BP as a linear temperament

Hi there.

Quite surprised I haven't seen this ever before.

While reading the articles on the BP site, I found nor a word about the
possibility of viewing the tempered version of BP as a "linear" temperament
with a generator close to 9/7 and a period of 3/1. Having done a few
experiments, one of my final ideas was a generator of ~439.81426942 cents
(i.e. 9/7 widened by 1/3 of the 245/243 comma), which makes both 9/7 and 5/3
detuned by the same amount (or, more precisely, 7/3 and 5/3) and therefore
has a pure 7/5 and 15/7.

The entire scale looks like this:

! bptemp.scl
!
Tempered version of the Bohlen-Pierce scale
13
!
142.69792
297.11635
439.81427
7/5
75/49
879.62854
1022.32646
49/25
15/7
1462.14073
1604.83865
1759.25708
3/1

Petr

On 3/12/06, Petr Pařízek <p.parizek@chello.cz> wrote:
[...]
> (i.e. 9/7 widened by 1/3 of the 245/243 comma), which makes both 9/7 and 5/3
> detuned by the same amount (or, more precisely, 7/3 and 5/3) and therefore
> has a pure 7/5 and 15/7.
[...]

So does that mean it has an "eigenmonzo" of 7/5? I freakin love that
word, dude. Eigenmonzo.

Keenan

--- In tuning@yahoogroups.com, Petr Paøízek <p.parizek@...> wrote:

> While reading the articles on the BP site, I found nor a word about the
> possibility of viewing the tempered version of BP as a "linear"
temperament
> with a generator close to 9/7 and a period of 3/1.

The linear temperament I associate to BP is the "bohpier" temperament;
the linear temperament obtained by tempering out 3125/3087 and
245/243, with wedgie <<13 19 23 0 0 0||. If the octave is a generator,
the other generator will be a sharp 27/25 of size about 3^(1/13); if
we want to use 3 as an equivalence instead, we get a period of
3^(1/13) as a generator, and can use the octave as the other
generator, hence we get the same generators looked at two different
ways. If we get rid of 2 altogether, we don't have a linear
temperament, but an equal temperament, obtained by dividing 3 into 13
equal parts.

On 3/12/06, Gene Ward Smith <genewardsmith@coolgoose.com> wrote:
> --- In tuning@yahoogroups.com, Petr Paøízek <p.parizek@...> wrote:
>
> > While reading the articles on the BP site, I found nor a word about the
> > possibility of viewing the tempered version of BP as a "linear"
> temperament
> > with a generator close to 9/7 and a period of 3/1.
>
> The linear temperament I associate to BP is the "bohpier" temperament;
> the linear temperament obtained by tempering out 3125/3087 and
> 245/243, with wedgie <<13 19 23 0 0 0||. If the octave is a generator,
> the other generator will be a sharp 27/25 of size about 3^(1/13); if
> we want to use 3 as an equivalence instead, we get a period of
> 3^(1/13) as a generator, and can use the octave as the other
> generator, hence we get the same generators looked at two different
> ways. If we get rid of 2 altogether, we don't have a linear
> temperament, but an equal temperament, obtained by dividing 3 into 13
> equal parts.

You are talking about two different linear temperaments. The "real" BP
one has no 2s and it doesn't temper out 3125/3087, only 245/243.

Keenan

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> On 3/12/06, Petr Pařízek <p.parizek@...> wrote:
> [...]
> > (i.e. 9/7 widened by 1/3 of the 245/243 comma), which makes both
9/7 and 5/3
> > detuned by the same amount (or, more precisely, 7/3 and 5/3) and
therefore
> > has a pure 7/5 and 15/7.
> [...]
>
> So does that mean it has an "eigenmonzo" of 7/5? I freakin love that
> word, dude. Eigenmonzo.

Bohpier, on the other hand, has a minimax eigenmonzo of 5; the
generator is 5^(1/19).

