List of 7-prime limit accidentals

[Xen-Gedankenwelt]

Ok, I finished the list of 7-limit ratios (including lower limits) with an exact Olympian representation:

7-limit Olympian set.pdf

(24.95 KiB) Downloaded 447 times

The list only contains up-accidentals, and I omitted cent values, systematic comma names, and very large ratios (monzos are still listed). If the data is fine, I'll update the list in my opening post.

However, I noticed that there seem to be two errors in sag_ji4.par:
- 6656/6480 should be reduced to 416/405
- 576/572 doesn't make sense: If reduced, it becomes 144/143, but that doesn't match with the specified accidental / cent range

In addition, and don't seem to be an accurate representation of 9:10. A double-apotome minus is which represents the 19-limit ratio 39/38, so has to be a 19-limit ratio, too. Still, it seems to be a close approximation for 10/9, but not completely accurate.

[Dave Keenan]

Ok, I finished the list of 7-limit ratios (including lower limits) with an exact Olympian representation:
7-limit Olympian set.pdf

Nicely done. You could include the proper Sagittal symbols in the PDF, at higher resolution than the forum smilies, if you install the Bravura font. Unfortunately, there isn't yet an easy way to type the symbols, as there is for the forum smiles. But it would be possible to make a Windows keyboard layout that would let us do so. Do you use Microsoft Windows?
[Edit: Easy typing of Sagittal symbols is now available. See: viewtopic.php?p=808#p808]

I note the potential for confusion of the vertical bar | with the digit 1. Some fonts are worse than others, for this. That's why I use square brackets for bra <...] and ket [...> vectors except when writing an inner product <... | ...>, in which case there will be a space on either side of the vertical bar. An alternative to using square brackets is to always include a space between the vertical bar and any digits.

However, I noticed that there seem to be two errors in sag_ji4.par:
- 6656/6480 should be reduced to 416/405
- 576/572 doesn't make sense: If reduced, it becomes 144/143, but that doesn't match with the specified accidental / cent range

Thanks for those. I'm pleased to see that we do have the reduced value 416/405 for in George's JI notation spreadsheet, the JI notation levels diagram and in footnote 19 on page 24 of http://sagittal.org/sagittal.pdf.

I see that 576/572 was erroneously given for in sag_ji4.par. George's spreadsheet gives the 13-limit comma 567:572 for that symbol. So it seems "67" was accidentally transposed to "76" when making sag_ji4.par.

I have now corrected both of those in http://sagittal.org/sag_ji4.par. Thanks.

In addition, and don't seem to be an accurate representation of 9:10. A double-apotome minus is which represents the 19-limit ratio 39/38, so has to be a 19-limit ratio, too. Still, it seems to be a close approximation for 10/9, but not completely accurate.

My apologies. You are correct that is not an exact representation of 9:10. That symbol represents a 19-limit ratio, as you say. When untempered they differ by only 0.003 cents. The only exact representations of 9:10 involve a change of nominal, e.g. C:D or C:E = C:E

[Xen-Gedankenwelt]

Thanks for your reply!

As a start, I separated the ratios into two lists, depending on whether they have an exact representation in the Olympian symbol set, or not.

Here is my current to-do list:

• Complete the ratios and accidentals in the first list (exact / Olympian)
• Add accurate dual-accidentals and approximate single-accidentals in the second list
• Add systematic comma names

I can also modify the monzos in the pdf for better readability. Maybe I'll also add proper Sagittal symbols later (yes, I'm using Windows), but I first want to complete the lists.


Just to make sure that I understand systematic comma names correctly:
25:28 = +-[4 0 -2 1> (not in Olympian set) would be 7:25MS+A, and 1792:2025 = +-[-8 4 2 -1> (contained in Olympian set) would be c7:25MS+A, correct?

Some of the ratios in the Olympian set have a somewhat high absolute value of 3-exponent, and the highest I found is [-43 24 1 1>. Is it guaranteed that there is at most one less complex ratio (in terms of absolute value of 3-exponent), or do I have to check if there are more? In the latter case, would I prepend cc, ccc and so on?

P.S.: Working with those lists is a great way for me to understand and learn Sagittal notation - thanks for your help so far!

