B-P recorder

I'm looking for information or ideas on designs for Bohlen-Pierce fipple
instruments. I'd like to make something like a recorder out of PVC or such,
since that should ensure pretty well matching (odd) harmonics in the timbre.
I keep thinking there must be some "trick" to getting Bohlen-Pierce to work
really well on it in a 2-tritave range, but haven't figured it out. In
particular, I'm wondering if there might be some way to ensure a clean
tritave jump. Anyone here tried something like this or know something of it
they don't mind sharing?

On Wed, Apr 20, 2011 at 8:27 PM, Daniel Nielsen <nielsed@...> wrote:
>
> I'm looking for information or ideas on designs for Bohlen-Pierce fipple instruments. I'd like to make something like a recorder out of PVC or such, since that should ensure pretty well matching (odd) harmonics in the timbre. I keep thinking there must be some "trick" to getting Bohlen-Pierce to work really well on it in a 2-tritave range, but haven't figured it out. In particular, I'm wondering if there might be some way to ensure a clean tritave jump. Anyone here tried something like this or know something of it they don't mind sharing?

I'm not sure there actually exists anyone in the world who's managed
to get themselves to hear tritaves as being actual equivalence
intervals, but if you don't get enough of a response here, you also
might want to try asking on the Facebook Xenharmonic Alliance group,
where a few BP aficionados hang out:

http://www.facebook.com/group.php?gid=2229924481

-Mike

Mike: "I'm not sure there actually exists anyone in the world who's managed
to get themselves to hear tritaves as being actual equivalence
intervals,"

That would make one pretty "odd" (:p).

"...but if you don't get enough of a response here, you also
might want to try asking on the Facebook Xenharmonic Alliance group,
where a few BP aficionados hang out..."

That sounds like a plan - thanks. I'm not bonded to BP or anything, although
it's not surprising that it's so popular (relatively). I may try to build
some simpler instruments first, just to make certain I have the basics
right.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> I'm not sure there actually exists anyone in the world who's managed
> to get themselves to hear tritaves as being actual equivalence
> intervals,

How would you test this? When I play the BP pentatonic 33331 with odd
harmonics-only timbre I sort of hear it. It's harder if you make double
or triple tritave jumps but I don't share your skepticism. The chords
sound much more consonant with all-harmonics timbres but then the
tritave equivalence is destroyed.

Kalle

--- In tuning@yahoogroups.com, Daniel Nielsen <nielsed@...> wrote:

> That sounds like a plan - thanks. I'm not bonded to BP or anything, although
> it's not surprising that it's so popular (relatively).

So far as I can see its major merit is that it forces you into xenharmonicity. Do you know of another?

>
> > That sounds like a plan - thanks. I'm not bonded to BP or anything,
> although
> > it's not surprising that it's so popular (relatively).
>
> So far as I can see its major merit is that it forces you into
> xenharmonicity. Do you know of another?
>

Do you mean another scale system that would work well on PVC recorder? I was
looking at the BP system as a sort of puzzle regarding applying overtone
blowing (or some other system) to the tritave, but I would probably be
happier with 19-TET as far as my own musical experimentation goes. I haven't
found anything like this online after a brief search, probably due to the
lack of fingers. I'm still working on growing these.

The warped canon example on Mr Miller's site hasn't sold me on BP, although
it does sound xenharmonic and, for lack of a better word, "consistent". My
ear isn't good enough to tell whether the octaves are retained in that
example, or whether it is replaced with the tritave, which I assume it is.
If it means anything, my daughter (2 y.o.) really didn't care for it, while
she did very much enjoy the 12-TET during the same listening "experiment".
Certain of the other performances I've seen online have given me a more
favorable impression of BP, though.

Gene, I really appreciate your and others' direction/ideas on this.

On Thu, Apr 21, 2011 at 4:10 AM, Kalle Aho <kalleaho@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > I'm not sure there actually exists anyone in the world who's managed
> > to get themselves to hear tritaves as being actual equivalence
> > intervals,
>
> How would you test this? When I play the BP pentatonic 33331 with odd
> harmonics-only timbre I sort of hear it. It's harder if you make double
> or triple tritave jumps but I don't share your skepticism. The chords
> sound much more consonant with all-harmonics timbres but then the
> tritave equivalence is destroyed.

I never said it was impossible, I'm just saying that I haven't heard
anyone actually able to hear it yet. Ron Sword can't, Paul Erlich
can't, I certainly can't, and I thought I remembered Elaine Walker
saying she couldn't really hear it either in one of her videos
(although maybe she's enlightened now). I just posted on her Facebook
so let's see if she can or not. But if Elaine can't hear it that way
after being all about BP for years now, I fear there may be no hope.

Either way, at this point in my understanding of things, I don't
believe that there's any part of anything that's not influenced or
subjected to learning, so I see no reason why you can't learn to hear
tritaves as "equivalent" in some sense. I just don't know anyone who
really claims to be able to do it.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Thu, Apr 21, 2011 at 4:10 AM, Kalle Aho <kalleaho@...> wrote:
> >
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > > I'm not sure there actually exists anyone in the world who's managed
> > > to get themselves to hear tritaves as being actual equivalence
> > > intervals,
> >
> > How would you test this? When I play the BP pentatonic 33331 with odd
> > harmonics-only timbre I sort of hear it. It's harder if you make double
> > or triple tritave jumps but I don't share your skepticism. The chords
> > sound much more consonant with all-harmonics timbres but then the
> > tritave equivalence is destroyed.
>
> I never said it was impossible, I'm just saying that I haven't heard
> anyone actually able to hear it yet. Ron Sword can't, Paul Erlich
> can't, I certainly can't, and I thought I remembered Elaine Walker
> saying she couldn't really hear it either in one of her videos
> (although maybe she's enlightened now). I just posted on her Facebook
> so let's see if she can or not. But if Elaine can't hear it that way
> after being all about BP for years now, I fear there may be no hope.
>
> Either way, at this point in my understanding of things, I don't
> believe that there's any part of anything that's not influenced or
> subjected to learning, so I see no reason why you can't learn to hear
> tritaves as "equivalent" in some sense. I just don't know anyone who
> really claims to be able to do it.

But what would count in your terms as hearing tritaves as equivalent?
When I play the BP pentatonic scale over many tritaves I don't hear
any kind of modulation going on. I hear the scale as repeating with
"more of the same". Isn't this enough for some kind of equivalence?

Kalle

On Thu, Apr 21, 2011 at 1:34 PM, Kalle Aho <kalleaho@...> wrote:
>
> But what would count in your terms as hearing tritaves as equivalent?
> When I play the BP pentatonic scale over many tritaves I don't hear
> any kind of modulation going on. I hear the scale as repeating with
> "more of the same". Isn't this enough for some kind of equivalence?

How about this scale, which repeats at the 3/2:

C D E F# G A B C# D E F# G# A B C# D# E ...

That is, it's just

C D E F# G

repeated over and over again at the 3/2, so then you get G A B C# D, etc.

I don't hear the 3/2 as a true interval of equivalence in the same way
that I hear the 2/1 - I hear notes that are separated by a 2/1 as
sharing the same chroma but differing in pitch height. This doesn't
happen with the above scale and it doesn't happen for me with BP
either.

Elaine has written back on Facebook and says:

"That happened to me only after being familiar with BP for 20 years
(1990 - 2010) and also at a point where I was actively composing with
it for a few months. It also has to be "tonal" music. If you are
playing random notes, there shouldn't be anything in our brains that
would assume that the tritave is the frame, other than it is the most
consonant interval."

I guess it's possible...

-Mike

This conversation confuses me.

>> But what would count in your terms as hearing tritaves as equivalent?

Why would one even want to hear tritaves as equivalent?

From Elaine's site:
-----
It all started in the early 1970s when Heinz Bohlen became curious as
to why musicians always use the same 12 note per octave tuning. After
getting some unsatisfactory answers from musicians, he took it upon
himself to research music and tonality. It was finally the
understanding of combination tones that lead him to believe he could
use the same method that lead to the 12 tone equal temperament, to
devise another tuning from a different framework. He understood that
the 12 tone scale was based around the a major triad, the inversion,
and then filling in the gaps. So, he started with a new triad that was
very harmonically pure, but not contained within the normal 12 tone
framework. He made the stunning discovery that a scale of thirteen
almost equal steps within the framework of an octave and a fifth
(which John Pierce later dubbed as a tritave) contained this very pure
triad. He recognized that this tuning shared a duality with the
traditional Western 12 tone tuning, it had harmonic value, and that it
would be a valid compositional tool.
-----

If I read that correctly, the "very harmonically pure" triad (I
presume that's the 3:5:7?) came first, and the BP scale came out of
it. But

(a) was it even the goal of the scale inventors to create a non-octave
scale? and

(b) was the goal of the scale to create something in which the first
and last notes *sound the same* the way an octave sounds the same?

If the answer to (a) is "yes", then how would it be different from,
say, Carlos Beta? If I hear Beta, I don't expect to hear a perfect
fourth as "equivalent" to some other non-octave pitch. If I play 0
cents, then 498 cents, and then either 996 or 1698 cents, the 1698
will sound equivalent to the 498, and the 996 won't -- regardless of
how Carlos designed the scale. If we know our ears hear equivalence
between multiples of 1200 cents, but not for multiples of 498 cents or
100 cents or other arbitrary numbers, why would it work for 1902
cents? Why would I even want it to?

If the answer to (b) is "yes", that seems pretty nutty to me. It would
be like training people to see blue as green -- even if it's possible,
what's the point?

As Gene said, the BP scale forces you to be xenharmonic, but I don't
see how making a non-octave scale suddenly makes non-octave intervals
"equivalent".

