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

User avatar
herman.miller
Posts: 37
Joined: Sun Sep 06, 2015 8:27 am

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

Post by herman.miller »

http://www.elvenminstrel.com/music/tuni ... gram01.htm

I'm wondering about the best way to notate this tuning. It looks the Spartan level is sufficient for smaller subsets of the tuning, and Athenian should be good for almost anything. Promethean gets you an extra level, but many of the symbols are different. Here's what I've got for different sizes of accidentals (in cents):

108.20 :(||~:
82.66 :||): or :)||~:
66.87 :(|\:
51.09 :/|):
41.33 :/ /|:
25.52 :|):
15.79 :~|(:
9.76 :)|(:
6.03 :|(:
3.73 :)|:

The similarity of :~|(: and :)|(: is a bit confusing, so the Promethean :|~: could perhaps be substituted in place of :~|(:. How does this look?
User avatar
Dave Keenan
Site Admin
Posts: 2180
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

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

Post by Dave Keenan »

Hi Herman. That's a tough one! Not intended to approximate JI. 12 generators before you get a decent fifth.

Can you tell me how many generators each alteration in your list corresponds to? And what mapping you have assumed, if any?
User avatar
herman.miller
Posts: 37
Joined: Sun Sep 06, 2015 8:27 am

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

Post by herman.miller »

I'm not assuming any mapping for now, just trying to find accidentals of approximately the right size. You could either use +10 generators or -11 generators for the 7th harmonic, for instance.

And I think I got the 108c one wrong, it should be :/||): .

Here's the list with a few more additions sorted by generators. Many of these will only be needed if you're using large subsets of the tuning or if you're using the chain of fifths every +12 generators to notate the unmarked nominals. (E.g. a chain of generators starting with C would be C F:\ \!: A:~|(: D:!): G:(!/: B:)!(:, etc.)

-5 (108.2039) :/||):
+8 (66.8737) :(|\:
-13 (41.3302) :/ /|:
+21 (25.5435) :|):
-26 (82.6604) :||): or :)||~:
+29 (92.4172) :||\:
-34 (15.7867) :~|(: or :|~:
+42 (51.0870) :/|):
-47 (57.1170) :(|):
+55 (9.7567) :)|(:
-60 (98.4472) :(||(:
+63 (76.6305) :~||(:
-68 (31.5735) :(|:
+76 (35.3002) :(|(:
-81 (72.9037) :)||(:
-89 (6.0300) :|(:
User avatar
רועיסיני
Posts: 63
Joined: Tue Apr 11, 2023 12:11 am
Real Name: Roee Sinai

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

Post by רועיסיני »

Actually, for the case it still interests someone, the fifth from this tuning is 700.4292¢ which falls in the validity range of the Trojan symbols! [Edit: that is the fifth of 233edo, which triggered my interest in using Trojan for this tuning in the beginning. The correct value is even lower than that at 700.3106¢. Thanks @Dave Keenan for the correction!] So it's possible to use them for this tuning. The apotome of +84 generators or 2¹³⁶⁻⁸⁴ᶲ splits into 4 parts which can be notated by :|~:, :/|\: and :/||~:, and in each quarter there is a 5-note MOS whose pitches fall less than 1 hundredth of an apotome away from 20ths of the apotome, which suggests they can just be notated by all the Trojan accidentals, and they indeed can (the only one that could still be a problem is :)/|: but the value is 0.0954915A > 1/11 A which means it's still in the correct range):
:~|(: -89 (6.03¢)
:)/|: +55 (9.75674¢)
:/|: -34 (15.7867¢)
:(|: +110 (19.5135¢)
:|~: +21 (25.5435¢)
:|): -68 (31.5735¢)
:|\: +76 (35.3002¢)
:/|~: -13 (41.3302¢)
:/|): +131 (45.057¢)
:/|\: +42 (51.087¢)
:(|\: -47 (57.117¢)

