periodicity block definition

Hello all,

The periodicity block must be one of the most useful
constructs with which to understand musical tuning.
The choice of unison vectors, the effects this choice
have on the resulting PBs, must be one of the most basic
areas of inquiry here. Here, I would like to present
some points for discussion (and/or clarification)...

() Do we consider any intervals valid as unison vectors,
even if they are very large? If so, do we have a name
for PBs that have small unison vectors (i.e. "well
formed" PBs...).

() Do we have precise ideas on what counts as "small"
and "large" when it comes to unison vectors? What
properties are associated with them? For example, I
think Erlich and I managed to show a while back that
PBs with unison vectors smaller than their smallest 2nds
share certain properties with MOS.

() How are the above affected by the decision to
temper out some or all of the unison vectors? For
example, what happens when there are commatic unison
vectors larger than any chromatic ones?

-Carl

I was playing around with an interval conversion calculator
I created in an Excel spreadsheet, and I happened to notice
that 5 enharmonic dieses [= (128/125)^5] are almost the
same size as a 9:8 whole-tone.

enharmonic diesis = (2^7)*(5^-3) = ~41.05885841 cents

5 enharmonic dieses = (2^35)*(5^-15) = ~205.294292 cents

9/8 = (2^-3)*(3^2) = ~203.9100017 cents

difference: ((128/125)^5) / (9/8) =

2^x 3^y 5^z

| 35 0 -15|
- |- 3 2 0|
-------------
| 38 -2 -15|

= (2^38)*(3^-2)*(5^-15) = ~1.384290297 cents = ~1&3/8 cents.

Has anyone ever noticed this before, or used it as a unison-vector?
Any comments? I'd like to see a periodicity-block derived from it.

-monz
http://www.monz.org
"All roads lead to n^0"

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On Wed, 4 Jul 2001 16:21:53 -0700, "monz" <joemonz@yahoo.com> wrote:

>= (2^38)*(3^-2)*(5^-15) = ~1.384290297 cents = ~1&3/8 cents.
>
>
>Has anyone ever noticed this before, or used it as a unison-vector?
>Any comments? I'd like to see a periodicity-block derived from it.

Hmm, I see that this is a unison vector in 25-TET, although the 5-step
"whole tone" is quite large at 240 cents, and there's a better
approximation of 9/8 at 192 cents. Of course this is also a unison vector
in 31-TET. Along a line from 31-TET to 56-TET (31+25) there is a number of
tempered scales that share this unison vector, and this line approaches
very close to 5-limit just (closer even than the line between 31 and 22,
which goes through 53!) I'm not sure what you can do with this, but it's a
start. Look at the chart at http://www.io.com/~hmiller/png/et-scales.png
and draw a line from 31 to 25: it looks like there could be some good
scales there.

--- In tuning-math@y..., carl@l... wrote:

> () How are the above affected by the decision to
> temper out some or all of the unison vectors? For
> example, what happens when there are commatic unison
> vectors larger than any chromatic ones?

Nothing too special, if you're tempering out the commatic ones. You
might call such a scale "artificial", if you believe all scales
should start out in JI and then evolve into a tempered form.

If the chromatic unison vector is larger than one of the scale steps,
you get an improper scale.

>> () How are the above affected by the decision to
>> temper out some or all of the unison vectors? For
>> example, what happens when there are commatic unison
>> vectors larger than any chromatic ones?

> /../
>
> If the chromatic unison vector is larger than one of the scale
> steps, you get an improper scale.

If the scale is just, then the difference between a commatic
and chromatic unison vector is one of a naming only, right?
So would your statement here be better put, "If any unison
vector which is left untempered is larger than one of the scale
steps, you get an improper scale."? Or does the difference
in naming actually affect propriety?

-Carl

--- In tuning-math@y..., carl@l... wrote:
> >> () How are the above affected by the decision to
> >> temper out some or all of the unison vectors? For
> >> example, what happens when there are commatic unison
> >> vectors larger than any chromatic ones?
>
> > /../
> >
> > If the chromatic unison vector is larger than one of the scale
> > steps, you get an improper scale.
>
> If the scale is just, then the difference between a commatic
> and chromatic unison vector is one of a naming only, right?

Right. But you said "temper out" above, so I was focusing on that
case.

> So would your statement here be better put, "If any unison
> vector which is left untempered is larger than one of the scale
> steps, you get an improper scale."?

Yes I think that's right.

> Or does the difference
> in naming actually affect propriety?

Well it would be kind of perverse to call something a _commatic_
unison vector if it's larger than one of the scale steps and it's not
tempered out . . . don't you think?

>> Or does the difference
>> in naming actually affect propriety?
>
>Well it would be kind of perverse to call something a _commatic_
>unison vector if it's larger than one of the scale steps and it's
>not tempered out . . . don't you think?

Yes.

-Carl