Pajara tuning examples - cast your vote

If anyone is interested, I'd appreciate your listening to some pajara
tuning comparisons and offering your opinion as to which you think is
best:

/tuning/topicId_67957.html#68285

--George

First of all, I find it incredibly hard to compare the JI and the ET versions. They sound really really different and good in their own ways. I wonder what the JI would sound like if you deliberately introduced some random tuning errors so that some slow beating would be introduced.

Group A

JI7a -- the best in group A, the utonal chord feels just right.
JI17a -- something sounds wrong, like losing pitch height

22a -- good, beats are pleasant, fifths a bit weird
34a -- gloomy
56a -- the slow beating is annoying
34WTa -- good
12ab -- not too bad actually, but maj 3rd is too sharp in final chord

Group B -- sounds better than group A overall

JI9b -- very good

22b -- really bad
34b -- okay
56b -- not relaxing
34WTb -- very good
12ab -- bad

Group C

JI17c -- good, very emotional
22c -- bad. inharmonic, like a gong/bell
34c -- uneven. some chords very close to JI others very far
56c -- bad. inharmonic, like a gong/bell
34WTc -- same problem as 34c, also has gong/bell sound

I'd take the JI versions if I could, but I know that's against the rules here so I'll pick 34WT as next best. I'd be interested to hear some meantone versions if the progressions are not impossible in meantone.

--Magnus

On Thu, 30 Nov 2006, George D. Secor wrote:

>
> If anyone is interested, I'd appreciate your listening to some pajara
> tuning comparisons and offering your opinion as to which you think is
> best:
>
> /tuning/topicId_67957.html#68285
>
> --George
>
>
>

--- In MakeMicroMusic@yahoogroups.com, "George D. Secor" <gdsecor@...>
wrote:
>
> If anyone is interested, I'd appreciate your listening to some pajara
> tuning comparisons and offering your opinion as to which you think is
> best:
>
> /tuning/topicId_67957.html#68285

Results of the survey are summarized here:

/tuning/topicId_67957.html#68521

My thanks to the 6 participants.

--George

What is Pajara? I've searched in the ATG, it's like trying to translate
a word from Chinese and being referred to a Mongolian dictionary.

-Cameron Bobro

Cameron Bobro wrote:
> What is Pajara? I've searched in the ATG, it's like trying to translate
> a word from Chinese and being referred to a Mongolian dictionary.

If that's even harder than using a Chinese dictionary, you really are in trouble :-S

Pajara's a rank 2 regular temperament with equal temperaments of 10, 12, 22, 34, 56, etc. notes to the octave as special cases.

Graham

--- In MakeMicroMusic@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Cameron Bobro wrote:
> > What is Pajara? I've searched in the ATG, it's like trying to
translate
> > a word from Chinese and being referred to a Mongolian dictionary.
>
> If that's even harder than using a Chinese dictionary, you really
are in
> trouble :-S
>
> Pajara's a rank 2 regular temperament with equal temperaments of 10,
>12,
> 22, 34, 56, etc. notes to the octave as special cases.
>
>
> Graham
>

Thanks! That's something someone outside the tuning group can
understand.

What are the tempering intervals/amounts, and on which
generating intervals, and why and how where they determined?

-Cameron Bobro

--- In MakeMicroMusic@yahoogroups.com, "Cameron Bobro"
<misterbobro@...> wrote:

> > Pajara's a rank 2 regular temperament with equal temperaments of
10,
> >12,
> > 22, 34, 56, etc. notes to the octave as special cases.
> >
> >
> > Graham
> >
>
> Thanks! That's something someone outside the tuning group can
> understand.

It is? I thought for sure Graham was going to totally snow you with
that.

Another way of saying it is that pajara tempers out both 50/49 and
64/63. Because 50/49 is tempered out, 7/5 is equated to 10/7 as a
neutral tritone of 600 cents, which is the period. Because 64/63 is
tempered out, the dominant seventh chord is equated with the ottonal
tetrad. The generator can be taken as a sharp fifth, or the semitone
you get by subtracting 600 cents from the sharp fifth.

> What are the tempering intervals/amounts, and on which
> generating intervals, and why and how where they determined?

That's what George has been on about. Paul Erlich likes 22-equal,
which has a very sharp fifth of 709 cents. This is nice in practice
since 22 isn't a large number, and it makes sense in theory if your
theory involves weighting all the 7-limit consonaces the same,
especially if you don't worry about the 9-limit. However, if you
think the 5-limit and its triads are more important, which apparently
most people's ears do, then you might want to weight it more heavily.
You then might get the 56 or even, pushing it, the 34 equal tuning.
The 700 cent fifth of 12-et, however, is far too flat a pajara fifth
however you slice it. But since we are familiar with 12, many aspects
of pajara seem familiar also.

>That's what George has been on about. Paul Erlich likes 22-equal,
>which has a very sharp fifth of 709 cents. This is nice in practice
>since 22 isn't a large number, and it makes sense in theory if your
>theory involves weighting all the 7-limit consonaces the same,
>especially if you don't worry about the 9-limit. However, if you
>think the 5-limit and its triads are more important, which apparently
>most people's ears do, then you might want to weight it more heavily.
>You then might get the 56 or even, pushing it, the 34 equal tuning.
>The 700 cent fifth of 12-et, however, is far too flat a pajara fifth
>however you slice it. But since we are familiar with 12, many aspects
>of pajara seem familiar also.

I should point out that Paul's TOP tuning uses a fifth of 705 cents
and an octave of 1197 cents.