Regards,
Jake

From what (little) I know of it, the motivation was more to extend the root
triad beyond the 5-limit - and thus to delete even factors from its ratios -
and expanding the period was a ready way to do this. Eliminating a "good"
2/1 then begged the question of whether it might be possible for 3/1 to
supplant 2/1 as our ears' anchor point. According to the quoted material,
this apparently is momentarily possible after a long period of
exposure/usage and a somewhat altered state of consciousness.

My guess would be that a piece would need a very wide pitch range that is
run across a lot quickly before the ear would hold on to 3/1 as a buoy. That
being the goal, then my solo recorder notion probably wouldn't do a great
job of that. However, that's not to say the ear doesn't get anything
meaningful out of the spread of notes chosen for BP.

To be clear, I think a BP recorder could be really cool, and I don't
mean to be a downer on the project. (In fact, the BP discussion
inspired me to listen to "No Brighter Sun, No Darker Night", my
favorite piece from the BP symposium, on my commute home.) I just
literally don't understand the talk of "equivalence".

Regards,
Jake

On 4/21/11, Daniel Nielsen <nielsed@...> wrote:
> From what (little) I know of it, the motivation was more to extend the root
> triad beyond the 5-limit - and thus to delete even factors from its ratios -
> and expanding the period was a ready way to do this. Eliminating a "good"
> 2/1 then begged the question of whether it might be possible for 3/1 to
> supplant 2/1 as our ears' anchor point. According to the quoted material,
> this apparently is momentarily possible after a long period of
> exposure/usage and a somewhat altered state of consciousness.
>
> My guess would be that a piece would need a very wide pitch range that is
> run across a lot quickly before the ear would hold on to 3/1 as a buoy. That
> being the goal, then my solo recorder notion probably wouldn't do a great
> job of that. However, that's not to say the ear doesn't get anything
> meaningful out of the spread of notes chosen for BP.
>

On Thu, Apr 21, 2011 at 5:00 PM, Jake Freivald <jdfreivald@...> wrote:
>
> (a) was it even the goal of the scale inventors to create a non-octave
> scale? and

Definitely yes.

> (b) was the goal of the scale to create something in which the first
> and last notes *sound the same* the way an octave sounds the same?

Right - tritave equivalence in place of octave equivalence.

> If the answer to (b) is "yes", that seems pretty nutty to me. It would
> be like training people to see blue as green -- even if it's possible,
> what's the point?

The hell if I know, but people claim that it's possible. I've seen
some research indicating that certain parts of octave equivalence are
built into the mammalian brain, but there may be other ways you can
learn to hear a "tritave" as equivalent.

-Mike

> > (a) was it even the goal of the scale inventors to create a non-octave
> > scale? and
>
> Definitely yes.
[Snip]
> Right - tritave equivalence in place of octave equivalence.

Okay, then I don't get it. Beta divides the 4/3 into equal parts, and
BP divides the tritave into equal parts. The only difference is in the
intent of the scale creator.

Regards,
Jake

On 4/21/11, Mike Battaglia <battaglia01@...> wrote:
> On Thu, Apr 21, 2011 at 5:00 PM, Jake Freivald <jdfreivald@...> wrote:
>>
>> (a) was it even the goal of the scale inventors to create a non-octave
>> scale? and
>
> Definitely yes.
>
>> (b) was the goal of the scale to create something in which the first
>> and last notes *sound the same* the way an octave sounds the same?
>
> Right - tritave equivalence in place of octave equivalence.
>
>> If the answer to (b) is "yes", that seems pretty nutty to me. It would
>> be like training people to see blue as green -- even if it's possible,
>> what's the point?
>
> The hell if I know, but people claim that it's possible. I've seen
> some research indicating that certain parts of octave equivalence are
> built into the mammalian brain, but there may be other ways you can
> learn to hear a "tritave" as equivalent.
>
> -Mike
>
>
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On Thu, Apr 21, 2011 at 6:33 PM, Jake Freivald <jdfreivald@...> wrote:
>
> Okay, then I don't get it. Beta divides the 4/3 into equal parts, and
> BP divides the tritave into equal parts. The only difference is in the
> intent of the scale creator.

The claim is that there is a way to perceive BP such that the 3/1
starts sounding like the interval of equivalence, not the 2/1. That's
about all I know.

-Mike

On Thu, Apr 21, 2011 at 6:35 PM, Mike Battaglia <battaglia01@...> wrote:
> On Thu, Apr 21, 2011 at 6:33 PM, Jake Freivald <jdfreivald@...> wrote:
>>
>> Okay, then I don't get it. Beta divides the 4/3 into equal parts, and
>> BP divides the tritave into equal parts. The only difference is in the
>> intent of the scale creator.
>
> The claim is that there is a way to perceive BP such that the 3/1
> starts sounding like the interval of equivalence, not the 2/1. That's
> about all I know.

Ron Sword is telling me on Facebook after the BP conference that he
was hearing it that way, that tritaves started sounding like new
octaves, and then afterward octaves sounded compressed. He said the BP
conference was 3 days of hours of hours of nothing but BP, and that's
how he ended up hearing things by the end of it.

As Ron has pretty extensive training in 12-equal, more so than
probably 99% of the rest of the community, I believe anecdotes from
folks like that. Elaine's also saying that the timbre doesn't matter
so much, so I guess it's more a matter of just cognitively adapting to
organize things differently.

-Mike

As a long-time friend and correspondent of XJ Scott, proprietor of the site nonoctave.com and author of the best microtuning software for the Mac OS ("Lil' Miss Scale Oven"), perhaps I can shed some light on non-octave scales and interval equivalence.

Okay, so we think we hear octaves as "equivalent", because we're used to thinking in note-names that treat them as such. Really, we hear octaves as "distinct but similar"--"that's the same note, but higher (or lower)". One of the important features of the octave in music is that we can use octave-doubling in arrangement to produce timbral/textural "thickening" of parts, because the octave is such a simple interval that, when tuned pure, it allows two notes to fuse very strongly. This high degree of fusion--a pure octave is almost seamless, really--is what allows us to treat it as "equivalent".

The next-simplest ratio after 2/1 is 3/1, and sure enough you can usually harmonize in parallel 3/1's to achieve a very similar effect as parallel 2/1's--textural thickening but no sensation of "added information" because the blend is so strong. With odd-harmonic-only timbres, this works especially well: if the partials of the tonic are 1:3:5:7:9:11:13:15:17:19 (etc.), the partials of a note 3/1 above it will be 3:9:15:21:27:33:39:45:51:57, which is really just like a stretched-out version of the octave relationship for full-harmonic timbres: 1:2:3:4:5:6:7:8:9:10 and 2:4:6:8:10:12:14:16:18:20. So while we may not (because of our conditioning) be able to here the tritave equivalence in BP, it should theoretically be possible to hear, because the tritave does retain *most* of the properties that make octave-equivalence work.

Now, I like BP for xenharmonic purposes, but I think its progenitors actually thought that it should work "just as naturally" as 12-TET (or at least meantone)--i.e., it wasn't *supposed* to sound xenharmonic and alien and freaky and weird, it was just supposed to sound a bit "different". This is emphatically *not* the case, and I've got many ideas as to why.

For starters, if you compare the Just BP "major scale" ("Lambda") with a Just major scale, something is pretty obvious:

BP:
1/1 0 cents
25/21 301 cents
9/7 435 cents
7/5 583 cents
5/3 884 cents
9/5 1017 cents
15/7 1319 cents
7/3 1467 cents
25/9 1768 cents
3/1 1902 cents

Diatonic Major:
1/1 0 cents
9/8 204 cents
5/4 386 cents
4/3 498 cents
3/2 702 cents
5/3 884 cents
15/8 1088 cents
2/1 1200 cents

In the BP scale, most of the dyadic consonances are much less concordant than the dyadic consonances in the major scale, and the dissonances are more numerous and more discordant. There are also a lot of consonances to be found in the major scale besides the usual 4:5:6 and 10:12:15, because there is a greater number of consonant triads you can construct with a 2, 3, 5 basis than a 3, 5, 7 basis; that's basic math, really. So right off the bat, there's going to be less consonance in BP than in meantone.

Scale-wise, there are more "semitones" in BP (Lambda is TssTsTsTs, the major scale is TTsTTTs), and this makes writing melodies more challenging, since there is a greater likelihood of two adjacent notes in the scale being discordant with one another.

Also, in the diatonic major scale, the two intervals that compose the major triad (5/4 and 6/5) are in the same interval-class, and the major and minor triads are an otonal-utonal pair. In BP Lambda, the two intervals that form the main 3:5:7 triad are not in the same interval-class: 5/3 is a BP "major fifth" and 7/5 is a BP 4th. In other words, parallel tonal harmony doesn't work the way we'd expect it in BP. Also, this makes it possible to form either an otonal 3:5:7 triad (1/1-5/3-7/3) OR an utonal 1/(3:5:7) (1/1-7/5-7/3) triad on many of the same roots; if you look at the Lambda mode in terms of which degree forms an otonal (o), utonal (u), both (o/u), or neither (n) triad, it looks like this:

I: o/u
II: u
III: o/u
IV: n
V: o/u
VI: o
VII: o/u
VIII: o
IX: u

That's four roots that are tonally ambiguous. Also note that the utonal BP triad, when examined as consecutive harmonics, is 15:21:35, which is significantly more complex than the otonal triad of 3:5:7 (vs. 4:5:6 and 10:12:15 in "regular" meantone harmony). So in the harmonic entropy component of discordance, the BP utonal triad is probably going to sound quite a bit more discordant (it would be interesting to see 3HE values for the two pairs of triads for comparison).

On the other hand, if we don't treat the utonal triad as the "minor" triad, but rather the one with the "minor" 5th of 75/49, the our minor chord now approximates harmonics 15:23:35, which isn't much worse than the utonal triad in complexity but is worse in terms of roughness, the ambiguity is gone and we now have minor triads on the II and IX degrees and unambiguous major triads on the others (except the IV), so that might be functionally a bit better...but it will be psychoacoustically worse, I suspect.

