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

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

Well that's a tough act to follow...

And Dave specifically told me not to do this. So, I'll do it just a bit

I did some comma-hunting, using some rudimentary bounds (like on the counts of unique primes, the max individual prime count, and product complexity), and then just scanning big output lists with my eyes for remarkable-looking results. From the Facebook post above, it looks like Mike Battaglia is working on some way more legit complexity measurements specialized to this situation, which could be used to much more effectively zero in on points of interest here, so I look forward to seeing what he comes up with that.
• five ~833.1 ¢ F1 (0+1ɸ) [; 1⟩ are 14.0 ¢ off from three ~1393.2 ¢ f5 √5 (-1+2ɸ) [; 0 1⟩, so that comma is [; -5 3⟩. In terms of nobles, we can say that four ~560.1 ¢ f5/F1 (-1+2ϕ)/(0+1ϕ) [; -1 1⟩ are 14.0 ¢ off from one ~2226.2 ¢ f5/f1 (-1+2ϕ)/(-1+1ϕ) [; 1 1⟩.
• four ~422.5 ¢ f11/f5 (-2+3ϕ)/(-1+2ϕ) [; 0 -1 1⟩ are close to three ~560.1 ¢ f5/F1 (-1+2ϕ)/(0+1ϕ) [; -1 1⟩. So that comma is ~9.7 ¢ (-5+10ɸ)/(3+5ɸ) [; 3 -7 4⟩. I'm not sure how to reduce it further.
• five ~339.3 ¢ f19/f11 (-3+4ϕ)/(-2+3ϕ) [; 0 0 -1 0 1⟩ are close to three ~560.1 ¢ f5/F1 (-1+2ϕ)/(0+1ϕ) [; -1 1⟩. So that comma is ~16.5 ¢ [; 3 -3 -5 0 5⟩ and I'm too lazy to get that into (a+bɸ)/(x+yɸ) form.
• one ~833.1 ¢ F1 (0+1ɸ) [; 1⟩ is 7.4 ¢ off from one ~840.5 ¢ 13/8 (13+0ϕ)/(2+0ϕ)³ [-3 0 0 0 0 0 1 ;⟩. So that comma is [-3 0 0 0 0 0 1; -1⟩.
• one ~833.1 ¢ F1 (0+1ɸ) [; 1⟩ is 15.8 ¢ off from one ~848.9 ¢ F79/F31 (-5+8ϕ)/(-3+5ϕ) [; 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 1⟩. So that comma is [; -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 1⟩.
• lots of commas like 3ɸ/5, 5ɸ/8, 8ɸ/13, or 3ɸ²/8, 5ɸ²/13, etc...
• 64/3f11². So it equates two f11/4 ~248.7 ¢ diminished thirds with 4/3, and these are very very close, off by only about half a cent: [6 -1 0 0 -2 ;⟩
• 20ϕ²f5/189 equates three f5/3 ~324.2 ¢ minor thirds with 7/4, off by only about 4¢ [2 -3 3 -1 ;⟩
• 48/f5³ ... so tempering this out equates three 4/f5 neutral 2nds ~173.8 ¢ each with 4/3, which is a porcupine-like situation. That's about 23 ¢ and is [4 1 -3 ;⟩.
• 243/F11³, a 3.5 ¢ comma between five compound fifths and three F11 at ~3168.7 ¢ each, so that's [0 5 0 0 0 -3 ;⟩.
Anyway, this idea of using merciful intonation and just intonation together is incredibly exciting. I told Dave over email that it feels sorta like the next Bohlen-Pierce. I can't wait to hear and make some music with this.
Last edited by cmloegcmluin on Mon Sep 26, 2022 3:00 am, edited 3 times in total.

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

Also, I begin to wonder about the silver ratio, δₛ = 1+√2 ≈ 2.414213562. It's another can of worms, but I'll bet a lot of this same structure could be generalized to it.

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

Thanks @cmloegcmluin. It would have been strange if this thread did not contain at least one example of a feudal comma. You have given several.

Clarifications

I'd like to clarify a few things that have come up in email and facebook conversations:

1. It is so far only a conjecture, I don't have a proof, that all noble numbers are the quotient of two feudal primes or units. But it is certainly the case that not all such quotients are noble.