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:

> You are talking about two different linear temperaments. The "real" BP
> one has no 2s and it doesn't temper out 3125/3087, only 245/243.

The "real" BP is 1/13 of a 3, and tempers out both. Bohpier is just
what you get by adding octaves to that, which you need to do if you
want to get a linear temperament. "Real" BP is an equal temperament
with no octaves.

> > > (i.e. 9/7 widened by 1/3 of the 245/243 comma), which makes
> > > both 9/7 and 5/3 detuned by the same amount (or, more
> > > precisely, 7/3 and 5/3) and therefore has a pure 7/5 and 15/7.
> >
> > So does that mean it has an "eigenmonzo" of 7/5? I freakin love
> > that word, dude. Eigenmonzo.

Me too.

> Bohpier, on the other hand, has a minimax eigenmonzo of 5; the
> generator is 5^(1/19).

I freakin hate that word. Bohpier.

What ever happened to beep?

-Carl

> The "real" BP is 1/13 of a 3, and tempers out both. Bohpier is just
> what you get by adding octaves to that, which you need to do if you
> want to get a linear temperament.

What a bizarre thing to do. You make a questionable terminology
distinction ("linear" <-> octaves) and then alter temperaments just
so you can use the word!

-Carl

Gene wrote:

> The "real" BP is 1/13 of a 3, and tempers out both. Bohpier is just
> what you get by adding octaves to that, which you need to do if you
> want to get a linear temperament. "Real" BP is an equal temperament
> with no octaves.

The "real" BP is the one which is in Manuel's archive as "bohlen-p.scl".

Petr

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> I freakin hate that word. Bohpier.
>
>
> What ever happened to beep?

We were using it for something else, but maybe it would be better for
this.

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > The "real" BP is 1/13 of a 3, and tempers out both. Bohpier is just
> > what you get by adding octaves to that, which you need to do if you
> > want to get a linear temperament.
>
> What a bizarre thing to do. You make a questionable terminology
> distinction ("linear" <-> octaves) and then alter temperaments just
> so you can use the word!

There's nothing bizarre about it. Bohlen-Pierce tempers out 245/243
and 3125/3087, so obviously there is a very very very very close
relationship between it and the linear temperament which tempers these
two out. What do you suggest--simply ignoring that fact?

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Gene wrote:
>
> > The "real" BP is 1/13 of a 3, and tempers out both. Bohpier is just
> > what you get by adding octaves to that, which you need to do if you
> > want to get a linear temperament. "Real" BP is an equal temperament
> > with no octaves.
>
> The "real" BP is the one which is in Manuel's archive as "bohlen-p.scl".

That's Bohlen's original scale, but it's a detempered version of the
equal division of 3 into 13 parts, which is, I think, what people are
mostly using in practice, though maybe I'm all wet in thinking that.
It was quickly noted, already by Bohlen, that the equal division works
well, and it has the usual advantanges. In any case, I don't see the
justification for tempering out 245/243 but not 3125/3087 and calling
that the "real" BP scale; but certainly you can do that also, and
treat it as Petr did, as a no-octaves linear temperament.

> > > The "real" BP is 1/13 of a 3, and tempers out both. Bohpier
> > > is just what you get by adding octaves to that, which you
> > > need to do if you want to get a linear temperament.
> >
> > What a bizarre thing to do. You make a questionable terminology
> > distinction ("linear" <-> octaves) and then alter temperaments just
> > so you can use the word!
>
> There's nothing bizarre about it. Bohlen-Pierce tempers out 245/243
> and 3125/3087, so obviously there is a very very very very close
> relationship between it and the linear temperament which tempers
> these two out. What do you suggest--simply ignoring that fact?

It isn't obvious to me, but if the complexity of bohpier isn't
significantly higher I'll eat a bug (but I get to pick which
kind).

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:

> It isn't obvious to me, but if the complexity of bohpier isn't
> significantly higher I'll eat a bug (but I get to pick which
> kind).

