Search found 1099 matches

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: 20
Views: 2822

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.o...
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: 20
Views: 2822

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

Here at last are the 6th-order-limit nobles as feudal 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 usef...
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: 20
Views: 2822

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: 20
Views: 2822

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: 20
Views: 2822

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: 20
Views: 2822

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 ...
by Dave Keenan
Wed Sep 21, 2022 12:39 am
Forum: The lounge
Topic: Noble frequency ratios as prime-count vectors in ℚ(√5)
Replies: 20
Views: 2822

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

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 nee...
by Dave Keenan
Wed Sep 21, 2022 12:33 am
Forum: The lounge
Topic: Noble frequency ratios as prime-count vectors in ℚ(√5)
Replies: 20
Views: 2822

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

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 , c...
by Dave Keenan
Wed Sep 21, 2022 12:23 am
Forum: The lounge
Topic: Noble frequency ratios as prime-count vectors in ℚ(√5)
Replies: 20
Views: 2822

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

T h e F e u d a l M a n i f e s t o https://forum.sagittal.org/download/file.php?mode=view&id=426 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 di...
by Dave Keenan
Sat Sep 17, 2022 4:21 pm
Forum: The lounge
Topic: Noble frequency ratios as prime-count vectors in ℚ(√5) [superseded]
Replies: 9
Views: 1673

Re: Noble frequency ratios as prime-count vectors in ℚ(√5) [superseded]

[Note: This thread is of historical interest only. Please see the updated version (using a better choice for the fundamental primes) at https://forum.sagittal.org/viewtopic.php?p=4613#p4613] There is much discussion of this topic on the facebook group Xenharmonic Alliance - Mathematical Theory . Ma...