## 15&22 (Porcupine) notation

Dave Keenan
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### 15&22 (Porcupine) notation

There are two main approaches to notating linear temperaments:
1. MOS nominals, where Latin letters other than A to G, or Greek letters, are used to represent the notes in a Moment of Symmetry scale having typically 4 to 10 notes. This requires a non-standard staff in many cases, and isn't discussed in this post.
2. Chain-of-fifths (CoF) nominals, which uses only the nominals A to G, as a chain of fifths F C G D A E B, or equivalently as a chain of fourths B E A D G C F. This uses a standard staff.

See https://www.prismnet.com/~hmiller/music ... upine.html.

Porcupine has a generator of 160 to 165 cents (called a quill). Porcupine is commonly embedded in 15-edo, 22-edo or (their sum) 37-edo, so we can't assume it will be embedded in any particular EDO. In fact we can't assume it will be a closed tuning at all. The first thing we need to know, to generate a CoF-nominals notation for any linear temperament, is how many generators there are to its approximation of a 2:3 fifth, or a 3:4 fourth. Three quills make a fourth. This tells us that a chain of quills will be notated as 3 interleaved chains of fourths (as you can see below).

Then we need an accidental symbol to indicate which of the three chains of fourths a note is on. We'd like that symbol to represent a comma for a low prime number, or a simple ratio if possible. It turns out that the 5-comma 81/80 will do the job, because it contains 3 to the power 4, and 5 to the power -1, and Porcupine has -3 generators to the prime 3, and -5 generators to the prime 5, and -3 x 4 + -5 x -1 = -7, and you can see below, that adding to a note corresponds to jumping 7 places backwards on the chain of generators.

The following shows vertically, 37 notes in a chain of Porcupine generators (quills) using chain-of-fifth nominals. I've offset some of them so you can see the 3 interleaved chains of fourths. The Porcupine[7] MOS scale with the simplest notation is shown with its nominals in red.

G
A
B
C
D
E
F
G
A
B
C
D
E
F
G
A
B
C
D
E
F
G
A
B
C
D
E
F
G
A
B
C
D
E
F
G
A

herman.miller
Posts: 12
Joined: Sun Sep 06, 2015 8:27 am

### Re: 15&22 (Porcupine) notation

The same accidentals can be used for MOS nominals if you're using a notation based on porcupine[7]. I like to use 36/35 instead of but it's the same in 7-limit porcupine because the 64/63 is tempered out. If you're using a porcupine[8] notation you might want an accidental for 21/20, which could be or .

The reason I like to use and is that I think it better approximates the size of the interval, and makes a sequence like A B C D E F G look more evenly spaced. If you're using 5-limit porcupine, 25/24 would be another option.

Dave Keenan
Site Admin
Posts: 1082
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

### Re: 15&22 (Porcupine) notation

Here, for Joe Monzo, is a notation for porcupine[37] in pure Sagittal. See two posts back for an explanation of the offsets and colours.

G
A
B
C
D
E
F
G
A
B
C
D
E
F
G
A
B
C
D
E
F
G
A
B
C
D
E
F
G
A
B
C
D
E
F
G
A