Noble frequency ratios as prime-count vectors in ℚ(√5)

Introductions, chit chat and off-topic banter.
User avatar
Dave Keenan
Site Admin
Posts: 2085
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

Noble frequency ratios as prime-count vectors in ℚ(√5)

Post by Dave Keenan »

The Feudal Manifesto



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 — 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 prime-count 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:
The standard "explanation" for the "ℤ" is that it stands for the German "Zahlen". But that explains nothing. In German, "Zahlen" refers to any kind of number. The integers are called "ganzen Zahlen", which starts with "G". In fact German mathematicians in the 1920s used "G" or "Γ" (gamma) for the integers for that reason. It is sometimes helpfully added that "zählen" is the verb "to count". But this is also powerless to explain "ℤ" for the integers, since "zählen Zahlen" are the "counting numbers", which are only the positive integers. The article Earliest Uses of Symbols of Number Theory tells us we have Edmund Landau (German) followed by the Bourbaki group (French) to thank for "ℤ", in the 1930's. But it offers only the standard non-explanation for why they chose "ℤ".
ℚ(√5) is a quadratic field where the rationals are adjoined with the irrational number "root 5". Conveniently for our purposes, the numbers of ℚ(√5) can be written in the form (a+bϕ)/(x+yϕ) where ϕ is the golden ratio, (1+√5)/2 ≈ 1.618, and a, b, x, y are integers (and x, y are not both zero). "ϕ" is the Greek symbol phi, pronounced in English like the "fy" in "modify".
Rant about ϕ:
It's a complete mess, the way that fonts treat GREEK PHI SYMBOL (U+03D5) versus GREEK SMALL LETTER PHI (U+03C6), because the Unicode committee screwed it up. In Greek language text, the small letter phi comes in two stylistic variants straight phi and curly phi ϕφ, in the same way the Latin letter gee comes in the stylistic variants open gee and closed gee ɡg. The choice of which one to use in a given font should be up to the font designer, and at present the curly form is more common in Greek text.

But in mathematics (prior to the Unicode stuffup) these two forms of phi were treated as distinct symbols. Straight phi was the standard mathematical form, and curly phi was a rarely-used variant. This is attested to by the popular mathematical typesetting system \((La)\TeX\), and the Wolfram Language (formerly Mathematica) and by pre-Unicode versions of the Adobe Symbol font. So Unicode should have provided code points for two "phi symbol" characters with fixed forms, in addition to the variable-form "small letter phi" character. Instead, they only provided one phi symbol character, and for its reference glyph they gave the rarely-used-as-a-symbol curly phi. In fact, in version 1 of Unicode, the description for U+03D5 was "GREEK SMALL LETTER SCRIPT PHI", where "script" implies curly. Many fonts were designed on that basis, and remain that way. But in version 3 of Unicode, instead of adding a new code point for a straight phi symbol, they compounded the problem, by changing the reference glyph for the single existing phi symbol from curly to straight!

So, depending on what device you're viewing this on, and depending on whether it's in a proportionally-spaced or monospaced section below, you may see the phi symbol as either straight, curly, or a "glyph not found" box . Sigh. Please treat all of them as equivalent, and try to imagine them all as straight phi, since that's the standard form used to represent the golden ratio. I look forward to a time when all fonts comply with the current Unicode standard, and everything looks as it should below, no matter what device you're viewing it on.

Those of you who are wise in the ways of mathematical typography may be wondering why I don't just use [latex]\phi[/latex] \(\phi\) or MATHE­MA­TI­CAL ITALIC PHI SYMBOL (U+1D719) 𝜙. The reason is that I prefer to use a non-italic (non-slanted) phi symbol, because the golden ratio is a constant, not a variable.

I have heard rumors that someone, perhaps as illustrious as Roger Penrose, has used capital letter phi or \Phi \(\Phi\) for the golden ratio, and the lowercase phi symbol or \phi \(\phi\) for its reciprocal, which would have been logical had it started out that way, but the lowercase is too entrenched as the golden ratio, to be able to change it now without confusion.
Both the rational numbers and the noble numbers are subsets of ℚ(√5). The rationals are numbers of the above form where b=0 and y=0. The nobles are those where, when we cross-multiply and subtract, we get ay-bx = ±1. I call the numbers in ℚ(√5) that are not noble "ignoble" (or low born) numbers. The rationals are included in the ignobles. And I call all of ℚ(√5) (noble and ignoble together), the "feudal" numbers, because, if we care about the difference between nobles and ignobles, we must be in a feudal system. And "feudal" sounds a bit like "phi-dal". And although this did not consciously occur to me when coining the term: It ties in nicely with the neo-medieval theme of Margo's work. And thanks to Ralph Hutchison for pointing out that another reason the name "feudal" is fitting, is that in chess, a knight's move is a distance of √5.

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.

Image Image

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 prime-count 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 prime-count vector (PC-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 PC-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

Dekker units and primes of ℚ(√5).png

[This article was first posted (using a different choice for the fundamental primes) here: viewtopic.php?f=21&t=555]

User avatar
Dave Keenan
Site Admin
Posts: 2085
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

Re: Noble frequency ratios as prime-count vectors in ℚ(√5)

Post by Dave Keenan »

Here is a table of the first 32 (superunison) noble numbers, working our way down the Stern-Brocot tree, shown as quotients of feudal primes (primes of ℚ(√5)). The alternating light grey and lighter grey backgrounds delineate the levels of the tree. The nobles whose cent values are shown in blue, can be heard in Margo Schulter's MI/JI cadence sampler. @mschulter

