### Luna/Hemithirds notation

Posted:

**Wed Jun 03, 2020 12:37 am**This is in response to a request in this facebook thread:

https://www.facebook.com/groups/9790550 ... 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: http://sagittal.org/Sagittal-SMuFL-Map.pdf

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:

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

https://www.facebook.com/groups/9790550 ... 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: 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