Porcusmine

Porcusmine is a 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. It is not chiral and it has maximum variety 3 for certain tunings only (otherwise it has maximum variety 4 in general). 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. 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. It was also later called the Pinetone diminished octatonic when independently discovered by Lillian Hearne. Similar Porcupine[8] detemperings have also been discussed by Kite Giedraitis as well as by Lillian Hearne.

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.