These accidentals are actually more than enough, as there are only 12 chains of fifths in this tuning, so if you remove the quarter apotome multiple accidentals and divide the remaining ones into adjacent pairs, the difference between them is a tempered Pythagorean comma of +144 generators (3.72674¢) so only one accidental from each pair (in addition to the quarter apotome multiples) is enough. If you like low generator–count for each accidental then the middle two in each group of 4 are probably the best, while if you like common accidentals and dislike blatant size reversals the ones which are usually used to notate 144edo can be used instead. However, if you want to avoid nominal crossings, for large enough scales you may need to use all of them.

If you also want an accidental for :/|: + :|):, or for :(|: + :|\:, you can use the schisma accent for -233 generators (2.30325¢) and so :'::/|): notates :/|: + :|): = -102 steps (47.3602¢) and :.::(|\: = notates :(|: + :|\: = 186 steps (54.8137¢). You may also need to use it on other accidentals for larger scales to avoid nominal crossings, and for even larger scales I suggest a smaller accent can be used for -610 generators, and then for -1597 etc. in a balanced almost-ternary notation.

Here are some selected accidentals ordered by the magnitude of their generator count:
-5 :~|||(:
+8 :/||:
-13 :/|~:
+21 :|~:
-26 :)||~:
+29 :)||):
-34 :/|:
+42 :/|\:
-47 :(|\:
+55 :)/|:
-60 :'::(||(:
+63 :/||~:
-68 :|):
+76 :|\:
-81 :'::||):
+84 :/||\: (102.174¢)
-89 :~|(:
+97 :||~: (60.8437¢)
-102 :'::/|): (47.3602¢)
+110 :(|:
+118 :||\: (86.3872¢)
+131 :/|):
+144 :.::~|(: (3.72674¢)
+152 :||): (70.6005¢)
+173 :(||(: (96.1439¢)
+186 :.::(|\: (54.8137¢)
-233 :'::|: (2.30325¢)

I don't mind the existence of another sagittal notation for the same tuning, but I'd like to point out that in the previous notation suggested there are some "apotome complement" sagittals that are not actually used to notate apotome complement intervals, such as
:|(: + :/||): = -89 - 5 = -94
://|: + :)||(: = -13 - 81 = -94
:(|(: + :~||(: = 76 + 63 = 139
:/|): + :(|\: = 42 + 8 = 50
Either :|): + :||): = 21 - 26 = -5 or :(|: + :)||~: = -68 - 26 = -94
Where the apotome is 84 generators (the sums differ from it by 178 = 89*2, 55, 34 and 89 generators).
The other direction is also not satisfied – 8 + 76 = 21 + 63 = 29 + 55 = 84, but none of these pairs are apotome complements in that notation.
Last edited by רועיסיני on Thu May 25, 2023 7:25 pm, edited 3 times in total.
User avatar
Dave Keenan
Site Admin
Posts: 2180
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

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

Post by Dave Keenan »

Brilliant. Yes, now that you mention it, the 144edo notation is the obvious solution. And 144 is a Fibonacci number. What is the largest MOS cardinality that can be notated with only the 144edo symbols without nominal crossings? I'm guessing 55, since 7×12 only gets you to 84.
User avatar
רועיסיני
Posts: 63
Joined: Tue Apr 11, 2023 12:11 am
Real Name: Roee Sinai

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

Post by רועיסיני »

Using the interpretations I suggested for these symbols, a chain of 12 generators starting from C is
C F:\!~: A:/|: D:~!: F:|||): C:~!!!(: E:\!/: G:~|||(: D:!!!): F:~|: B:\!: D:/|~:,
which means that +8 generators above D:!!!): is C:|||):. This still doesn't mean we can't notate some of the 13-note MOSes using this notation, and the MOS starting from C (or any other diatonic note) works, but a 21-note MOS has 8-steps from 13 intervals that come from a contiguous chain of generators and so at least one of them has to be some form of :!): which will appear higher than+8 generators above it.