Table 1

Stern-	Simplest	Noble	Cents	Quotient of	Approximation	Alternative
Brocot	adjacent	mediant		feudal primes	in 12edo	form
order	mediends			or units			for units
	1 (wts)	ϕ
-----------------------------------------------------------------------------------
1	1/1	2/1	1.6180	 833.1	( 1+0ϕ)/(-1+1ϕ)			1/ϕ⁻¹ = ϕ
2	2/1	3/1	2.6180	1666.2	( 0+1ϕ)/(-1+1ϕ)			ϕ/ϕ⁻¹ = ϕ²
3	3/2	4/3	1.3820	 560.1	(-1+2ϕ)/( 0+1ϕ)			(-1+2ϕ)/ϕ
3	3/1	4/1	3.6180	2226.2	(-1+2ϕ)/(-1+1ϕ)			(-1+2ϕ)/ϕ⁻¹
4	4/3	5/4	1.2764	 422.5	(-2+3ϕ)/(-1+2ϕ)
4	5/3	7/4	1.7236	 942.5	(-1+3ϕ)/(-1+2ϕ)
4	5/2	7/3	2.3820	1502.6	(-1+3ϕ)/( 0+1ϕ)	minor tenth	(-1+3ϕ)/ϕ
4	4/1	5/1	4.6180	2648.7	(-2+3ϕ)/(-1+1ϕ)			(-2+3ϕ)/ϕ⁻¹
5	5/4	6/5	1.2165	 339.3	(-3+4ϕ)/(-2+3ϕ)
5	7/5	10/7	1.4198	 606.9	(-1+4ϕ)/(-1+3ϕ)	tritone
5	8/5	11/7	1.5802	 792.1	(-2+5ϕ)/(-1+3ϕ)
5	7/4	9/5	1.7835	1001.6	(-3+5ϕ)/(-2+3ϕ)	minor seventh
5	7/3	9/4	2.2764	1424.1	(-3+5ϕ)/(-1+2ϕ)
5	8/3	11/4	2.7236	1734.6	(-2+5ϕ)/(-1+2ϕ)
5	7/2	10/3	3.3820	2109.4	(-1+4ϕ)/( 0+1ϕ)			(-1+4ϕ)/ϕ
5	5/1	6/1	5.6180	2988.1	(-3+4ϕ)/(-1+1ϕ)			(-3+4ϕ)/ϕ⁻¹
6	6/5	7/6	1.1780	 283.6	(-4+5ϕ)/(-3+4ϕ)
6	9/7	13/10	1.2957	 448.5	(-1+5ϕ)/(-1+4ϕ)
6	11/8	15/11	1.3672	 541.4	(-3+7ϕ)/(-2+5ϕ)
6	10/7	13/9	1.4393	 630.4	(-4+7ϕ)/(-3+5ϕ)
6	11/7	14/9	1.5607	 770.6	(-5+8ϕ)/(-3+5ϕ)
6	13/8	18/11	1.6328	 848.9	(-3+8ϕ)/(-2+5ϕ)
6	12/7	17/10	1.7043	 923.0	(-2+7ϕ)/(-1+4ϕ)
6	9/5	11/6	1.8220	1038.6	(-5+7ϕ)/(-3+4ϕ)
6	9/4	11/5	2.2165	1378.0	(-5+7ϕ)/(-2+3ϕ)
6	12/5	17/7	2.4198	1529.9	(-2+7ϕ)/(-1+3ϕ)
6	13/5	18/7	2.5802	1641.0	(-3+8ϕ)/(-1+3ϕ)
6	11/4	14/5	2.7835	1772.3	(-5+8ϕ)/(-2+3ϕ)
6	10/3	13/4	3.2764	2054.5	(-4+7ϕ)/(-1+2ϕ)
6	11/3	15/4	3.7236	2276.0	(-3+7ϕ)/(-1+2ϕ)
6	9/2	13/3	4.3820	2557.9	(-1+5ϕ)/( 0+1ϕ)			(-1+5ϕ)/ϕ
6	6/1	7/1	6.6180	3271.7	(-4+5ϕ)/(-1+1ϕ)			(-4+5ϕ)/ϕ⁻¹

The "simplest adjacent mediends" (pronounced "mee-dee-ENDS") of a noble, or its "SAMs" for short, are the simplest pair of rationals, that are directly connected on the Stern-Brocot tree, that you can take the noble-mediant of, to obtain the noble. in other words, they are the two rationals at the top of the zigzag for a given noble. The one at the very top, the simplest, can be called the "1SAM", and the other one the "2SAM" (short for "second-simplest adjacent mediend"). The ordinary mediant of the noble's 1SAM and 2SAM is its 3SAM, and so on down.

It is important to realise that there is no octave-equivalence among nobles. None of the above differ by 1200 cents. Nor is there tritave equivalence (1902c) or fifth-equivalence (702c). And perhaps more surprising, there is no phitave equivalence among nobles. Only one pair of nobles differ by 833c, namely ϕ and ϕ² (833c and 1666c).

When the unison is added to the above list, and all of the 528 pairwise cent differences are calculated, the most any difference-interval occurs is 3 times. Those difference-intervals are the simple nobles near: 339c, 560c, 607c, 792c, 849c, 923c, 943c. There are some other difference-intervals that approximately reoccur 3 or more times. Notably 138c, 242c, 501c. These point to small "feudal commas" that may be exploited in future.

One might say that nobility is fragile. It rarely survives multiplication. Very few pairs of nobles can breed successfully. 😉 And when nobles attempt to multiply with rationals, the result is usually an irrational ignoble.

Below is the complete set of 15 primes and 3 units that occur in the above table. Notice that they don't have golden parts greater than 8 or negative wooden parts greater than 5. And 6 does not occur as a golden part, despite the existence of the primes (-5+6ϕ) and (-1+6ϕ) which do occur in higher-order nobles.

Units and primes of the first 32 nobles:

Table 2

-1+1ϕ, 1+0ϕ, 0+1ϕ   [the units ϕ⁻¹, ϕ⁰, ϕ¹]
-1+2ϕ
-2+3ϕ
-1+3ϕ
-3+4ϕ
-1+4ϕ
-4+5ϕ
-3+5ϕ
-2+5ϕ
-1+5ϕ
-5+7ϕ
-4+7ϕ
-3+7ϕ
-2+7ϕ
-5+8ϕ
-3+8ϕ

So we'd need PC-vectors with 16 entries, just to represent the first 32 nobles! Mind you, these are probably the only nobles of any interest musically. Their simplest adjacent mediends have an ordinary prime limit of 13.

If we limit ourselves to the first 16 nobles, we only need 8 entries. You might think we'd need another 6 entries to represent the 13-prime-limit rationals, but it turns out we only need another 4 entries, for a total of 12.

That's because we have a tiny win when we come to represent rational numbers in the feudal system, as some of the ordinary primes are feudal composites, and are composed of the same primes that make up the simplest nobles.

Of the ordinary primes, only 2, and those whose decimal representation ends in 3 or 7, are feudal primes. Which means that 5, and those ending in 1 or 9, are composite. More about that soon.

So the following ordinary primes are prime in ℚ(√5):

Table 3

 2+0ϕ
 3+0ϕ
 7+0ϕ
13+0ϕ
17+0ϕ
23+0ϕ
37+0ϕ
43+0ϕ
47+0ϕ
53+0ϕ
67+0ϕ
73+0ϕ
...

They can be called "wooden primes", as they are feudal primes with a zero golden part. A wooden prime is a feudal prime that is also an ordinary prime. Notice that the feudal primes that occur as the numerator or denominator of a noble number above, are all "non-wooden primes", as they all have a non-zero golden part.

The following ordinary primes are composite in ℚ(√5), with the given prime factorisation:

Table 4

 5+0ϕ = (-1+2ϕ)(-1+2ϕ)
11+0ϕ = (-2+3ϕ)(-1+3ϕ)
19+0ϕ = (-3+4ϕ)(-1+4ϕ)
29+0ϕ = (-4+5ϕ)(-1+5ϕ)
31+0ϕ = (-3+5ϕ)(-2+5ϕ)
41+0ϕ = (-5+6ϕ)(-1+6ϕ)  [Note: (-5+6ϕ) and (-1+6ϕ) are not used in the first 32 nobles]
59+0ϕ = (-5+7ϕ)(-2+7ϕ)
61+0ϕ = (-4+7ϕ)(-3+7ϕ)
71+0ϕ = (-7+8ϕ)(-1+8ϕ)  [Note: (-7+8ϕ) and (-1+8ϕ) are not used in the first 32 nobles]
79+0ϕ = (-5+8ϕ)(-3+8ϕ)
...

It is known that ordinary primes that are composite in ℚ(√5) always factorise into exactly two unequal primes, except for 5 which is the square of a prime. These can be called "split primes". See what patterns you can spot in the above factorisations before reading on.

The golden parts of the prime factors are always the same, and are the floor of the square root of the ordinary prime. The wooden parts sum to give the negative of the golden part. The product of the wooden parts plus the product of the golden parts is the ordinary prime.

And notice that the prime factors of these ordinary primes include all 15 prime factors of the first 32 nobles.

If you want to confirm any of these factorisations, you can do the multiplication. You will obtain some ϕ²'s, but these can be eliminated by using the identity ϕ² = 1+ϕ. Or you can just use the formula in the last line below.

