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"

_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at http://mail.yahoo.com

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