1:54 am East Australian time]

Dave Keenan wrote:
Wed May 20, 2020 12:25 pm
That principle is that, up to some point, it's easier to deal with fractions greater than half than to deal with fractions subtracted from the whole. Or putting it another way: At some point it's better to stop making symbols for larger fractions and instead symbolise it as subtracting a small fraction from the whole. So where is that point?

In the case of symbols within the apotome, we settled on something close to 60% for reasons of convenience having to do with square roots of 2,3-commas. In the case of minas within the schisma, in Olympian, the figure is about 62.5% since we go from for 50% to instead of for 75%, with the capture zone boundary roughly halfway between.
Oh! I had meant to say that: when you posted this, it was an incredible coincidence, because I had just taken a note to myself to ask you over on the 140th mina topic why exactly you stopped where you stopped. To have answered me without me asking, and so immediately, and for a completely different reason, you must have a darn good simulation of me running on your end!

Dave Keenan
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### Re: Magrathean diacritics

[Post by Ash9903b4 Tue May 26, 2020 4:31 am East Australian time]

I've only used tina diacritics in one context, and that was to make it clear that the mina accent represents different schisminas at the Olympian level, not just the 455n. Maybe we could check the differences between existing symbols with and without mina accents and just default to those values, since, after all, Sagittal already supports those ratios, though imprecisely at the Magrathean level.

For example, Olympian (49:55s) is sharper than plain (19s) by the 49:1045n, or about 3.92 tinas. Now, would we use a 4-tina diacritic for the 49:55s in Magrathean instead? And if so, would there be an issue if we defined the 4-tina diacritic as the 49:1045n?

Dave Keenan
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### Re: Magrathean diacritics

[Post by Ash9903b4 » Tue May 26, 2020 4:58 am East Australian time]

Update: It seems like @cmloegcmluin has already gone ahead and calculated all the attested values of the mina accents in Olympian in the 37-limit thread. And if the number of Olympian symbols that use a particular schismina value is anything to go by, then no; the 4-tina diacritic should stand for the 7:3025n, attested in three accidentals: .

Dave Keenan
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### Re: Magrathean diacritics

[Originally posted Tue May 26, 2020 9:53 am]

Ash9903b4 wrote: I've only used tina diacritics in one context, and that was to make it clear that the mina accent represents different schisminas at the Olympian level, not just the 455n. Maybe we could check the differences between existing symbols with and without mina accents and just default to those values, since, after all, Sagittal already supports those ratios, though imprecisely at the Magrathean level.
That's brilliant! Thanks. It's obvious once you say it. But it hadn't occurred to me, despite the fact that cmloegcmluin was also pushing in that direction when he wrote:
cmloegcmluin wrote: In determining commas for the tinas, is it not also very important to consider what these commas produce when combined with the existing commas in Sagittal? This question would seem to become increasingly pressing with each new precision level.
Sorry I summarily dismissed that, cmloegcmluin.
Ash9903b4 wrote: Update: It seems like @cmloegcmluin has already gone ahead and calculated all the attested values of the mina accents in Olympian in the 37-limit thread. And if the number of Olympian symbols that use a particular schismina value is anything to go by, then no; the 4-tina diacritic should stand for the 7:3025n, attested in three accidentals: .
Makes sense. I retract my earlier rejection of 7:3025n (4.07 tinas), which was because its 5-schisma complement is more than 1/6th tina away from 10 tinas. We don't need to specify the comma for 10 tinas in the SMuFL documentation. And if and where we do specify the 10 tina comma, we can either note that 7:605n misses out by only 0.004 cents, or we can choose a comma that is not the 5s-complement of the 4 tina comma.

So we've nailed down 3, 4, 6, 8 and 9 tinas. To complete SMuFL, we only need 1, 2, 5, 7 and either 0.5 or 0.33.

Any thoughts about whether the dot should be ½ tina or ⅓ tina?