(a+bϕ)×(x+yϕ)
= ax + (ay+bx)ϕ + byϕ²
= ax + (ay+bx)ϕ + by(1+ϕ)
= (ax+by) + (ay+bx+by)ϕ
= (ax+by) + (by+bx+ay)ϕ [reordered so the two uses of "by" are close together]

Here's feudal integer multiplication shown graphically:

Figure 5
        ___________
       |   ______  |
    a  | +|  b   | |  ϕ
    :`.|  |,':   | |
   _:  `.,'  :_  | |
  | :  ,'`.  : | | |
  | :,'    `.: | | |
  | x    +   y | | |  ϕ
  |  __________| | |
  | |          | | |
  v v          v v v
(ax+by)  +  (by+bx+ay)ϕ

User avatar
Dave Keenan
Site Admin
Posts: 2085
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

Re: Noble frequency ratios as prime-count vectors in ℚ(√5)

Post by Dave Keenan »

I've been somewhat glossing over the idea of "units" — those integers of ℤ[ϕ] (and hence ℚ(√5)) that Dirk Dekker and I have been colouring red. Why are they called units? And if they are all powers of ϕ then why isn't ϕ considered to be just another prime in this system? After all, we need an entry for the ϕ-count in our prime-count vectors,

A hint comes above when I note that -1+1ϕ (a unit that appears in the denominator of some nobles above) is ϕ-1. The standard definition of a negative power is as the reciprocal of the corresponding positive power, x-n = 1/xn. This means that ϕ1 x ϕ-1 = 1. So we can check whether (0+1ϕ)(-1+1ϕ) = 1 using the multiplication algorithm above.

Figure 6
        ___________
       |   ______  |
    0  | +|  1   | |  ϕ
    :`.|  |,':   | |
   _:  `.,'  :_  | |
  | :  ,'`.  : | | |
  | :,'    `.: | | |
  | -1   +   1 | | |  ϕ
  |  __________| | |
  | |          | | |
  v v          v v v
 (0+1)    +  (1+-1+0) ϕ
= 1+0ϕ
= 1

So we see that (-1+1ϕ) really is the reciprocal of (0+1ϕ).

It is a rare thing that an integer has a reciprocal that is also an integer. 1/2, 1/3, 1/4, ... are not integers. In fact, that is the definition of a "unit":

"A unit is an integer whose reciprocal is also an integer."


Among the ordinary integers, only 1 and -1 have that property. They happen to be their own reciprocals. e.g. 1/-1 = -1. As soon as you have a pair of integers that multiply to give 1, e.g -1 × -1 = 1, then you can break the naive definition of a prime number as "a number divisible only by 1 and itself". e.g. we can write 3 = 3 × -1 × -1 so that 3 appears to no longer be prime. And we can break the naive version of the fundamental theorem of arithmetic that says that "any integer can be represented as a product of primes in only one way", because we can write 6 as 2 × 3 or -2 × -3.

But we don't actually want to treat -2×-3 as a different factorisation from 2×3 because it's obvious that -2 and 2 are related, as are -3 and 3. We call them "associated primes" or simply "associates". The only difference between -2 and 2 is multiplication by a unit (an integer whose reciprocal is also an integer).

So we can amend the naive definition of a prime to become:

"A prime is an integer divisible only by units and its associates (which are copies of itself multiplied by different units)".

And within any set of associated primes, we can always designate one of them as the fundamental prime (or the normal form of the prime). Among the ordinary integers these are the positive primes. Among the feudal integers, for this thread I've chosen the fundamental primes so that:
(a) the prime factorisations of rational numbers do not contain any unit (other than 1), and
(b) the prime factorisations of noble numbers do not contain any unit (other than 1) unless this is forced by (a).
This means that: A fundamental prime is either a wooden-prime with a zero golden part, or it has a negative wooden part whose absolute value is less that its positive golden part. This also ensures that the absolute values of its parts, are as small as possible, and that the factors of split primes are as close to the square root of the ordinary prime as possible.

Then we can amend the naive version of the fundamental theorem to become:

"Any nonzero integer can be represented as a product of zero-or-more fundamental primes, and one unit, in only one way (ignoring the order of the factors)."


A number system that obeys this form of the fundamental theorem is called a unique factorisation domain (UFD). Fortunately for us, the feudal integers form a unique factorisation domain, which means that every feudal number (not only the feudal integers) has a unique representation as a prime-count vector. However the vector must have an entry for the unit, in addition to entries for the primes, so strictly speaking, it is a unit-and-prime-count vector.

I mentioned that all integer powers of ϕ are feudal units. Their negations, -ϕⁿ are the only other feudal units, but we don't need negative feudals for tuning theory. Recognition of feudal units is aided by the fact that their wooden and golden parts are successive Fibonacci numbers, including "negative Fibonacci numbers" obtained by "running the rule backwards", as shown below.

Here's a list of some (positive) feudal units:

Table 5
...
ϕ-9	-55+34ϕ		Notice how the minus sign alternates 
ϕ-8	 34-21ϕ		between wooden and golden parts
ϕ-7	-21+13ϕ		for powers of ϕ with negative exponents
ϕ-6	 13-8ϕ
ϕ-5	 -8+5ϕ
ϕ-4	  5-3ϕ
ϕ-3	 -3+2ϕ
ϕ-2	  2-1ϕ
ϕ-1	 -1+1ϕ
ϕ0 = 1	  1+0ϕ
ϕ1 = ϕ	  0+1ϕ
ϕ2	  1+1ϕ
ϕ3	  1+2ϕ
ϕ4	  2+3ϕ
ϕ5	  3+5ϕ
ϕ6	  5+8ϕ
ϕ7	  8+13ϕ
ϕ8	 13+21ϕ
ϕ9	 21+34ϕ
...

I note that the procedure given in the earlier post, for "reducing a noble number to lowest terms", only serves to factor out common units in the numerator and denominator, leaving any primes as fundamental primes. It does not cancel any common primes that may occur in the numerator and denominator. However the latter has not proven necessary, because every noble examined so far has been either a ratio of two primes, a single prime, or a unit. But we should no longer call it "reducing to lowest terms" but rather "normalising".

Each step of the reduction procedure is equivalent to dividing top and bottom by ϕ. To see this, notice that dividing by ϕ is equivalent to multiplying by ϕ-1 = -1+1ϕ, and (a+bϕ)(-1+1ϕ) = (-a+b)+(b+-b+a)ϕ = (a-b)+aϕ.

User avatar
Dave Keenan
Site Admin
Posts: 2085
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

Re: Noble frequency ratios as prime-count vectors in ℚ(√5)

Post by Dave Keenan »

Here's a 2D table of the small feudal integers we've been discussing. Only the units and fundamental-primes are labelled. As usual, units are shown in red, and primes in black or green. Those in black are the fundamental-primes that occur in the first 32 nobles (first 6 levels of the Stern-Brocot tree).

Primes labelled with an upper or lowercase "f" followed by an ordinary prime number, are factors of that ordinary prime, as well as being numerators or denominators of various simple nobles. Both upper and lowercase "f" stand for both "factor of" and "feudal prime". The phrase "f11 and F11" can be read as "eff eleven and big eff eleven". "f11" can also be called "small eff eleven" if necessary. "Small" and "big" relate to their real values as well as their letters. So "f5" should not be called "small eff five" since 5 only has one factor, namely √5 = -1+2ϕ.