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 (which in my original suggestion was notated by :)/|:) and :|): for +76 generators (for which I suggested :|\:). Then all the 21-steps will only result a change in accidental, and the -34-steps may either result a change in accidental or a change of nominal, but always in the correct direction. However, 55-steps can sometimes change a nominal in the wrong direction, when they complement a -34-step that changes a nominal to a 21-step, which means that 55-note MOSes are the largest ones possible, so your guess was correct! (these values, when notated as I suggested in my previous comment, are what I called "the middle two in each group of 4").

If you want to avoid the use of :)/|: and :/|: together when the former is smaller than the latter, and want to use the symbols according to their Trojan apotome fraction, but want to notate more than 13-note MOSes, you can use :~|(:, :/|: :|~:, :|\:, :/|~: and :/|\: (with their values from my previous comment), and then any 21-note MOS can be notated without nominal crossings. However, 34-note MOSes already pose a problem, since +21 generators above C:~!!!(: is B:/|: (and a similar problem will exist for any 34-note MOS). So maybe the best solution will be to use these symbols most of the time and introduce the others only when needed, for lines with very small steps which otherwise will have undesired nominal crossings.
User avatar
Dave Keenan
Site Admin
Posts: 2180
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

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

Post by Dave Keenan »

I get 700.3106 ¢ for the 12-generator fifth. That's (12 × n(1\2)) mod 1200 ¢ = ((12 × n(1/2)) mod 1)×1200 ¢ = (12/ϕ² mod 1)×1200 ¢.

I would expect a chain of 12 generators to give 13 notes, so I think you are missing a final G.

I agree we need not be constrained by the Trojan capture zones, and should avoid :)/|: . I think we should simply notate the 144-note MOS (from -71 g to +72 g) the same as 144edo and derive generator counts for each accidental from that. I don't see larger MOS as being of any musical interest. If such a notation were to be misinterpreted as 144edo, all notes of the MOS would be within ±1.9 ¢.

Here are the generator counts I get for the 144edo accidentals:

Acc	Gens	Cents
:~|(:	+55	9.76
:/|:	-34	15.79
:|~:	+21	25.54
:|):	-68	31.57
:/|~:	-13	41.33
:/|\:	+42	51.09
:#:	+84	102.17
D:A	+12	700.31

Centering the notation on D minimises the number of sharps and flats.
User avatar
רועיסיני
Posts: 63
Joined: Tue Apr 11, 2023 12:11 am
Real Name: Roee Sinai

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

Post by רועיסיני »

Dave Keenan wrote: Wed May 24, 2023 11:10 am I get 700.3106 ¢ for the 12-generator fifth. That's (12 × n(1\2)) mod 1200 ¢ = ((12 × n(1/2)) mod 1)×1200 ¢ = (12/ϕ² mod 1)×1200 ¢.
That's correct. I don't know where I got the other number from.

Dave Keenan wrote: Wed May 24, 2023 11:10 am I would expect a chain of 12 generators to give 13 notes, so I think you are missing a final G.
I meant a generator chain of 12 notes, not a chain of 12 generators. It just makes it easier to see how we notate an interval a certain number of generators away from every note.

Dave Keenan wrote: Wed May 24, 2023 11:10 am I agree we need not be constrained by the Trojan capture zones, and should avoid :)/|: .
I don't know what made you think that I don't like the Trojan capture zones. I think they are quite handy and a valid consideration when the fifth is between 699.1831¢ and 700.7552¢, which indeed holds in this case, and all of my proposals except one agreed with them. I however understand that you dislike the use of :)/|: here and therefore all of the proposals in the previous comment tried to avoid it.

