New Olympian diacritics

User avatar
Dave Keenan
Site Admin
Posts: 352
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

New Olympian diacritics

Post by Dave Keenan »

Many thanks to Xen-Gedankenwelt for helping to get the full set of Sagittal symbols up as "smilies" on this forum.
View more smilies

None of the recently uploaded symbols will be of any interest to those of you using the "mixed" Sagittal notation, as this only requires the single-shaft symbols, and all those recently uploaded were 3-shaft and X-shaft symbols, equivalent to single-shaft symbols added to sharps and double-flats, or subtracted from flats and double-sharps.

But anyone interested in the Olympian (extreme precision) JI notation (mixed or pure), should note the new section at the very end entitled :olympian:, with four tiny diacritical marks. These are an alternative representation for one or two "minas" up or down. A mina (pronounced meena) is approximately 0.4 cents in size. The mina was first defined as the difference between the primary (or exact) meaning of :(|\: as the 35-large-diesis (8505/8192), and its secondary (or approximate) meaning as the 13-large-diesis (27/26).

When it was first decided that we should have a diacritical mark to symbolise this difference, and thereby allow an exact symbol for the 13-large-diesis, it was expedient to simply re-use the grave and acute diacritics that we used on the left for 5-schisma alterations (~2.0 cents), by declaring that if they appeared to the right of a symbol they were to be interpreted as mina diacritics (~0.4 cents). So we had
Diesis	Symbol	Smiley code
35L	:(|\:	:(|\:
13L	:(|\::':	:(|\::':
125L	:(|\::.:	:(|\::.:
7L	:.::(|\:	:.::/|):
625L	:.::(|\::.:	:.::(|\::.:
See the right-hand end of http://sagittal.org/SagittalJI.gif for their ratios.

However this violates the rule that the smallest alteration should be the furthest from the notehead, and allows for left/right confusion. So when Tony Salinas (J.A.Martin Salinas) asked for a way to notate even smaller alterations, for compositions that involved precisely-controlled beating effects, we decided the mina should have a diacritic whose shape and size differed from that of the schisma diacritic, and so could be placed on the left of a symbol, and to the left of any schisma diacritic. It should also allow for a possible future extension using other diacritic shapes, for alterations in multiples of about 0.14 cents that we call "tinas" -- the mythical Magrathean extension.

After ridiculous amounts of discussion and testing, George and I agreed (3 years ago) that the new mina diacritics should look like a slanted breve -- something like a miniature version of this symbol :/|): without its shaft, and only about half the height of the schisma diacritic. So we now have:
Diesis	Symbol	Smiley code
35L	:(|\:	:(|\:
13L	:`::(|\:	:`::(|\:
125L	:,::(|\:	:,::(|\:
7L	:.::(|\:	:.::/|):
625L	:,::.::(|\:	:,::.::(|\:

The smiley codes for the four Olympian extension diacritics use the back-quote (up) and the comma (down):
:`:   :`:
:,:   :,:
:``:  :``:
:,,:  :,,:

Their Sagispeak pronunciations are:
:`: mi
:,: mo
:``: mimi
:,,: momo

[Edit: Thanks to cmloegcmluin for this updated version of the following paragraphs, as discussed here.]

As a symbol, :`::|: mai :,::!: mao, the mina's primary role is the 5.7.13-schismina, 4095:4096, 0.423c, and it has a capture zone spanning from 0.211 to 0.773c (from 1.5 to 5.5 steps of 809EDA). Secondary roles within this capture zone include the 5⁴.7-schismina, 4374:4375, 0.396c. As a diacritic component of other symbols ( :`: mi- :,: mo- ) the mina changes the primary role of the core symbol by one of these two commas, or another comma of a similar size.

Similarly, :``::|: mimai :,,::!: momao, the 2 mina symbol, has a primary role as a (5.13):(7.11)-schismina, 2079:2080, 0.833c, and it has a capture zone spanning from 0.773 to 1.194c (from 5.5 to 8.5 steps of 809EDA). Secondary roles within this capture zone include the 13:5³-schismina, 255879:256000, 0.818c. As a diacritic component of other symbols ( :``: mimi- :,,: momo- ) the double mina changes the primary role of the core symbol by one of these two commas, or another comma of a similar size.

User avatar
Dave Keenan
Site Admin
Posts: 352
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

Re: New Olympian diacritics

Post by Dave Keenan »

Magrathean Diacritics

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. :oops: 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. : D The Magratheans of Douglas Adams' comedy sci-fi The Hitchhikers Guide to the Galaxy were ideal for this.

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
Magrathean-GS17.gif (2.64 KiB) Viewed 1847 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
Posts: 8
Joined: Sun Mar 15, 2020 11:27 am

Re: New Olympian diacritics

Post by 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
Posts: 8
Joined: Sun Mar 15, 2020 11:27 am

Re: New Olympian diacritics

Post by 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
Last edited by Ash9903b4 on Wed Apr 01, 2020 3:54 pm, edited 1 time in total.

User avatar
Dave Keenan
Site Admin
Posts: 352
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

Re: New Olympian diacritics

Post by 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
Posts: 8
Joined: Sun Mar 15, 2020 11:27 am

Re: New Olympian diacritics

Post by Ash9903b4 »

Dave Keenan wrote:
Wed Apr 01, 2020 3:34 pm
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].

User avatar
Dave Keenan
Site Admin
Posts: 352
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

Re: New Olympian diacritics

Post by 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
Posts: 8
Joined: Sun Mar 15, 2020 11:27 am

Re: New Olympian diacritics

Post by Ash9903b4 »

Dave Keenan wrote:
Wed Apr 01, 2020 4:10 pm
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?
Dave Keenan wrote:
Wed Apr 01, 2020 4:10 pm
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.

User avatar
Dave Keenan
Site Admin
Posts: 352
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

Re: New Olympian diacritics

Post by Dave Keenan »

Ash9903b4 wrote:
Thu Apr 02, 2020 12:24 am
How do you define a canonical form for edas? Do you just use the just intonated apotome that results from stacking seven fifths?
That's correct. [-11 7>
Dave Keenan wrote:
Wed Apr 01, 2020 4:10 pm
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.
Cool! Is that because Q(√5) is the smallest field that contains the noble numbers, and all elements of Q(√5) can be obtained as products and quotients of 0+√5, 1+√5, 2+√5, 3+√5, 4+√5, ... which are the primes of Q(√5)?

Could it be, that to get the nobles, we only need the odd primes: 1+√5, 3+√5, 5+√5, ... ? That would be because nobles are of the form (i+mϕ)/(j+nϕ) where ϕ = (1+√5)/2 and i, j, m, n are whole numbers such that | in-jm | = 1.
Similarly, you can use the field Q(√2) and map √2 to 8539 half-tinas, I think.
I assume Q(√2) doesn't relate to the nobles, but to the "silver aristocratic" numbers (where the nobles are the gold aristocratics). Is that correct?
https://en.xen.wiki/w/Metallic_MOS
I'd be happy to just have the nobles, as I understand they maximally avoid nearby simple ratios.

User avatar
Dave Keenan
Site Admin
Posts: 352
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

Re: New Olympian diacritics

Post by 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 archetype being 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.

Post Reply