The conjecture goes further to say that the primes involved are only, and all, non-wooden primes, which are the factors of the ordinary primes that split or ramify in the feudal domain, i.e. those primes named with "f" and "F" above. And that except in the case of the noble ϕ, which requires the unit 1, the units involved are only ϕ⁻¹ and ϕ, which can be called f1 and F1. And that except in the case of the nobles ϕ and ϕ², the numerator is never a unit and the the quotient always consist of a small f divided by a small f, or a big F divided by a big F, where f5 can also be written as F5 since they are both equal to √5, and the units f1 and F1 are included as potential denominators. And if this all be true, we know there is still a further restriction on which of these quotients are noble, given by the "unimodular" requirement, whereby, when the wooden and golden parts of the numerator and denominator are treated as a 2×2 matrix, its determinant must be ±1.

2. MI (or anti-JI) dyads (dyads approximating simple noble number frequency ratios) should not be expected to be tuned by ear by the slowing of beats or "locking in" of partials in the same way that JI dyads or CT (combination tone) triads can. Their whole shtick is in maximally avoiding coinciding harmonics. So there should be no advantage in tuning them precisely, provided they don't stray so far that they cease to be approximate simple nobles and become approximate simple rationals instead. So there should be no need to give decimal places of cents for nobles, but they probably need to be tuned within say ±5  ¢.

However, Mike Battaglia responds:
You claim in this that noble numbers are not sensitive to mistuning in the sense that there are no partials to try to get to cohere, but I don't really agree with that. In particular the thing you are now calling n5/4 seems to be pretty fragile. There is a real sweet spot somewhere in the 330-340 cent range where things are maximally "stanky," let's call it, and then things kind of fall off pretty steeply, particularly as you tune sharp.
I guess we should expect that, as with JI dyads, the more complex an MI dyad is, the more accurately it will need to be tuned, if it is to retain its identity at all. But perhaps there is something happening there that is not predicted by our simple model of coinciding partials.

3. I contend that humanity has always used MI with JI. It's only RTT (regular temperament theory) that hasn't had a way to deal with MI until now. And frankly it's not that important, because near-enough approximations of MI usually turn up anyway.

People used to argue over whether the minor seventh interval in the dominant seventh chord in 12-edo was an approximation of 4:7 or 5:9 or 9:16. The Noble Mediant paper says it's none of those. Its purpose is instability. Notice that 9:16 is the mediant of 4:7 and 5:9. I claim that the ideal tuning of that minor seventh is the limit as you keep taking mediants 4:7 5:9 9:16 14:25 23:41 .... Mind you, because of the lack of a requirement for accurate tuning of mercifuls, and the lack of audible justness once rationals become too complex, either of those last two rationals would be perfectly good tunings of that noble (which is f31/f11 = n7/4 ≈ 1002 ¢). The "n7/4" notation is described in the following post.

If you ever see ratios involving prime numbers like 17, 19 and above, as explanations for the tuning of some interval, it's worth considering whether a better explanation might be that it is approximating a simple noble number, for the purpose of maximally non-coinciding harmonics. Another possibility is that it may have nothing to do with harmonics, coinciding or otherwise, neither JI nor MI. It may instead have to do with combination tone effects. The give-away for CT harmony is that the effect is easier to hear with sine waves than with harmonic timbres. And even easier when distortion, such as asymmetrical soft clipping, is applied to the sine waves after they have been combined.

4. MI is not CT. The golden ratio has a reputation for producing interesting combination tone effects. So you could be forgiven for assuming that all noble numbers are capable of producing similar combination tone effects, but perhaps less strongly as they get more complex. But this is not the case, except in in the sense that any interval is capable of producing combination tone effects. It only needs to be combined with the right, precisely-defined, third note, or rather one of a very few such notes.

The golden ratio (833.1  ¢) is only special in relation to CT harmony because the right third note for it happens to be obtained by stacking another golden ratio interval above or below it.

Other "celebrity irrationals" that have this CT-self-stacking property are the silver ratio and to a lesser extent the bronze ratio. These are in general called metallic ratios. These are very different from the noble numbers of MI. The golden ratio ϕ just happens to be a member of both clubs.

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

Naming Nobles

About that n7/4 or $$\text{n}\frac74$$ notation for noble numbers shown in Figure 8 and Tables 14 and 15. The small "n" can be considered the ennoblement function. When applied to any rational number, small "n" gives a unique noble number. Not to be confused with big "N" which gives the arithmetic norm of a feudal number. Small "n" gives a one-to-one correspondence* between rationals and nobles. So these n(n/d) notations, with the parentheses omitted, can be used as unique identifiers for the noble numbers. The fp₁/fp₂ and Fp₁/Fp₂ forms don't have this property, because most such ratios are not noble, and you can't tell by inspection which ones are.

