What the heck is that diagram about? And why would anyone want to represent a noble frequency ratio as a prime-count vector? And what does that even mean?
“Mercy without justice is the mother of dissolution; justice without mercy is cruelty.” — Thomas Aquinas
A musical interval or dyad, whose frequency ratio is a simple noble number, is maximally ambiguous and a source of tension, wanting to resolve to one of the justly-intoned intervals on either side of it. Near examples of these in 12edo are the tritone, the minor seventh and the minor tenth. In a play on the word "just", I like to call simple noble-number intervals "mercifully-intoned". They have also been called "anti-JI" or metastable intervals. It is rational to be just, and noble to be merciful.
The explosion of musical temperaments that occurred in the first decade of the 21st century, was largely the result of applying the tools of linear-algebra (matrix math) to justly-intoned interval ratios represented as prime-count vectors (also called "PC-vectors", or simply "vectors") — a form of scale engineering that we now call Regular Temperament Theory (RTT).
My hope is that by expanding the repertoire of intervals that can be represented as vectors, we will be able to apply the methods of RTT to find and work with musical temperaments that target the simple noble frequency ratios of merciful intonation (MI), in addition to the simple rational frequency ratios of just intonation (JI).
There's no guarantee this will lead to anything useful, but I hope fellow students of tuning-math will at least find the following series of posts entertaining.
I dedicate this series to Margo Schulter @mschulter, whose explorations in neo-medieval harmony serve to ground this work. Please listen to her sampler of MI/JI cadences.
It's not clear whether this will ever have any relationship to the Sagittal microtonal notation, but the use of ℚ(√5) for prime factorisations of noble numbers, was suggested to me by @Ash9903b4 in the Magrathean diacritics thread on the Sagittal forum. And the Sagittal forum seems like a good place to record the results of my research.
Feudal Numbers
The blackboard bold "ℚ" (for "quotient") represents the set of rational numbers, i.e. the usual ratios or fractions of the form n/d where n and d are integers (and d is not zero). The set of integers is represented by the blackboard bold "ℤ".
Rant about ℤ:
Rant about ϕ:
I note that some ignobles are irrational and some ignobles are rational, but all nobles are irrational, which is why they can't be represented as ordinary prime-count vectors in ℚ.
We can say that a feudal number is a quotient of two "feudal integers", a numerator and denominator, both having the form a+bϕ where I call a the "wooden part", and b the "golden part". a is called the "wooden part" because it remains an ordinary dull integer, while b is golden because it gets multiplied by the golden ratio. Notice that it's not bϕ that is the "golden part", but only b. This is analogous to the imaginary part of a complex number. e.g. the feudal integer 3+2ϕ has wooden part 3 and golden part 2. The set of feudal integers is represented as "ℤ[ϕ]".
Why ℤ[ϕ] and not ℤ[√5]? It turns out that because 5 modulo 4 = 1, ℤ[√5] is not the largest set of integer-like numbers (the largest ring) in ℚ(√5). It's only half of it. It turns out we still have a ring if we take all the numbers midway "horizontally" (integer part) between the numbers of ℤ(√5), provided we also offset them by 1/2 "vertically" (√5 part). And ϕ = 1/2 + √5/2. See Dirk Dekker's plot at the end of this post.
So a feudal integer is like a work of gold inlay, or a golden afternoon spent in a wooden boat, telling tales of feudal times.
I'll be talking a lot about feudal integers and feudal primes, so I'll just be calling them "integers" and "primes" in this context, and when I need to refer to the ordinary pre-feudal kind, I'll call them "ordinary integers" and "ordinary primes". Elsewhere you may see the ordinary kind called "rational integers" and "rational primes", but I find that terminology confusing.
