## Luna/Hemithirds notation

Dave Keenan
Posts: 878
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
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### Luna/Hemithirds notation

Luna is a complex but extremely accurate 5-limit temperament shown on page 175 of Paul Erlich's Middle Path paper.
It has an octave period, and a generator of 193.2 cents, with 2 generators to the 4:5 major third and -15 generators (octave-reduced) to the 2:3 perfect fifth.

Luna is almost a reductio-ad-absurdum for a chain-of-fifths-based notation. i.e. If you want your perfect fifths spelled as conventional perfect fifths. Given that it has 15 such chains, it needs 7 up/down pairs of Sagittals to distinguish the chains of fifths. I found a suitable set of symbols by extending the temperament to this 13-limit generator map ⟨0 -15,  2   5 -22, 23] with errors of the order of 3 cents in the primes above 5.

I record the set of symbols here for posterity, but I don't think it's very useful. So I won't bother notating Luna[25] with them. It's almost as complicated as a JI notation. It's way too many symbols, particularly when the scale under consideration only has 25 notes. They each represent their primary comma which you can see here: http://sagittal.org/Sagittal-SMuFL-Map.pdf

symbol	gens	cents	comma
---------------------------------------
-105	114.0 ¢	apotome
87	  8.4 ¢	5/7-kleisma
-31	 10.8 ¢	1/(11.13)-comma
-62	 21.6 ¢	1/5-comma
25	 30.0 ¢	1/7-comma
-6	 40.8 ¢	5/11-small-diesis
-124	 43.2 ¢	1/25-small-diesis
-37	 51.6 ¢	11-medium-diesis


Paul Erlich mentioned that there is a 7 note MOS, albeit uneven. The 6 Note MOS is more even. But no matter which you use for your nominals, the obvious chroma is -6 generators (40.8 cents), which we already have a symbol for above, namely  . This symbol is pronounced "janai", and the downward variant "janao". But since there is only the one symbol pair, you can simply call them "up" and "down".

The 7-note MOS can be notated with the "compound nominals": A B C  D  E  F G
Note that all the 4:5 major thirds have their conventional (meantone) spelling in this chain of 6 generators — one chain of thirds being A C E G and the other B D F . Then Luna[25], -16 to +8, can be notated:

E
E
E

F
F
F
F

G	A	(alternative spellings)
G	A	(alternative spellings)
G	A	(alternative spellings)
G	A	(alternative spellings)
G	A	(alternative spellings)

B
B
B
B

C
C
C
C

D
D
D
D

E


The gaps show where the 70.8 cent steps are. All the other steps are 40.8 cents. A single symbol that corresponds to -12 generators, and so can substitute for is 81.6 ¢, the 11/7-small-semitone.

21 of the 24 major thirds can be spelled correctly. Only the following 3 are spelled as diminished fourths:

F : B
F : B
F : B

8 of the 10 perfect fifths can be spelled as a kind of fifth, although they appear diminished. The other 2 are spelled as augmented fourths:

D : A
E : B
F# : C
G# : D
B : E (spelled as an augmented fourth)
C : F (spelled as an augmented fourth)
D : A
E : B
F : C
G : D

herman.miller
Posts: 12
Joined: Sun Sep 06, 2015 8:27 am

### Re: Luna notation

This looks like a good solution for a difficult temperament to notate. I looked over my notes on Sagittal notation to see if I had anything for luna, and the only thing I had is full of schisma accents.

Dave Keenan