So exactly which noble does n(n/d) give you? It gives you the noble that is the noble mediant of n/d with the simplest of its two children on the Stern-Brocot tree.

A reminder: The noble mediant is a weighted mediant where the simpler of the two ratios (in this case n/d) gets a weight of 1, while the more complex ratio (in this case its simplest child) gets a weight of ϕ. So n(n/d) = (n+nₛ꜀ϕ)/(d+dₛ꜀ϕ).

So "n" gives the noble for which the given rational is at the top of the noble's zigzag. For example n1/1 = ϕ, n2/1 = ϕ², n3/2 = F5/F1, n3/1 = f5/f1.

Douglas Blumeyer created this wonderful diagram, showing the one-to-one correspondence between rationals and nobles provided by this ennoblement function.

This is one of several ennoblement functions described by Mike Battaglia in the facebook thread. It is the only one with the property of giving a noble for every rational, and a different noble from each rational, and thereby serving as a name for that noble. Mike suggested a postfix tilde operator, e.g. (7/4)~. Mike also suggested that neither ϕ nor 1/ϕ should be written as (1/1)~, that they should instead be left as ϕ or 1/ϕ. That points to the one tiny glitch with this ennoblement function:

* Provided you're only dealing with noble numbers greater than 1, as is the case here, there is no problem. If you want to include noble numbers less than 1, in most cases it's fine. Normally n(d/n) = 1 / n(n/d). For example n1/2 = 1/ϕ². But that means that both 1/ϕ and ϕ want to be called n1/1. My approach is to say that the name "n1/1" strictly belongs to ϕ, and 1/ϕ = ϕ⁻¹ can be referred to as 1/n(1/1) or (n1/1)⁻¹. I note that it can't be referred to as n⁻¹(1/1) as that would (annoyingly) be the inverse function, not the reciprocal.

1/1 is at the root of the Stern-Brocot tree and is the only node whose two children, 1/2 and 2/1, are equally simple, where simplicity is the reciprocal of complexity, and complexity is given by the product of the numerator and denominator. So I need to modify the above definition to say that:

n(n/d) gives the noble mediant of n/d with the simplest of its two children on the Stern-Brocot tree, or in the (one) case where both children are equally simple, the greater of its two children.

However noble numbers less than 1 are typically only used as noble fractions of an octave, for the purpose of generating MOS scales, and these are usually written with a backslash. And there is no problem with defining n1\1 as 1/ϕ of an octave and n1\2 as 1/ϕ² of an octave and so on.
Attachments
sbtree nobles - refined 3.png

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

Modular Inverse

I've attached an Excel spreadsheet with a formula that blew my mind the other day.

In the feudal facebook thread, Mike Battaglia came up with a cool way of factoring an (almost*) arbitrary feudal integer into a product of nobles and units (assuming that's always possible, as I conjecture). I note that's NOT primes and units, but nobles and units, since that's what interested Mike at the time. *The only requirement is that the wooden and golden parts must be relatively prime. If not, ordinary common factors should be extracted first.

To do it, Mike used the unimodular property of nobles, and ran it backwards via a concept called the "modular (multiplicative) inverse".

Given that xy mod m = 1. Given x and m, find y.

Wolfram have it here: https://reference.wolfram.com/language/ ... verse.html

But I'm more comfortable in Excel, so I looked up how to compute it in Excel, and found this forum thread (no need to read it as I'm about to summarise it): https://www.excelforum.com/excel-formul ... ction.html
in which someone asks my exact question (only badly because they made it sound like they wanted the functional inverse rather than the multiplicative inverse). Why more people don't use the word "reciprocal" instead I don't know.

Anyway, clearly there's no built-in Excel function for it, because someone kindly provides some visual basic code to compute it. And then this guy called daddylonglegs posts the following formula, where I've substituted the variables x y and m from the above, for the corresponding cell references.

y=MATCH(1,INDEX(MOD(ROW(INDIRECT("1:"&m))*x,m),0),0)

And it works! This man is a god among Excel programmers. All bow down and worship. I still don't fully understand how it works. But somehow it's generating an array (not in any cells of the spreadsheet, just in "mid air") that lists the products of x with the integers 1 to m, and then looking it up to find which of them is equal to 1. I didn't even know you could do that in Excel, generate arrays on the fly that are not in any cells. It's basically tricking Excel into doing that. Excel thinks this array is a list of cell references, more-specifically row references. Mind blown.