On the third hand, we can look at 5:7:9 as a harmonic basis instead of 3:5:7; it's less concordant, but perhaps more familiar sounding (since it's not so wide). However, it doesn't improve tonal ambiguity (5 triads are still ambiguous) at all or lead to the two component intervals sharing a class, so again, it's not going function like meantone here either.

In summary, while tritave equivalence and a 3:5:7 basic harmonic unit might be viable alternatives to the octave and a 4:5:6 harmonic unit in and of themselves, the diatonic scale and tonal structure that Bohlen and Pierce came up with to support these harmonic bases fails to be analogous to meantone harmony in many important ways, and that's probably why most music in BP sounds like something that came from deep space. That's also why I don't think it's "just" our conditioning that keeps us from being able to intelligize BP in ways analogous to meantone music.

As an aside, I've noticed that the few pieces of music that *do* sound fairly intelligible in BP actually seem to ignore the basis of BP and run in other directions with the scale. For instance, I've noticed that Elaine Walker has a tendency to "meantone-ize" BP in her playing, treating 25/21 as 6/5, and making BP "minor" triads that are basically 5:6:15's, and occasionally using 75/49 as a pseudo-3/2 for root-movement in chord progressions.

And this concludes my analysis of the Bohlen-Pierce scale.

-Igliashon

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:
>
> This conversation confuses me.
>
> >> But what would count in your terms as hearing tritaves as equivalent?
>
> Why would one even want to hear tritaves as equivalent?
>
> From Elaine's site:
> -----
> It all started in the early 1970s when Heinz Bohlen became curious as
> to why musicians always use the same 12 note per octave tuning. After
> getting some unsatisfactory answers from musicians, he took it upon
> himself to research music and tonality. It was finally the
> understanding of combination tones that lead him to believe he could
> use the same method that lead to the 12 tone equal temperament, to
> devise another tuning from a different framework. He understood that
> the 12 tone scale was based around the a major triad, the inversion,
> and then filling in the gaps. So, he started with a new triad that was
> very harmonically pure, but not contained within the normal 12 tone
> framework. He made the stunning discovery that a scale of thirteen
> almost equal steps within the framework of an octave and a fifth
> (which John Pierce later dubbed as a tritave) contained this very pure
> triad. He recognized that this tuning shared a duality with the
> traditional Western 12 tone tuning, it had harmonic value, and that it
> would be a valid compositional tool.
> -----
>
> If I read that correctly, the "very harmonically pure" triad (I
> presume that's the 3:5:7?) came first, and the BP scale came out of
> it. But
>
> (a) was it even the goal of the scale inventors to create a non-octave
> scale? and
>
> (b) was the goal of the scale to create something in which the first
> and last notes *sound the same* the way an octave sounds the same?
>
> If the answer to (a) is "yes", then how would it be different from,
> say, Carlos Beta? If I hear Beta, I don't expect to hear a perfect
> fourth as "equivalent" to some other non-octave pitch. If I play 0
> cents, then 498 cents, and then either 996 or 1698 cents, the 1698
> will sound equivalent to the 498, and the 996 won't -- regardless of
> how Carlos designed the scale. If we know our ears hear equivalence
> between multiples of 1200 cents, but not for multiples of 498 cents or
> 100 cents or other arbitrary numbers, why would it work for 1902
> cents? Why would I even want it to?
>
> If the answer to (b) is "yes", that seems pretty nutty to me. It would
> be like training people to see blue as green -- even if it's possible,
> what's the point?
>
> As Gene said, the BP scale forces you to be xenharmonic, but I don't
> see how making a non-octave scale suddenly makes non-octave intervals
> "equivalent".
>
> Regards,
> Jake
>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Thu, Apr 21, 2011 at 5:00 PM, Jake Freivald <jdfreivald@...> wrote:
> >
> > (a) was it even the goal of the scale inventors to create a non-octave
> > scale? and
>
> Definitely yes.
>
> > (b) was the goal of the scale to create something in which the first
> > and last notes *sound the same* the way an octave sounds the same?
>
> Right - tritave equivalence in place of octave equivalence.

Back in my salad days I experimented with scales repeating at 3, with the idea that maybe it wouldn't sound so much like a key change if you repeated a theme 3 higher in such a scale. This idea turned out to be wrong, and 3/2 didn't work like that either. It never occurred to me that octaves might be the problem and I made no attempt to excise them, but I've never been convinced by the equivalency in BP music. But there are possibilities, particularly with temperaments with small periods, of repeating at some other point. I wouldn't try it with the idea the interval of repetition is going to work like an octave, but it's a viable way to make music.

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> Okay, so we think we hear octaves as "equivalent", because we're used to thinking in note-names that treat them as such.

Do you have evidence this is so--what do people unfamiliar with note-names say?

> In the BP scale, most of the dyadic consonances are much less concordant than the dyadic consonances in the major scale, and the dissonances are more numerous and more discordant.

Removing an important prime from your group of JI generators always has this effect, and 2 is as important as it gets.

On Thu, Apr 21, 2011 at 7:37 PM, cityoftheasleep
<igliashon@...> wrote:
>
> As a long-time friend and correspondent of XJ Scott, proprietor of the site nonoctave.com and author of the best microtuning software for the Mac OS ("Lil' Miss Scale Oven"), perhaps I can shed some light on non-octave scales and interval equivalence.

Is it better than Scala?

> The next-simplest ratio after 2/1 is 3/1, and sure enough you can usually harmonize in parallel 3/1's to achieve a very similar effect as parallel 2/1's--textural thickening but no sensation of "added information" because the blend is so strong.

OK, I dig it. That makes sense, and fits well with the rule that you
can omit the fifth in a chord without altering its quality.

> With odd-harmonic-only timbres, this works especially well: if the partials of the tonic are 1:3:5:7:9:11:13:15:17:19 (etc.), the partials of a note 3/1 above it will be 3:9:15:21:27:33:39:45:51:57, which is really just like a stretched-out version of the octave relationship for full-harmonic timbres: 1:2:3:4:5:6:7:8:9:10 and 2:4:6:8:10:12:14:16:18:20. So while we may not (because of our conditioning) be able to here the tritave equivalence in BP, it should theoretically be possible to hear, because the tritave does retain *most* of the properties that make octave-equivalence work.

Elaine claims that she can hear it for full harmonic timbres as well.
Ron Sword was telling me that after 3 days of nothing but BP, he
started hearing tritaves as equivalent and octaves as "compressed,"
and that it was a trip. As the brain is basically a machine that
continually adapts and finds ways to extricate order from the
information coming into it, I don't see why this shouldn't be
possible.

> Now, I like BP for xenharmonic purposes, but I think its progenitors actually thought that it should work "just as naturally" as 12-TET (or at least meantone)--i.e., it wasn't *supposed* to sound xenharmonic and alien and freaky and weird, it was just supposed to sound a bit "different". This is emphatically *not* the case, and I've got many ideas as to why.

> That's four roots that are tonally ambiguous. Also note that the utonal BP triad, when examined as consecutive harmonics, is 15:21:35, which is significantly more complex than the otonal triad of 3:5:7 (vs. 4:5:6 and 10:12:15 in "regular" meantone harmony). So in the harmonic entropy component of discordance, the BP utonal triad is probably going to sound quite a bit more discordant (it would be interesting to see 3HE values for the two pairs of triads for comparison).

Aha. So maybe it would be worthwhile to apply some regular mapping
optimizations to the 3.5.7 subgroup then; pick some chords and
optimize scales around them for lowest Graham complexity. I'll mess
around with Graham's temperament finder later on and see what comes
out. Has this search been done before?

Maybe one idea is to find a chord such that its inverse can be
-tempered- to some lower-complexity ratio, i.e. if 91/90 vanishes, the
inverse of 6:7:9 is 10:13:15, both of which aren't very complex at
all.

> In summary, while tritave equivalence and a 3:5:7 basic harmonic unit might be viable alternatives to the octave and a 4:5:6 harmonic unit in and of themselves, the diatonic scale and tonal structure that Bohlen and Pierce came up with to support these harmonic bases fails to be analogous to meantone harmony in many important ways, and that's probably why most music in BP sounds like something that came from deep space. That's also why I don't think it's "just" our conditioning that keeps us from being able to intelligize BP in ways analogous to meantone music.

So maybe the BP Lambda scale just sucks, but we have the entire 3.5.7
subgroup to screw around with...

> As an aside, I've noticed that the few pieces of music that *do* sound fairly intelligible in BP actually seem to ignore the basis of BP and run in other directions with the scale. For instance, I've noticed that Elaine Walker has a tendency to "meantone-ize" BP in her playing, treating 25/21 as 6/5, and making BP "minor" triads that are basically 5:6:15's, and occasionally using 75/49 as a pseudo-3/2 for root-movement in chord progressions.
>
> And this concludes my analysis of the Bohlen-Pierce scale.

Hear hear!

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Aha. So maybe it would be worthwhile to apply some regular mapping
> optimizations to the 3.5.7 subgroup then; pick some chords and
> optimize scales around them for lowest Graham complexity. I'll mess
> around with Graham's temperament finder later on and see what comes
> out. Has this search been done before?

Of course it has been. But that doesn't mean it isn't mostly unexplored territory, starting with everything above the 7 no-2s limit, which BP doesn't do with as much accuracy as one might like.

> > In summary, while tritave equivalence and a 3:5:7 basic harmonic unit might be viable alternatives to the octave and a 4:5:6 harmonic unit in and of themselves, the diatonic scale and tonal structure that Bohlen and Pierce came up with to support these harmonic bases fails to be analogous to meantone harmony in many important ways, and that's probably why most music in BP sounds like something that came from deep space. That's also why I don't think it's "just" our conditioning that keeps us from being able to intelligize BP in ways analogous to meantone music.
>
> So maybe the BP Lambda scale just sucks, but we have the entire 3.5.7
> subgroup to screw around with...

