Magrathean diacritics

[Dave Keenan]

I received the following query by email:

... I noticed SMuFL had a range for Magrathean Sagittal. Just wondering about the current state of the project, and what it would be like.

I had difficulty finding what George and I had agreed on, about 5 years ago. So I thought I should put the answer here, so the work is not lost.

The main reason we have never before published it, is that it is kind of embarrassing that it even exists. Some people already complain that Sagittal has too many symbols, often failing to appreciate that 99% of what anyone ever wants to do microtonally can be notated with the Spartan subset of 8 symbols.

But since we are dealing here with symbols (specifically diacritical marks or accents) having a resolution of a half-tina or 0.07 cents (completely inaudible except in very special circumstances), we needed a name for some beings that were more powerful than the Olympian gods, and yet were ridiculous. The Magratheans of Douglas Adams' comedy sci-fi The Hitchhikers Guide to the Galaxy were ideal for this.

[Edit: "Magrathean" is pronounced MAG-ra-ΘEE-ən. Hear it here: https://youtu.be/6broMc3j8OQ?t=88 and here: https://youtu.be/6broMc3j8OQ?t=176]

To ensure that the symbols were as distinct as possible given their necessarily-small size, George and I used the discipline of forcing ourselves to represent them in as few pixels as possible. You can see the final result here, at two different resolutions, one with 6 pixels between staff-line centres, and one with 8. The 3 and 6 tina symbols are identical to the 1 and 2 mina symbols described in the previous post, and the 9 tina symbol can also be used for 3 minas. To these we add 1 and 2 tina symbols and a dot to represent a half-tina. The intention is that, in a proper outline font, the 1, 2 and 3 tina symbols should be something like miniature versions of the following symbols without their shafts , and the half-tina dot should be circular.

Magrathean-GS17.gif

(2.64 KiB) Downloaded 3288 times

George came up with the following definitions, with the help of a comma list produced by Gene Ward Smith. The ratios below have had their powers of 2 and 3 removed. These can be found if necessary, since we know the approximate size in cents. The trailing "n" is the symbol for "schismina" (our name for commas smaller than 1.8075 cents).

0.5 tina: 49:9765625n
1 tina: 121:1225n
1.5 tinas: 121:5359375n
2 tinas: 13:6655n
2.5 tinas: 343:15625n
3 tinas, 1 mina: 455n (~4375n)
3.5 tinas: 11:47n
4 tinas: 7:3025n
4.5 tinas: (no definition)
5 tinas: 25:2401n (~343:14641n)
5.5 tinas: (no definition)
6 tinas, 2 minas: 65:77n (~13:125n)
6.5 tinas: 13:37n
7 tinas: 143:1715n
7.5 tinas: 3125:5929n
8 tinas: 420175n
8.5 tinas: (no definition)
9 tinas: 539n
9.5 tinas: (no definition)

[Ash9903b4]

I'd like to reopen this thread to suggest a different, somewhat simpler interpretation for the single-tina diacritic, and one that actually came up when studying temperaments in 494: the 55:343-schismina (14765025303/14763950080). It's the difference between the 7:11-comma () and the 5:49 comma, or about 0.126 cents.

An even simpler but higher-limit mapping would be the 13:119-schismina (0.117 cents, 117448461/117440512), or the difference between the 17-comma () and the 7:13-comma.

[Ash9903b4]

Here are some relatively simple 19-limit mappings for each Magrathean diacritic:

tempered out: 1990625n ≈ 169:1925n ≈ 104975n

0.5 tina: 17:9295n

1 tina: 65:833n ≈ 5:10241n ≈ 232925n ≈ 25:14399n

1.5 tinas: 5:2023n ≈ 25:15379n

2 tinas: 5831n ≈ 35:1573n ≈ 13:6655n ≈ 5:218491n ≈ 187:665n ≈ 475:1001n

2.5 tinas: 85:121n ≈ 343:15625n

3 tinas: 455n ≈ 4375n ≈ 11:4225n ≈ 7:1615n ≈ 13:67375n

3.5 tinas: 119:143n ≈ 209:595n

4 tinas: 17:425n ≈ 7:3025n ≈ 17:6875n ≈ 49:1045n ≈ 85:637n ≈ 49:1859n

4.5 tinas: 265625n ≈ 247:625n ≈ 289:325n

5 tinas: 25:2401n ≈ 11:2375n ≈ 5:2431n ≈ 625:833n ≈ 95:2401n ≈ 11:9025n

5.5 tinas: 24565n ≈ 17:11011

6 tinas: 13:125n ≈ 65:77n ≈ 2358125n ≈ 169:475n ≈ 124355n

6.5 tinas: 7:2125n ≈ 625:5979n

7 tinas: 7:425n ≈ 11:119n ≈ 1729n ≈ 5:1729n ≈ 143:1715n ≈ 1890625n ≈ 605:637n ≈ 25:2527n