Anyway, the attached spreadsheet uses the above formula to show what Mike did.
Starting with a+bϕ he computed the feudal integer below that as
ModularInverse(-b, a) + ModularInverse(a, b) × ϕ
and so on for the feudal integer below that, until you can't do modular inverse any more.

I note that daddylonglegs original formula does not handle the case where m = 1. Wolfram confirms that ModularInverse(x, 1) = 0 for any x. So I modified the formula as follows:

y=IF(m=1,0,MATCH(1,INDEX(MOD(ROW(INDIRECT("1:"&m))*x,m),0),0))

An even more complete version, that handles negative moduli, is the following:

y=IF(ABS(m)=1,0,SIGN(m)*MATCH(1,INDEX(MOD(ROW(INDIRECT("1:"&ABS(m)))*SIGN(m)*x,ABS(m)),0),0))
Attachments
ModularInverse.xlsx

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

Approximating Nobles with Complex Rationals

In discussions in the facebook thread, Mike Battaglia suggested approximating the nobles using phi-weighted geometric means. This would allow us to have only ordinary primes in our basis, but the entries in the vectors would become feudal integers instead of ordinary integers. This is equivalent to having only integer entries in the vectors while doubling the number of basis elements. Every prime in the basis would be accompanied by another basis element that is the phi-th root of that prime. e.g. 2.21/ϕ.3.31/ϕ.5.51/ϕ.7.71/ϕ.11.111/ϕ.13.131/ϕ. Or it may turn out to be better to use some non-integer power other than 1/ϕ in approximating the nobles. [Edit: Indeed it does. See two posts down: viewtopic.php?p=4628#p4628]

This idea for approximating the nobles, suggested to me another way of approximating them, namely with complex rationals. After all, that's effectively what Margo Schulter has done, right from the beginning of her neo-medieval work.

Here's one possibility below. The idea is that the primes above 13 are only used in approximating nobles for MI/anti-JI purposes, and are not used in rationals for JI. And every noble is approximated by a rational that contains at least one prime above 13. For each noble below, I've used the first of its convergents that contains a prime above 13. This leads to errors of more than 7 cents in two cases, but these are in intervals wider than 2100 cents.

