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

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Dave Keenan
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Re: Notation for Fibonacci tuning (Wilson's horogram #22)?

Post by Dave Keenan »

[I had not read your above post when I posted the following,]
רועיסיני wrote: Wed May 24, 2023 4:23 pm I don't know what made you think that I don't like the Trojan capture zones.
I did not think you don't like them. But you seemed to allow that they might not be strictly adhered to, when you wrote:
However, if you want to just borrow the accidentals from 144edo, and you don't intend to use them as a part of a larger system or care about the Trojan symbols boundaries when applied to the Fibonacci tuning itself, you can use :~|(: to notate +55 generators ...

You wrote:
I think they are quite handy and a valid consideration when the fifth is between 699.1831¢ and 700.7552¢ ...
I agree, and I'm impressed by how much you have read of this forum.
Why did you choose to use :|): for -68 steps and not for +76? It's indeed a smaller number in absolute value, but +76 gives you an apotome complement of +8 generators, which is much better than the +152 generators that -68 gives you. Also, having :|): as an accidental for -68 and not an accidental for +76 gives you the problem in the beginning of my last post, where D:!!!): is below C:|||): and the distance between them is only 8 generators.
I strongly prefer having an accidental for +76 generators, ...
I now accept that we need accidentals for both -68 g (31.57 ¢) and +76 g. (35.30 ¢). The Trojan boundaries and flag arithmetic give these as :|): and :|\: respectively.

I also accept that we need both 55 g (9.76 ¢) and -89 g (6.03 ¢). My solution for these is to introduce a non-Trojan symbol :)~|: (11.13-comma, 144/143) for 55 g (justified here), and use :~|(: for -89 g as you suggest.

I find that in order to avoid nominal crossings, symbols for the following numbers of generators must be introduced in the following order, as the generator chain grows in both the positive and negative direction, up to 144 notes. This was done with 0 g notated as D (the point of symmetry in the chain of fifths), but I'm not sure if that matters.

First	Value of
occurs	accidental	Accidental
(g)	(g)	(¢)
----------------------------------
1	-13	41.33	:/|~:
2	-34	15.79	:/|:
3	21	25.54	:|~:
4	-68	31.57	:|):
-4	76	35.30	:|\:
5	55	9.76	:)~|:
6	42	51.09	:/|\:
-17	-89	6.03	:~|(:
-30	-102	47.36	:/|):
-38	110	19.51	:(|:
-51	-123	21.82	
-59	131	45.06	

Symbols for -123 g and 131 g would need to use accent marks, so I prefer not to allow them, but instead to say that we can only notate up to the 89-note MOS of the Golden tuning.
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Re: Notation for Fibonacci tuning (Wilson's horogram #22)?

Post by רועיסיני »

I'm glad that we both found :)~|: to be the best symbol for +55g, but I think your calculation is pessimistic – for a start, :|): can be pushed way down, to at least 28, as :|\: can take its place in the beginning. Similarly, :~|(: can be probably pushed to at least 31. I think the problem is that when choosing between two accidentals you always select the one whose generator count has smaller absolute value, which is not necessarily the best one. You need to choose the accidental that takes the smallest number of steps in the EDO where the apotome vanishes, because every accidental can be added or subtracted from any whole number of apotomes (which is easier to see in revo as you literally "get for free" 3 more sagittals). That EDO in this case is 21-edo, which gives 2 steps for :|): = -68g and -1 step for :|\: = 76g, and so because |2|>|-1|, :|\: should be preferred.

I also think that the whole 144-note MOS can be notated with monotonic nominals and without accents, if you use :/|): for +131g (-2 steps in 21edo) instead of -102g (3 steps in 21edo). The way to do this is:
  1. Notate the middle 55-note MOS inside it with only :)~|:, :/|:, :|~:, :|\:, :/|~: and :/|\:, which I'll call the simple accidentals. Note that for every simple accidental there are accidentals 55g away from it in each direction.
  2. Every -34g step divides into +55g and -89g, notate the middle note with the accidental that is 55g from the fitting note in the correct direction.
  3. Every +21g step divides into two +55g steps and one -89g step, and crucially the -89 step is always between the two 55g steps. That's because two 55g steps from any note in the small MOS will get to at least -27 + 55*2 = 83 > 72 steps from the middle note, which means it will not be in the large MOS. Therefore, you can notate each of the two middle notes with the accidental that is 55g away from the note close to it.
However, for the 233-note MOS we have to use accents, as +84g above D has to be D:/||\: and -81g above D has to be E:!!!~:, but the former is higher than the latter. I'm not afraid of accent marks but I agree that they should not be used when possible. However, if someone feels they need more accidentals, I think the schisma accent :': for -233g is a good choice, as it allows you to write, for example, -102g as :'::/|):, -123g as :'::(|: and -81g as :'::||):.
Last edited by רועיסיני on Thu May 25, 2023 8:10 pm, edited 1 time in total.
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Re: Notation for Fibonacci tuning (Wilson's horogram #22)?