You people are dreamers. Getting rid of 2 is a drastic change, and you can't paper over it no matter what you do.

On Thu, Apr 21, 2011 at 8:40 PM, genewardsmith
<genewardsmith@...> wrote:
>
> Of course it has been. But that doesn't mean it isn't mostly unexplored territory, starting with everything above the 7 no-2s limit, which BP doesn't do with as much accuracy as one might like.

So, uh, what are the results of said search?

> > So maybe the BP Lambda scale just sucks, but we have the entire 3.5.7
> > subgroup to screw around with...
>
> You people are dreamers. Getting rid of 2 is a drastic change, and you can't paper over it no matter what you do.

Of course we're dreamers! I spend most of my free time talking on a
yahoo mailing list about microtonal music! Jeez...

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Is it better than Scala?

It's better than Scala's Mac implementation, and it's a less-geeky more-cheeky interface.

> Aha. So maybe it would be worthwhile to apply some regular mapping
> optimizations to the 3.5.7 subgroup then; pick some chords and
> optimize scales around them for lowest Graham complexity. I'll mess
> around with Graham's temperament finder later on and see what comes
> out. Has this search been done before?

Yes. See some of Paul's photos on the facebook Xenharmonic Alliance page. He put a bunch of 3.5.7 horagrams up. Believe it or not, Lambda is actually about as good as it gets in terms of plenitude of target triads.

> Maybe one idea is to find a chord such that its inverse can be
> -tempered- to some lower-complexity ratio, i.e. if 91/90 vanishes, the
> inverse of 6:7:9 is 10:13:15, both of which aren't very complex at
> all.

Interesting thought...

> So maybe the BP Lambda scale just sucks, but we have the entire 3.5.7
> subgroup to screw around with...

Indeed, but 13-ED3 dominates it pretty heavily. See Paul's 2D scale tree for 3.5.7 on facebook. Maybe using 5/1 or 7/1 as an equivalence interval would yield better scales (though definitely weaker psychoacoustic equivalence, if any).

-Igs

On Thu, Apr 21, 2011 at 9:04 PM, cityoftheasleep
<igliashon@...> wrote:
>
> Indeed, but 13-ED3 dominates it pretty heavily. See Paul's 2D scale tree for 3.5.7 on facebook. Maybe using 5/1 or 7/1 as an equivalence interval would yield better scales (though definitely weaker psychoacoustic equivalence, if any).

Using 7/1 would be ridiculous, you'd end up with 3 and a half 7-aves
covering the entire human hearing range. Using 5 gets you 4 and a
third 5-taves, which may be better. Maybe 7/2 or 5/2 would work. But
if you can get 5/2 to sound like it's adding no information, you've
really done something magical, as most of us are pretty conditioned to
hear ratios of 5 as indicating whether the chord is highly tonal
(4:5:6) or less tonal (10:12:15). Maybe in magic temperament you can
get 5/4 to take on that role.

-Mike

On Thu, Apr 21, 2011 at 8:40 PM, genewardsmith
<genewardsmith@...> wrote:
> >
> > So maybe the BP Lambda scale just sucks, but we have the entire 3.5.7
> > subgroup to screw around with...
>
> You people are dreamers. Getting rid of 2 is a drastic change, and you can't paper over it no matter what you do.

Also, why do we need to cut 2 out at all? You don't need to eliminate
2 from the math entirely to form scales that don't have 2/1 in them
and that repeat at the 3/1. All you have to do is treat 3/1 as the
period, have something else as the generator, and pick a temperament
in which 2/1 has really high Graham complexity.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Thu, Apr 21, 2011 at 8:40 PM, genewardsmith
> <genewardsmith@...> wrote:
> >
> > Of course it has been. But that doesn't mean it isn't mostly unexplored territory, starting with everything above the 7 no-2s limit, which BP doesn't do with as much accuracy as one might like.
>
> So, uh, what are the results of said search?

You can't run my programs, but anyone may turn the crank on the Graham machine. For 3.5.7 BP comes out on top at a 5.0 error, with [<1 1 1|, <0 3 5|] second. That one tempers out 3125/3087, the gariboh comma ("boh" from Bohlen-Pierce) and BP tempers out 245/243, sensamagic. Put the commas together in the 3.5.7 subgroup and you get the <13 19 23| sval. Put them together in the full 7-limit and you get bohpier temperament. Reduce the error to 1.0 and now [<1 3 3|, <0 -5 -4|] comes out on top, tempering out 16875/16807, the mirkwai comma. With an error of 5.0 and the 3.5.7.11 subgroup, the top score belongs to the temperament tempering out both 245/243 and 1331/1323. This belongs to the BP family but is not an extension of BP; for that, try tempering out 77/75 as well as 245/243.

There's also this page:

http://www.nonoctave.com/tuning/twelfth.html

Less obvious in some ways, more in others. See also here:

http://www.facebook.com/group.php?gid=2229924481&v=photos&so=30

Slim pickings, if you ask me.

-Igs
--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > On Thu, Apr 21, 2011 at 8:40 PM, genewardsmith
> > <genewardsmith@> wrote:
> > >
> > > Of course it has been. But that doesn't mean it isn't mostly unexplored territory, starting with everything above the 7 no-2s limit, which BP doesn't do with as much accuracy as one might like.
> >
> > So, uh, what are the results of said search?
>
> You can't run my programs, but anyone may turn the crank on the Graham machine. For 3.5.7 BP comes out on top at a 5.0 error, with [<1 1 1|, <0 3 5|] second. That one tempers out 3125/3087, the gariboh comma ("boh" from Bohlen-Pierce) and BP tempers out 245/243, sensamagic. Put the commas together in the 3.5.7 subgroup and you get the <13 19 23| sval. Put them together in the full 7-limit and you get bohpier temperament. Reduce the error to 1.0 and now [<1 3 3|, <0 -5 -4|] comes out on top, tempering out 16875/16807, the mirkwai comma. With an error of 5.0 and the 3.5.7.11 subgroup, the top score belongs to the temperament tempering out both 245/243 and 1331/1323. This belongs to the BP family but is not an extension of BP; for that, try tempering out 77/75 as well as 245/243.
>

On Thu, Apr 21, 2011 at 9:17 PM, genewardsmith
<genewardsmith@...> wrote:
>
> You can't run my programs, but anyone may turn the crank on the Graham machine. For 3.5.7 BP comes out on top at a 5.0 error, with [<1 1 1|, <0 3 5|] second. That one tempers out 3125/3087, the gariboh comma ("boh" from Bohlen-Pierce) and BP tempers out 245/243, sensamagic. Put the commas together in the 3.5.7 subgroup and you get the <13 19 23| sval. Put them together in the full 7-limit and you get bohpier temperament. Reduce the error to 1.0 and now [<1 3 3|, <0 -5 -4|] comes out on top, tempering out 16875/16807, the mirkwai comma. With an error of 5.0 and the 3.5.7.11 subgroup, the top score belongs to the temperament tempering out both 245/243 and 1331/1323. This belongs to the BP family but is not an extension of BP; for that, try tempering out 77/75 as well as 245/243.

I can't tell if this is the same one you found, but Graham's 3.5.7.11
winner seems to be

http://x31eq.com/cgi-bin/rt.cgi?ets=7_12&limit=3_5_7_11&key=6_-3_-2_1_5_0_1&error=5.0

Is it just coincidence that this happens to be the tritave equivalent
of suprapyth, in that it forms MOS's at 5, 7, 12, 17...?

-Mike

On Thu, Apr 21, 2011 at 9:35 PM, cityoftheasleep
<igliashon@...> wrote:
>
> There's also this page:
>
> http://www.nonoctave.com/tuning/twelfth.html
>
> Less obvious in some ways, more in others. See also here:
>
> http://www.facebook.com/group.php?gid=2229924481&v=photos&so=30
>
> Slim pickings, if you ask me.
>
> -Igs

This is why I'm saying not to nix 2/1, but just to not use it as the
period. Then you can have intervals like 5/4 in your scales while
still not having a 2/1 in there or establishing it as the period.

-Mike

On Thu, Apr 21, 2011 at 9:35 PM, cityoftheasleep
<igliashon@...> wrote:
>
> There's also this page:
>
> http://www.nonoctave.com/tuning/twelfth.html
>
> Less obvious in some ways, more in others. See also here:
>
> http://www.facebook.com/group.php?gid=2229924481&v=photos&so=30
>
> Slim pickings, if you ask me.

In fact, here you go. What about 1108.33 cents as a generator, 3/1 as
a period? Generates a 5L2s scale that's basically a stretched out
meantone, and there's 2:5:6 chords in there and 5:6:15 chords as well.
Since the 2:5:6 chords are 1/1 5/2 3/1, 2 is implicit in the math
since you have an interval like 5/2 that you're considering to exist
at all. However, you still don't have an octave in this scale, nor is
2/1 the period.

This is why I'm saying - if you want to explore nonoctave scales, it
doesn't make sense to limit yourself to subgroups that don't have 2.
You can have 2 involved mathematically while still constructing scales
that don't have 2/1 in them and don't use 2/1 as the period.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> > Slim pickings, if you ask me.
> >
> > -Igs
>
> This is why I'm saying not to nix 2/1, but just to not use it as the
> period. Then you can have intervals like 5/4 in your scales while
> still not having a 2/1 in there or establishing it as the period.
>
> -Mike
>

I was more referring to the ED3's on nonoctave.com. 13 really does nail it pretty hard, nothing even comes close until you get to 30 and 32. 30 is basically 19-EDO with a slight stretch, I've looked at it before and it's kind of interesting as a BP temperament. But as far as MOS scales, it doesn't seem to have anything better to add than you'd find in 13 (although generally-speaking, melodies will probably sound better in 30-ED3 because of improved pentachordal concordance...only thing is the pseudo-octave is really in-tune, and that could be problematic.

