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Luna/Hemithirds notation

Posted: Wed Jun 03, 2020 12:37 am
by Dave Keenan
This is in response to a request in this facebook thread: ... 285089228/

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:

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: B: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:b: C E G:#: and the other B:b: D F:#: . Then Luna[25], -16 to +8, can be notated:

G:#::(!(:	A:b::(!(::(!(:	(alternative spellings)
G:#:	A:b::(!(:	(alternative spellings)
G:#::(|(:	A:b:	(alternative spellings)
G:#::(|(::(|(:	A:b::(|(:	(alternative spellings)
G:#::(|(::(|(::(|(:	A:b::(|(::(|(:	(alternative spellings)

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

Re: Luna notation

Posted: Wed Jun 03, 2020 11:39 am
by herman.miller
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.

Re: Luna/Hemithirds notation

Posted: Wed Jun 03, 2020 3:17 pm
by Dave Keenan
It will work fine for the Hemithirds (7-limit) temperament too, since this has generator map ⟨0 -15, 2 5], and almost the same optimal generator. In fact I don't know why Luna isn't just called 5-limit Hemithirds or vice versa.

All of the 4:7 subminor sevenths (5 generators) can be notated as subminor sevenths.