Post by רועיסיני »

In fact, if you're only using accidentals for -89, +55, +34, +110, +21, -68, +76, -13, -102 and +42 generators without an accidental for +131 generators you can't notate the 144-note MOS with monotonic accidentals. That's because if you take the diatonic note that's in the 12 middle ones and assume without loss of generality that it is D, then the note 59 generators before it has to be notated as A:b::\!~: (as that's the only accidental that is 1 or 11 modulo 12, an accidental for +131g will fix that) and the note 51 generators after it has to be notated as G:#::!~: (as that's the only accidental that is 3 or 9 modulo 12, an accidental for -123g may fix that), but the former is lower than the latter and therefore the nominals must cross between them.
I'm not sure if having an accidental for -123g will fix the entire MOS or just this problem, but an accidental for +131g will fix it without needing one for -102g, as I showed in the previous comment. That's because then you'll cover exactly all the numbers that are 0, 5, 8, 13 or 16 modulo 21.
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Re: Notation for Fibonacci tuning (Wilson's horogram #22)?

Post by Dave Keenan »

Thanks for your corrections to my proposed 34-MOS notation. In making further suggestions you made a typo where you wrote: "A:b::|\: (or A:!!/:)". That should be A:\!!: . A mnemonic is that the slope of a lone barb never changes in going between evo and revo. This is true for both left and right barbs, and for any number of shafts.

It is good that our different approaches are converging. I see now, that -102 g is not required to notate any MOS up to and including the 144-MOS, as its potential uses can be replaced by uses of 131 g, which is required for the 144-MOS anyway (as are 110 g and -123 g).

However, I can't accept :/|): as a notation for 131 g as it violates the flag arithmetic that has :/|: + :|): = -34 + -68 = -102 g. So 131 g needs to be accented :.::/|): . And therefore it seems better to use the unaccented -102 g in place of 131 g where possible.

I haven't managed to understand your procedure for how to notate the 144-MOS with monotonic nominals without accents, even if we allow 131 g to be :/|): . It seems to me that both of 110 g and -123 g (or their apotome complements) are required. And one of those must be an accented form of the other.

In fact, flag arithmetic says that :(|: must be 110 g as follows. :(|: = :(|): - :|): = (:#: - :/|\:) - :|): = (84 - 42) - -68 = 42 + 68 = 110 g. And so -123 g must be :'::(|: .

I agree we can assign -233g to :': .

I see that the introduction of the -68 g accidental :|): can be delayed by additional uses of 76 g :|\: . However :|): notates simpler, and therefore more audible, harmonies than :|\: and so I prefer to introduce it as soon as we go beyond the 13-MOS.

I have been assuming that the canonical notation for the smaller MOS will be the same as the notation they would get as a subset of the 144-MOS notation when centered on 0 g = D (for odd cardinality) or centered on the generator between 0 g and 1 g = D:G:\!~: (for even cardinality).

I should have mentioned earlier: I corrected a bunch of places where I had "D:G" when it should have been "D:A". I took the liberty of correcting them in your post too, as it was my error that simply got replicated.
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Re: Notation for Fibonacci tuning (Wilson's horogram #22)?

Post by Dave Keenan »

Because this tuning is so thoroughly divorced from a chain of fifths that it requires 6 up/down pairs of accidentals for just 13 notes, a composer might well choose to use MOS nominals instead of chain-of-fifths nominals.

Miller's law says this would have to be based on either the 5-MOS or the 8-MOS. The 5-MOS has the advantage of being able to use standard staves with 7 positions to the octave (skipping 2 positions).

The notes of the 5-MOS could be named (in generator order) F A D G B. The A and the G are more like A semi-sharp and G semi-flat. And so could be written on the staff using Stein-Zimmermann accidentals — something I refer to as "24edo-based pseudo-nominals". F A:>: D G:<: B. Then, to extend the chain we need to find Sagittal accidentals for ±5, 10, 15, 20, ... generators. Or we can just use the accidentals for ±5 g and stack them side by side. e.g.

F:~|||(::~|||(: A:>::~|||(::~|||(: D:~|||(::~|||(: G:<::~|||(::~|||(: B:~|||(::~|||(: F:~|||(: A:>::~|||(: D:~|||(: G:<::~|||(: B:~|||(: F A:>: D G:<: B F:~!!!(: A:>::~!!!(: D:~!!!(: G:<::~!!!(: B:~!!!(: F:~!!!(::~!!!(: A:>::~!!!(::~!!!(: D:~!!!(::~!!!(: G:<::~!!!(::~!!!(: B:~!!!(::~!!!(:

If using the 8-MOS as nominals, they could be, in pitch order, I J K L M N O P. With :/||: for +8 generators. In generator order (-3 g to +4 g): K N I L O J M P. So L here behaves like D does in the chain-of-fifths case.

