Tuning-Math Archive Section 1: 651

Message: 651 - Contents - Hide Contents Date: Mon, 06 Aug 2001 19:44:08 Subject: Re: Another BP linear temperament From: Paul Erlich --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> Hi Dave, > > <<Ok. But for this to be a "generalised meantone" in any sense other > than the one which is much better described by the term "linear > temperament", then a chain of _four_ 9/7's would have to be > tritave-equivalent to a 7/6. It needs a chain of five according to > your comma.>> > > I disagree, and I think your looking at this in much too narrow a way. > The process of determining a comma and fractionalizing and > distributing it is what's being generalized...

But no one ever called schismic temperament, which seems to have predated meantone, a "generalized meantone" . . .

> > Again, it's a generalization of meantone because the "tone" and the > "mean" are generalized!

Ah . . . this may be what Dave and I are missing. What are the JI "tones" here? Anyway, let's not let squabbles over terminology (Margo suggested some nice ideas for this) blind us to the fact that all of the others on this list have great pools of insight from which they are drawing.

Message: 671 - Contents - Hide Contents Date: Fri, 17 Aug 2001 19:56:20 Subject: Re: Hi! Seeking advice From: Paul Erlich --- In tuning-math@y..., BobWendell@t... wrote:

> Thanks so much, Paul! I feel very fortunate indeed to have run across > you all here. I was hoping for some kind of more coherent > presentation format

I trust you've read the _Gentle Introduction to Fokker Periodicity Blocks_ (A gentle introduction to Fokker periodicity bl... * [with cont.] (Wayb.))?

Message: 674 - Contents - Hide Contents Date: Sat, 18 Aug 2001 03:06:12 Subject: Re: Microtemperament and scale structure From: genewardsmith@j... Apparently the math stuff belongs here, so I am reposting it in case anyone wants to follow up. --- In tuning@y..., kalleaho@m... wrote:

> I understand that with microtemperament one can achieve greater > number of consonant intervals in a scale that is originally tuned in > strict Just Intonation. > This can be a great method for generating new and interesting scales > but doesn't it destroy the structure of the original scale?

When you finally get to the point where you can't tell the difference, the question becomes moot. However, the answer is yes-- introducing approximations increases the flexibility of harmonic relationships and thereby changes the structure. The question is really best understood in terms of group theory. The intervals of any form of just intonation are by definition positive rational numbers, and so an element of the abelian group of positive rationals under multiplication (since our hearing, and hence musical structure, is multiplicative.) We are never really interested in all rational numbers, but can content ourselves with a finitely generated subgroup--for one easy example, the group G of all the numbers of the form 2^a * 3^b * 5^c, where a, b and c are integers, are the numbers generated by the first three primes, 2, 3 and 5. An approximate tuning system can very often be seen as a (subset of a) group homomorphism to a group of smaller rank, and most significantly to a group of rank 1. For another easy example, the rank 3 free group G can be sent to a rank 1 free group by a homomorphism h12(2) = 12, h12(3) = 19, h12(5) = 28. A subset of 88 contiguous elements of this group associated to notes tuned as usual by setting g12(1) = 440 hz, g12(2) = 2*440 hz, g12(3) = 2^19/12*440 hz, g12(5) = 2^28/12*440 hz is the standard keyboard. Any such homomorphism is defined by its kernel, which are the elements sent to the identity. In the case of h12, the kernel is spanned by 81/80 (the diatonic comma) and 128/125 (the great diesis), where we have h12(81/80) = h12(128/125) = 0. A tuning system which does not contain the diatonic comma in its kernel (and this includes just intonation!) will have a structure quite different that which musicians normally expect. On the other hand one that does, such as what we get from the 19 or 31 tone system, will seem more "normal". Consider the system h72(2) = 72, h72(3) = 114, h72(5) = 167, h72(7) = 202, h72(11) = 249 (the last two values are irrelevant here, but they do no harm and they cover the range of primes which makes the 72 system interesting.) This 72 system has a 12 system tuning embedded in it, so we could suppose that structurally they are very similar. However, intersecting the kernels shows they are remotely related; in particular h72(81/80) = 1 and h72(128/125) = 3, the second is not so important but the first shows that the 72 system is fundamentally different in structure from the 12 system. In contrast, h31(2) = 31, h31(3) = 49, h31(5) = 72, h31(7) = 87 and h31(11) = 107 *does* have the property that h31(81/80) = 0; and while h31(128/125) = 1 we still find h31 is much closer in structre to h12 than is h72.