## Definition of the tina reviewed

FloraC
Posts: 15
Joined: Sat Nov 07, 2020 4:19 am

### Definition of the tina reviewed

So I'm considering the opposite practice of finding commas for the tina. I'm finding the tina for the commas.

You know, 8269edo is quite comparable with 8539edo, despite that the latter is more famous(?). Both are prime. Both are 27-odd-limit-consistent (and provide nothing new over 2460edo). But anyway I don't actually like using prime edos to measure the interval size.

Notice 8269 + 8539 = 16808. 16808edo is, as you've probably known if you're reading this, a 31-limit monster, using the val [16808 26640 39027 47186 58146 62197 68702 71399 76032 81653 83270>. So why don't we define the tina as 2\16808? Since it's equivalent to 1\8404 in 2.3.7.11.17.23.31 subgroup, you may as well say it's 1\8404.

Forget the eda stuff.

You may view it as a linear temperament of 8269 & 8539. 8269edo and 8539edo are flexible approximations thereof. Like what I posted before – a comparison of 224edo and 270edo – they're wonderful "edo twins".

Notation-wise, it doesn't really matter. All default ratios would remain the same using any of 1\8269, 1\8539 or 2\16808. Some would gain larger errors, compensated by others getting smaller.

1\: 10241/10240
2\: 5832/5831
3\: 4096/4095
4\: 3025/3024
5\: 2401/2400
6\: 2080/2079
7\: 1701/1700
8\: 382976/382725
9\: 131072/130977

except for the fractional-tina ratio 1515591/1515520 though, but it's easy to find one for 1\16808 (way more elegant than 1\17078).

For 8269edo, the schisma is mapped to 13 steps, not 14. In case of 16808edo, the schisma is 27 steps. That induces larger error for the schisma, yet I find it fascinating that the 5/7k is split into exactly 3 schismas, one for 5s, two for 7s (=5/7k - 5s), three for 5/7k, and four for 25/7k (225/224). It's also the case in 2460edo.

If what's already there can't be de-established, how about this? I might call 1\8269 the "major tina", 1\8539 the "minor tina" (the default), and 2\16808, the "mean tina".

Quick reference

Relative errors (prime 3 – 37)
• 8269edo: -5.49, -2.34, -1.79, -4.01, +6.40, -23.02, -11.26, -33.38, +35.52, -24.93, +3.01
• 8539edo: +0.52, +5.60, -0.37, -8.66, -5.48, +15.48, -5.30, +30.45, -29.97, +11.77, +47.77
• 16808edo: -4.97*, +3.26, -2.15, -12.66, +0.92, -7.54, -16.56, -2.94, +5.54, -13.16, -49.22
TE simple badness (prime limit 3 – 37, patent vals score the best in all cases)
• 8269edo: 17.32, 14.55, 13.13, 11.76, 15.45, 22.27, 21.20, 27.00, 38.44, 38.20, 37.08
• 8539edo: 1.64, 10.68, 10.29, 15.62, 15.32, 19.97, 19.41, 27.62, 33.67, 32.72, 39.66
• 16808edo: 15.68*, 18.98, 16.46, 19.08, 18.27, 17.18, 18.54, 17.66, 18.37, 18.06, 28.49
* contorted order-8.

Dave Keenan
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Location: Brisbane, Queensland, Australia
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### Re: Definition of the tina reviewed

Hi @FloraC. You show an impressive grasp of this terminology and the mathematics behind it. It's a shame you didn't turn up on this forum about 8 months ago.

But nothing about tinas is set in stone. In our SMuFL submission we were careful not to embed any comma definitions into their glyph names (or the mina glyph names), as we had (perhaps foolishly) done for all previous Sagittal glyph names. Once published, SMuFL glyph names can never be changed. The mina and tina comma definitions are only in their SMuFL descriptions, which can be changed.

The fractional-tina dot is particularly flexible, since we've said that it can represent any reciprocal-integer fraction of a tina. The 1/2 tina comma in its description is only the default.

It is a basic principle of Sagittal, that while every symbol must have a primary or default definition as a rational comma, they can always be redefined as representing so-called secondary commas, even in purely rational (untempered) intonations. Any secondary comma should of course be close to the primary in size.

Another basic principle of Sagittal is that symbols can represent tempered versions of their commas. For example, in notating EDOs. So those are two ways to view what you are doing here. And as such it is perfectly valid.

Dave Keenan
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Location: Brisbane, Queensland, Australia
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### Re: Definition of the tina reviewed

Responding more to the specifics of what you wrote:

Although 8539 and 8269 may not increase the prime or odd consistency limit over 2460edo, they do extend consistency to higher powers of the lower primes, e.g. 3 and 5. We've seen that this is more important than simple prime or odd limit, from our analysis of the Scala archive statistics on ratio popularity.

I think your definitions of "major tina", "minor tina" and "mean tina" are fine.

What are the units of your "relative errors"? Relative to what? I was expecting numbers less than 1.0.

FloraC
Posts: 15
Joined: Sat Nov 07, 2020 4:19 am

### Re: Definition of the tina reviewed

What are the units of your "relative errors"? Relative to what? I was expecting numbers less than 1.0.
It's "percent of edostep". Sorry for forgetting about the percentage sign.

