Notation for Fibonacci tuning (Wilson's horogram #22)?

[Dave Keenan]

The Trojan accidentals are not very good at flag arithmetic, so I wasn't very worried about that, but if it's important to you we may try to fix it! I haven't found a good accidental between and in the graph, but I think I have a solution: use instead of for -68g, and thus becomes free of meaning and can be used as a flag in = +131g. You can also use as a synonym for = +165g, if you like, but I'm not sure I'd advise to do that as is actually about 0.7¢ larger than for the golden fifth size. This proposition makes some problems with the flag arithmetic considerations you made afterwards, but I don't think we should place a lot of weight on them, as they involve double shaft accidentals, which don't have flag arithmetic in Trojan anyway, and which isn't used here. What do you think? Do you have another solution?

I spent a ridiculous amount of time investigating the possibilities for this Fibonacci tuning notation, with a number of spreadsheets. I found the following 37-limit map that gives no error greater than 1.6 cents in any prime
⟨0 12 82 44 30 -61 -149 -36 93 -92 191 66].
I applied it to every comma up to the half-apotome that has an Olympian notation.

There is no unaccented symbol corresponding to 131 generators. And there is no accented version of corresponding to 131 generators, which is equivalent to saying that there is no secondary comma for that corresponds to 131 generators. Therefore there is nothing to be gained by not using for -68g.

And it is preferable to use rather than for -68g because notates simpler ratios (ratios of 7 rather than 25).

I have only two suggestions for notating 131g. My preference is for . This is not valid according to any Olympian comma for that symbol, but only becomes valid when we define as -233g by fiat, simply because it is such a useful thing to do for this notation and it does at least give a size that is similar to its untempered size.

Another possibility is to use the fact that the 11/17-small-diesis corresponds to 131g with the above mapping, and its symbol is which falls between and in untempered size order. The accent could be dropped, which is equivalent to saying we could use with a secondary comma definition of 11/17S.

The final spreadsheet is attached.

[רועיסיני]

I spent a ridiculous amount of time investigating the possibilities for this Fibonacci tuning notation, with a number of spreadsheets. I found the following 37-limit map that gives no error greater than 1.6 cents in any prime
⟨0 12 82 44 30 -61 -149 -36 93 -92 191 66].

, very impressive.

My preference is for .

It's a bit sad that the 144-note MOS cannot be notated with monotonic accidentals and without accents (although the 89-note one can), but if you prefer that I'll go with it.

[Dave Keenan]

That map was primarily chosen to preserve the generator counts of those Trojan-derived symbols that we had already agreed on, and which we agreed on. In the end, the generator counts for primes greater than 23 were not relevant, nor was the generator count for prime 19. But they were chosen to have the smallest absolute values that gave errors smaller than (or not much larger than) those of the primes whose mappings were already determined by the agreed symbols.

You say it is sad that the 144-note MOS cannot be notated with monotonic accidentals and without accents. So why do you not argue for the use of as the symbol for 131g?

I think we owe it to future readers to consider that option more thoroughly. So I just attempted to assign generator counts, and hence cents values, to the flags under such a scheme. It is possible to do so consistently, but it requires that be assigned a large negative value of cents, namely -19.5 ¢. Negative values for upward-pointing flags are a bad idea. I will use <...> as brackets here to indicate that only the flags are being referenced, in cases where the symbol is not allowed in the notation.

This occurs because <> = - = 45.1 - 19.5 = 25.5 ¢ and <> = - <> = 6.0 - 25.5 = -19.5 ¢

Whereas, if we don't use the values of <> and <> are underconstrained and can be given the more reasonable assignments of <> = 0 ¢ and <> = 6.0 ¢ (-89g), thereby also implying a reasonable value of <> = 3.7 ¢ (144g).

Another reason not to use or any other unaccented symbol for 131g is that once we agree that nothing is gained by not using for -68g, we might as well also use for -102g because although it is not required, it is useful in avoiding having both sharps and flats applied to D. Or to put it another way, to use no more than 13 pseudo-nominals Ab Eb ... C# G# in a chain of fifths. And once we have = -102g and = -233g introduced for other reasons, it seems better to combine them to give a symbol for 131g rather than introduce a new symbol.

