15&22 (Porcupine) notation

[Dave Keenan]

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]

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]

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

[cmloegcmluin]

This stuff's just on the periphery of my ability to understand. If I read and re-read every sentence five times slowly and ponder it deeply I can usually figure out what is meant and why. I suppose if I had a bunch of pretty diagrams I could make sense of it more easily. This is my second attempt at understanding what's happening here.

What's the relationship between the notation given here and the EDO notations on the periodic table? It looks like this would check out in 15, 22, or 37, i.e. that porcupine[7] will always be A B C D E F G whether it's embedded in 15, 22, or 37. I believe this thread predates the periodic table. Perhaps similar guiding principles which led to the notation described here led to the notations selected for 15, 22, and 37 in the periodic table.

Is the main thrust of this topic more about the porcupine scales which are not closed, AKA (maybe?) ones which aren't actually an EDO because their generator is e.g. some JI interval which will never loop back to an exact octave, but you could still use this notation there?

[herman.miller]

Yes, you could use this notation for, e.g., a tuning with a generator based on the golden ratio like (2 phi + 1) / (15 phi + 7) (about 162.56 cents), which is within the range of tunings consistent with porcupine temperament. The thing that characterizes porcupine temperament is the mapping [<1, 2, 3, 2], <0, -3, -5, 6]>, or equivalently, the commas that are tempered out (e.g. 250/243). An EDO like 15p also tempers out other commas, e.g. 256/243 (blackwood temperament). So you might expect to see some overlap between porcupine notation and notations for EDOs that support porcupine temperament, but they won't necessarily be the same.

[cmloegcmluin]

Per Dave's suggestion, I have broken this topic off at the point it veered away from Porcupine-specific discussion toward other linear temperaments, and linear temperaments in general. You can find the continuation of that discussion here.