[Edit: It might be better to avoid the letter I and instead use J K L M N O P Q as in oneirotonic notation. 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.]
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Re: Notation for Fibonacci tuning (Wilson's horogram #22)?

Post by רועיסיני »

Dave Keenan wrote: Thu May 25, 2023 12:20 pm Thanks for your corrections to my proposed 34-MOS notation. In making further suggestions you made a typo where you wrote: "A:b::|\: (or A:!!/:)". That should be A:\!!: . A mnemonic is that the slope of a lone barb never changes in going between evo and revo. This is true for both left and right barbs, and for any number of shafts.
Thanks too! I edited my post with a correction. I remembered the slopes remains the same when adding to a sharp or a flat but I probably god confused when I subtracted an accidental from a sharp or a flat and forgot to reverse the slope there. I also had some more mistakes in the 55-note NOS which I corrected as well.
Dave Keenan wrote: Thu May 25, 2023 12:20 pm It is good that our different approaches are converging. I see now, that -102 g is not required to notate any MOS up to and including the 144-MOS, as its potential uses can be replaced by uses of 131 g, which is required for the 144-MOS anyway (as are 110 g and -123 g).
-123g is not required for the 144-note MOS, as it can be notated with (monotonic nominals and) only accidentals for -89, +55, -34, +110, +21, -68, +76, -13, +131 and +42 generators. There may be other sets that would allow you to notate the 144-note MOS, which may not include, say, +131g but will have lo include accidentals for other values, like -123g. I'm not sure which sets work and which don't, I will probably check that later today.
Dave Keenan wrote: Thu May 25, 2023 12:20 pm However, I can't accept :/|): as a notation for 131 g as it violates the flag arithmetic that has :/|: + :|): = -34 + -68 = -102 g. So 131 g needs to be accented :.::/|): . And therefore it seems better to use the unaccented -102 g in place of 131 g where possible.
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?
Dave Keenan wrote: Thu May 25, 2023 12:20 pm I haven't managed to understand your procedure for how to notate the 144-MOS with monotonic nominals without accents, even if we allow 131 g to be :/|): . It seems to me that both of 110 g and -123 g (or their apotome complements) are required. And one of those must be an accented form of the other.
Which part of the procedure didn't you understand? I edited the last point to make it clearer, maybe this will help.
Dave Keenan wrote: Thu May 25, 2023 12:20 pm I see that the introduction of the -68 g accidental :|): can be delayed by additional uses of 76 g :|\: . However :|): notates simpler, and therefore more audible, harmonies than :|\: and so I prefer to introduce it as soon as we go beyond the 13-MOS.

I have been assuming that the canonical notation for the smaller MOS will be the same as the notation they would get as a subset of the 144-MOS notation when centered on 0 g = D (for odd cardinality) or centered on the generator between 0 g and 1 g = D:G:\!~: (for even cardinality).
I see. I'm not sure how relevant this becomes if we use ://|: instead of :|): anyway, but I'm fine with having notations for smaller MOSes that are different from their notations as the middle of larger ones, as long as they are all compatible.

The MOS nominals idea is good too IMO, but I think the CoF nominals are still worth considering, especially because we already found quite a good solution, we just need to improve it a bit for the point we are both fine with it.
Last edited by רועיסיני on Thu May 25, 2023 10:06 pm, edited 1 time in total.
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Re: Notation for Fibonacci tuning (Wilson's horogram #22)?

Post by Dave Keenan »

רועיסיני wrote: Thu May 25, 2023 8:09 pm The Trojan accidentals are not very good at flag arithmetic, ...
I believe that all 12n-EDOs up to 204edo, and 240edo, have consistent flag arithmetic. It was a Trojan design criterion to maximise this.

Tomorrow I will investigate your ://|: proposal and try again to understand how the 144-MOS does not need -123 g.
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Re: Notation for Fibonacci tuning (Wilson's horogram #22)?