Yes, it's a shame we don't get to the 144-MOS without accents, but we must not allow the notations for more complex scales/tunings to make the notations for simpler scales/tunings more complicated than they need to be.

... it seems much more intuitive when I'm writing text (and not sheet music) to have the accent to the right and not to the left of the accidental, much like writing an accidental to the right and not to the left of the note name. I corrected my previous post but maybe now when the accidentals are differentiated only by shape and not by place it's possible to make an exception for people like me that when writing text the accents can come after the accidental, but the accidental and the accents still have to come in monotonic order.

That's fine, since you have a good reason for doing it. I just thought you might have been following the old convention.

But there is also a school of thought that says that an accented symbol should be thought of as a single symbol that has only been broken into multiple characters to minimise the number of characters in the font, like diacritics on letters, like "é".

There is even a sense in which combinations like and particularly are like a single symbol whose geometrical beauty and rapid recognition is lost when rearranged as and  . So nowadays I am inclined to write them in text in the same order as on the staff. i.e. staff o text E . It's tempting to go all the way and write E.

[רועיסיני]

Okay, now I fully agree that the correct symbol is And not . It's more of a hassle but it's worth it.
Also, I didn't really think about evo symbols when writing the order exception but I agree they break it.

[Dave Keenan]

@רועיסיני Please check the following summary for accuracy.

The extended Trojan notation we have agreed on, as a chain-of-fifths notation for the Fibonacci tuning, is:

Acc	Gens	Cents
----------------------
D:A	+12	700.31
 = 	+84	102.17
	-89	6.03
	+55	9.76
	-34	15.79
	110	19.51
	+21	25.54
	-68	31.57
	+76	35.30
	-13	41.33
	-102	47.36
	+42	51.09

Symbols for larger alterations are obtained in the usual manner, by adding these symbols, or their inverted versions, to the left of sharps or flats (evo), or by using the standard apotome-complements as shown in figure 13 of https://sagittal.org/sagittal.pdf (revo).

Any of these can be raised or lowered by 2.30 ¢ (-233 generators) by the use of schisma accents and to the left of all other symbols.

Some notable accented symbols are:

Acc	Gens	Cents
----------------------
	-233	2.30
	144	3.73
	-123	21.82
	131	45.06

Some notable revo symbols for possible use in MOS-nominals notations are:

Acc	Gens	Cents
----------------------
	-5	108.20
	+8	66.87
	+16	133.75 [Edit]

@herman.miller in case you're still interested.

[רועיסיני]

Looks good to me.

[רועיסיני]

I just noticed that +16 generators can be notated using = , so when using 8-note MOS nominals there is also a symbol for the double chroma.

Also, speaking about the 8-note MOS notation, I saw you notated the 8-note MOS as L O J M P K N Q in generator order:

In that case they would be L O J M P K N Q in generator order and M would behave like D in the chain-of-fifths case.

However, in oneirotonic notation Q Is a generator away from L, not from N, so I think it's better to stay consistent with that and call the nominals Q L O J M P K N in generator order, which means that J K L M N O P Q, just like in usual oneirotonic, notates the LsLLsLsL MOS. That may also mean that J, and not M, will be treated like D.

[Dave Keenan]

I just noticed that +16 generators can be notated using = , so when using 8-note MOS nominals there is also a symbol for the double chroma.

Thanks! I added it to the table above.

... in oneironic notation Q Is a generator away from L, not from N, so I think it's better to stay consistent with that and call the nominals Q L O J M P K N in generator order, which means that J K L M N O P Q, just like in usual oneironic, notates the LsLLsLsL MOS. That may also mean that J, and not M, will be treated like D.

You are so right! I don't know why I failed to realise that Fibonacci tuning does not merely have the same number of MOS nominals as oneirotonic tunings, it actually is an oneirotonic tuning, because its generator size is within the oneirotonic range.