Table 16

Stern-	Noble	      Second-simplest	Cents	Non-13-limit	  Prime-count vector
order	based on	mediend			approx-	 Error	     3     7    13     19    29    41
1SAM (wt 1)	2SAM (wt ϕ)		imation	 cents	  2     5    11   ; 17    23    31    53
----------------------------------------------------------------------------------------------------------
1	n1/1		 2/1		 833c	34/21	+1.1	[ 1 -1  0 -1  0  0;  1  0  0  0  0  0  0 ⟩
2	n2/1		 3/1		1666c	34/13	-1.8	[ 1  0  0  0  0 -1;  1  0  0  0  0  0  0 ⟩
3	n3/2		 4/3		 560c	29/21	-1.3	[ 0 -1  0 -1  0  0;  0  0  0  1  0  0  0 ⟩
3	n3/1		 4/1		2226c	29/8	+3.3	[-3  0  0  0  0  0;  0  0  0  1  0  0  0 ⟩
4	n4/3		 5/4		 422c	23/18	+1.9	[-1 -2  0  0  0  0;  0  0  1  0  0  0  0 ⟩
4	n5/3		 7/4		 943c	19/11	+3.7	[ 0  0  0  0 -1  0;  0  1  0  0  0  0  0 ⟩
4	n5/2		 7/3		1503c	19/8	-5.1	[-3  0  0  0  0  0;  0  1  0  0  0  0  0 ⟩
4	n4/1		 5/1		2649c	23/5	-6.8	[ 0  0 -1  0  0  0;  0  0  1  0  0  0  0 ⟩
5	n5/4		 6/5		 339c	17/14	-3.2	[-1  0  0 -1  0  0;  1  0  0  0  0  0  0 ⟩
5	n7/5		10/7		 607c	17/12	-3.9	[-2 -1  0  0  0  0;  1  0  0  0  0  0  0 ⟩
5	n8/5		11/7		 792c	19/12	+3.5	[-2 -1  0  0  0  0;  0  1  0  0  0  0  0 ⟩
5	n7/4		 9/5		1002c	41/23	-0.8	[ 0  0  0  0  0  0;  0  0 -1  0  0  1  0 ⟩
5	n7/3		 9/4		1424c	41/18	+1.1	[-1 -2  0  0  0  0;  0  0  0  0  0  1  0 ⟩
5	n8/3		11/4		1735c	19/7	-5.9	[ 0  0  0 -1  0  0;  0  1  0  0  0  0  0 ⟩
5	n7/2		10/3		2109c	17/5	+9.2	[ 0  0 -1  0  0  0;  1  0  0  0  0  0  0 ⟩
5	n5/1		 6/1		2988c	17/3   +14.9	[ 0 -1  0  0  0  0;  1  0  0  0  0  0  0 ⟩
6	n6/5		 7/6		 284c	20/17	-2.2	[ 2  0  1  0  0  0; -1  0  0  0  0  0  0 ⟩
6	n9/7		13/10		 448c	22/17	-2.1	[ 1  0  0  0  1  0; -1  0  0  0  0  0  0 ⟩
6	n11/8		15/11		 541c	26/19	+1.6	[ 1  0  0  0  0  1;  0 -1  0  0  0  0  0 ⟩
6	n10/7		13/9		 630c	23/16	-2.2	[-4  0  0  0  0  0;  0  0  1  0  0  0  0 ⟩
6	n11/7		14/9		 771c	64/41	+0.3	[ 6  0  0  0  0  0;  0  0  0  0  0 -1  0 ⟩
6	n13/8		18/11		 849c	31/19	-1.3	[ 0  0  0  0  0  0;  0 -1  0  0  1  0  0 ⟩
6	n12/7		17/10		 923c	17/10	+4.4	[-1  0 -1  0  0  0;  1  0  0  0  0  0  0 ⟩
6	n9/5		11/6		1039c	31/17	+1.5	[ 0  0  0  0  0  0; -1  0  0  0  1  0  0 ⟩
6	n9/4		11/5		1378c	31/14	-1.8	[-1  0  0 -1  0  0;  0  0  0  0  1  0  0 ⟩
6	n12/5		17/7		1530c	17/7	+6.2	[ 0  0  0 -1  0  0;  1  0  0  0  0  0  0 ⟩
6	n13/5		18/7		1641c	31/12	+2.1	[-2 -1  0  0  0  0;  0  0  0  0  1  0  0 ⟩
6	n11/4		14/5		1772c	64/23	-0.5	[ 6  0  0  0  0  0;  0  0 -1  0  0  0  0 ⟩
6	n10/3		13/4		2055c	23/7	+4.9	[ 0  0  0 -1  0  0;  0  0  1  0  0  0  0 ⟩
6	n11/3		15/4		2276c	41/11	+1.7	[ 0  0  0  0 -1  0;  0  0  0  0  0  1  0 ⟩
6	n9/2		13/3		2558c	57/13	+1.0	[ 0  1  0  0  0 -1;  0  1  0  0  0  0  0 ⟩
6	n6/1		 7/1		3272c	53/8	+1.8	[-3  0  0  0  0  0;  0  0  0  0  0  0  1 ⟩


The main point of using a basis that only approximates the nobles, instead of a feudal-prime basis that can represent them exactly, is to reduce the number of entries in the vectors (the number of dimensions), to make it more likely that we can find good temperaments that combine JI and MI, where a good temperament is one that combines low rank, low complexity and low error. By doing so, we are effectively fixing certain commas in advance.

However, as Steve Martin pointed out, if you only want 11-limit JI and the 8 simplest nobles, it's hard to beat the exact basis 2.3.f5.7.f11.F11.ϕ. Only 7 dimensions. That's only two more than 11-limit JI, and you can do a lot with those 8 nobles. And of course, if you want 13-limit JI and the 8 simplest nobles, it's only 8 dimensions. But beyond the 8 simplest nobles, for every two additional nobles, you need an additional dimension (on average), so the exact feudal prime basis for representing 13-limit JI and the 32 simplest nobles requires 20 dimensions.

The vectors in the above table can represent 13-limit JI and the 32 simplest nobles (approximately), but they only require 13 dimensions. One of the dimensions (prime 53) is only used for the last and widest noble (3272c). So reducing it to 12 dimensions and 31 nobles wouldn't really be a problem.

