BP questions

I have a couple of, probably pretty easy, BP questions...

In the first place, there is some talk on the BP website of the
"diesis"

http://members.aol.com/bpsite/names.html

as the difference between the sizes of the BP semitones. Is this how
this term is generally used?? I'm confused, since I thought it had
to do with the TEMPERING of a string of 12 Pythagorean fifths... but
maybe I had that wrong in the first place.

The BP scale, of course, is entirely just with no tempering...

I was looking at Monz' generally excellent dictionary, but it simply
said that a "diesis" could be a term for a whole bunch of very small
intervals...

And shouldn't it have something to do with 12?? This scale consists
of 13 steps... What's with this??

AND, regarding the generation of the BP scale:

http://members.aol.com/bpsite/scales.html

"Let's do the experiment right here. Four values of the assumed scale
are present from the very beginning: 1/1 as the base tone,
then 5/3 and 7/3 to form the 7/5/3 triad with the base tone, and
finally 3/1 as the top tone (first phase). Now we want to play
the triad so that it ends with the top tone (we could also say that
we want to suspend it from the top tone). This then leads to
two new tones: 9/7 and 15/7 (second phase)."

What exactly does this mean "suspending it from the top tone??" Does
it mean creating another triad from the top note of the scale?? In
that sense, the BP scale is created through triads the way a
"classic" JI scale is created (????)

I'm not clear exactly on the math on that... sure, I can see that I
can get a 15 from 3x5 and a 9 from 3x3, but I'm "fuzzy" on the
specifics.... How is this process done exactly??

Thanks!

_______ ____ ____ __
Joseph Pehrson

Hi Joseph,

I interpret Heinz's use of diesis indeed as a generic term for 'small'
intervals. In this case his minor and major diesis are exactly the unison
intervals that can be used as generators of a 13 sized periodicity block. In
that sense they can be compared with 81/80 and 128/125 spanning a regular 12
sized block in octave based tuning.

Suspending from the top tone means forming a 3/5/7 triad where the '7'
coincides with the original '3'. It's best to visualize in a square lattice
diagram:
The basic triad is:

7

1 5

The added one would have it's '7' to on the '1', so divide all elements by
7.

7

1 5

1/7 5/7

I left out the powers of 3 to show the basic relations.
The next reasoning is not completely clear to me (Heinz?), but the result in
the diagram is:

7/5 7

1/5 1 5

1/7 5/7

The next process of 'filling in the gaps' results in:

49/25

7/25 7/5 7

1/25 1/5 1 5 25

1/7 5/7 25/7

25/49

Which is a possible periodicity block.
It helps to fill in the 13eq equivalents:

8

11 4 10

1 7 0 6 12

3 9 2

5

If you create a scale in the traditional JI way, you get 'my' major BP
scale:

http://www.kees.cc/music/scale13/scale13.html

Which is a 'tryhill' scale, without filling in the gaps.

Hope that helps.

Kees

> -----Original Message-----
> From: Joseph Pehrson [mailto:pehrson@pubmedia.com]
> Sent: Wednesday, January 10, 2001 9:02 AM
> To: tuning@egroups.com
> Subject: [tuning] BP questions
>
>
> I have a couple of, probably pretty easy, BP questions...
>
> In the first place, there is some talk on the BP website of the
> "diesis"
>
> http://members.aol.com/bpsite/names.html
>
> as the difference between the sizes of the BP semitones. Is this how
> this term is generally used?? I'm confused, since I thought it had
> to do with the TEMPERING of a string of 12 Pythagorean fifths... but
> maybe I had that wrong in the first place.
>
> The BP scale, of course, is entirely just with no tempering...
>
> I was looking at Monz' generally excellent dictionary, but it simply
> said that a "diesis" could be a term for a whole bunch of very small
> intervals...
>
> And shouldn't it have something to do with 12?? This scale consists
> of 13 steps... What's with this??
>
> AND, regarding the generation of the BP scale:
>
> http://members.aol.com/bpsite/scales.html
>
> "Let's do the experiment right here. Four values of the assumed scale
> are present from the very beginning: 1/1 as the base tone,
> then 5/3 and 7/3 to form the 7/5/3 triad with the base tone, and
> finally 3/1 as the top tone (first phase). Now we want to play
> the triad so that it ends with the top tone (we could also say that
> we want to suspend it from the top tone). This then leads to
> two new tones: 9/7 and 15/7 (second phase)."
>
> What exactly does this mean "suspending it from the top tone??" Does
> it mean creating another triad from the top note of the scale?? In
> that sense, the BP scale is created through triads the way a
> "classic" JI scale is created (????)
>
> I'm not clear exactly on the math on that... sure, I can see that I
> can get a 15 from 3x5 and a 9 from 3x3, but I'm "fuzzy" on the
> specifics.... How is this process done exactly??
>
> Thanks!
>
> _______ ____ ____ __
> Joseph Pehrson
>
>
>
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>

Joseph Pehrson wrote,

>"diesis"

>http://members.aol.com/bpsite/names.html

>as the difference between the sizes of the BP semitones. Is this how
>this term is generally used??

Well, it's usually used to mean the interval 128:125, or the amound by which
a stack of three pure major thirds falls short of an octave.

>I'm confused, since I thought it had
>to do with the TEMPERING of a string of 12 Pythagorean fifths... but
>maybe I had that wrong in the first place.

You did, but no matter.

>I was looking at Monz' generally excellent dictionary, but it simply
>said that a "diesis" could be a term for a whole bunch of very small
>intervals...

That's correct. Bohlen uses the term "diesis" in this sense.

>And shouldn't it have something to do with 12?? This scale consists
>of 13 steps... What's with this??

>AND, regarding the generation of the BP scale:

Bohlen's derivation is rather idiosyncratic. And the "classic" JI scale is a
musico-historical myth anyway. Just think of it like this. 12-tET, 12 equal
steps per 2:1, gives an excellent approximation (for an ET with that few
notes) to all ratios of integers up to 6 -- hence allowing for 4:5:6:8
chords and their mirrorings. The Pierce scale, 13 equal steps per 3:1, gives
an excellent approximation to all ratios of _odd_ numbers up to 9, hence
allowing for 3:5:7:9 chords and their mirrorings. In both cases,
de-tempering the set of 12 or 13 to JI means that you have to choose where
to put some "wolves" or out-of-tune intervals -- a choice with elements of
arbitrariness in both cases.