Search found 1093 matches

by Dave Keenan
Sat Oct 01, 2022 12:23 pm
Forum: The lounge
Topic: Noble frequency ratios as prime-count vectors in ℚ(√5)
Replies: 14
Views: 659

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 conje...
by Dave Keenan
Fri Sep 30, 2022 9:23 am
Forum: The lounge
Topic: Noble frequency ratios as prime-count vectors in ℚ(√5)
Replies: 14
Views: 659

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 ...
by Dave Keenan
Thu Sep 29, 2022 6:26 pm
Forum: The lounge
Topic: Noble frequency ratios as prime-count vectors in ℚ(√5)
Replies: 14
Views: 659

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 pr...
by Dave Keenan
Fri Sep 23, 2022 9:17 pm
Forum: The lounge
Topic: Noble frequency ratios as prime-count vectors in ℚ(√5)
Replies: 14
Views: 659

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

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

Many thanks to Mike Battaglia for these extensive discussions.
by Dave Keenan
Fri Sep 23, 2022 8:04 pm
Forum: The lounge
Topic: Noble frequency ratios as prime-count vectors in ℚ(√5)
Replies: 14
Views: 659

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

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.or...
by Dave Keenan
Fri Sep 23, 2022 5:22 pm
Forum: The lounge
Topic: Noble frequency ratios as prime-count vectors in ℚ(√5)
Replies: 14
Views: 659

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

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...
by Dave Keenan
Thu Sep 22, 2022 6:30 pm
Forum: The lounge
Topic: Noble frequency ratios as prime-count vectors in ℚ(√5)
Replies: 14
Views: 659

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

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, onl...
by Dave Keenan
Thu Sep 22, 2022 3:20 pm
Forum: The lounge
Topic: Noble frequency ratios as prime-count vectors in ℚ(√5)
Replies: 14
Views: 659

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

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...
by Dave Keenan
Thu Sep 22, 2022 12:55 pm
Forum: The lounge
Topic: Noble frequency ratios as prime-count vectors in ℚ(√5)
Replies: 14
Views: 659

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

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 co...
by Dave Keenan
Wed Sep 21, 2022 11:02 pm
Forum: The lounge
Topic: Noble frequency ratios as prime-count vectors in ℚ(√5)
Replies: 14
Views: 659

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

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 ...