regarding L and s

Regarding the L and s issues, I found this draft from a response
I never sent to a prior thread.

>
> >Ah ha. Well then it's done -
> [Paul H. Erlich:]
> Far from it --
>

In an et that supports LLsLLLs where L and s are exactly one size
each, it is easy to determine what a sharp and flat are and how
transposition works. # = -b = L-s.

In the case of LL'sLL'Ls where L' is a pinch smaller than L, the
problem is quite different, even if the desired results (useage
model) is the same.

The best I came up with was that a sharp was a vector such that
the following could occur.

L L' s L L' L s
+ 0 0 L-s 0 L-L' 0 0
-------------------------------------------
L L' L s L L' s

...so the familiar transposition from C to G has occurred (F#
causes an adjustment of A by a syntonic comma?). Flats had a
similar behavior with the vector being subtracted.

Unfortunately, as interesting as it was, it didn't seem to work.
[I can't remember the particulars, but melodic and harmonic
minors started having some bizarre behavior as sharps and flats
were added at the same time, with new and fascinating intervals
appearring in the middle of the scale.]

I finally concluded that dealing with 53 was going to be
significantly different than dealing with 31 for music that
wants exact transposition.

Bob Valentine

Robert C Valentine wrote,

>I finally concluded that dealing with 53 was going to be
>significantly different than dealing with 31 for music that
>wants exact transposition.

Exactly. With respect to standard notation, we may say:

Diatonic tunings: 12, 19, 26, 31, 43, 50, 55ETs, meantone

Non-diatonic ETs: 15, 22, 27, 34, 41, 53ETs, JI

See Blackwood's _The Structure of Recognizable Diatonic Tunings_ for an
excessively rigorous presentation of all this.

In this regard, Pythagorean tuning may not work, or it may, depending on
whether you use the schismatic 5-limit approximations, or not.

> [Bob Valentine, TD 431.2]
>
> In an et that supports LLsLLLs where L and s are exactly one
> size each, it is easy to determine what a sharp and flat are
> and how transposition works. # = -b = L-s.

Without having looked into it too closely, it seems to me
that this could be more rigorously defined. Isn't all of
'# = -b = L-s' true only if s = L/2 ?

> In the case of LL'sLL'Ls where L' is a pinch smaller than L,
> the problem is quite different,
> <snip>
> Unfortunately, as interesting as it was, it didn't seem to
> work. [I can't remember the particulars, but melodic and
> harmonic minors started having some bizarre behavior as sharps
> and flats were added at the same time, with new and fascinating
> intervals appearring in the middle of the scale.]

Too bad you don't have more to post on this - it *does*
sound interesting!

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
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Bob Valentine wrote,

>> In an et that supports LLsLLLs where L and s are exactly one
>> size each, it is easy to determine what a sharp and flat are
>> and how transposition works. # = -b = L-s.

Joe Monzo wrote,

>Without having looked into it too closely, it seems to me
>that this could be more rigorously defined. Isn't all of
>'# = -b = L-s' true only if s = L/2 ?

Nope -- Bob is absolutely correct. For example, in 31-equal, L=5, s=3, and # = -b
= L-s = 2. For the rigorous presentation, see for example Blackwood's _Structure
of Recognizable Diatonic Tunings_.

Me
> > In an et that supports LLsLLLs where L and s are exactly one
> > size each, it is easy to determine what a sharp and flat are
> > and how transposition works. # = -b = L-s.
>
> Joe Monzo :
> Without having looked into it too closely, it seems to me
> that this could be more rigorously defined. Isn't all of
> '# = -b = L-s' true only if s = L/2 ?
>

I wasn't saying that N# = (N+1)b, I was saying that the size
of a # and a b were the same.

19tet : L = 3, s = 2, # = -b = L-s = 1
31tet : L = 5, s = 3, # = -b = L-s = 2

> Too bad you don't have more to post on this - it *does*
> sound interesting!

I'll go hunting for my notes. It was a vacation in a very sunny place
and, not liking to spend many hours in the sun, I spent some time
indoors in "microtonal numerology" mode. (Don't worry, I also went
snorkelling and actually heard(!) a fish eating).

Bob Valentine