15&22 (Porcupine) notation

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Dave Keenan
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15&22 (Porcupine) notation

Post by Dave Keenan » Tue Oct 11, 2016 2:28 pm

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:/|::b:
 C
D:\!:
  E:/|::b:
 F
G:\!:
  A:/|::b:
 B:b:
C:\!:
  D:/|::b:
 E:b:
F:\!:
  G:/|::b:
 A:b:

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herman.miller
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Re: 15&22 (Porcupine) notation

Post by herman.miller » Sun Oct 16, 2016 11:22 am

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.

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Dave Keenan
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Re: 15&22 (Porcupine) notation

Post by Dave Keenan » Sun Dec 01, 2019 11:36 am

Here, for Joe Monzo, is Herman's notation for porcupine[37] in pure Sagittal. See two posts back for Herman's 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:b:
C:\!:
  D:!!/:
 E:b:
F:\!:
  G:!!/:
 A:\!!/:

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