Table 6

  23 |  			23
   … |
  17 |  			17
   … |
  13 |  			13
   … |
   7 |				7
   6 |
-----+---------------------------------------------------------------------------------------------
     |				g o l d e n   p a r t
   + |	-3ϕ	-2ϕ	-1ϕ	0ϕ	1ϕ	2ϕ	3ϕ	4ϕ	5ϕ	6ϕ	7ϕ	8ϕ
-----+---------------------------------------------------------------------------------------------
   5 |	ϕ⁻⁴											ϕ⁶
w  4 |
o  3 |				3					ϕ⁵
o  2 |			ϕ⁻²	2			ϕ⁴
d  1 |				ϕ⁰	ϕ²	ϕ³
e  0 |				.	ϕ¹							
n -1 |					ϕ⁻¹	f5	F11	F19	F29	F41		F71
  -2 |							f11		F31		F59	
p -3 |						ϕ⁻³		f19	f31		F61	F79
a -4 |									f29		f61	
r -5 |										f41	f59	f79
t -6 |												
  -7 |												f71
  -8 |									ϕ⁻⁵

The above table is effectively also a plot of these fundamental primes and units on the feudal plane. A dot marks the origin. However the axes are transposed from the conventional arrangement with wooden part horizontal. On it, you can see the two hyperbola-like curves of units (orangish-red for even exponents and purplish-red for odd) as described by Henrik Lindquist Ljungstrand in the Xenharmonic Alliance Mathematical Theory facebook group.

The common asymptote that is approached by negative exponents of ϕ (both odd and even) — a sloping line (not shown) passing through the origin — is the boundary between the positive and negative feudals. We have no use for negative feudals (or feudal zero) in tuning theory, since we have no use for non-positive frequency ratios.

The "f" and "F" can be thought of as "almost square root" operators. In the case of f5 they give the actual square root. You can extract the feudal integer from the number after the "f" or "F" with a little mental arithmetic as follows. Call this number p. Find the greatest square number that is not greater than p, call it s. Its square root is the golden part of both factors, call it b. So b² = s and b = ⌊√p⌋. Subtract s from p to obtain a remainder. Factorise this remainder p-s into two factors (not negative, but may include 0 or 1) that sum to b, call these factors a and A, where a is the smaller factor. So we need a×A = p-s and a+A = b = √s. Then fp = -A+bϕ and Fp = -a+bϕ. You may be surprised that big A goes to small "f" and vice versa. That's because they get negated, and -A is less than -a.

For example, consider f31 and F31. The greatest square not greater than 31 is 25 = 5². So s = 25 and b = 5. The remainder is p-s = 31-25 = 6, which can be factored as 2×3 so that 2+3 = 5 = b. So f31 = -3+5ϕ and F31 = -2+5ϕ.

It is a nice consequence of this, that "f1" and "F1" can be used as alternative names for ϕ⁻¹ and ϕ. Try running the above algorithm on them. But we must remember they are units, not primes, and we only need a (big) F1-count entry in our vectors, not an f1-count.

You can run it the other way too. If you see "-4+5ϕ", you can figure out what its "f" number is. Ignore the minus sign. Square the 5 and add the product of 4 with its complement 5-4 = 1. That's 5² + (4 × 1) = 29. It's got the -4 while its partner has the -1, so it's the smaller of the two. So it's f29. [Note for advanced readers: The number after the "f" is the negation of the norm of the original number. -N(a+bϕ) = -(a²+ab-b²) = b²-ab-a² = b² - (a(b+a)) = b² + (-a(b - -a))]

User avatar
Dave Keenan
Site Admin
Posts: 2085
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

Re: Noble frequency ratios as prime-count vectors in ℚ(√5)

Post by Dave Keenan »

In RTT it is convenient that we have an obvious ordering of primes, so that unless we are told otherwise, a vector like [-2 0 0 1 can be interpreted as 2⁻²×3⁰×5⁰×7¹ = 7/4. And it's convenient that this ordering agrees with the typical order of introduction of primes, as temperaments become more complex.

It's not so obvious what the standard ordering should be for the feudal primes, since they are arranged on a plane, not a line. And their order of introduction is very different depending on whether you're adding a prime in order to facilitate new nobles, or new rationals. Some rationals require wooden primes. Nobles never do.

A standard ordering also needs to be easy to remember.

Lexicographic order of the a+bϕ form would be easy to remember, but unfortunately it's useless. You end up having to list the infinite set of split-prime factors (used mostly by nobles) before you can start on the unsplit or wooden primes (used by rationals). It's still useless if you rearrange them as bϕ+a.

Ordering them by their real value would at least progress through both split and unsplit, but this value is not available to inspection.

One option is to place them in order of the ordinary prime that they are a factor of. And for split primes, listing the small "f" factor before the big "F" factor.

red for units, green for primes not used by the first 32 nobles.

Table 8

 0+1ϕ =  F1  (ϕ)
 2+0ϕ =   2
 3+0ϕ =   3
-1+2ϕ =  f5
 7+0ϕ =   7
-2+3ϕ = f11
-1+3ϕ = F11
13+0ϕ =  13
17+0ϕ =  17
-3+4ϕ = f19
-1+4ϕ = F19
23+0ϕ =  23  [primes beyond here are likely only wanted for nobles, not rationals]
-4+5ϕ = f29	
-1+5ϕ = F29
-3+5ϕ = f31	
-2+5ϕ = F31
37+0ϕ =  37  [fairly useless]
-5+6ϕ = f41	
-1+6ϕ = F41
43+0ϕ =  43  [fairly useless]
43+0ϕ =  47  [fairly useless]
53+0ϕ =  53  [fairly useless]
-5+7ϕ = f59
-2+7ϕ = F59
-4+7ϕ = f61
-3+7ϕ = F61
67+0ϕ =  67  [fairly useless]
-7+8ϕ = f71
-1+8ϕ = F71
76+0ϕ =  76  [fairly useless]
-5+8ϕ = f79
-3+8ϕ = F79

This gives the nearest thing to backwards-compatibility with the existing rational-only PC-vectors. The problem with it is the inclusion of many unwanted high wooden primes among the high non-wooden primes wanted for nobles.

I think the solution is to provide a marker that says, "No more wooden primes beyond this point". I suggest a semicolon.

So we first list, in ordinary-prime-order, all the feudal primes that are used to encode rational numbers in a given temperament, followed by a semicolon, followed by the unit (ϕ-count) followed by the primes that are only used to encode nobles, in ordinary-prime-order. The semicolon (followed by a space) separates the two sections, whether or not commas (or only spaces) are used to separate the other counts.

For a 13-prime-limit-plus-nobles temperament, the order would be:

Table 9

 2+0ϕ =   2
 3+0ϕ =   3
-1+2ϕ =  f5  [This will be used for nobles as well as being doubled for the ordinary prime 5]
 7+0ϕ =   7
-2+3ϕ = f11  [This will be used for nobles as well as the ordinary prime 11]
-1+3ϕ = F11  [This will be used for nobles as well as the ordinary prime 11]
13+0ϕ =  13
; semicolon separator
 0+1ϕ =  F1  (ϕ)  [primes from here on will only be used for nobles, not rationals]
-3+4ϕ = f19
-1+4ϕ = F19
-4+5ϕ = f29	
-1+5ϕ = F29
-3+5ϕ = f31	
-2+5ϕ = F31
-5+6ϕ = f41	
-1+6ϕ = F41
-5+7ϕ = f59
-2+7ϕ = F59
-4+7ϕ = f61
-3+7ϕ = F61
-7+8ϕ = f71
-1+8ϕ = F71
-5+8ϕ = f79
-3+8ϕ = F79

This would be the default way to interpret a feudal vector. But of course you are free to use a different basis; you just have to spell out what it is. For example, if we don't need to encode any nobles beyond the first 32 (the "6th-order-limit"), we might decide to omit the last four green primes from the nobles-only section, rather than carry around a bunch of entries that are always zero.

