Semaphore / godzilla

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keenanpepper
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Joined: Mon Apr 26, 2021 11:25 am
Real Name: Keenan Pepper

Semaphore / godzilla

Post by keenanpepper »

Hi, I have some questions about how to notation semaphore / godzilla with sagittal. By this I mean the linear temperament tempering out 49/48, with 81/80 probably also tempered out but not too sure about the mapping of any other primes.

I think the way you would notate it with 7 Pythagorean nominals is like this:

...E:b: C:|): B:b: G:|): F D:|): C A:|): G E:|):=F:!): D B:|):=C:!): A G:!): E D:!): B A:!): F:#: E:!): C:#:...

So, my first question is: is this correct? And my second question is, is there some way to notate this with only 5 nominals? That seems more natural to me because semaphore doesn't have a 7-note MOS but it does have a 5-note MOS. But I can't figure out any obvious way to do it because there's no symbol for :|)::|): or equivalently for the limma. Thoughts?

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Dave Keenan
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Re: Semaphore / godzilla

Post by Dave Keenan »

Hi Keenan. Welcome to the Sagittal forum.

Here's a mapping from Graham's temperament finder.

Reduced Mapping
2 3 5 7 11 13
[ ⟨ 1 2 4 3 6 6 ]
⟨ 0 -2 -8 -1 -12 -11 ] ⟩

TE Generator Tunings (cents)
⟨1204.4822, 254.5504]

Semafour+, the 13-limit extension of semafour [sic], has:

Reduced Mapping
2 3 5 7 11 13
[ ⟨ 1 2 4 3 3 6 ]
⟨ 0 -2 -8 -1 2 -11 ] ⟩

TE Generator Tunings (cents)
⟨1207.0465, 255.2153]

They differ only in their mapping of prime 11. So we should avoid symbols that involve prime 11. And we probably won't need to go as high as 13.

Generators for use with pure octaves are given here: https://en.xen.wiki/w/Semaphore_and_Godzilla

Your notation is apparently using the octave-inversion of that approximately 251 cents subminor-third/supermajor-second generator, the approximately 949 cent supermajor-sixth/subminor-seventh generator, so the bottom row of the mapping (which is all that matters for the notation, given that the period is an octave) becomes easier to think about, with mostly positive numbers, as:

⟨ 0 2 8 1 12 11 ] (godzilla) or ⟨ 0 2 8 1 -2 11 ] (semafour+) so we should try to do the notation using only:

2 3 5 7
⟨ 0 2 8 1 ]

Yes. Your 7-nominals chain-of-fifths notation is valid. And :|): is a good choice for the accidental.

It can be seen as two interleaved chains of fifths (because there are two generators to the fifth) with the accidental distinguishing the chains.

There is a symbol equivalent to :|)::|): , the double septimal comma or 1/49-medium-diesis 4096/3969. It is :(/|: . Unfortunately we couldn't make symbols with two-arcs-on-the-same-side "work" graphically. Of course you can just use the two symbols together :|)::|): . I see that 4096/3969 is -10 generators.

The symbol :||\: has been used for the limma, when the limma can't be notated by a change of nominal, although that is not the primary role of that symbol. I see the limma is -10 generators.

I agree it makes sense to have a 5-nominals notation because it has a 5 note proper MOS (and none larger below 14). But surely that would be 5 nominals in a chain of generators, not a chain of fifths? I favour what I call pseudo-nominals where we use notes chosen from 24edo, notated using conventional notation plus :<: and :>: . That way you can still use a standard 5-line staff. So for our "nominals" we'd have:

G E:>: D C:<: A

Then we need sagittal accidentals for 5 generators and 10 generators (about -55 cents and -110 cents). As you suggest, we could use

+5 gens :!):
+10 gens :(\!: or :!)::!):

So a chain of generators could be notated:

G:|)::|): E:>::|)::|): D:|)::|): C:<::|)::|): A:|)::|): G:|): E:>::|): D:|): C:<::|): A:|): G E:>: D C:<: A G:!): E:>::!): D:!): C:<::!): A:!): G:!)::!): E:>::!)::!): D:!)::!): C:<::!)::!): A:!)::!):

It's often considered desirable for (the commas of) the chosen symbols to have untempered sizes that are not too different from their tempered sizes. The untempered 7-comma is only about half the size that it is in godzilla/semafour. Some 7-limit commas that map to -5 generators and have untempered size around 55c are:
36/35 49c :/|): 35-medium-diesis
28/27 63c :.::(|\: 7-large-diesis
This 13-limit comma is also valid as -5 gens.
27/26 65c :`::(|\: 13-large-diesis
If you used either of the last two, you could drop the accents, and so the symbol could represent both. But it seems simpler to use :/|): .

My spreadsheet isn't set up to calculate the generator counts of multi-shaft sagittal commas. But you can find their 7-limit monzos listed here:
viewtopic.php?p=578#p578

@herman.miller can probably suggest some -10 gen symbols with untempered size around 110 cents.

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