-Igs

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> In fact, here you go. What about 1108.33 cents as a generator, 3/1 as
> a period? Generates a 5L2s scale that's basically a stretched out
> meantone, and there's 2:5:6 chords in there and 5:6:15 chords as well.
> Since the 2:5:6 chords are 1/1 5/2 3/1, 2 is implicit in the math
> since you have an interval like 5/2 that you're considering to exist
> at all. However, you still don't have an octave in this scale, nor is
> 2/1 the period.

Nor is there a half-decent 3:5:7 chord. I thought the point was to find a scale with the same basis as BP but that worked better?

> This is why I'm saying - if you want to explore nonoctave scales, it
> doesn't make sense to limit yourself to subgroups that don't have 2.
> You can have 2 involved mathematically while still constructing scales
> that don't have 2/1 in them and don't use 2/1 as the period.

Insofar as I've explored the territory, it's been in the form of looking at MOS's in ED3's, whether or not ratios of 2 are approximated hasn't really been a consideration. I really just haven't found any scales that give much in the way of 3:5:7 triads...it all keeps pointing back to Lambda, but Lambda ain't ever gonna do what Bohlen and Pierce thought it should. But by all means keep fishing. I haven't done any exhaustive searching.

-Igs

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> I can't tell if this is the same one you found, but Graham's 3.5.7.11
> winner seems to be
>
> http://x31eq.com/cgi-bin/rt.cgi?ets=7_12&limit=3_5_7_11&key=6_-3_-2_1_5_0_1&error=5.0

There's no "the winner" in Cangwu World, but this is indeed the temperament tempering out 245/243 and 1331/1323.

> Is it just coincidence that this happens to be the tritave equivalent
> of suprapyth, in that it forms MOS's at 5, 7, 12, 17...?

I get that it has MOS at 5, 7, 12, 17, 22, 39, 56, 95... but these are MOS inside of 3, not 2, so I don't see any real connection to superpyth.

On Thu, Apr 21, 2011 at 10:08 PM, cityoftheasleep
<igliashon@...> wrote:
>
> I was more referring to the ED3's on nonoctave.com. 13 really does nail it pretty hard, nothing even comes close until you get to 30 and 32. 30 is basically 19-EDO with a slight stretch, I've looked at it before and it's kind of interesting as a BP temperament. But as far as MOS scales, it doesn't seem to have anything better to add than you'd find in 13 (although generally-speaking, melodies will probably sound better in 30-ED3 because of improved pentachordal concordance...only thing is the pseudo-octave is really in-tune, and that could be problematic.

OK, I'll kick things off:

5-ED3 = Magic temperament
15-ED3 = I think you see where this is going
5L5s in 15-ED3 = MagicblackBPwood

There are 1:3:5 triads in abundance in the above scale, and the
pattern is the same as blackwood, which since you're familiar with
should be pretty trippy. Try 1:3:5:9 triads and transposing them
decatonically up and down the scale.

6-ED3 = Hanson temperament
18-ED3 = Your powerful brain should be working now
6L6s in 18-ED3 = HansonHexBPSomething

There are 2:5:6:11 tetrads all over this scale, and it's a beautiful
sound. You could chop it off at 2/1 as well to get one of those
sub-periodic scales I've been talking about on tuning-math, which
would be some kind of Hanson MODMOS. (Although some tweaking would be
necessary, I think). You won't find this scale at all if you're
ignoring 2, however, because you need 2 for 2:5:6:11.

-Mike

On Thu, Apr 21, 2011 at 10:26 PM, Mike Battaglia <battaglia01@...> wrote:
>
> OK, I'll kick things off:

Also, check out 4L4s, the 1 2 1 2 1 2 1 2 mode of 12-ED3 - that's
another fantastic scale, with 2:5:6 chords everywhere. It's basically
blackwood, from 15-TET, kind of.

-Mike

> perhaps I can shed some light on non-octave scales and interval equivalence.

Yes, you can. Igs, you freaking rock. Thanks very much for taking the time.

I don't have BP instruments, but I can still try to use the tritave as an equivalence if I use 19-ED3, which divides the tritave into approximately 100-cent steps. And for convenience, I think I'll temper the 3/1 to exactly 1900 cents. Kinda like what Mike Battaglia was talking about with various ED3 temperaments, only, you know, cheating.

Scala generates MOSs for me, and there are 19 interesting ones that don't have the octave in them, e.g., the MOS with generator 15 and 5L+4S:

0
300
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I mean, what the hell, I may never hear the tritave as equivalent, but this is a different approach from what I'm doing now, and I have instruments that are already capable of playing in the scale. And part of the point is to try new things, right?

Thanks for the discussion, all.

Regards,
Jake

> Scala generates MOSs for me, and there are 19 interesting ones that don't
> have the octave in them, e.g., the MOS with generator 15 and 5L+4S:

Actually, that one was a boneheaded one to pick -- octaves are all over the place in it. Here are the ones without octaves in them, all 5- and 7-note scales. Some of them actually sound kind of interesting.

0
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Igs wrote:

> Insofar as I've explored the territory, it's been in the form
> of looking at MOS's in ED3's, whether or not ratios of 2 are
> approximated hasn't really been a consideration. I really just
> haven't found any scales that give much in the way of 3:5:7
> triads...it all keeps pointing back to Lambda, but Lambda ain't
> ever gonna do what Bohlen and Pierce thought it should.

I think lambda is due to Bohlen alone. It gives six 3:5:7
triads in 9 notes/tritave, in JI. That's not too shabby if you
ask me. Where does it fall short for you?

> But by all means keep fishing. I haven't done any exhaustive
> searching.

Stretched 8-ET (val <13 19 23|) would be a good place to start.
Also check MOS in 27-ET (using my spreadsheet perhaps) using
the best approx in 27 but keeping in mind the accuracy will
improve once stretched.

-Carl

Not trying to cap this BP spill - it's good stuff - just making a minor
update: This pipe sounds very good, and the 2nd harmonic is really easy to
get, but I haven't made any holes. I'm thinking I may go for 14-TET, since I
wanted to look at it more anyhow.

I wrote:
> Stretched 8-ET (val <13 19 23|) would be a good place to start.

Derp, that's BP, sorry. -C.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> 5-ED3 = Magic temperament
> 15-ED3 = I think you see where this is going
> 5L5s in 15-ED3 = MagicblackBPwood
>
> There are 1:3:5 triads in abundance in the above scale, and the
> pattern is the same as blackwood, which since you're familiar with
> should be pretty trippy. Try 1:3:5:9 triads and transposing them
> decatonically up and down the scale.

You could do this in 19-EDO, too, more or less. 15-ED3 is basically the Negri generator as a rank-1 temperament. But I thought we were looking for 3:5:7 triads?

> 6-ED3 = Hanson temperament
> 18-ED3 = Your powerful brain should be working now
> 6L6s in 18-ED3 = HansonHexBPSomething

Eh, not too thrilled with this one.

Again, this seems like a tangent. I thought we were supposed to find something better to achieve 3:5:7 harmony in a 3/1 period? Or are you giving up on that?

Anyway, this is missing one of the main points I made for why I don't think BP works as a meantone analogue: "root tonality ambiguity" (for lack of a better term). One of the defining features of meantone, as I see it, is that playing the same step-pattern (i.e. 1-3-5) on each root produces *either* an otonal or an utonal triad. Same deal with Blackwood, Porcupine[7] (though not 8), Pajara[10], and Mavila[7] (among other noteworthy scales). Not saying that's important in and of itself, just that it's one of the defining features of meantone. So if we wanted to find a tuning system where we have a 1:3 instead of a 1:2, and triads that approximate 3:5:7 instead of 4:5:6, but everything else works pretty much as it would in meantone, well, we currently don't have that and it may be impossible.

-Igs

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

> I think lambda is due to Bohlen alone. It gives six 3:5:7
> triads in 9 notes/tritave, in JI. That's not too shabby if you
> ask me. Where does it fall short for you?

Did you read my initial post on the subject? I gave quite a detailed explanation.

-Igs

Gene wrote:

> > Okay, so we think we hear octaves as "equivalent", because
> > we're used to thinking in note-names that treat them as such.
>
> Do you have evidence this is so--what do people unfamiliar with
> note-names say?

I dunno about 3:1, but the King of ear training, David L Burge,
says that fifths and octaves are the intervals most commonly
mixed up by beginners.

Paul Erlich has pointed out that when amateurs sing in a group,
they sometimes sing at a fifth or fourth when they mean to sing
in unison.

The BP music of Charles Carpenter sounds perfectly serviceable
to me. He has three tracks from his album Splat! up here

http://www.charlescarpenter.net/discography.html

-Carl

On Fri, Apr 22, 2011 at 12:31 AM, cityoftheasleep
<igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> You could do this in 19-EDO, too, more or less. 15-ED3 is basically the Negri generator as a rank-1 temperament. But I thought we were looking for 3:5:7 triads?

I thought we were looking for usable harmonies that repeat at the tritave?

> > 6-ED3 = Hanson temperament
> > 18-ED3 = Your powerful brain should be working now
> > 6L6s in 18-ED3 = HansonHexBPSomething
>
> Eh, not too thrilled with this one.
>
> Again, this seems like a tangent. I thought we were supposed to find something better to achieve 3:5:7 harmony in a 3/1 period? Or are you giving up on that?
>
> Anyway, this is missing one of the main points I made for why I don't think BP works as a meantone analogue: "root tonality ambiguity" (for lack of a better term). One of the defining features of meantone, as I see it, is that playing the same step-pattern (i.e. 1-3-5) on each root produces *either* an otonal or an utonal triad. Same deal with Blackwood, Porcupine[7] (though not 8), Pajara[10], and Mavila[7] (among other noteworthy scales). Not saying that's important in and of itself, just that it's one of the defining features of meantone. So if we wanted to find a tuning system where we have a 1:3 instead of a 1:2, and triads that approximate 3:5:7 instead of 4:5:6, but everything else works pretty much as it would in meantone, well, we currently don't have that and it may be impossible.

