Notation for metallic scales

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Dave Keenan
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Re: Notation for metallic scales

Post by Dave Keenan »

It just occurred to me that noble ratios could be notated by using two accidentals against the same note, with a sign between them (say a small letter N) to indicate that you should take the noble mediant of the two rationals indicated, which would be two consecutive rationals on the Stern-Brocot tree zigzag that corresponds to the desired noble. This would be the simplest pair of consecutive zigzag rationals that can be notated using the same nominal. And perhaps their accidentals should also be required to point in the same direction (both sharpening or both flattening).

I wouldn't want to add a sagittal symbol for this noble-mediant operator. I'm sure we could re-purpose some existing "N"-like SMuFL symbol.

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volleo6144
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Re: Notation for metallic scales

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Dave Keenan wrote: Wed Oct 21, 2020 12:47 pm ... the simplest pair of consecutive zigzag rationals that can be notated using the same nominal.
Exactly or just in-range?

1:φ (833¢) gives us a :||\:N:/|:m6 if the former, assuming that double-sharps and other far-out symbols aren't allowed.
A random guy who sometimes doodles about Sagittal and microtonal music in general in his free time.

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Re: Notation for metallic scales

Post by Dave Keenan »

volleo6144 wrote: Sat Oct 24, 2020 7:34 am
Dave Keenan wrote: Wed Oct 21, 2020 12:47 pm ... the simplest pair of consecutive zigzag rationals that can be notated using the same nominal.
Exactly or just in-range?

1:φ (833¢) gives us a :||\:N:/|:m6 if the former, assuming that double-sharps and other far-out symbols aren't allowed.
Good question. As you know, the zigzag ratios for φ are the ratios of consecutive Fibonacci numbers:
1/1 2/1 3/2 5/3 8/5 13/8 21/13 ...

The reader must, at least initially, assume the symbols represents their default commas at the specified JI precision level, so that would suggest they must be exactly notated. But there might be cases where you could get away with one of them being a ratio that could only be notated approximately at that precision level, if it was obvious that the default value did not have a noble mediant with the other (exactly notated) ratio because the diagonal products differ by more than 1.

But in this case, wouldn't it be the same notation either way?

I assume you've chosen 5/3 and 8/5 because 3/2 can't reasonably be notated as any kind of sixth, nor 5/3 as any kind of fifth, whereas 5/3 is a major sixth and 8/5 is a minor sixth. 13/8 is a neutral sixth.

We should probably also standardise which ratio should have it's accidental closer to the notehead. I suggest it should be the one that is closest to the desired noble, i.e. the more complex one, as you have done, so that if the "N" and the more distant accidental are ignored, the error is minimised.

We could use something like "~" instead of "N".

The choice of the two ratios might also be affected by whether we are using the Evo or Revo flavour. I think :\!::#:~:/|:m6 could be confusing. We also have to consider whether people would tend to read something like :/|: ~:\!::#:m6 as (:/|: ~:\!::#:)m6 or (:/|: ~:\!:):#:m6.

Even in Revo, :||\:~:/|:m6 isn't going to work on the staff unless the 1/1 is between A and D# on the chain of fifths, so the m6 has no sharps or flats. With 1/1 as C, 5/3 is :\!:A and 8/5 is :/|::b:A or :\!!:A and 13/8 is (approximately) :(!/:A.

So in Revo you could write :\!:~:\!!:A unambiguously, but in Evo, instead of the possibly-ambiguous :\!:~:/|::b:A, I at first thought we might prefer something further down the zigzag. But on examining the possibilities I think we have to simply define "~" as binding less tightly than sharps or flats, and so go with the direct equivalent of the Revo, namely :\!:~:/|::b:A.

So I don't think the choice of ratios should depend on Revo vs Evo.

You also raised the issue of whether "double-sharps and other far-out symbols aren't allowed". I think they should be allowed, but I think the two ratios should not be allowed to differ by more than 25/24 or about 71 cents, which is what 5/3 and 8/5 differ by.

I suspect that the rule "no more than 71 cents between them" also ensures they can be notated with the same nominal.

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Re: Notation for metallic scales

Post by Dave Keenan »

Some other ideas:

Require that the accidental nearest the notehead notates a ratio that is in the same capture zone as the noble.

Omit the accidental furthest from the notehead, with the convention that the intended noble is always the one whose alteration of the nominal is greater in magnitude than the alteration given.

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