[Dave Keenan]

Just to make sure that I understand systematic comma names correctly:
25:28 = +-[4 0 -2 1> (not in Olympian set) would be 7:25MS+A, and 1792:2025 = +-[-8 4 2 -1> (contained in Olympian set) would be c7:25MS+A, correct?

Those are absolutely correct. Well done.

Some of the ratios in the Olympian set have a somewhat high absolute value of 3-exponent, and the highest I found is [-43 24 1 1>. Is it guaranteed that there is at most one less complex ratio (in terms of absolute value of 3-exponent), or do I have to check if there are more? In the latter case, would I prepend cc, ccc and so on?

I think you have to check if there are more. I think it is guaranteed that all single-shaft symbols are the least complex in their size category. But when you add them to an apotome (3-shaft), or subtract them from one or two apotomes (2-shaft and X-shaft), you may end up with some that are not the least complex. It's possible that someone could prove there is at most one less complex ratio for these, but I don't know of such a proof.

I have an Excel spreadsheet that automatically names any ratio up to the double-apotome. I'd be happy to make it available after some tidying up. Would this be of any use to you? Do you have a recent version of Excel?

It presently uses:
complex (c)
supercomplex (sc)
hypercomplex (hc)
ultracomplex (uc)
5-complex (5c)
6-complex (6c)
...

But if you think the ordering of super, hyper, ultra is not sufficiently standardised, you could use 2c, 3c, 4c instead of sc, hc, uc.

P.S.: Working with those lists is a great way for me to understand and learn Sagittal notation - thanks for your help so far!

My pleasure. It is good to have someone else who understands the mathematical details underlying Sagittal.

[cmloegcmluin]

It's possible that someone could prove there is at most one less complex ratio for these, but I don't know of such a proof.

These boundaries were carefully chosen so that commas can be named systematically using only their prime factors greater than 3

Interesting!

I had been wondering for some time what exactly was meant by "carefully chosen".

Trivially, we could make a comma more complex by adding, say, [ -50508 31867 ⟩, which is only 0.012577¢, and therefore small enough in size that it's unlikely to change its size category. But of course we'd notice right away if any of our commas had astronomically large prime exponents like that.

To do the reverse — check that none of the commas have a simpler representation (small absolute 3-exponent) — would require considering every 3-limit comma less than the area between the lower and upper bound of that size category.

For instance, since the small diesis size category ranges from 33.38249264¢ to 45.11249784¢, any 3-limit comma less than 11.7300052¢* could potentially be added to a given comma to get another comma which under Sagittal comma-naming rules would be given the same name; and if a comma thus found ever had a smaller absolute 3-exponent than the original comma, we would have found a new simplest comma that should be represented by that name. And if that comma name had been used in Sagittal but to represent the more complex comma with a different value, we would have thus discovered an exception in Sagittal.

If the above assessment is accurate, it should be straightforward to script up a test that Sagittal commas are in their simplest form. I also predict that they already are (i.e. that the Sagittalsmiths have already gotten it right).

I have an Excel spreadsheet that automatically names any ratio up to the double-apotome. I'd be happy to make it available after some tidying up. Would this be of any use to you? Do you have a recent version of Excel?

Interested, yes.

*Half the Pythagorean comma, as well as the upper limit of the kleisma size category.

[Dave Keenan]

I agree with your argument. But there's no need to check. That property was guaranteed by generating candidates with steadily increasing 3-exponents, starting from zero and working in both positive and negative directions.

Here's the Comma Namer spreadsheet.

[cmloegcmluin]

Trying to reverse engineer this. Only 2 examples of complex ('c'-prefixed) names, nothing more complex than that, and only 1 of those 2 examples has a non-complex example to foil against. So at this point I guess one 'c' level corresponds to a movement by one schisma (Mercatur's comma)?

[Dave Keenan]

When aa wrote that spreadsheet back in 2004, only me an' Gaad knew how it worked. Now ...

Yes., the dearth of "complex" commas shows how effective our choice of size-category boundaries is. My attempt to explain this on the Yahoo tuning group when it was first developed, severely broke down. I made the mistake of temporarily adopting the spelling "komma" for the purpose of my exposition, to distinguish the generic term from the specific size category "comma". Despite the fact that I explained this at the start of my exposition, people who seemed determined to disagree with anything I wrote, seized on this as a reason to express their outrage and read no further. It's all in your resurrected archive in all its horror.