Well, since you're the king of subgroups around here, I thought you
might be down to use the same subgroup-based approach for scales that
repeat at the 3/1 instead of the 2/1. The 6L6s scale above has a bunch
of interesting harmonies in it, although they're not stereotypical BP
3:5:7 chords. But I say so what?

-Mike

--- "cityoftheasleep" <igliashon@...> wrote:

> > I think lambda is due to Bohlen alone. It gives six 3:5:7
> > triads in 9 notes/tritave, in JI. That's not too shabby if
> > you ask me. Where does it fall short for you?
>
> Did you read my initial post on the subject? I gave quite
> a detailed explanation.

Sorry, I'm reading these out of order, saving the longest
ones for... now.

Your first point is that 3.5.7 isn't as concordance as 2.3.5.
You yourself, and later Gene, point out that it's kind of
inevitable.

Your second point about there being more semitones, I don't
really buy.

Your third point I definitely buy. But xenharmonic scales
of any stripe that behave this way are hard to come by.

That brings us to:
> That's four roots that are tonally ambiguous.

It's a point, but I think chord coverage is more important,
and the scale is covered by 3:5:7 triads.

> On the third hand, we can look at 5:7:9 as a harmonic basis
> instead of 3:5:7

Didja know Heinz Bohlen developed a scale for that too?

!
Heinz Bohlen's 8-tone scale based on 5:7:9 and 2:1.
8
!
10/9
6/5
9/7
7/5
14/9
5/3
9/5
2/1
!

> In summary, while tritave equivalence and a 3:5:7 basic
> harmonic unit might be viable alternatives to the octave and
> a 4:5:6 harmonic unit in and of themselves, the diatonic scale
> and tonal structure that Bohlen and Pierce came up with to
> support these harmonic bases fails to be analogous to meantone
> harmony in many important ways,

I agree. Good analysis.

> That's also why I don't think it's "just" our conditioning
> that keeps us from being able to intelligize BP in ways
> analogous to meantone music.

This may be a bit strong for me. I think the Carpenter stuff
I linked to sounds intelligible and has a tonal/functional
logic to it that I like.

-Carl

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> Also, in the diatonic major scale, the two intervals that compose the
major triad (5/4 and 6/5) are in the same interval-class, and the major
and minor triads are an otonal-utonal pair. In BP Lambda, the two
intervals that form the main 3:5:7 triad are not in the same interval-
class: 5/3 is a BP "major fifth" and 7/5 is a BP 4th. In other words,
parallel tonal harmony doesn't work the way we'd expect it in BP.

Note that in BP pentatonic the same pattern of scale steps produces two
3:5:7 chords and two 1/1:7/5:7/3 chords in 4 out of 5 notes of the
scale. The scale has huge large steps around ~9/7 but you could
constantly modulate. It also works as a pentad.

Kalle

In response to your lovely comments here, a bit of a BP piece from some time ago. I was writing with live instruments in mind so it's a score-sketch, but then I realized how unlikely it is that the BP community, whose enthuiastic cooperation would be required in order to realize the score, would touch the thing with a ten-foot pole, and so it sits on a shelf.

http://soundcloud.com/cameron-bobro/runny-hunny-cbobro

The title at least is a keeper, as it jokingly refers to the tuning.

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> As a long-time friend and correspondent of XJ Scott, proprietor of the site nonoctave.com and author of the best microtuning software for the Mac OS ("Lil' Miss Scale Oven"), perhaps I can shed some light on non-octave scales and interval equivalence.
>
> Okay, so we think we hear octaves as "equivalent", because we're used to thinking in note-names that treat them as such. Really, we hear octaves as "distinct but similar"--"that's the same note, but higher (or lower)". One of the important features of the octave in music is that we can use octave-doubling in arrangement to produce timbral/textural "thickening" of parts, because the octave is such a simple interval that, when tuned pure, it allows two notes to fuse very strongly. This high degree of fusion--a pure octave is almost seamless, really--is what allows us to treat it as "equivalent".
>
> The next-simplest ratio after 2/1 is 3/1, and sure enough you can usually harmonize in parallel 3/1's to achieve a very similar effect as parallel 2/1's--textural thickening but no sensation of "added information" because the blend is so strong. With odd-harmonic-only timbres, this works especially well: if the partials of the tonic are 1:3:5:7:9:11:13:15:17:19 (etc.), the partials of a note 3/1 above it will be 3:9:15:21:27:33:39:45:51:57, which is really just like a stretched-out version of the octave relationship for full-harmonic timbres: 1:2:3:4:5:6:7:8:9:10 and 2:4:6:8:10:12:14:16:18:20. So while we may not (because of our conditioning) be able to here the tritave equivalence in BP, it should theoretically be possible to hear, because the tritave does retain *most* of the properties that make octave-equivalence work.
>
> Now, I like BP for xenharmonic purposes, but I think its progenitors actually thought that it should work "just as naturally" as 12-TET (or at least meantone)--i.e., it wasn't *supposed* to sound xenharmonic and alien and freaky and weird, it was just supposed to sound a bit "different". This is emphatically *not* the case, and I've got many ideas as to why.
>
> For starters, if you compare the Just BP "major scale" ("Lambda") with a Just major scale, something is pretty obvious:
>
> BP:
> 1/1 0 cents
> 25/21 301 cents
> 9/7 435 cents
> 7/5 583 cents
> 5/3 884 cents
> 9/5 1017 cents
> 15/7 1319 cents
> 7/3 1467 cents
> 25/9 1768 cents
> 3/1 1902 cents
>
> Diatonic Major:
> 1/1 0 cents
> 9/8 204 cents
> 5/4 386 cents
> 4/3 498 cents
> 3/2 702 cents
> 5/3 884 cents
> 15/8 1088 cents
> 2/1 1200 cents
>
> In the BP scale, most of the dyadic consonances are much less concordant than the dyadic consonances in the major scale, and the dissonances are more numerous and more discordant. There are also a lot of consonances to be found in the major scale besides the usual 4:5:6 and 10:12:15, because there is a greater number of consonant triads you can construct with a 2, 3, 5 basis than a 3, 5, 7 basis; that's basic math, really. So right off the bat, there's going to be less consonance in BP than in meantone.
>
> Scale-wise, there are more "semitones" in BP (Lambda is TssTsTsTs, the major scale is TTsTTTs), and this makes writing melodies more challenging, since there is a greater likelihood of two adjacent notes in the scale being discordant with one another.
>
> Also, in the diatonic major scale, the two intervals that compose the major triad (5/4 and 6/5) are in the same interval-class, and the major and minor triads are an otonal-utonal pair. In BP Lambda, the two intervals that form the main 3:5:7 triad are not in the same interval-class: 5/3 is a BP "major fifth" and 7/5 is a BP 4th. In other words, parallel tonal harmony doesn't work the way we'd expect it in BP. Also, this makes it possible to form either an otonal 3:5:7 triad (1/1-5/3-7/3) OR an utonal 1/(3:5:7) (1/1-7/5-7/3) triad on many of the same roots; if you look at the Lambda mode in terms of which degree forms an otonal (o), utonal (u), both (o/u), or neither (n) triad, it looks like this:
>
> I: o/u
> II: u
> III: o/u
> IV: n
> V: o/u
> VI: o
> VII: o/u
> VIII: o
> IX: u
>
> That's four roots that are tonally ambiguous. Also note that the utonal BP triad, when examined as consecutive harmonics, is 15:21:35, which is significantly more complex than the otonal triad of 3:5:7 (vs. 4:5:6 and 10:12:15 in "regular" meantone harmony). So in the harmonic entropy component of discordance, the BP utonal triad is probably going to sound quite a bit more discordant (it would be interesting to see 3HE values for the two pairs of triads for comparison).
>
> On the other hand, if we don't treat the utonal triad as the "minor" triad, but rather the one with the "minor" 5th of 75/49, the our minor chord now approximates harmonics 15:23:35, which isn't much worse than the utonal triad in complexity but is worse in terms of roughness, the ambiguity is gone and we now have minor triads on the II and IX degrees and unambiguous major triads on the others (except the IV), so that might be functionally a bit better...but it will be psychoacoustically worse, I suspect.
>
> On the third hand, we can look at 5:7:9 as a harmonic basis instead of 3:5:7; it's less concordant, but perhaps more familiar sounding (since it's not so wide). However, it doesn't improve tonal ambiguity (5 triads are still ambiguous) at all or lead to the two component intervals sharing a class, so again, it's not going function like meantone here either.
>
> In summary, while tritave equivalence and a 3:5:7 basic harmonic unit might be viable alternatives to the octave and a 4:5:6 harmonic unit in and of themselves, the diatonic scale and tonal structure that Bohlen and Pierce came up with to support these harmonic bases fails to be analogous to meantone harmony in many important ways, and that's probably why most music in BP sounds like something that came from deep space. That's also why I don't think it's "just" our conditioning that keeps us from being able to intelligize BP in ways analogous to meantone music.
>
> As an aside, I've noticed that the few pieces of music that *do* sound fairly intelligible in BP actually seem to ignore the basis of BP and run in other directions with the scale. For instance, I've noticed that Elaine Walker has a tendency to "meantone-ize" BP in her playing, treating 25/21 as 6/5, and making BP "minor" triads that are basically 5:6:15's, and occasionally using 75/49 as a pseudo-3/2 for root-movement in chord progressions.
>
> And this concludes my analysis of the Bohlen-Pierce scale.
>
> -Igliashon
>
> --- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@> wrote:
> >
> > This conversation confuses me.
> >
> > >> But what would count in your terms as hearing tritaves as equivalent?
> >
> > Why would one even want to hear tritaves as equivalent?
> >
> > From Elaine's site:
> > -----
> > It all started in the early 1970s when Heinz Bohlen became curious as
> > to why musicians always use the same 12 note per octave tuning. After
> > getting some unsatisfactory answers from musicians, he took it upon
> > himself to research music and tonality. It was finally the
> > understanding of combination tones that lead him to believe he could
> > use the same method that lead to the 12 tone equal temperament, to
> > devise another tuning from a different framework. He understood that
> > the 12 tone scale was based around the a major triad, the inversion,
> > and then filling in the gaps. So, he started with a new triad that was
> > very harmonically pure, but not contained within the normal 12 tone
> > framework. He made the stunning discovery that a scale of thirteen
> > almost equal steps within the framework of an octave and a fifth
> > (which John Pierce later dubbed as a tritave) contained this very pure
> > triad. He recognized that this tuning shared a duality with the
> > traditional Western 12 tone tuning, it had harmonic value, and that it
> > would be a valid compositional tool.
> > -----
> >
> > If I read that correctly, the "very harmonically pure" triad (I
> > presume that's the 3:5:7?) came first, and the BP scale came out of
> > it. But
> >
> > (a) was it even the goal of the scale inventors to create a non-octave
> > scale? and
> >
> > (b) was the goal of the scale to create something in which the first
> > and last notes *sound the same* the way an octave sounds the same?
> >
> > If the answer to (a) is "yes", then how would it be different from,
> > say, Carlos Beta? If I hear Beta, I don't expect to hear a perfect
> > fourth as "equivalent" to some other non-octave pitch. If I play 0
> > cents, then 498 cents, and then either 996 or 1698 cents, the 1698
> > will sound equivalent to the 498, and the 996 won't -- regardless of
> > how Carlos designed the scale. If we know our ears hear equivalence
> > between multiples of 1200 cents, but not for multiples of 498 cents or
> > 100 cents or other arbitrary numbers, why would it work for 1902
> > cents? Why would I even want it to?
> >
> > If the answer to (b) is "yes", that seems pretty nutty to me. It would
> > be like training people to see blue as green -- even if it's possible,
> > what's the point?
> >
> > As Gene said, the BP scale forces you to be xenharmonic, but I don't
> > see how making a non-octave scale suddenly makes non-octave intervals
> > "equivalent".
> >
> > Regards,
> > Jake
> >
>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> I thought we were looking for usable harmonies that repeat at the tritave?