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

Choosing pairs of Rationals for the Geometric Mean Approximation Scheme

The following table gives more insight into the process that led to the choices in the previous post, for approximating nobles by complex rationals. It is also useful in deciding which pairs of rationals to use in approximating nobles by a weighted geometric mean of two rationals, using a basis of the form 2.21/ϕ.3.31/ϕ.5.51/ϕ.7.71/ϕ.11.111/ϕ.13.131/ϕ (or other non-integer powers of the primes) as suggested by Mike Battaglia.

It shows the series of convergents (SAMs) for each noble — the ratios at the corners of its zigzag on the Stern-Brocot tree — up to the first non-13-limit ratio (the one chosen for use above). Any consecutive pair prior to the last one, is a potential pair for use in the weighted-geometric-mean scheme. For those shown in green, the choice is made for us, and in two cases we are forced to use a non-adjacent mediend (those in parenthesis) if we want to stay within the 13-prime-limit.

Table 17

Noble	Cents	Convergents up to first non-13-limit	(level of SB-tree)
1	2	3	4	5	6	7	8	9	10	11
----------------------------------------------------------------------------------------------------
n1/1	 833c	1/1	2/1	3/2	5/3	8/5	13/8	21/13	34/21

n2/1	1666c		2/1	3/1	5/2	8/3	13/5	21/8	34/13

n3/2	 560c			3/2	4/3	7/5	11/8	18/13	29/21
n3/1	2226c			3/1	4/1	7/2	11/3	18/5	29/8

n4/3	 422c				4/3	5/4	9/7	14/11	23/18
n5/3	 943c				5/3	7/4	12/7	19/11
n5/2	1503c				5/2	7/3	12/5	19/8
n4/1	2649c				4/1	5/1	9/2	14/3	23/5

n5/4	 339c					5/4	6/5	11/9	17/14
n7/5	 607c					7/5	10/7	17/12
n8/5	 792c					8/5	11/7	19/12
n7/4	1002c					7/4	9/5	16/9	25/14	41/23
n7/3	1424c					7/3	9/4	16/7	25/11	41/18
n8/3	1735c					8/3	11/4	19/7
n7/2	2109c					7/2	10/3	17/5
n5/1	2988c					5/1	6/1	11/2	17/3

n6/5	 284c						6/5	7/6	13/11	20/17
n9/7	 448c						9/7	13/10	22/17
n11/8	 541c						11/8	15/11	26/19
n10/7	 630c						10/7	13/9	23/16
n11/7	 771c						11/7	14/9	25/16	39/25	64/41
n13/8	 849c						13/8	18/11	31/19
n12/7	 923c				(5/3)		12/7	17/10
n9/5	1039c						9/5	11/6	20/11	31/17
n9/4	1378c						9/4	11/5	20/9	31/14
n12/5	1530c				(5/2)		12/5	17/7
n13/5	1641c						13/5	18/7	31/12
n11/4	1772c						11/4	14/5	25/9	39/14	64/23
n10/3	2055c						10/3	13/4	23/7
n11/3	2276c						11/3	15/4	26/7	41/11
n9/2	2558c						9/2	13/3	22/5	35/8	57/13
n6/1	3272c						6/1	7/1	13/2	20/3	33/5	53/8


Dave Keenan
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Joined: Tue Sep 01, 2015 2:59 pm
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### Re: Noble frequency ratios as prime-count vectors in ℚ(√5)

Here's a proposal for standard pairs of convergents (in green) for approximating each noble using Mike Battaglia's geometric mean approximation scheme, with a "prime" basis of 2.3.5.7.11.13.2χ.3χ.5χ.7χ.11χ.13χ.17χ where χ = ϕ/√5 ≈ 0.7236.

Mike proved that this constant, for which I'm using the Greek small letter chi (pronounced in English as "kai", rhymes with "sky"), is the ideal value for this exponent, when the nobles are approximated using pairs of successive convergents that are sufficiently complex. Numerical optimisations of various kinds using simple convergents, support values in the range 0.69 to 0.75 which is centred on that value. So I'm using the exponent of χ ≈ 0.7236 in calculating the errors below.

The chosen ratios are all within the 18-integer limit and therefore within the 17-prime-limit. All but two nobles (n12/7 and n12/5) have both ratios within the 13-prime limit. The table lists the usual 32 nobles from the first 6 levels of the SB-tree, in order of their size in cents, and shows the errors caused by the chosen approximations.

