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.
| 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.