Dave Keenan wrote: Wed May 24, 2023 11:10 am Here are the generator counts I get for the 144edo accidentals:

Acc	Gens	Cents
:~|(:	+55	9.76
:/|:	-34	15.79
:|~:	+21	25.54
:|):	-68	31.57
:/|~:	-13	41.33
:/|\:	+42	51.09
:#:	+84	102.17
D:A	+12	700.31
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, so if you insist on using the accidentals for 144edo, the notation will be

Acc	Gens	Cents
:~|(:	+55	9.76
:/|:	-34	15.79
:|~:	+21	25.54
:|):	+76	35.3
:/|~:	-13	41.33
:/|\:	+42	51.09
:#:	+84	102.17
D:A	+12	700.31

However, I think this can be improved. The Trojan symbols that fits 35.3¢ (or 0.3455A) is actually :|\:, and I don't see a reason to not use it here, other than lack of compatibility with 144edo, so I think a better notation will be

Acc	Gens	Cents
:~|(:	+55	9.76
:/|:	-34	15.79
:|~:	+21	25.54
:|\:	+76	35.3
:/|~:	-13	41.33
:/|\:	+42	51.09
:#:	+84	102.17
D:A	+12	700.31

There is also a problem of :~|(: notating an interval of 9.76¢ (0.0955A) which is in the capture zone of :)/|:. Now, I won't suggest to replace the symbols, but I think that making :~|(: notate -89 generators is a solution of this discrepancy that is worth considering. It gives you a quite handy accidental of :~|||(: for -5 generators, and keeps both +8 generators and -13 generators from every note fine in regards to nominal crossings. Therefore an alternative suggestion (which was also the last one on my previous post) is

Acc	Gens	Cents
:~|(:	-89	6.03
:/|:	-34	15.79
:|~:	+21	25.54
:|\:	+76	35.3
:/|~:	-13	41.33
:/|\:	+42	51.09
:#:	+84	102.17
D:A	+12	700.31

This also has the advantage that it can be extended, if needed, to the system I introduced in my first post on this thread.
User avatar
Dave Keenan
Site Admin
Posts: 2180
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

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

Post by Dave Keenan »

[I wrote almost all of the following before your above message arrived.]

Ah. I see now, that we can't simply notate the 144-note MOS as for 144edo with monotonic (non-crossing) nominals. That results in inconsistent generator counts in some cases. For each 144edo accidental, I have chosen the generator count with the smallest magnitude. But I see that with only these 6 symbols, it's monotonic only to the 8-note MOS, -3 g to +4 g. In pitch order:
D E:!~: F:\!: G:\!~: G:#::|): A:/|~: B:/|: C:|~:
This breaks if we try to add a ninth note at -4 g, as follows:
D E:!~: F:\!: G:\!~: A:b::!): G:#::|): A:/|~: B:/|: C:|~:

This is solved by adding a symbol for +76 g. This should certainly be :|\: as you have it, as it obeys both flag arithmetic (42 g - -34 g) and Trojan capture zones (35.30 ¢).

Then the above becomes:
D E:!~: F:\!: G:\!~: G:#::!/: G:#::|): A:/|~: B:/|: C:|~:

:~|(: and :/|\: have not been used so far.

Excuse me for using the Evo flavour. I don't think well in Revo. George Secor would have been pleased to see you using Revo. :)

The addition of :|\: for +76 g, to the six 144edo symbols gets us to the 34-note MOS, -16 g to 17 g.
D D:/|~: D:#::!): E:b::~|(: E:b::|\: E:!~: E:/|: E:/|~: F:\!: F:|~: F:/|\: F:#::~!(: F:#::/|: G:\!~: G G:|~: G:#::!/: G:#::~!(: G:#::|): A:!~: A A:/|~: B:b::\!: B:b::~|(: B:\!/: B:!~: B:/|: C:\!~: C:\!: C:|~: C:#::!/: C:#::~!(: D:b::|): D:\!~:

[Edit: The yellow hilited notations are wrong and should be E:b::!): and C:#::|): . And the cyan hilited notation could be replaced with A:b::|\: to give a more symmetrical notation. Thanks @רועיסיני.]