Post by רועיסיני »

Dave Keenan wrote: Thu May 25, 2023 8:48 pm
רועיסיני wrote: Thu May 25, 2023 8:09 pm The Trojan accidentals are not very good at flag arithmetic, ...
I believe that all 12n-EDOs up to 204edo, and 240edo, have consistent flag arithmetic. It was a Trojan design criterion to maximise this.
Yes, but only for single-shaft symbols. Also, according to the spreadsheet, in Olympian :)|: + :|(: = :,::)|(: so I thought it was fine if flag arithmetic was correct only up to accents, but perhaps this example is too extreme to be relevant here.
Tomorrow is Shavout, a Jewish holiday, and after that it's Sabbath, so don't expect any response from me on this thread until Sunday (In a complete coincidence sunset in Israel is about the same time as midnight in Australia so the Jewish-Gregorian calendar difference and our different time zones almost cancel out).
Also, it seems like a +131g accidental is necessary for the notation of the 144-note MOS, unless you're doing it in a really weird and roundabout way, since without it 59g after D has to be A:\!!!~: and 59g before D has to be G:/|||~:, but the former is lower than the latter.
Last edited by רועיסיני on Sun May 28, 2023 3:47 am, edited 1 time in total.
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Re: Notation for Fibonacci tuning (Wilson's horogram #22)?

Post by Dave Keenan »

Thanks for the heads-up re Shavout and Sabbath. Interesting re our calendars and time zones.
רועיסיני wrote: Thu May 25, 2023 10:23 pm
I believe that all 12n-EDOs up to 204edo, and 240edo, have consistent flag arithmetic. It was a Trojan design criterion to maximise this.
Yes, but only for single-shaft symbols.
Multi-shaft symbols, and more generally symbols above the half-apotome, are completely determined by the choice of single-shaft symbols below the half-apotome. And since the evo flavour uses only single-shaft symbols, it makes sense to only be concerned that the single shaft symbols have valid flag arithmetic. However it often works out that double shaft symbols (apotome complements) have valid flag arithmetic where the second shaft has the same value as the symbol :(|): = :/||\: - :/|\:
Also, according to the spreadsheet, in Olympian :)|: + :|(: = :)|(::,: so I thought it was fine if flag arithmetic was correct only up to accents, but perhaps this example is too extreme to be relevant here.
It should really be called flag-and-accent arithmetic or symbol-element arithmetic (which can also include shafts). Yes, it's difficult to maintain symbol-element arithmetic with Olympian, which is on a par with 2460edo. Yes, too extreme to be relevant here.

I note that there are no longer any right accents. There was a period of time when we used the same glyphs for both schisma accents and mina accents, but now that we have distinct glyphs for them we place them all on the left. If a symbol has both mina and schisma accents, the minas (the smallest alterations) are placed furthest from the core symbol. Feel free to volunteer to update the old documents and diagrams that still show right accents. :-)
Also, it seems like a +131g accidental is necessary for the notation of the 144-note MOS ...
I always thought that it was necessary. Thanks for confirming.

I can now see that a -123 g symbol is not necessary to notate the 144-MOS with monotonic nominals. It can start off as follows:

D
D:)~|:
D:/|:
D:|~:
D:|):
D:/|~:
D:/|\:
D:'::(|\: (or whatever we decide for D:#: - 131 g)
D:#::!/:
D:#::!~:
E:b::\!:
E:b::~!(:
E:b:

I was unnecessarily limiting myself to the 13-note chain of fifths from Ab to G#. To avoid -123 g it is necessary to extend this to the 15 note chain from Db to D#. It is unusual to have both sharps and flats on the same nominal, but not unreasonable.
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Re: Notation for Fibonacci tuning (Wilson's horogram #22)?

Post by רועיסיני »

Dave Keenan wrote: Thu May 25, 2023 11:34 pm
רועיסיני wrote: Thu May 25, 2023 10:23 pm
I believe that all 12n-EDOs up to 204edo, and 240edo, have consistent flag arithmetic. It was a Trojan design criterion to maximise this.
Yes, but only for single-shaft symbols.
Multi-shaft symbols, and more generally symbols above the half-apotome, are completely determined by the choice of single-shaft symbols below the half-apotome. And since the evo flavour uses only single-shaft symbols, it makes sense to only be concerned that the single shaft symbols have valid flag arithmetic. However it often works out that double shaft symbols (apotome complements) have valid flag arithmetic where the second shaft has the same value as the symbol :(|): = :/||\: - :/|\:
Indeed, but I noticed that the double shaft flag flag arithmetic usually only works (for a not too small number of symbols) only when the fifth is very close to just, which is not the case in Trojan. Even 72edo's notation has :||): = 4 but :/||\: - :/|\: + :|): = 5.
Dave Keenan wrote: Thu May 25, 2023 11:34 pm Yes, too extreme to be relevant here.
Then another, maybe more relevant, is the notation we agreed on for 252edo and 300edo where :/|: + :|): = :'::/|):, just like I suggested here. There is also a comparable number of accidentals in both notations.
Dave Keenan wrote: Thu May 25, 2023 11:34 pm I note that there are no longer any right accents.
I know, but 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.


What about using ://|: for -68g of the phi tuning? Have any thoughts about it?
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