7.5 tinas: 19:21875n ≈ 11:3757n

8 tinas: 19:385n ≈ 420175n ≈ 175:187n ≈ 343:1625n ≈ 95:169n ≈ 34969n ≈ 17:5225n

8.5 tinas: 11:40625n ≈ 177023n

9 tinas: 539n ≈ 125:2401n ≈ 41503n

9.5 tinas: 1328125n ≈ 425:637n ≈ 245:1859n ≈ 247:3125n ≈ 354025n ≈ 143:78125n

[Dave Keenan]

Wow! Thanks for those, Ash.

Are they all based on a single consistent mapping from primes to half-tinas? If so, can you give the number of half-tinas for each prime. I know 809-EDA is highly consistent, but I have no idea about 1618-EDA. The dots for half tinas were an expedient afterthought when Dr Tony Martin-Salinas asked if we could do better than tinas.

[Ash9903b4]

Wow! Thanks for those, Ash.

Are they all based on a single consistent mapping from primes to half-tinas? If so, can you give the number of half-tinas for each prime. I know 809-EDA is highly consistent, but I have no idea about 1618-EDA. The dots for half tinas were an expedient afterthought when Dr Tony Martin-Salinas asked if we could do better than tinas.

No, I just rounded them to the nearest half tina. The issue is that the patent val for 8539edo is so accurate that none of the primes in the 19-limit actually map to odd multiples of the half-tina, so the “between the cracks” intervals I posted all come from accumulation of tiny errors. In any case the patent val in half-tinas is ⟨17078 27068 39654 47944 59080 63196 69806 72546].

[Dave Keenan]

Ah. OK. What about the subtle difference between 8539edo and 809eda? Does that change which intervals are between the cracks?

Here's a wild thought. What if the odd half-tinas were not used to notate ratios, but to notate noble numbers? How could that be made to work?

[Ash9903b4]

Ah. OK. What about the subtle difference between 8539edo and 809eda? Does that change which intervals are between the cracks?

I don't think so, unless the interval is incredibly close to a quarter tina (like the 121:1225n), but I'd have to check. How do you define a canonical form for edas? Do you just use the just intonated apotome that results from stacking seven fifths?

Here's a wild thought. What if the odd half-tinas were not used to notate ratios, but to notate noble numbers? How could that be made to work?

Well, you'd now have a new series of primes involving the square root of five. In particular, 5 itself would now be a composite number, and √5 would map to 19827 half-tinas. That's one way to get half-tinas, I guess, but then you'd have to figure out what 1 + √5, 2 + √5, etc. correspond to.

Similarly, you can use the field Q(√2) and map √2 to 8539 half-tinas, I think.

[Edit by Dave Keenan: The notation of noble ratios is discussed further here: Notation for metallic scales]

[Dave Keenan]

I just want to flag here, that if and when we finally get around to nailing down everything Magrathean:
I suggest that the ASCII (both short and long) for the various tina and half-tina diacritics should be "l" (lowercase-L) for down and "@" for up, followed by the number of tinas. e.g. 5 tinas up would be "@5", and 1.5 tinas down would be "l1.5". The reason for using those characters is that they are the only ones that haven't been used yet, in the short or long ASCII*. And Magrathean is the least useful part of Sagittal, so it should get the dregs. And we have a rule that when an up/down pair consists of a letter and a special character, the letter is down and the special character is up. The archetypes being b# and v ^.

If more than ASCII is available, the script ℓ could be used. e.g. 1 tina down "ℓ1".

*We haven't used []{}": either, but we'd like to keep them as delimiters, or to stand for non-sagittal accidentals. We've only used "p" and "c" as alternatives to the period . and comma , respectively, for representing downward schisma and mina diacritics in a context where the period or comma is a delimiter.

[Dave Keenan]

The above posts were moved here from the New Olympian diacritics topic.

[Dave Keenan]

How do you define a canonical form for edas? Do you just use the just intonated apotome that results from stacking seven fifths?

Yes. [-11 7>