I think I've got more to say about smufl. In fact, although the code name in smufl can't be changed, the descriptions can. They just changed the description of an accidental in HEJI. It was a 23- or 29-comma, now 23 only, since the addition of the supplement block.

Another thing I'm not sure about is adding a lot of descriptions for each accidental. For example, the accSagittal5CommaUp has this description:

5 comma up, (5C), 1° up [22 27 29 34 41 46 53 96 EDOs], 1/12-tone up

Basically it states three things: its role in ji, that in common edos, and that in 12edo-relatives.

But I'm not sure why simply "5 comma up" isn't sufficient for the description section. Does presenting the common edos and "1/12-tone up" (when its real size is far from that) make things clearer? The common edo enumeration isn't exhaustive, is it? Furthermore, if the 5C is a 1/12-tone, it implies 72edo, so why isn't the "1/12-tone up" part assimilated into the previous part?

Now imagine one is to use Sagittal for edos. The user will figure out a list of accidentals for a given edo (and not the opposite). Even if they've studied the system, they have to look up the website anyway. Like how do they know 1\58edo is 5C? And 2\34edo? 2\41edo? Otherwise, if one has no knowledge that the Sagittal accidental is intended for three different usages, they can't acquire it by looking at the description of a specific accidental.

Then the MuseScore team imported all the descriptions to their ui, and it clutters the scene. Should I report to them that I don't feel like seeing all of those?

Dave Keenan
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Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
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### Re: Definition of the tina reviewed

FloraC, you might be able to help us with something else, although it is rather off-topic. Two days ago we implemented transport layer security for this forum, and so its URL changed from http: to https:.

Since then, my co-admin, cmloegcmluin, has experienced problems where he is asked to log in every time he comes back to the forum. The "Remember me" checkbox apparently having no effect.

Are you experiencing this behaviour?

Another symptom is that, even when logged in, the URLs of forum pages have something like
?sid=13334c8df1887826fc9b47d88a0abc4d
added to the end of them.

Are you experiencing this?

FloraC
Posts: 15
Joined: Sat Nov 07, 2020 4:19 am

### Re: Definition of the tina reviewed

I'm often auto logged out after a short period of time, and I do see something like

&sid=259dd11a2b55f7d5891efc7c5867b8df

I didn't know those are bugs.

Dave Keenan
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Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
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### Re: Definition of the tina reviewed

Those symptoms would also occur if you have cookies blocked for "forum.sagittal.org" (or disabled for all domains).

Have you blocked or disabled cookies?

FloraC
Posts: 15
Joined: Sat Nov 07, 2020 4:19 am

### Re: Definition of the tina reviewed

Dave Keenan
Posts: 1574
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
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### Re: Definition of the tina reviewed

Thanks for that help with our networking problems, FloraC. We're working on it.
FloraC wrote: Mon Jan 11, 2021 5:32 pm I think I've got more to say about smufl. In fact, although the code name in smufl can't be changed, the descriptions can.
Yes. I mentioned that above.
Another thing I'm not sure about is adding a lot of descriptions for each accidental. For example, the accSagittal5CommaUp has this description:

5 comma up, (5C), 1° up [22 27 29 34 41 46 53 96 EDOs], 1/12-tone up

Basically it states three things: its role in ji, that in common edos, and that in 12edo-relatives.
That's exactly right. It is designed to inform 3 kinds of people with 3 different ways of approaching microtonality.
But I'm not sure why simply "5 comma up" isn't sufficient for the description section. Does presenting the common edos and "1/12-tone up" (when its real size is far from that) make things clearer?
For someone interested in JI, the common EDOs and 12-relative info adds nothing.
The common edo enumeration isn't exhaustive, is it?
That's correct. SMuFL descriptions should be kept relatively short. So we limited ourselves to some common EDOs.
Furthermore, if the 5C is a 1/12-tone, it implies 72edo, so why isn't the "1/12-tone up" part assimilated into the previous part?
Because the kind of person who thinks in 12-relative terms, may not recognise 1°72-edo as equivalent to 1/12-tone. But we figured EDO folks would recognise that, so we saved some space by avoiding that redundancy.
Now imagine one is to use Sagittal for edos. The user will figure out a list of accidentals for a given edo (and not the opposite). Even if they've studied the system, they have to look up the website anyway. Like how do they know 1\58edo is 5C? And 2\34edo? 2\41edo?
Yes. That is true.
Otherwise, if one has no knowledge that the Sagittal accidental is intended for three different usages, they can't acquire it by looking at the description of a specific accidental.
That may be so. But they should be able to aquire it by reading the SMuFL "Implementation note" at the bottom of the first page of Sagittals.
https://w3c.github.io/smufl/gitbook/tab ... ntals.html
Then the MuseScore team imported all the descriptions to their ui, and it clutters the scene. Should I report to them that I don't feel like seeing all of those?
Sure. The fact that represents a 5-comma is certainly the most important fact about it. Instead of including the SMuFL descriptions (for any SMuFL symbols, not only Sagittal), they could instead include a link to the relevant page of the SMuFL documentation.

We have also found that the SMuFL glyph names (which are in lower camel case) can serve as useful short descriptions if they are broken up into their component words by inserting spaces and uncapitalising.