With this format, the conversion of an existing 13-limit rationals-only PC-vector into a feudal PC-vector would look like this:

Figure 7

[ a  b  c  d  e  f 
  |  |  |  |  |\  \ 
[ a  b 2c  d  e  e  f;  0  0  0 ... 

i.e. The 5-count must be doubled and the 11-count duplicated.

If we were interested in, at most, the two nobles that can be composed from f5, f11 and F11 (in addition to the 13-limit rationals), we would still need to put a semicolon at the end of the vector, so the reader would know that 5 and 11 have been split. Like this: [a b 2c d e e f;

When we are representing nobles independent of any rational prime limit, the PC-vectors will begin with a semicolon, and will list (non-wooden) primes in ordinary-prime order. Assuming 6th-order-limit nobles, the vectors will be of the form:

[; α β γ δ ε ζ η θ ι κ λ μ ν ξ π ρ

Where α (alpha) is the count of units, β (beta) is the count of prime f5 (-1+2ϕ), γ (gamma) is the count of prime f11 (-2+3ϕ), etc, up to the 16th entry ρ (rho) as the count of prime F79 (-3+8ϕ).

User avatar
Dave Keenan
Site Admin
Posts: 2085
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

Re: Noble frequency ratios as prime-count vectors in ℚ(√5)

Post by Dave Keenan »

I've had some fun naming the orders of nobles on the Stern-Brocot tree. To get enough names, I've rationalised multiple European nobility ranking systems, but I've used English words. The result is closest to the French system. I made up the adjectives for the middle three orders, as I couldn't find any existing. I note that "marquis" is pronounced "mar-KEE" in the original French, but has been anglicised as "MAR-kwis".

Table 10

Singular, plural, adjective:
1. prince, princes, princely
2. duke, dukes, ducal
3. marquis, marquises, marquial
4. count, counts, countial
5. viscount, viscounts, viscountial
6. baron, barons, baronial
7. knight, knights, knightly
8. lord, lords, lordly

If you wanted to use the corresponding female terms, they could be applied to the left half of the Stern-Brocot tree, which contains the reciprocals of the right half that we use here. These are not usually used as frequency ratios, but are used as fractions of an octave in generating noble MOS scales.

I designed the terminology used in this thread so as not to clash with the terminology used in Douglas Blumeyer's article Metallic MOS. I note that feudalism and aristocracy are distinct medieval sociopolitical systems.

Here are the first 32 nobles on the Stern-Brocot tree (the 6th-order-limit nobles), in cents, and as feudal numbers in normal form. Units are shown as ϕ⁻¹, 1, ϕ, ϕ² but I remind you they can also be written as -1+1ϕ, 1+0ϕ, 0+1ϕ, 1+1ϕ. The noble identifiers beginning with "n" are explained in a later post.

Figure 8   Stern-Brocot tree (sideways, with nobles)

Order 6      5          4              3               2     1
     barons viscounts  counts         marquises       duke prince
                                                           1:1 \
               n6/5                                          / |
 7:6 \        f29/f19                                        | .
     284c (-4+5ϕ)/(-3+4ϕ)                                    | .
        6:5 \                                                | .
      /     339c (-3+4ϕ)/(-2+3ϕ)                             |
11:9 /        \      f19/f11                                 |
               \      n5/4                                   |
                 5:4 \                                       |
               /      \          n4/3                        |
14:11\        /        \        f11/f5                       |
      \      /         422c (-2+3ϕ)/(-1+2ϕ)                  |
        9:7 /            \                                   |
     448c (-1+5ϕ)/(-1+4ϕ) \                                  |
13:10/        F29/F19      \                                 |
               n9/7         \                                |
                              4:3 \                          |
              n11/8         /      \                         |
15:11\        F61/F31      /        \                        |
     541c (-3+7ϕ)/(-2+5ϕ) /          \          n3/2         |
       11:8 \            /            \         f5/F1        |
      /      \          /             560c (-1+2ϕ)/ϕ         |
18:13/        \        /                \                    |
               \      /                  \                   |
                 7:5 /                    \                  |
               /      n7/5                 \                 |
17:12\        /      F19/F11                \                |
      \     607c (-1+4ϕ)/(-1+3ϕ)             \               |
       10:7 /                                 \              |
     630c (-4+7ϕ)/(-3+5ϕ)                      \             |
13:9 /        f61/f31                           \            |
              n10/7                              \           |
                                                   3:2 \     |
              n11/7                              /     |     |
14:9 \        f79/f31                           /      |     |
     771c (-5+8ϕ)/(-3+5ϕ)                      /       |     |
       11:7 \                                 /        |     |
      /     792c (-2+5ϕ)/(-1+3ϕ)             /         |     |
19:12/        \      F31/F11                /          |     |
               \      n8/5                 /           |     |
                 8:5 \                    /            |     |
               /      \                  /             |     |  n1/1
21:13\        /        \                /              |     |  F1/1
      \      /          \              /               |   833c ϕ
       13:8 /            \            /                |     |  1/f1
     849c (-3+8ϕ)/(-2+5ϕ) \          /                 |     |
18:11/        F79/F31      \        /                  |     |
              n13/8         \      /                   |     |
                              5:3 /                    |     |
              n12/7         /                          |     |
17:10\        F59/F19      /                           |     |
     923c (-2+7ϕ)/(-1+4ϕ) /                            |     |
       12:7 \            /                             |     |
      /      \         943c (-1+3ϕ)/(-1+2ϕ)            |     |
19:11/        \        /        F11/f5                 |     |
               \      /          n5/3                  |     |
                 7:4 /                                 |     |
               /      n7/4                             |     |
16:9 \        /      f31/f11                           |     |
      \    1002c (-3+5ϕ)/(-2+3ϕ)                       |     |
        9:5 /                                          |     |
    1039c (-5+7ϕ)/(-3+4ϕ)                              |     |
11:6 /        f59/f19                                  |     |
               n9/5                                    \     /
Order 6      5          4              3                 2:1
               n9/4                                     /
11:5 \        f59/f11                                  |
    1378c (-5+7ϕ)/(-2+3ϕ)                              |
        9:4 \                                          |
      /    1424c (-3+5ϕ)/(-1+2ϕ)                       |
16:7 /        \      f31/f5                            |
               \      n7/3                             |
                 7:3 \                                 |
               /      \          n5/2                  |
19:8 \        /        \        F11/F1                 |
      \      /        1503c (-1+3ϕ)/ϕ                  |
       12:5 /            \                             |
    1530c (-2+7ϕ)/(-1+3ϕ) \                            |
17:7 /        F59/F11      \                           |
              n12/5         \                          |
                              5:2 \                    |
              n13/5         /      \                   |
18:7 \        F79/F11      /        \                  |
    1641c (-3+8ϕ)/(-1+3ϕ) /          \                 |  n2/1
       13:5 \            /            \                |  F1/f1
      /      \          /              \             1666c ϕ²
21:8 /        \        /                \              |
               \      /                  \             |
                 8:3 /                    \            |
               /      n8/3                 \           |
19:7 \        /      F31/f5                 \          |
      \    1735c (-2+5ϕ)/(-1+2ϕ)             \         |
       11:4 /                                 \        |
    1772c (-5+8ϕ)/(-2+3ϕ)                      \       |
14:5 /        f79/f11                           \      |
              n11/4                              \     |
                                                   3:1 /
              n10/3                              /
13:4 \        f61/f5                            /
    2055c (-4+7ϕ)/(-1+2ϕ)                      /
       10:3 \                                 /
      /    2109c (-1+4ϕ)/ϕ                   /
17:5 /        \      F19/F1                 /
               \      n7/2                 /
                 7:2 \                    /
               /      \                  /
18:5 \        /        \                /
      \      /          \            2226c (-1+2ϕ)/ϕ⁻¹
       11:3 /            \            /         f5/f1
    2276c (-3+7ϕ)/(-1+2ϕ) \          /          n3/1
15:4 /        F61/f5       \        /
               n11/3        \      /
                              4:1 /
               n9/2         /
13:3 \        F29/F1       /
    2558c (-1+5ϕ)/ϕ       /
        9:2 \            /
      /      \        2649c (-2+3ϕ)/ϕ⁻¹
14:3 /        \        /        f11/f1
               \      /          n4/1
                 5:1 /
               /      n5/1
11:2 \        /      f19/f1
      \    2988c (-3+4ϕ)/ϕ⁻¹	                               .
        6:1 /                                                  .
    3272c (-4+5ϕ)/ϕ⁻¹	                                       .
 7:1 /        f29/f1                                           |
               n6/1                                            \
                                                             1:0
     barons viscounts  counts         marquises       duke prince
Order 6      5          4              3               2     1