If we want to go further than that, I'm in trouble, because we'd need a symbol for -89 g (6.03 ¢) and that's smaller than my :~|(: and I don't want to have :)/|: with :/|: . A -89 g symbol would get us to the 55-note MOS, after which we'd need symbols for -102 g (47.36 ¢) and 110 g (19.51 ¢).
User avatar
רועיסיני
Posts: 63
Joined: Tue Apr 11, 2023 12:11 am
Real Name: Roee Sinai

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

Post by רועיסיני »

I'm glad we got to the same conclusion about :|\: notating +76g.
However I think you have some mistakes in the MOS you notated: 2 steps (+8g) is E:!!!): (E:b::!):) and not D:#::!):, which is +152 generators, and the same for -2 steps, which suggests it may be better to notate them as D:||\: and D:!!/:. This means that if you change G:#::|): to A:b::|\: (or A:!!/:) you can notate the 34-note MOS with just :)|~:, :/|:, :|~:, :|\:, :/|~: and :/|\: (without :|):), as
D D:/|~: D:||\: E:(!!(: E:\!!: E:!~: E:/|: E:/|~: F:\!: F:|~: F:/|\: F:(||(: F:/|||: G:\!~: G G:|~: G:/||: G:(||(: A:!!/: A:!~: A A:/|~: B:\!!!: B:(!!(: B:\!/: B:!~: B:/|: C:\!~: C:\!: C:|~: C:/||: C:(||(: D:!!/: D:\!~:
And even the 55-note MOS as
D D:|~: D:/|~: D:||\: D:(||(: E:(!!(: E:\!!: E:\!/: E:!~: E E:/|: E:/|~: F:\!~: F:\!: F:~|(: F:|~: F:/|\: F:/||~: F:(||(: F:/|||: G:\!!: G:\!~: G:\!: G G:|~: G:/|~: G:/||: G:(||(: A:(!!(: A:!!/: A:\!~: A:!~: A A:/|: A:/|~: A:/||: B:\!!!: B:(!!(: B:\!!: B:\!/: B:!~: B:~|(: B:/|: B:/|~: C:\!~: C:\!: C C:|~: C:/|\: C:/||: C:(||(: D:(!!(: D:!!/: D:\!~: D:!~:
[Edit (25/5/23): The notes highlighted in yellow should have the barb to the left and not to the right, thanks @Dave Keenan! The notes highlighted in cyan are also wrong, and should be replaced by E:\!!!:, B:\!!~:, B:~!(:, and C:/|||: respectively.]
[Edit (14/7/23)] Adding staff notations to these MOSes as well:


[/Edit]

Regarding the symbol for -89 generators, if you prefer using :~|(: for +55 it seems like either :)|(: or :|(:, as both :'::)|(: and :'::|(: are smaller than :~|(: for fifths between 700 and 701 cents, and they are also smaller JI symbols. If instead you want :~|(: to notate -89g I think :)~|: could be a good choice for +55g as its value of ~8¢ for a fifth size of 700¢ is quite close to the correct value and for the golden fifth its even closer, and its size reversal with :~|(: is not as apparent as :)/|:'s with :/|:. So in my opinion the best choice would be:

Basic system:
Acc	Gens	Cents
:)~|:	+55	9.76
:/|:	-34	15.79
:|~:	+21	25.54
:|\:	+76	31.57
:/|~:	-13	41.33
:/|\:	+42	51.09
:#:	+84	102.17
D:A	+12	700.31

Extension:
Acc	Gens	Cents
:~|(:	-89	9.76
:(|:	+110	15.79
:|):	-68	31.57
:/|):	+131	41.33
And people can also have :': for -233g and smaller accents for -610g and -1597g if they want even finer divisions.
Last edited by רועיסיני on Fri Jul 14, 2023 8:29 pm, edited 2 times in total.
Post Reply