Well if that's the case, sonny-jim, the sky's the limit! As you've noticed, those aren't at all hard to find.

> Well, since you're the king of subgroups around here, I thought you
> might be down to use the same subgroup-based approach for scales that
> repeat at the 3/1 instead of the 2/1. The 6L6s scale above has a bunch
> of interesting harmonies in it, although they're not stereotypical BP
> 3:5:7 chords. But I say so what?

So nothing. Tritave-repeating scales with good harmony are a cinch. There's almost more of them than there aren't. What the "holy grail" would be, however, is a scale that out-BP's BP, i.e. uses the same basis as BP but actually manages to work something like a diatonic scale.

-Igs

--- In tuning@yahoogroups.com, "lobawad" <lobawad@...> wrote:
> http://soundcloud.com/cameron-bobro/runny-hunny-cbobro
>
> The title at least is a keeper, as it jokingly refers to the tuning.
>

This sounds pretty natural, actually. But then, so does 8-EDO to me (there's an 8-EDO piece on my next album, actually, which should be out in a few more months). I don't get what the title says about the tuning, though.

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, "lobawad" <lobawad@> wrote:
> > http://soundcloud.com/cameron-bobro/runny-hunny-cbobro
> >
> > The title at least is a keeper, as it jokingly refers to the tuning.
> >
>
> This sounds pretty natural, actually. But then, so does 8-EDO to me (there's an 8-EDO piece on my next album, actually, which should be out in a few more months). I don't get what the title says about the tuning, though.
>
> -Igs
>

I suspect that BP is actually no different from some kind of Fokker block or something. If you play it that way- a sensible array of Just intervals- rather than trying to squeeze some kind of pseudo-meantone thing out of it, it's going to sound as natural as any other decent Just array of about that complexity.

Didn't you yourself once say that you find yourself using the whole tuning when you're doing BP?

As far as the name, I'll wait a bit and see if anyone else guesses the joke.

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

> http://www.charlescarpenter.net/discography.html
>
> -Carl

Wow, that's awesome! It sounds like music written entirely in the diminished scale, but somewhat more xenharmonic. I think it's time I tested my own criticisms of BP by trying to compose in it myself. Perhaps the problem with music in BP has had nothing to do with the tuning, but with the composers? This Charles Carpenter guy is doing stuff with it I've never heard anyone do before, and it sounds pretty great.

-Igs

On Fri, Apr 22, 2011 at 11:52 AM, cityoftheasleep
<igliashon@...> wrote:
> > Well, since you're the king of subgroups around here, I thought you
> > might be down to use the same subgroup-based approach for scales that
> > repeat at the 3/1 instead of the 2/1. The 6L6s scale above has a bunch
> > of interesting harmonies in it, although they're not stereotypical BP
> > 3:5:7 chords. But I say so what?
>
> So nothing. Tritave-repeating scales with good harmony are a cinch. There's almost more of them than there aren't. What the "holy grail" would be, however, is a scale that out-BP's BP, i.e. uses the same basis as BP but actually manages to work something like a diatonic scale.

That's fine, but what I'm getting at is that you don't have to exclude
2 from the subgroup to do it. In fact, excluding 2 might be harmful -
as you pointed out in your analysis, lots of BP lambda intervals are
pretty discordant, whereas the diatonic scale is concordant almost
everywhere. If you exclude 2, then intervals like 5/4, 6/5, 7/4, etc
will never appear in your scales, because they involve powers of 2. If
you want to create a tritave scale based around 3:5:7, which is really
1:3:5:7, it would be nice to fill up the space with as many concordant
intervals as possible, which means NOT excluding nice sounding
intervals like the above. Excluding all factors of two from the
subgroup is like taking a sledgehammer to the whole thing, it's a bit
too harsh.

Maybe this means you'll end up with 2:6:7:10 instead of 1:3:5:7, but
have a better scale than Lambda. Is that acceptable? Now if I could
only figure out how to use Graham's temperament finder to explore the
7-limit with period at 3/1...

-Mike

Mike Battaglia <battaglia01@...> wrote:

> Maybe this means you'll end up with 2:6:7:10 instead of
> 1:3:5:7, but have a better scale than Lambda. Is that
> acceptable? Now if I could only figure out how to use
> Graham's temperament finder to explore the 7-limit with
> period at 3/1...

3.2.5.7

That's exactly how I regard it. look here:

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

On Fri, Apr 22, 2011 at 9:53 AM, lobawad <lobawad@...> wrote:

>
>
> I suspect that BP is actually no different from some kind of Fokker block
> or something. If you play it that way- a sensible array of Just intervals-
> rather than trying to squeeze some kind of pseudo-meantone thing out of it,
> it's going to sound as natural as any other decent Just array of about that
> complexity.
>
>

Say, you're Kees van Prooijen! It's good to have you here. For those who don't know, KvP invented what we call "BP" in the same era as B and P did, only later did everyone find out about these three synchonicities. And of course the tuning is kind of an prototypical/ancenstral? entity for the tuning work done here.

I'd be honored if you'd take a listen to the piece I posted and say whether it bears any resemblance to what you imagined back then. I ignored scales altogether and basically based the whole thing on interlocking motifs, any scales would then be theoretical constructs derived afterward.

--- In tuning@yahoogroups.com, Kees van Prooijen <keesvp@...> wrote:
>
> That's exactly how I regard it. look here:
>
> http://www.kees.cc/music/scale13/scale13.html
>
>
>
> On Fri, Apr 22, 2011 at 9:53 AM, lobawad <lobawad@...> wrote:
>
> >
> >
> > I suspect that BP is actually no different from some kind of Fokker block
> > or something. If you play it that way- a sensible array of Just intervals-
> > rather than trying to squeeze some kind of pseudo-meantone thing out of it,
> > it's going to sound as natural as any other decent Just array of about that
> > complexity.
> >
> >
>

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> I don't get what the title says about the tuning, though.

I don't get why I've tried twice at different times and can't get the player to play music. Maybe a different browser...

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> That's fine, but what I'm getting at is that you don't have to exclude
> 2 from the subgroup to do it.

I never said you did, though I have no idea what Bohlen or Pierce would have thought of that idea. Maybe Kees can shed some light on the idea.

> In fact, excluding 2 might be harmful -
> as you pointed out in your analysis, lots of BP lambda intervals are
> pretty discordant, whereas the diatonic scale is concordant almost
> everywhere. If you exclude 2, then intervals like 5/4, 6/5, 7/4, etc
> will never appear in your scales, because they involve powers of 2. If
> you want to create a tritave scale based around 3:5:7, which is really
> 1:3:5:7, it would be nice to fill up the space with as many concordant
> intervals as possible, which means NOT excluding nice sounding
> intervals like the above. Excluding all factors of two from the
> subgroup is like taking a sledgehammer to the whole thing, it's a bit
> too harsh.

I couldn't agree more.

> Maybe this means you'll end up with 2:6:7:10 instead of 1:3:5:7, but
> have a better scale than Lambda. Is that acceptable?