Notice that, once you get beyond the first two levels (which don't involve primes, only units), the nobles all consist of big "F" over big "F", or small "f" over small "f". The few apparent violations all involve f5, but consider that f5 could equally be called F5, since they both correspond to the square root of 5.

If you don't have the above diagram at hand, and you are given a noble in the form e.g. F31/F11, you can obtain the two simplest Stern-Brocot-adjacent ratios whose noble-mediant it is, as follows. First use the method described two posts back, which might be described as the greatest-square-remainder-complementary-factorisation (GSRCF) method, on the numerator and denominator, to get it to the form (-2+5ϕ)/(-1+3ϕ), then "fibonacci" the numerator and denominator forward exactly 3 steps*. In this case we get:

-2  5  3  8 11
-- -- -- -- --
-1  3  2  5  7

So F31/F11 is the noble mediant of 8/5 and 11/7. Of course it's also the noble mediant of any other pair of successive ratios in that sequence (and its continuation). But 8/5 and 11/7 are the pair at the start of the zigzag. Prior to them there was a non-adjacency, a discontinuity, a "teleport". 3/2 and 8/5 are not directly connected on the SB-tree.

*Another feudal reference (thanks to Douglas Blumeyer): The Holy Hand Grenade of Antioch. I note that the prince of nobles, ϕ, must be written as 1/f1 for it to take 3 steps. When written as F1/1, it only takes 2 steps.

User avatar
Dave Keenan
Site Admin
Posts: 2085
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

Re: Noble frequency ratios as prime-count vectors in ℚ(√5)

Post by Dave Keenan »

We like to visualise the common factors among a set of rational pitches by displaying them on a prime lattice. However a lattice may not be the best way to visualise the common factor relationships between nobles, because there are so many prime dimensions, and the count of each prime is so low, only -1, 0 or 1 (and only -2, -1, 0, 1, 2 for the unit ϕ).

Here's what might be called an "order-limit triangle" analogous to an "odd-limit diamond". This one is limited to nobles up to the 6th-order. And of course it needs to contain units as well as primes. The primes are listed in order of the ordinary prime of which they are a factor, which is also lexicographic order of their "fF" identifier.

Table 11

	    numerator |	f1 ϕ⁻¹	1  ϕ⁰	F1 ϕ¹	f5	f11	F11	f19	F19	f29	F29	f31	F31	f59	F59	f61	F61	f79	F79
		      |	0.618	1.000	1.618	2.236	2.854	3.854	3.472	5.472	4.090	7.090	5.090	6.090	6.326	9.326	7.326	8.326	7.944	9.944
    denominator	    \ |	 1+0ϕ	 0+1ϕ	 1+1ϕ	-1+2ϕ	-2+3ϕ	-1+3ϕ	-3+4ϕ	-1+4ϕ	-4+5ϕ	-1+5ϕ	-3+5ϕ	-2+5ϕ	-5+7ϕ	-2+7ϕ	-4+7ϕ	-3+7ϕ	-5+8ϕ	-3+8ϕ
  --------------------+-----------------------------------------------------------------------------------------------------------------------------------------------
  f1 ϕ⁻¹0.618	 1+0ϕ |		 833c	1666c	2226c	2649c		2988c		3272c									
  1  ϕ⁰	1.000	 0+1ϕ |			 833c															
  F1 ϕ¹	1.618	 1+1ϕ |				 560c		1503c		2109c		2558c								
  f5	2.236	-1+2ϕ |					 422c	 943c					1424c	1735c			2055c	2276c		
  f11	2.854	-2+3ϕ |							 339c				1002c		1378c				1772c	
  F11	3.854	-1+3ϕ |								 607c				 792c		1530c				1641c
  f19	3.472	-3+4ϕ |									 284c				1039c					
  F19	5.472	-1+4ϕ |										 448c				 923c				
  f31	5.090	-3+5ϕ |															 630c		 771c	
  F31	6.090	-2+5ϕ |																 541c		 849c

Perhaps it's better transposed, so as not to require horizontal scrolling, even though this puts the denominators above the numerators:

Table 12

	 denominator  |	f1 ϕ⁻¹	1  ϕ⁰	F1 ϕ¹	f5	f11	F11	f19	F19	f31	F31
		      |	0.618	1.000	1.618	2.236	2.854	3.854	3.472	5.472	5.090	6.090
  numerator	    / |	 1+0ϕ	 0+1ϕ	 1+1ϕ	-1+2ϕ	-2+3ϕ	-1+3ϕ	-3+4ϕ	-1+4ϕ	-3+5ϕ	-2+5ϕ
  --------------------+------------------------------------------------------------------------------
  f1 ϕ⁻¹0.618	 1+0ϕ |										
  1  ϕ⁰	1.000	 0+1ϕ |	 833c									
  F1 ϕ¹	1.618	 1+1ϕ |	1666c	 833c								
  f5	2.236	-1+2ϕ |	2226c		 560c							
  f11	2.854	-2+3ϕ |	2649c			 422c						
  F11	3.854	-1+3ϕ |			1503c	 943c						
  f19	3.472	-3+4ϕ |	2988c				 339c					
  F19	5.472	-1+4ϕ |			2109c			 607c				
  f29	4.090	-4+5ϕ |	3272c						 284c			
  F29	7.090	-1+5ϕ |			2558c					 448c		
  f31	5.090	-3+5ϕ |				1424c	1002c					
  F31	6.090	-2+5ϕ |				1735c		 792c				
  f59	6.326	-5+7ϕ |					1378c		1039c			
  F59	9.326	-2+7ϕ |						1530c		 923c		
  f61	7.326	-4+7ϕ |				2055c					 630c	
  F61	8.326	-3+7ϕ |				2276c						 541c
  f79	7.944	-5+8ϕ |					1772c				 771c	
  F79	9.944	-3+8ϕ |						1641c				 849c

And here the units and primes are listed in order of their size as a real number:

Table 13
	 denominator  |	f1 ϕ⁻¹	1  ϕ⁰	F1 ϕ¹	f5	f11	f19	F11	f31	F19	F31
		      |	0.618	1.000	1.618	2.236	2.854	3.472	3.854	5.090	5.472	6.090
  numerator	    / |	 1+0ϕ	 0+1ϕ	 1+1ϕ	-1+2ϕ	-2+3ϕ	-3+4ϕ	-1+3ϕ	-3+5ϕ	-1+4ϕ	-2+5ϕ
  --------------------+------------------------------------------------------------------------------
  f1 ϕ⁻¹0.618	 1+0ϕ |										
  1  ϕ⁰	1.000	 0+1ϕ |	 833c									
  F1 ϕ¹	1.618	 1+1ϕ |	1666c	 833c								
  f5	2.236	-1+2ϕ |	2226c		 560c							
  f11	2.854	-2+3ϕ |	2649c			 422c						
  f19	3.472	-3+4ϕ |	2988c				 339c					
  F11	3.854	-1+3ϕ |			1503c	 943c						
  f29	4.090	-4+5ϕ |	3272c					 284c				
  f31	5.090	-3+5ϕ |				1424c	1002c					
  F19	5.472	-1+4ϕ |			2109c				 607c			
  F31	6.090	-2+5ϕ |				1735c			 792c			
  f59	6.326	-5+7ϕ |					1378c	1039c				
  F29	7.090	-1+5ϕ |			2558c						 448c	
  f61	7.326	-4+7ϕ |				2055c				 630c		
  f79	7.944	-5+8ϕ |					1772c			 771c		
  F61	8.326	-3+7ϕ |				2276c						 541c
  F59	9.326	-2+7ϕ |							1530c		 923c	
  F79	9.944	-3+8ϕ |							1641c			 849c

If we wanted to use this triangle of nobles as a target interval set, for the tuning of some noble temperament, we might decide to truncate it to remove intervals larger than say 13/4 (2041 cents) (shown violet above) on the basis that these are so wide as to occur rarely, and to not require accurate tuning when they do. This leaves 24 nobles. We could then call it the "truncated 6th-order-limit triangle" and abbreviate it to the "6-TOLT", moving the "T" for "truncated" after the 6 to obtain a pronounceable acronym. This is analogous to the truncated integer-limit triangle (TILT) for rational intervals. [In future, the TILT article may be accessible via this link: TILT]

More likely than a purely noble temperament, would be a noble/rational temperament, in which case the target interval set could be the union of a TOLT and a TILT.

It's unfortunate that "order-limit" and "odd-limit" have the same abbreviation, so we are relying here on the fact that the corresponding odd-limit set is a diamond (OLD) rather than a triangle (TOLT, TILT).

User avatar
Dave Keenan
Site Admin
Posts: 2085
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

Re: Noble frequency ratios as prime-count vectors in ℚ(√5)

Post by Dave Keenan »

Here at last are the 6th-order-limit nobles as PC-vectors. The first version of these tables gave the quotients in the form (a+bϕ)/(n+mϕ). But Douglas Blumeyer @cmloegcmluin emailed me a table in which he replaced those with the f11/f5 form, which made me realise that these are a far more useful form, and so I have adopted them here. This also prompted me to add them to the Stern-Brocot tree in the earlier post, and to describe the GSRCF (Greatest Square Remainder Complementary Factorisation) mental-arithmetic algorithm for extracting the -2+3ϕ form from the f11 form, and the HHG (Holy Hand Grenade) method for obtaining the simplest adjacent mediends (SAMs).

Here are the 6th-order-limit nobles as PC-vectors in Stern-Brocot-tree reading-order of their SAMs. The noble identifiers beginning with "n" are explained in a later post.

Table 14

Stern-	Noble	      Second-simplest	Cents	Quotient    Prime count vector
Brocot	identifier	adjacent		of feudal
order	based on	mediend			primes	    ϕ  f5   F11   F19   F29   F31   F59   F61   F79
	1SAM (wt 1)	2SAM (wt ϕ)		or units    F1   f11   f19   f29   f31   f59   f61   f79
------------------------------------------------------------------------------------------------------------
1	n1/1		 2/1		 833c F1/1 1/f1	[;  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 
2	n2/1		 3/1		1666c	 F1/f1	[;  2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 
3	n3/2		 4/3		 560c	 f5/F1	[; -1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0 
3	n3/1		 4/1		2226c	 f5/f1	[;  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0 
4	n4/3		 5/4		 422c	f11/f5	[;  0 -1  1  0  0  0  0  0  0  0  0  0  0  0  0  0 
4	n5/3		 7/4		 943c	F11/f5  [;  0 -1  0  1  0  0  0  0  0  0  0  0  0  0  0  0 
4	n5/2		 7/3		1503c	F11/F1	[; -1  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0 
4	n4/1		 5/1		2649c	f11/f1	[;  1  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0 
5	n5/4		 6/5		 339c	f19/f11	[;  0  0 -1  0  1  0  0  0  0  0  0  0  0  0  0  0 
5	n7/5		10/7		 607c	F19/F11	[;  0  0  0 -1  0  1  0  0  0  0  0  0  0  0  0  0 
5	n8/5		11/7		 792c	F31/F11	[;  0  0  0 -1  0  0  0  0  0  1  0  0  0  0  0  0 
5	n7/4		 9/5		1002c	f31/f11	[;  0  0 -1  0  0  0  0  0  1  0  0  0  0  0  0  0 
5	n7/3		 9/4		1424c	f31/f5	[;  0 -1  0  0  0  0  0  0  1  0  0  0  0  0  0  0 
5	n8/3		11/4		1735c	F31/f5	[;  0 -1  0  0  0  0  0  0  0  1  0  0  0  0  0  0 
5	n7/2		10/3		2109c	F19/F1	[; -1  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0 
5	n5/1		 6/1		2988c	f19/f1	[;  1  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0 
6	n6/5		 7/6		 284c	f29/f19	[;  0  0  0  0 -1  0  1  0  0  0  0  0  0  0  0  0 
6	n9/7		13/10		 448c	F29/F19	[;  0  0  0  0  0 -1  0  1  0  0  0  0  0  0  0  0 
6	n11/8		15/11		 541c	F61/F31	[;  0  0  0  0  0  0  0  0  0 -1  0  0  0  1  0  0 
6	n10/7		13/9		 630c	f61/f31	[;  0  0  0  0  0  0  0  0 -1  0  0  0  1  0  0  0 
6	n11/7		14/9		 771c	f79/f31	[;  0  0  0  0  0  0  0  0 -1  0  0  0  0  0  1  0 
6	n13/8		18/11		 849c	F79/F31	[;  0  0  0  0  0  0  0  0  0 -1  0  0  0  0  0  1 
6	n12/7		17/10		 923c	F59/F19	[;  0  0  0  0  0 -1  0  0  0  0  0  1  0  0  0  0 
6	n9/5		11/6		1039c	f59/f19	[;  0  0  0  0 -1  0  0  0  0  0  1  0  0  0  0  0 
6	n9/4		11/5		1378c	f59/f11	[;  0  0 -1  0  0  0  0  0  0  0  1  0  0  0  0  0 
6	n12/5		17/7		1530c	F59/F11	[;  0  0  0 -1  0  0  0  0  0  0  0  1  0  0  0  0 
6	n13/5		18/7		1641c	F79/F11	[;  0  0  0 -1  0  0  0  0  0  0  0  0  0  0  0  1 
6	n11/4		14/5		1772c	f79/f11	[;  0  0 -1  0  0  0  0  0  0  0  0  0  0  0  1  0 
6	n10/3		13/4		2055c	f61/f5	[;  0 -1  0  0  0  0  0  0  0  0  0  0  1  0  0  0 
6	n11/3		15/4		2276c	F61/f5	[;  0 -1  0  0  0  0  0  0  0  0  0  0  0  1  0  0 
6	n9/2		13/3		2558c	F29/F1	[; -1  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0 
6	n6/1		 7/1		3272c	f29/f1	[;  1  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0 

And here we list them in order of their size in cents:

Table 15

Stern-	Noble	      Second-simplest	Cents	Quotient    Prime count vector
Brocot	identifier	adjacent		of feudal
order	based on	mediend			primes	    ϕ  f5   F11   F19   F29   F31   F59   F61   F79
	1SAM (wt 1)	2SAM (wt ϕ)		or units    F1   f11   f19   f29   f31   f59   f61   f79
------------------------------------------------------------------------------------------------------------
6	n6/5		 7/6		 284c	f29/f19	[;  0  0  0  0 -1  0  1  0  0  0  0  0  0  0  0  0 
5	n5/4		 6/5		 339c	f19/f11	[;  0  0 -1  0  1  0  0  0  0  0  0  0  0  0  0  0 
4	n4/3		 5/4		 422c	f11/f5	[;  0 -1  1  0  0  0  0  0  0  0  0  0  0  0  0  0 
6	n9/7		13/10		 448c	F29/F19	[;  0  0  0  0  0 -1  0  1  0  0  0  0  0  0  0  0 
6	n11/8		15/11		 541c	F61/F31	[;  0  0  0  0  0  0  0  0  0 -1  0  0  0  1  0  0 
3	n3/2		 4/3		 560c	 f5/F1	[; -1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0 
5	n7/5		10/7		 607c	F19/F11	[;  0  0  0 -1  0  1  0  0  0  0  0  0  0  0  0  0 
6	n10/7		13/9		 630c	f61/f31	[;  0  0  0  0  0  0  0  0 -1  0  0  0  1  0  0  0 
6	n11/7		14/9		 771c	f79/f31	[;  0  0  0  0  0  0  0  0 -1  0  0  0  0  0  1  0 
5	n8/5		11/7		 792c	F31/F11	[;  0  0  0 -1  0  0  0  0  0  1  0  0  0  0  0  0 
1	n1/1		 2/1		 833c F1/1 1/f1	[;  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 
6	n13/8		18/11		 849c	F79/F31	[;  0  0  0  0  0  0  0  0  0 -1  0  0  0  0  0  1 
6	n12/7		17/10		 923c	F59/F19	[;  0  0  0  0  0 -1  0  0  0  0  0  1  0  0  0  0 
4	n5/3		 7/4		 943c	F11/f5  [;  0 -1  0  1  0  0  0  0  0  0  0  0  0  0  0  0 
5	n7/4		 9/5		1002c	f31/f11	[;  0  0 -1  0  0  0  0  0  1  0  0  0  0  0  0  0 
6	n9/5		11/6		1039c	f59/f19	[;  0  0  0  0 -1  0  0  0  0  0  1  0  0  0  0  0 
6	n9/4		11/5		1378c	f59/f11	[;  0  0 -1  0  0  0  0  0  0  0  1  0  0  0  0  0 
5	n7/3		 9/4		1424c	f31/f5	[;  0 -1  0  0  0  0  0  0  1  0  0  0  0  0  0  0 
4	n5/2		 7/3		1503c	F11/F1	[; -1  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0 
6	n12/5		17/7		1530c	F59/F11	[;  0  0  0 -1  0  0  0  0  0  0  0  1  0  0  0  0 
6	n13/5		18/7		1641c	F79/F11	[;  0  0  0 -1  0  0  0  0  0  0  0  0  0  0  0  1 
2	n2/1		 3/1		1666c	 F1/f1	[;  2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 
5	n8/3		11/4		1735c	F31/f5	[;  0 -1  0  0  0  0  0  0  0  1  0  0  0  0  0  0 
6	n11/4		14/5		1772c	f79/f11	[;  0  0 -1  0  0  0  0  0  0  0  0  0  0  0  1  0 
6	n10/3		13/4		2055c	f61/f5	[;  0 -1  0  0  0  0  0  0  0  0  0  0  1  0  0  0 
5	n7/2		10/3		2109c	F19/F1	[; -1  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0 
3	n3/1		 4/1		2226c	 f5/f1	[;  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0 
6	n11/3		15/4		2276c	F61/f5	[;  0 -1  0  0  0  0  0  0  0  0  0  0  0  1  0  0 
6	n9/2		13/3		2558c	F29/F1	[; -1  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0 
4	n4/1		 5/1		2649c	f11/f1	[;  1  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0 
5	n5/1		 6/1		2988c	f19/f1	[;  1  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0 
6	n6/1		 7/1		3272c	f29/f1	[;  1  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0 

User avatar
Dave Keenan
Site Admin
Posts: 2085
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

Re: Noble frequency ratios as prime-count vectors in ℚ(√5)

Post by Dave Keenan »

By searching on "2, 3, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73" (wooden primes) in the OEIS, I found a reference to this book:
F. W. Dodd, Number Theory in the Quadratic Field with Golden Section Unit, Polygon Publishing House, Passaic, NJ 07055, 1983.
It is online here:
https://archive.org/details/numbertheor ... d/mode/2up
You just need to sign up for an account to "borrow" the book for an hour at a time, provided no one else is doing so.

There's some interesting stuff in there from a pure math perspective, beyond the things I've described above. Notably the concepts of conjugates and arithmetic norms (unfortunately very different from the vector norms used in advanced RTT). But I haven't yet found this additional material useful for our application. Although it did make me realise that when ordinary primes are composite in the feudal system, their two prime factors are complements, in the sense that if one is -a+bϕ, the other is -(b-a)+bϕ. e.g. the prime factors of 31 are -3+5ϕ and -(5-3) + 5ϕ = -2+5ϕ.

Conjugates have a simple form when the feudal basis is (1, √5), but when using a basis of (1, ϕ) = (1, (1+√5)/2), as is most convenient for representing our nobles, conjugates are messy, unless you introduce a symbol such as ϕ̅ (phi bar) for the negative unit -1/ϕ = -ϕ⁻¹ = 1-1ϕ = (1-√5)/2 as Dodd does. But this leads to other problems such as the appearance that 5 has two different prime factors (2+1ϕ)(2+1ϕ̅), i.e. 2+1ϕ and its conjugate, when in fact (2+1ϕ̅) is an associate of (2+1ϕ) since (2+1ϕ̅) = (2+1ϕ)ϕ⁻². This is why I've defined the complement above. The complement is simply the conjugate multiplied by ϕ².

The arithmetic norm (pronounced AR-ith-MET-ik norm) has symbol "N" and is defined such that N(a+bϕ) = a²+ab-b². It has only rational values (which may be negative, unlike vector norms) and has the properties that
N(αβ) = N(α)N(β)
N(α/β) = N(α)/N(β) when β≠0
N(fp) = N(Fp) = -p
and N(ϕⁿ) = (-1)ⁿ for all ordinary integers n.

What we need now, is for some clever volunteers to do computer searches for noble commas, and noble/rational commas, that can form null-space bases for temperaments. A first test might be to see if you can find a feudal-comma basis for Margo Schulter's Zest-24 tuning as used in her sampler of noble/rational cadences. [Not you, Douglas Blumeyer. We have other fish to fry. :) ]

Paul Erlich pointed out that 72-edo should be fertile ground to look for feudal commas.

As I said at the start of this series, there's no guarantee this will lead to anything useful. But I hope some fellow students of tuning-math find it interesting enough to investigate, if only as a form of recreational math.

User avatar
Dave Keenan
Site Admin
Posts: 2085
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

Re: Noble frequency ratios as prime-count vectors in ℚ(√5)

Post by Dave Keenan »

There is much discussion of this topic on the facebook group Xenharmonic Alliance - Mathematical Theory.

Many thanks to Mike Battaglia for these extensive discussions.

Post Reply