What does your claim even mean? You are comparing the complexity of
something with octaves to something without octaves. One way of
looking at it is that the complexity is the same, of course.

> > It isn't obvious to me, but if the complexity of bohpier isn't
> > significantly higher I'll eat a bug (but I get to pick which
> > kind).
>
> What does your claim even mean? You are comparing the complexity
> of something with octaves to something without octaves. One way of
> looking at it is that the complexity is the same, of course.

You'll need more notes to map those 2s. Since when is complexity
octave-equivalent?

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > > It isn't obvious to me, but if the complexity of bohpier isn't
> > > significantly higher I'll eat a bug (but I get to pick which
> > > kind).
> >
> > What does your claim even mean? You are comparing the complexity
> > of something with octaves to something without octaves. One way of
> > looking at it is that the complexity is the same, of course.
>
> You'll need more notes to map those 2s. Since when is complexity
> octave-equivalent?

Graham complexity is, obviously. We've used other OE complexity
measures also.

> > > > It isn't obvious to me, but if the complexity of bohpier isn't
> > > > significantly higher I'll eat a bug (but I get to pick which
> > > > kind).
> > >
> > > What does your claim even mean? You are comparing the complexity
> > > of something with octaves to something without octaves. One way
> > > of looking at it is that the complexity is the same, of course.
> >
> > You'll need more notes to map those 2s. Since when is complexity
> > octave-equivalent?
>
> Graham complexity is, obviously.

How so?

-Carl

Gene Ward Smith wrote:
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
> >>I freakin hate that word. Bohpier.
>>
>>
>>What ever happened to beep?
> > > We were using it for something else, but maybe it would be better for
> this.

I agree with that. I'll suggest "cicada" as the name for what we were calling "beep", as it's related to "bug", and tends to be somewhat dissonant.

>>>>> It isn't obvious to me, but if the complexity of bohpier
>>>>> isn't significantly higher I'll eat a bug (but I get to
>>>>> pick which kind).
>>>>
>>>> What does your claim even mean? You are comparing the
>>>> complexity of something with octaves to something without
>>>> octaves. One way of looking at it is that the complexity
>>>> is the same, of course.
>>>
>>> You'll need more notes to map those 2s. Since when is
>>> complexity octave-equivalent?
>>
>> Graham complexity is, obviously.
>
> How so?

Perhaps you're referring to the fact that, in the case of
nonlinear R2 temperaments, Graham multiplies by the number
of periods in an octave. I'm not sure how/if Graham would
generalize this to a non-octave temperament like BP, but
obvious generalizations do exist.

-Carl

Carl Lumma wrote:

> Perhaps you're referring to the fact that, in the case of
> nonlinear R2 temperaments, Graham multiplies by the number
> of periods in an octave. I'm not sure how/if Graham would
> generalize this to a non-octave temperament like BP, but
> obvious generalizations do exist.

I use the equivalence interval in place of the octave. That's only valid for comparing temperaments with the same equivalence interval. But it's easy enough to scale it so that it's in terms of octaves if you have the actual size of the equivalence interval. So for Bohlen-Pierce you multiply by log(2)/log(3).

A fully general measure would be to divide by the actual size of the period instead of multiplying by the number of periods to an octave.

Graham

> > Perhaps you're referring to the fact that, in the case of
> > nonlinear R2 temperaments, Graham multiplies by the number
> > of periods in an octave. I'm not sure how/if Graham would
> > generalize this to a non-octave temperament like BP, but
> > obvious generalizations do exist.
>
> I use the equivalence interval in place of the octave. That's
> only valid for comparing temperaments with the same equivalence
> interval. But it's easy enough to scale it so that it's in
> terms of octaves if you have the actual size of the equivalence
> interval. So for Bohlen-Pierce you multiply by log(2)/log(3).

