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!

## Hexagonal 37

- Dave Keenan
- Site Admin
**Posts:**2272**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 hexa

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.

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 hexa

*gram*, 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
- Site Admin
**Posts:**2272**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
- Site Admin
**Posts:**1786**Joined:**Tue Feb 11, 2020 3:10 pm**Location:**San Francisco, California, USA**Real Name:**Douglas Blumeyer (he/him/his)-
**Contact:**

### Re: Hexagonal 37

Hi @cicada,

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

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: \(1200 × \log_2(3/2) = 702 ¢\).

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

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.

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

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

Dave's notation

my notation

cents

JI ratio

prime count vector

Please let me know if you have any questions!!

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 2^{1}5^{1}/3^{2}. 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!!