But even after we got over that: To this day, I don't understand why Paul Erlich (not one of those described above) is freaked out by this comma size category proposal. I have never managed to understand what he's afraid of, unless it's the whole idea of defining boundaries for the historical names, no matter how useful or well founded, or how consistent they are with the historical names. How many commas in that spreadsheet have "schisma", "kleisma", "diesis", "limma" or "apotome" in their common names, but not in their systematic names? I omit "comma" from this list, as one can always take "comma" in its generic sense. At least Joe Monzo likes the category boundaries. It completed a project that he began.

Of course every right square-bracket in that spreadsheet ] should now be replaced with a right angle-bracket ⟩ so the monzos have their modern form.

Do you know about those leading commas in the monzos, like [,5 7⟩ ? That tells you that the exponents for primes 2 and 3 have been omitted, i.e. that it's a 2,3-free monzo. This is a standard that was invented by George Secor, and approved by the Monz himself, but not widely used. It includes indicating a 2-free monzo by a comma after the first exponent, e.g. [3,5⟩ . So the comma is always after the 3-exponent. It optionally includes using a comma for every third exponent from then on. e.g. [2 3,5 7 11,13 17 19,23 29 31,37 41 43, ...⟩ . I'm using the primes here to stand for their own exponents.

[cmloegcmluin]

When aa wrote that spreadsheet back in 2004, only me an' Gaad knew how it worked. Now ...

I know alllllllllllll too well how you feel...

This is a perfect example of where automated regression tests could protect your achievements. I can run a one-time check of comma complexity, of course, but I can also capture it as a test we can rerun whenever, so if we ever want to change anything later we'll be told instantly if we regress w/r/t this property (or any other). And if we want to tweak the conditions, the code will tell us unambiguously what our intentions were before.

Yes., the dearth of "complex" commas shows how effective our choice of size-category boundaries is. My attempt to explain this on the Yahoo tuning group when it was first developed, severely broke down. I made the mistake of temporarily adopting the spelling "komma" for the purpose of my exposition, to distinguish the generic term from the specific size category "comma". Despite the fact that I explained this at the start of my exposition, people who seemed determined to disagree with anything I wrote, seized on this as a reason to express their outrage and read no further. It's all in your resurrected archive in all its horror.

Found it. Sad. I know some of these folks personally.

How many commas in that spreadsheet have "schisma", "kleisma", "diesis", "limma" or "apotome" in their common names, but not in their systematic names?

I know the question was rhetorical, but the answer is: two. The tridecimal schisma is known in Sagittal as THE schismina, and the small BP diesis is a kleisma. So I think the systematic names are pretty well justified.

Do you know about those leading commas in the monzos, like [,5 7⟩ ? That tells you that the exponents for primes 2 and 3 have been omitted, i.e. that it's a 2,3-free monzo. This is a standard that was invented by George Secor, and approved by the Monz himself, but not widely used. It includes indicating a 2-free monzo by a comma after the first exponent, e.g. [3,5⟩ . So the comma is always after the 3-exponent. It optionally includes using a comma for every third exponent from then on. e.g. [2 3,5 7 11,13 17 19,23 29 31,37 41 43, ...⟩ . I'm using the primes here to stand for their own exponents.

The [,5 7⟩ ones were immediately obvious to me, for what it's worth. Though I think I had seen such abbreviations on Tonalsoft before, so I had been primed *ba-dum tshh*

I did not know about [3,5⟩ or [2 3,5 7 11,13 17 19,23 29 31,37 41 43, ...⟩, but that's a lovely thing indeed. Added to my toolbelt. Thanks for explaining!

[Dave Keenan]

It also allows things like writing single-prime-factor-above-3 commas (like in the Sagittal Prime Factor notation) as:

[2 3,5⟩
[2 3,0 7⟩
[2 3,0 0 11⟩
[2 3,,13⟩
[2 3,,0 17⟩
[2 3,,0 0 19⟩
[2 3,,,23⟩
[2 3,,,0 29⟩
[2 3,,,0 0 31⟩
[2 3,,,,37⟩

etc.
But that's probably pushing things.