In regular temperament theory (RTT), when we represent a rational number (a ratio of two ordinary integers) as a vector (a list of ordinary integers between square and angle brackets [...⟩ ), we first reduce the ratio to lowest terms, then we prime-factorise its numerator and denominator, and count the numbers of each prime (2, 3, 5, 7, 11 etc), and record these counts as entries in the vector, with a minus sign applied to the counts of primes in the denominator. For example \(\frac85\) is represented as [3 0 -1⟩, which is to say 2³×3⁰×5⁻¹ or \(\frac{2^3}{5^1}\).
It turns out we can do exactly the same thing when we want to represent a noble number as a vector. It's just that our reduction-to-lowest-terms procedure is different, and our primes turn out to be feudal integers like -1+2ϕ, -1+3ϕ, -2+3ϕ, -1+4ϕ, -3+4ϕ, -2+5ϕ, -3+5ϕ. But the entries in the vector are still just ordinary integers.
If you're looking for a pattern in the list above, forget it. Feudal primes are just as random as ordinary primes, except that many of them come in complementary pairs. More on that later.
In addition to the primes, we also have "units", like -1+1ϕ, 1+0ϕ, 0+1ϕ. These are analogous to the numbers 1 and -1 in the rational numbers. We don't use negative rationals in RTT, so our vectors haven't needed a "unit count" entry. But we will need a unit-count entry for our noble-number vectors, even though we don't care about negative noble numbers either. All three of those (feudal) units shown above are positive. But we only need a single entry to cover all of them, because it turns out they correspond to powers of ϕ. The 3 above are ϕ⁻¹, ϕ⁰, ϕ¹. All integer powers of ϕ are units in the feudal number system, but so far, I find we only need those three for the noble numbers.
It's interesting that so far, I've never had to prime-factorise the numerator or denominator of a noble number after it has been reduced to lowest-terms. After reducing to lowest terms, every noble so far, is simply a single prime or unit, divided by another single prime or unit. That's good in one way, because, as you can probably imagine, factorising feudal integers is not an easy thing to do by hand, or even with a spreadsheet. But it's bad in another way, because it means we'll need a lot of primes for a small number of nobles, and therefore we'll need a lot of entries in our vectors.
So how does this reduction to lowest terms work? A common way to obtain a noble number is as a noble mediant (a kind of phi-weighted mediant) between two adjacent rationals on the Stern-Brocot tree. See Erv Wilson's Scale Tree. For an alternative treatment see Continued Fractions by Jordi Solà-Soler, 2014.
Consider the 422.5 ¢ supermajor third that is the noble mediant of 4/3 and 5/4 and is therefore the limit of these fibonacci-like sequences of numerators and denominators that zigzag down the Stern-Brocot tree:
Figure 1 ----> 4 5 9 14 -- -- -- -- ... 3 4 7 11 ---->
I say fibonacci-like because each new ordinary integer is the sum of the previous two in the same row.
This noble is commonly written as (4+5ϕ)/(3+4ϕ), but that is not in lowest terms.
To put it in lowest terms we run the fibonacci-like sequences backwards, starting with 5/4 and 4/3. So each new ordinary integer is the difference of the previous two, working from right to left, like this:
Figure 2
<----
-2 3 1 4 5
... -- -- -- -- --
-1 2 1 3 4
<----
As soon as we get a negative ordinary integer (on the top or the bottom, or both), we stop. Then we write the noble number in lowest terms, as the noble mediant of the last two fractions, as:
Figure 3 -2+3ϕ ---- -1+2ϕ
And it turns out both of these (feudal) integers are prime.
How do I know? Professor Dirk Dekker wrote a wonderful paper in 2003 entitled Primes in quadratic fields. And it has a wonderful appendix 2 in which he has mapped the feudal primes (blue) and units (red), although of course he doesn't call them "feudal" but rather ℚ(√5).
It took me a while to figure out how to read it. When I did, I enlarged the relevant section and annotated it. Click the thumbnail below to see a larger version of the image we started this article with:
Figure 4
[This article was first posted (using a different choice for the fundamental primes) here: viewtopic.php?f=21&t=555]
And when nobles attempt to multiply with rationals, the result is usually an irrational ignoble.
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