Search me! I wonder if there's anywhere to find the original publications of Bohlen and/or Pierce where they describe their motivations in depth? Note that I never said I didn't *like* Lambda, just that I don't think it is analogous to meantone in any real functional sense. It's very analogous to Semaphore/Godzilla, in terms of the "root-tonality ambiguity", and while that makes it hard to do meantone-like things, it makes it easy to do lots of other cool things that you can't do in meantone, like ascend in parallel otonal harmony and then descend in parallel utonal harmony, or make split-"middle interval" chords that are both otonal and utonal, but have no critical-band dissonance (since the otonal "middle interval" and the utonal "middle interval" are much further apart than a semitone).

Also, you can approach BP like a compressed 8-EDO with better harmonies, and do 8-EDO-ish things with it that randomly sound a little smoother. Take the 2 2 1 2 1 Father pentatonic from 8-EDO, tack on an extra 2 2 1 "tetrachord" to fill out the 3/1, and voila, you have something that's pretty neat melodically. Or treat every other note of it like a compressed 4-EDO and do a kind of 2 2 2 2 2 2 1 scale, much like every other note of 13-EDO is sort of a compressed 6-EDO. There is all kinds of neat stuff you could probably do with 13-ED3, and one of these days I'll get around to exploring it; the only point was that it's not going to behave like meantone.

-Igs

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Now if I could
> only figure out how to use Graham's temperament finder to explore the
> 7-limit with period at 3/1...

The results of the temperament finder, aside from the way the mapping is given, are independent of whether you think of 2 or 3 as the period, so the thing to do is fiddle with the mapping (by elementary row operations) to make 3 a period. Of course sometimes, as with meantone, you don't even need to fiddle.

On Fri, Apr 22, 2011 at 1:05 PM, Graham Breed <gbreed@...> wrote:
>
> Mike Battaglia <battaglia01@...> wrote:
>
> > Maybe this means you'll end up with 2:6:7:10 instead of
> > 1:3:5:7, but have a better scale than Lambda. Is that
> > acceptable? Now if I could only figure out how to use
> > Graham's temperament finder to explore the 7-limit with
> > period at 3/1...
>
> 3.2.5.7

There you go! Surprises everywhere.

-Mike

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Mike Battaglia <battaglia01@...> wrote:
>
> > Maybe this means you'll end up with 2:6:7:10 instead of
> > 1:3:5:7, but have a better scale than Lambda. Is that
> > acceptable? Now if I could only figure out how to use
> > Graham's temperament finder to explore the 7-limit with
> > period at 3/1...
>
> 3.2.5.7

Nice and simple! But the pure-octaves tuning appears to be computed incorrectly, as it should be "pure whatever-the-first-on-the-list-is" tuning.

On Fri, Apr 22, 2011 at 2:06 PM, cityoftheasleep
<igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > That's fine, but what I'm getting at is that you don't have to exclude
> > 2 from the subgroup to do it.
>
> I never said you did, though I have no idea what Bohlen or Pierce would have thought of that idea. Maybe Kees can shed some light on the idea.

Most people seem to be already using the scale this way when they play
really stretched out major and minor chords; e.g. they treat them like
2:5:6, with 6 containing a factor of 2.

> or make split-"middle interval" chords that are both otonal and utonal, but have no critical-band dissonance (since the otonal "middle interval" and the utonal "middle interval" are much further apart than a semitone).

Oh, you mean ASSes?

> Also, you can approach BP like a compressed 8-EDO with better harmonies, and do 8-EDO-ish things with it that randomly sound a little smoother. Take the 2 2 1 2 1 Father pentatonic from 8-EDO, tack on an extra 2 2 1 "tetrachord" to fill out the 3/1, and voila, you have something that's pretty neat melodically. Or treat every other note of it like a compressed 4-EDO and do a kind of 2 2 2 2 2 2 1 scale, much like every other note of 13-EDO is sort of a compressed 6-EDO. There is all kinds of neat stuff you could probably do with 13-ED3, and one of these days I'll get around to exploring it; the only point was that it's not going to behave like meantone.

Well, a preliminary search of the 3.2.5.7 subgroup on the temperament
finder yields the winner as...

http://x31eq.com/cgi-bin/rt.cgi?ets=8_11&limit=3_2_5_7&key=1_-4_-13_1_0_4_10&error=5.0

meantone. But, it's the tritave version of meantone, which gives the
whole thing a kaleidoscopic feel to it. It forms MOS's at 8 and 11
notes, these being tritavized versions of meantone[5] and meantone[7]
respectively. But, it doesn't have much 7 involved, so we move next to

http://x31eq.com/cgi-bin/rt.cgi?ets=30_35&limit=3_2_5_7&key=1_2_-1_5_0_1_12&error=5.0

magic, which was already mentioned, and also doesn't have much 7. So
next, we get

http://x31eq.com/cgi-bin/rt.cgi?ets=19_35&limit=3_2_5_7&key=2_11_12_1_0_-2_-2&error=5.0

Pajara - surely that won't let us down! And it doesn't, forming a
tritave MOS at 13 notes and 16 notes, which are the equivalents of
pajara[8] and pajara[16]. There are lots of 1:3:5:7's and
octave-equivalents everywhere, so maybe that's good. The 16-note
really shines as furthermore, they turn into utonal tetrads, which is
also good. 4:5:7:12 seems to be the tetrad of choice. Maybe 16 notes
is a bit too many for you, but since we're dealing with tritaves now
(of which only four covers the entire span of human hearing), maybe
that isn't so bad.

Then there's negri, which splits the tritave into two 950 cent
intervals approximating 7/4:

http://x31eq.com/cgi-bin/rt.cgi?ets=30_16&limit=3_2_5_7&key=2_7_4_2_1_2_3&error=5.0

The 16-note MOS makes 4:7:10:12 is a good choice.

Lastly, there's semaphore, which forms a beautifully kaleidoscopic 14-note MOS:

http://x31eq.com/cgi-bin/rt.cgi?ets=30_8&limit=3_2_5_7&key=1_-4_2_2_0_8_1&error=5.0

There are only two 1:3:5:7 tetrads in the MOS, but then they turn into
things like major seventh chords, giving the whole scale a very
pleasantly "tonal" feel (transpose them up and down the scale and
you'll see what I mean.

Thus ends my preliminary SITREP.

-Mike

On Fri, Apr 22, 2011 at 2:26 PM, genewardsmith
<genewardsmith@...> wrote:
>
> Nice and simple! But the pure-octaves tuning appears to be computed incorrectly, as it should be "pure whatever-the-first-on-the-list-is" tuning.

Yeah, and the scala files it spits out are also octave-equivalent
versions as well, but you can't have everything I suppose.

-Mike

My original idea was pretty much purely theoretical. Along the lines
mentioned here, if you have an odd spectrum, 13 equal would be just as
natural as 12 equal for an ordinary spectrum.

Cameron, I think your piece sound pretty natural to me.

I dropped Heinz a line to take a look at this thread. Pierce never showed
any interest to communicate with either of us. And now that Max Mathews died
(yesterday!) there's probably no source of information in that corner
anymore.

I do remember Heinz saying he actually liked ordinary (even) spectra for BP
music. His words were something like: "it gives it a bite" :-)

Kees

Sad about Max Matthews, a stellar figure in digital music- there are a lot of nice tributes in the Csound mailing list.

Thanks for listening! People in general, in my experience, are much more into moods and associations than anything else, and the "civilian" response seems to be almost always based on that, as long as the piece sounds like it was done on purpose. My seven-year old flinches at what I'd call clumsy writing, in any kind of music, but otherwise is open to anything unless its truly scary (loves "dark metal", that's not scary in the slightest).

I also like full spectra with the "tritave". Have you heard the specially made clarinets, though? Love to have one of those!

--- In tuning@yahoogroups.com, Kees van Prooijen <keesvp@...> wrote:
>
> My original idea was pretty much purely theoretical. Along the lines
> mentioned here, if you have an odd spectrum, 13 equal would be just as
> natural as 12 equal for an ordinary spectrum.
>
> Cameron, I think your piece sound pretty natural to me.
>
> I dropped Heinz a line to take a look at this thread. Pierce never showed
> any interest to communicate with either of us. And now that Max Mathews died
> (yesterday!) there's probably no source of information in that corner
> anymore.
>
> I do remember Heinz saying he actually liked ordinary (even) spectra for BP
> music. His words were something like: "it gives it a bite" :-)
>
> Kees
>

--- "cityoftheasleep" <igliashon@...> wrote:

> > http://www.charlescarpenter.net/discography.html

> Wow, that's awesome! It sounds like music written entirely
> in the diminished scale, but somewhat more xenharmonic. I
> think it's time I tested my own criticisms of BP by trying to
> compose in it myself. Perhaps the problem with music in BP
> has had nothing to do with the tuning, but with the composers?
> This Charles Carpenter guy is doing stuff with it I've never
> heard anyone do before, and it sounds pretty great.

Yeah, his is the only stuff I've heard that uses it like
the regular temperament it was intended to be, rather than
just some 'I want to be weird' scale.

I notice his albums seem to be a victim of the end of
the CD era. The tracks can't even be previewed on Amazon.
And Carpenter seems to have repositioned himself as a
12-ET film/ad composer.

-Carl

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Well, a preliminary search of the 3.2.5.7 subgroup on the temperament
> finder yields the winner as...
>
> http://x31eq.com/cgi-bin/rt.cgi?ets=8_11&limit=3_2_5_7&key=1_-4_-13_1_0_4_10&error=5.0
>
> meantone.

The 3.2.5.7 "winners" should be precisely the same as the 2.3.5.7 ones. Decrease your error number and miracle should creep in there. Go even farther, and the dreaded ennealimmal pops up.

I'm trying now to work with metal. Since running across some nice, shiny
piping at the used hardware store, I'm wanting to make something in phi. It
just doesn't seem right to use one of the "alloyed" rings of the golden
horogram, so I'm thinking it best to use a scale in ring 13, probably made
of 6 or 7 notes (also, there is no ring 6 or 7 in the horogram). Is there a
sort of "standard" scale that might be suggested?