## Hexagonal 37

Posts: 1
Joined: Thu Jan 26, 2023 4:15 pm

### Hexagonal 37

Hello. I am looking to better understand how to notate the Hexagonal 37 system in Dorico using Sagittal. As a scala file the system presents as such.

! h.scl
hexagonal37
37

70.6724
111.731
133.238
141.345
182.404
203.91
223.463
244.969
274.582
315.641
364.807
386.314
427.373
456.986
498.045
519.551
568.717
590.224
609.776
631.283
680.449
701.955
743.014
772.627
813.686
835.193
884.359
925.418
955.031
976.537
996.09
1017.6
1058.66
1066.76
1088.27
1129.33
2/1

Dorico asks the user to define the Nominals which I have calculated in the following way according to how I am sonically hearing and approaching the system.

A 240 B
B 710 C
C 1820 D
D 2450 E
E 930 F
F 1820 G
G 2230 A

The software then asks the user to define further pitch deltas for the system.

I am unclear at what point as i am defining these pitch deltas that I presume I would consider the note sharp or flat as it relates to the Sagittal symbol. Any answers that would point me in the direction of expanding my understanding as a whole would be greatly appreciated.

Many Thanks!
Dave Keenan
Posts: 2189
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

### Re: Hexagonal 37

Hi @cicada. Welcome to the Sagittal forum.

I'm sorry it has taken me so long to respond to this. It was quite a puzzle. I could not find any online resources describing "Hegagonal37", it's purpose or structure. And now that I have analysed it enough to know how it should be notated I am mystified by two things:
1. Why is it given in cents in the Scala file instead of ratios, since it is in 5-limit just intonation? Ratios would have made it much easier to understand its structure.
2. Why is it called "hexagonal" when the shape it makes on the 5-limit lattice is not a hexagon but a hexagram, a 6-pointed star, as shown below?

Anyway, here's a Sagittal notation for it. I've chosen D as the tonic because it gives a symmetrical notation, but I'm sure you can transpose it to A, C or G if you prefer.

The order of the accidentals on the staff should be the reverse of the order they appear here in text. I'm sorry I don't know much about how to achieve this notation in Dorico.

						D
141

B	F
955	 457

F	 C	 G	 D	 A	 E	 B
365	1067	 569	  71	 773	 275	 977

A	  E	  B	 F	 C	 G
680	 182	 884	 386	1088	 590

C		  G		  D		  A		  E
996	 498	  0		 702	 204

A	 E	 B	  F	  C	  G
610	 112	 814	 316	1018	 520

F	 C	 G	 D	 A	 E	 B
223	 925	 427	1129	 631	 133	 835

B	F
743	 245

D
1059

Dave Keenan
Posts: 2189
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

### Re: Hexagonal 37

I noticed that the diagram above has some notations out of place when viewed in some browsers, due to slightly different widths relative to the tabs I used. If the above is messed up in your browser, the following should look better, and vice versa.

						D
141

B		F
955	 457

F	 C	 G	 D	 A	 E	 B
365	1067	 569	  71	 773	 275	 977

A	  E	  B	 F		 C		 G
680	 182	 884	 386	1088	 590

C		  G		  D		  A		  E
996	 498	  0		 702	 204

A		 E		 B		  F	  C	  G
610	 112	 814	 316	1018	 520

F	 C	 G	 D	 A	 E	 B
223	 925	 427	1129	 631	 133	 835

B	F
743	 245

D
1059

cmloegcmluin
Posts: 1712
Joined: Tue Feb 11, 2020 3:10 pm
Location: San Francisco, California, USA
Real Name: Douglas Blumeyer (he/him/his)
Contact:

### Re: Hexagonal 37

I've taken the diagram from Dave's response to you and added a few new features.

JI ratios

In addition to the raw cents values as you found them in the hexagonal37.scl file in the Scala archives, just below I've also shown the 5-limit JI ratio, which is the more fundamental piece of information. As you know, you can find the size of an interval in cents using this formula: $$1200 × \log_2(𝐢)$$, for example: $$1200 × \log_2(3/2) = 702 ¢$$.

prime-count vectors

To help you see the pattern between these JI ratios, I've also provided their prime-count vectors. These appear just below the JI ratios. They're the things that look like [ a b c ⟩.

These, naturally enough, are vectors that give the counts of the primes in the JI ratio. The vector is an efficient way to convey this information, when we make the assumption that the first element gives the count of the first prime, 2, and each element after that gives the count of the next highest prime: 3, then 5, 7, 11, etc. A positive count means that prime is found in the numerator, and a negative count means that prime is found in the denominator. For example, [1 -2 1⟩ is 10/9, because 10/9 = (2×5)/(3×3) or 2151/32. So you can see that the counts of these primes can also be understood as exponents on these primes.

