## Notation for metallic scales

Dave Keenan
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### Notation for metallic scales

Ash9903b4 wrote: Thu Apr 02, 2020 12:24 am How do you define a canonical form for edas? Do you just use the just intonated apotome that results from stacking seven fifths?
That's correct. [-11 7>
Dave Keenan wrote: Wed Apr 01, 2020 4:10 pm Here's a wild thought. What if the odd half-tinas were not used to notate ratios, but to notate noble numbers? How could that be made to work?
Well, you'd now have a new series of primes involving the square root of five. In particular, 5 itself would now be a composite number, and √5 would map to 19827 half-tinas. That's one way to get half-tinas, I guess, but then you'd have to figure out what 1 + √5, 2 + √5, etc. correspond to.
Cool! Is that because Q(√5) is the smallest field that contains the noble numbers, and all elements of Q(√5) can be obtained as products and quotients of 0+√5, 1+√5, 2+√5, 3+√5, 4+√5, ... which are the primes of Q(√5)?

Could it be, that to get the nobles, we only need the odd primes: 1+√5, 3+√5, 5+√5, ... ? That would be because nobles are of the form (i+mϕ)/(j+nϕ) where ϕ = (1+√5)/2 and i, j, m, n are whole numbers such that | in-jm | = 1.
Similarly, you can use the field Q(√2) and map √2 to 8539 half-tinas, I think.
I assume Q(√2) doesn't relate to the nobles, but to the "silver aristocratic" numbers (where the nobles are the gold aristocratics). Is that correct?
https://en.xen.wiki/w/Metallic_MOS
I'd be happy to just have the nobles, as I understand they maximally avoid nearby simple ratios.

Dave Keenan
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### Re: Notation for metallic scales

I investigated how many tinas (1/809ths of an apotome) were in the alterations required for the most noble number of all — the golden ratio, ϕ = (1+√5)/2 [Edit: as a frequency ratio for harmony (833.09¢)], for various choices of Pythagorean nominal plus sharps or flats. Disappointingly, it turns out that they are all closer to a whole number of tinas than an odd number of half-tinas.

With G = 1/1.
ϕ = Eb+40.9103¢ (+291.124 tinas)
ϕ = D#+17.4503¢ (+124.179 tinas)
ϕ = Fb-49.3147¢ (-350.931 tinas)

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### Re: Notation for metallic scales

We've got a couple parallel threads going on here, which I fine I think — I just wanted to note that I am very interested in the problem of how, using Sagittal, to notate scales based on metallic numbers, considering that it was just such a scale that brought me to Sagittal in the first place (and thank you Dave for linking my metallic MOS wiki post earlier!)

Some folks are going to be more interested in acoustic phi (833 cents) than logarithmic such we're talking about here (741 cents), and they may be completely different notational problems. I haven't even begun to consider it.

Dave Keenan
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### Re: Notation for metallic scales

Actually, I was talking about "acoustic" phi, i.e. phi as a frequency ratio, not a phi-th of an octave. I've edited to clarify that now. Thanks.

[rant]
I can't help thinking of the frequency ratio ϕ as the true ϕ. After all, if I write 3/2 (in the context of tuning) no one asks me if I mean "acoustic" 3/2 or "logarithmic" 3/2. i.e. No one asks me if I mean a frequency ratio of 3/2 (702¢) or 2/3rds of an octave (800¢)*. Everyone knows I mean the former, because 2/3rds of an octave is written as 2°3 or 2\3 or 2^(2/3) or 1200×2/3 cents. So "logarithmic" phi should be written 1°ϕ or 1\ϕ or 2^(1/ϕ) or 1200/ϕ cents, not as ϕ.

*Or would "logarithmic 3/2" mean 3/5ths of an octave, so the two parts of the octave are in a 2:3 ratio of cents? It doesn't matter with ϕ since 1 : ϕ = ϕ : (1+ϕ). Fortunately, we don't have to decide, since no one talks of "logarithmic 3/2".
[/rant]

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### Re: Notation for metallic scales

The above posts were moved here from the topic about New Olympian diacritics. This started when @Ash9903b4 suggested some rational mappings for the Magrathean-level tina and half-tina intervals, then @Dave Keenan had a wild thought that they could instead be mapped to metallic numbers — those least rational of numbers.

The conversation about tina mappings has gone elsewhere. Here, let's center the conversation around strategies for notating phenomena like phi (not logarithmic phi) and its recursive combination and difference tones. We don't have to bring tinas into it, though they may turn up.

Dave Keenan
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### Re: Notation for metallic scales

Thanks for that, @cmloegcmluin.

Of course metallic pitches can be notated according to what capture zone they fall into, in some JI precision level, just like anything else. But my hope was that, because tinas are so good at consistently approximating simple rationals, the presence of a half-tina dot might be able to indicate the notation of a simple noble number, or other "celebrity" irrational.

The trouble is, that while ϕ itself is 5928.399 tinas and so rounds to an odd number of half-tinas, once you start multiplying or dividing by powers of 2 and 3, to reduce it to a comma alteration (less than 68.6 cents) from some not-too-distant Pythagorean, the nobility is lost. Unlike simple-rationality, nobility is no respecter of octave reduction or octave inversion, let alone changing by fifths. Or perhaps I should say that octave reduction etc are no respecters of nobility.

I wonder how "phitave" reduction would affect nobles? But then what would be the equivalent of the chain of fifths to map to nominals?

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### Re: Notation for metallic scales

I'm intimidated by the awesomeness of this question and feel unprepared to give an educated response. But I don't want to prevent me from continuing the conversation. Let me take a baby step: might we be looking for a recursive notation for combination tones?

Dave Keenan
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### Re: Notation for metallic scales

cmloegcmluin wrote: Sat Apr 11, 2020 10:27 am might we be looking for a recursive notation for combination tones?
I'm not sure what you mean by "a recursive notation". Or do you mean "a notation for recursive combination tones"?

This facebook thread may be relevant. I know you've seen it before.
In it, I make the point that there is nothing special about phi, or any other celebrity irrationals, in regard to chords that depend on combination tones for their effect. When making use of combination tones, one usually wants a sine-wave timbre, and to apply distortion to the combination of sines.

I am thinking instead, of ordinary harmonic timbres, without distortion, but using noble-number frequency-ratios as a way of maximally avoiding coinciding harmonics. What Margo Schulter and I have called "merciful intonation" in contrast to "just intonation", which dovetailed nicely with Erv Wilson's earlier description of the phi neutral sixth (833 ¢) as, "the worstest of the worst — and yet somehow with divinity imbued, Lord have mercy!".

Margo produced a lovely sampler (on medieval.org, I think) with a series of cadences that go from an MI chord to a JI chord. But I can't find it ATM. I'm imagining notating those cadences.

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### Re: Notation for metallic scales

Whoa. Okay. Evidence certainly suggests I have enjoyed/appreciated/fancied the material on that link , but I have no clear memory of it. I want to understand everything on it! But I'll have to take a note to dig in later.

For what it's worth, even though I don't quite understand the math/music yet, based on the evidence below, I think "merciful" is a perfect name. Justice and mercy both fall into that same category of primordial virtues.

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