I hadn't thought of that. I was thinking of multiplying the
minimum continuous number of generators with the minimum
continuous number of periods necessary to complete the given
map.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:

> Perhaps you're referring to the fact that, in the case of
> nonlinear R2 temperaments, Graham multiplies by the number
> of periods in an octave. I'm not sure how/if Graham would
> generalize this to a non-octave temperament like BP, but
> obvious generalizations do exist.

It's clear how to define it for R2 no-octaves temperaments. To make
that comparable to R2 octave temperaments with the same number of
approximated primes, you might introduce a log scaling.

In this case, we have a no-octaves equal temperament, and we would
want to compare it to 5-limit equal temperaments. Here we have < * 13
19 23|,
and that we might compare to <8 13 19 23|, or <8 13 19| in the
5-limit. Dividing by log2(3) gives <* 8.20 11.99 14.51|.

> > Perhaps you're referring to the fact that, in the case of
> > nonlinear R2 temperaments, Graham multiplies by the number
> > of periods in an octave. I'm not sure how/if Graham would
> > generalize this to a non-octave temperament like BP, but
> > obvious generalizations do exist.
>
> It's clear how to define it for R2 no-octaves temperaments. To
> make that comparable to R2 octave temperaments with the same
> number of approximated primes, you might introduce a log scaling.
>
> In this case, we have a no-octaves equal temperament, and we
> would want to compare it to 5-limit equal temperaments. Here
> we have < * 13 19 23|, and that we might compare to
> <8 13 19 23|, or <8 13 19| in the 5-limit. Dividing by log2(3)
> gives <* 8.20 11.99 14.51|.

You people have octaves on the brain. (Yes, I know, literally
too, but there's no reason a model of temperaments need dictate
psychoacoustics.) What's wrong with multiplying by the number
of peroids in the IE? One must normalize to the number of
target basis elements (ie, {2 3 5 7} > {3 5 7}), but that's it.
Though I was only speaking of R2 temperaments here, it's
conceivable that even the rank of the temperament might be
taken care of in the increased accuracy we'd expect from higher
ranks.

-Carl

Carl Lumma wrote:

> You people have octaves on the brain. (Yes, I know, literally
> too, but there's no reason a model of temperaments need dictate
> psychoacoustics.) What's wrong with multiplying by the number
> of peroids in the IE? One must normalize to the number of
> target basis elements (ie, {2 3 5 7} > {3 5 7}), but that's it.
> Though I was only speaking of R2 temperaments here, it's
> conceivable that even the rank of the temperament might be
> taken care of in the increased accuracy we'd expect from higher
> ranks.

It depends on how seriously you take the equivalence interval. If the 3:1 really takes the place of an octave for the Bohlen-Pierce scale, then you can directly compare the number of notes within a 3:1 to the number of notes to an octave for a normal temperament. But it also makes sense to consider the number of notes you need for a given pitch range, and the octave's as good a standard as any. It's only the choice of equivalence interval that really matters.

I think weighted complexities automatically sort this out. If the weighting's by octaves or cents, then you can directly compare different equivalence intervals. The units would be steps/octave or whatever. Weighted wedgie measures should also work regardless of the prime intervals, and don't require an equivalence interval.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Carl Lumma wrote:
>
> > You people have octaves on the brain. (Yes, I know, literally
> > too, but there's no reason a model of temperaments need dictate
> > psychoacoustics.) What's wrong with multiplying by the number
> > of peroids in the IE? One must normalize to the number of
> > target basis elements (ie, {2 3 5 7} > {3 5 7}), but that's it.
> > Though I was only speaking of R2 temperaments here, it's
> > conceivable that even the rank of the temperament might be
> > taken care of in the increased accuracy we'd expect from higher
> > ranks.
>
> It depends on how seriously you take the equivalence interval.
> If the 3:1 really takes the place of an octave for the
> Bohlen-Pierce scale, then you can directly compare the number
> of notes within a 3:1 to the number of notes to an octave for
> a normal temperament.

Thank you.

> But it also makes sense to consider the number of notes you
> need for a given pitch range,

I can't see how that makes sense.

-Carl