Table 18

Noble	Noble	Error	Convergents (SAMs) up to first non-13-limit	(level of SB-tree)
ID	cents	cents	1	2	3	4	5	6	7	8	9	10	11
------------------------------------------------------------------------------------------------------------
n6/5	 284c	-0.6						6/5	7/6	13/11	20/17
n5/4	 339c	-0.7					5/4	6/5	11/9	17/14
n4/3	 422c	-0.9				4/3	5/4	9/7	14/11	23/18
n9/7	 448c	+0.5						9/7	13/10	22/17
n11/8	 541c	-0.5						11/8	15/11	26/19
n3/2	 560c	-0.9			3/2	4/3	7/5	11/8	18/13	29/21
n7/5	 607c	+1.0					7/5	10/7	17/12
n10/7	 630c	+0.9						10/7	13/9	23/16
n11/7	 771c	-0.9						11/7	14/9	25/16	39/25	64/41
n8/5	 792c	-1.0					8/5	11/7	19/12
n1/1	 833c	+0.9	1/1	2/1	3/2	5/3	8/5	13/8	21/13	34/21
n13/8	 849c	+0.4						13/8	18/11	31/19
n12/7	 923c	-0.4					12/7	17/10
n5/3	 943c	+0.5				5/3	7/4	12/7	19/11
n7/4	1002c	+0.4					7/4	9/5	16/9	25/14	41/23
n9/5	1039c	+2.0						9/5	11/6	20/11	31/17
n9/4	1378c	-2.2						9/4	11/5	20/9	31/14
n7/3	1424c	-0.5					7/3	9/4	16/7	25/11	41/18
n5/2	1503c	-0.4				5/2	7/3	12/5	19/8
n12/5	1530c	+0.6						12/5	17/7
n13/5	1641c	-0.6						13/5	18/7	31/12
n2/1	1666c	+1.0		2/1	3/1	5/2	8/3	13/5	21/8	34/13
n8/3	1735c	+2.0					8/3	11/4	19/7
n11/4	1772c	+1.6						11/4	14/5	25/9	39/14	64/23
n10/3	2055c	-1.9						10/3	13/4	23/7
n7/2	2109c	-1.7					7/2	10/3	17/5
n3/1	2226c	+0.9			3/1	4/1	7/2	11/3	18/5	29/8
n11/3	2276c	+1.5						11/3	15/4	26/7	41/11
n9/2	2558c	-1.3						9/2	13/3	22/5	35/8	57/13
n4/1	2649c	+0.7				4/1	5/1	9/2	14/3	23/5
n5/1	2988c	+4.9					5/1	6/1	11/2	17/3
n6/1	3272c	+4.3						6/1	7/1	13/2	20/3	33/5	53/8


Dave Keenan
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Location: Brisbane, Queensland, Australia
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### Re: Noble frequency ratios as prime-count vectors in ℚ(√5)

I'm linking to this old post by Dan Stearns because it seems like it might be relevant.
https://yahootuninggroupsultimatebackup ... html#77465

There are other useful posts in that thread, such as this one with a list of noble and near-noble triads at its end (quoted below).
https://yahootuninggroupsultimatebackup ... html#77421

Dave Keenan, 2008, in the Yahoo group: tuning, wrote: There are no saturated strictly-noble chords smaller than an octave.
However I found the following approximate one (and its inverse) which
should still qualify as saturated metastable.
0-338-944 0-606-944 cents

The simplest saturated noble triad is
0-833-1666 cents and is its own inverse.
Some others narrower than this (inverse pairs) are:
0-560-1503 0-943-1503 cents
0-422-1424 0-1002-1424
0-339-1378 0-1039-1378
0-607-1378 0-771-1378

volleo6144
Posts: 81
Joined: Mon May 18, 2020 7:03 am
Location: Earth
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### Re: Noble frequency ratios as prime-count vectors in ℚ(√5)

Well, this is certainly something to come back to. I unfortunately don't think I have the attention span to read all of that at once.

One thing did stand out to me, though...
Dave Keenan wrote: Thu Sep 22, 2022 12:55 pm
Table 8
(...)
76+0ϕ =  76  [fairly useless]

I take it that was meant to be 73?
I'm in college (a CS major), but apparently there's still a decent amount of time to check this out. I wonder if the main page will ever have 59edo changed to green...