In the prime count vectors I provided here, I put a _ instead of the count of prime 2's; that's because they won't help you understand the structure of this hexagram-shaped lattice, where the horizontal axis corresponds to counts of prime 3, and the diagonal bottom-left to top-right axis corresponds to counts of prime 5. The only role that prime 2 is playing here is in octave-reducing the ratios, so that they fall between 1/1 and 2/1. Since these concepts are new to you, it may be valuable for you to work out the first element in each of these vectors for yourself.

Another notation option

In addition to the Sagittal notation Dave provided you, I've provided an alternative Sagittal notation of my own. Neither one is right or wrong; they both use the building blocks of Sagittal in different ways, with different advantages and disadvantages, so which one you choose just depends on your style and way of thinking about these things and what you think is right for your music and your performers.

Dave's approach has the advantage of being very straightforward: every factor of 5 in your JI ratios is represented by a single flag on a sagittal (in particular, the flag that we call a left barb: the straight diagonal line out to the left from the main vertical shaft). But Dave's notation has the disadvantage that it requires up to three accidentals for a given pitch. And in some cases the accidentals even counteract each other, i.e. one adjusts pitch up while another adjusts pitch down.

Now, to be clear, I think Dave was absolutely right to suggest this notation to you, as a beginner. But I think it may also be valuable for your learning to get exposed to another way to approach this notation problem. So that's why in the notation I provided, I consolidated all of Dave's accidentals down into a single sagittal. This is much more compact and nice to look at. But anyone who uses it will need to take a little time to familiarize themselves with the meanings of a total of six sagittals, and their relationship to the count of prime 5's is not obvious. (This notation is very similar to the one Mark Johnson provided you over on that Steinberg forum thread you shared with me, which I plan to reply to soon. A lot of the differences are accounted for by him centering his on C, while Dave and I chose to center on D, which has the advantage of being symmetrical w/r/t sharps and flats, b/c D is in the middle of FCGDAEB)

So each block in this diagram goes:

Dave's notation
my notation
cents
JI ratio
prime count vector

Please let me know if you have any questions!!

 D D 141 ¢ 625/576 [ _ -2 4⟩ B F B F 955 ¢ 457 ¢ 125/72 125/96 [ _ -2 3⟩ [ _ -1 3⟩ F C G D A E B F C G D A E B 365 ¢ 1067 ¢ 569 ¢ 71 ¢ 773 ¢ 275 ¢ 977 ¢ 100/81 50/27 25/18 25/24 25/16 75/64 225/128 [ _ -4 2⟩ [ _ -3 2⟩ [ _ -2 2⟩ [ _ -1 2⟩ [ _ 0 2⟩ [ _ 1 2⟩ [ _ 2 2⟩ A E B F C G A E B F C G 680 ¢ 182 ¢ 884 ¢ 386 ¢ 1088 ¢ 590 ¢ 40/27 10/9 5/3 5/4 15/8 45/32 [ _ -3 1⟩ [ _ -2 1⟩ [ _ -1 1⟩ [ _ 0 1⟩ [ _ 1 1⟩ [ _ 2 1⟩ C G D A E C G D A E 996 ¢ 498 ¢ 0 ¢ 702 ¢ 204 ¢ 16/9 4/3 1/1 3/2 9/8 [ _ -2 0⟩ [ _ -1 0⟩ [ _ 0 0⟩ [ _ 1 0⟩ [ _ 2 0⟩ A E B F C G A A A F C G 610 ¢ 112 ¢ 814 ¢ 316 ¢ 1018 ¢ 520 ¢ 64/45 16/15 8/5 6/5 9/5 27/20 [ _ -2 -1⟩ [ _ -1 -1⟩ [ _ 0 -1⟩ [ _ 1 -1⟩ [ _ 2 -1⟩ [ _ 3 -1⟩ F C G D A E B F C G D A E B 223 ¢ 925 ¢ 427 ¢ 1129 ¢ 631 ¢ 133 ¢ 835 ¢ 256/225 128/75 32/25 48/25 36/25 27/25 81/50 [ _ -2 -2⟩ [ _ -1 -2⟩ [ _ 0 -2⟩ [ _ 1 -2⟩ [ _ 2 -2⟩ [ _ 3 -2⟩ [ _ 4 -2⟩ B F B F 743 ¢ 245 ¢ 192/125 144/125 [ _ 1 -3⟩ [ _ 2 -3⟩ D D 1059 ¢ 1152/625 [ _ 2 -4⟩