Porcusmine

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Porcusmine is a max-variety-4 or max-variety-3, generator-offset scale with step pattern 4L 3M 1s. It is equivalent to the porcupine (7L 1s) mos with 4 of the large steps made bigger and the other 3 made smaller, or to the smitonic (4L 3s) with an extra-small step added. Porcusmine is not chiral. 19edo is the smallest edo to support a non-degenerate porcusmine.

Porcusmine can be tuned as a 5-limit JI scale or a tempered version thereof, where L represents 10/9, M represents 27/25, and s represents 25/24.

"Porcusmine" is a name given by cellularAutomaton, who discovered it as a detempering of porcupine, although the scale has also been discussed by Kite Giedraitis and possibly others. The name is a portmanteau of "porcupine" and "smitonic", since its step sizes are intermediate between those of the 8-note porcupine and 7-note smitonic mosses.

Modes

Porcusmine has 8 modes. Names are provided based on porcupine (tuning-neutral) and smitonic (for tunings where s is small), as well as a unique portmanteau name. When M and s are close, the scale approaches diminshed (4L 4s), which has only two distinct modes and is therefore not useful for naming. The porcupine mode names refer to William Lynch's names, and the smitonic mode names refer to Alexandru Ianu's names. Below are the modes arranged in cyclic order:

Modes in cyclic order
LMLMLMLs

Porcusmine Octopus

Porcusmine Sothic Ascending

Octzothic

MLMLMLsL

Porcusmine Mantis

Porcusmine Dagothic

Mandagothic

LMLMLsLM

Porcusmine Dolphin

Porcusmine Lorkhanic

Dolkhanic

MLMLsLML

Porcusmine Crab

Porcusmine Almalexian

Crablexian

LMLsLMLM

Porcusmine Tuna

Porcusmine Vivecan

Tunivecan

MLsLMLML

Porcusmine Salmon

Porcusmine Kagrenacan

Salmenacan

LsLMLMLM

Porcusmine Starfish

Porcusmine Nerevarine

Starevarine

sLMLMLML

Porcusmine Whale

Porcusmine Sothic Descending

Whothic

Tunings

Porcusmine is MV3 when L+s=M+M, as reflected in the table below.

Porcusmine tunings
Tuning L:M:s Good JI approximations Other notes
JI (5-limit AGS interpretation) 2.581:1.885:1 just 3/2, 6/5, 5/4, 9/8, and 10/9
19edo 3:2:1 excellent pental thirds, fairly good 3-limit MV3
23edo 4:2:1 excellent 6/5 and 9/8, fairly good 7/5
26edo 4:3:1 excellent 10/9, 11/8, and 7/4; fairly good 5/4 and 3/2
27edo 4:3:2 excellent 10/9, 7/6, 6/5, and 7/5; fairly good 9/7 and 3/2 MV3
27edo 5:2:1 excellent 5:6:7; fairly good 7/4, 11/9, and 3/2
30edo 5:3:1 excellent 15/14, 9/8, 6/5, and 9/7; fairly good 15/8, 7/4, and 11/8 MV3
31edo 5:3:2 excellent 15/14 or 16/15, 7/6, 5/4, and 7/5; fairly good 6/5, 9/7, 3/2 meantone fifths
31edo 6:2:1 excellent 7/4, 7/6, 11/9, 5/4, 7/5; fairly good 6/5, 3/2, and 11/8 single meantone fifth
33edo 5:4:1 excellent 10/9; fairly good 11/6 and 11/8
34edo 5:4:2 very close to 5-limit just tuning gentle fifths
34edo 6:3:1 excellent 6/5; fairly good 9/8, 11/9, and 14/11
41edo 6:5:2 excellent 3-limit; good 7-limit; fairly good 5-limit

5-limit AGS interpretation

Porcusmine can be interpreted as a member of an AGS series AGS(9/5, 50/27), which is constant structure at 5, 6, 7, 8, 15, 23, 38, 61, and 99 notes. It is noteworthy for an AGS series to retain constant structure to this size.

In this AGS interpretation, Whothic is the "root mode" of Porcusmine. The 5-limit tuning of Whothic generated by AGS(9/5, 50/27) is:

1/1 25/24 125/108 5/4 25/18 3/2 5/3 9/5 2/1

In this tuning, the interval 25/18 occurs frequently, and the scale could be more consonant by tempering out 126/125 to equate this interval with 7/5.