-Carl

Cameron Bobro wrote:
> --- In MakeMicroMusic@yahoogroups.com, Graham Breed <gbreed@...> wrote:

>> Pajara's a rank 2 regular temperament with equal temperaments of 10, >> 12, >> 22, 34, 56, etc. notes to the octave as special cases.
> > Thanks! That's something someone outside the tuning group can > understand. > > What are the tempering intervals/amounts, and on which > generating intervals, and why and how where they determined?

One generator (the period) has to be a half-octave. You know this because each equal temperament has an even number of notes to the octave.

The other generator is a perfect fifth which approximates 3:2. It's typically sharp of JI. The tempering amount is what's under discussion, but somewhere around 56 seems to be preferred, and anywhere from 22 to 34 is acceptable given the context.

Because of the symmetry, you can also have generators of a semitone, fourth, etc.

Graham

--- In MakeMicroMusic@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:

> It is? I thought for sure Graham was going to totally snow you
>with that.

With such poor judgement of how others think, I sure hope you don't
vote. :-) Hahaha!

His answer was quite straightforward. A search for the term "pajara"
at the Alternative Tuning Group, looking for the oldest references,
immediately reveals, first of all, arguments, then branches like a
crazed ivy into a hall of terminology mirrors, sporting a .gif image
completely illegible on my computer and wearing chains of broken
links.

It would be very easy to view the tuning groups here as functioning
to obscure and bog down tuning developments in the wide world, or as
lost in a world of empty games whose irrelevance to actual practice
is conveniently obscured by masonic lingo.

In fact, judgements along these lines have been passed by those who
lurk and don't post.

Now, I know for certain there is far more going on here than empty
games, because I discovered the groups after doing quite a bit of
reinventing the wheel, alone in the dark, and have tested things by
the brute method of listening to laypeople sing along, completely
unaware that anything "xeno" is going on at all.

Anyway, on with the show.

>Because 64/63 is tempered out, the dominant seventh chord is
>equated with the ottonal tetrad.

Which sounds like a trait common to many tunings- doesn't this tame
the dominant?

> That's what George has been on about. Paul Erlich likes 22-equal,
> which has a very sharp fifth of 709 cents.

The "sharpness" of a fifth (or any other interval, as long as we're
not working with pure sines) isn't completely bound to its size. For
example, there is a rational fifth around 709 cents which has a
darker and more traditionally fifth-like character than other fifths
of around, say, 707 cents. When I get time, I'll finish a little
Csound program that runs through high fifths and you can hear for
yourself. (I believe that you posted a list of high fifths, have to
find that). This also illustrates that "justifying" and rating
intervals by sheer proximity to JI is an insufficient approach once
you've gone beyond a cent or two.

> But since we are familiar with 12, many aspects
> of pajara seem familiar also.

In the case of 17s at or near equal, there are also difference tones
which land almost exactly on 12-EDO intervals, which I find amusing
because when I tried noodling a bit in 12-EDO yesterday, for the
first time in months; the M3 made me cringe, yet the 401 cent
difference tone which appears in my main tuning works very nicely as
a harmony because it is subsumed into the harmonics and isn't
pretending to be a "third".

--- In MakeMicroMusic@yahoogroups.com, Graham Breed <gbreed@...>
wrote:

> One generator (the period) has to be a half-octave. You know this
> because each equal temperament has an even number of notes to the
>octave.
> The other generator is a perfect fifth which approximates 3:2.
>It's
> typically sharp of JI. The tempering amount is what's under
>discussion,
> but somewhere around 56 seems to be preferred, and anywhere from
>22 to 34 is acceptable given the context.

Does "pajara" refer to only one tempering of 3/2? In a well-
temperament like Secor's there are patently more than one version of
3/2, and I see (and hear) nothing wrong with using in the same
tuning several tempered 3/2's which would usually be associated with
other octave divisions.

-Cameron Bobro

--- In MakeMicroMusic@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> I should point out that Paul's TOP tuning uses a fifth of 705 cents
> and an octave of 1197 cents.
>
> -Carl

Which should work out very nicely, as, for example, the difference
tones of the tempered 3/2 with the 1/1 are continually
suggesting that beating tempered octave- you could probably tweak
those cents a tiny bit and get a perfect relationship as far as the
beats.

-Cameron Bobro

--- In MakeMicroMusic@yahoogroups.com, "Cameron Bobro"
<misterbobro@...> wrote:

> >Because 64/63 is tempered out, the dominant seventh chord is
> >equated with the ottonal tetrad.
>
> Which sounds like a trait common to many tunings- doesn't this tame
> the dominant?

It's common in the sense that 12-edo is common, and it turns up in a
number of the less-accurate 7-limit temperaments. Aside from pajara,
there's dominant, blackwood, augene, porcupine, superpyth and beatles.
Of these, porcupine has garnered the most notice, I think. The Kees
tuning of dominant, which is simply the Pythagorean tuning, which you
then use with the grotty sharp major thirds and sevenths, is pretty
familiar in the guise of its 12-edo version, I guess. It's meantone
where you pretend a V7 is really an otonal tetrad.

> > That's what George has been on about. Paul Erlich likes 22-equal,
> > which has a very sharp fifth of 709 cents.
>
> The "sharpness" of a fifth (or any other interval, as long as we're
> not working with pure sines) isn't completely bound to its size. For
> example, there is a rational fifth around 709 cents which has a
> darker and more traditionally fifth-like character than other fifths
> of around, say, 707 cents.

I've suggested making use of sharp wolf intervals precisely tuned to
rational ones such as 26/17 or 32/21 as a way of slightly taming the
wolf, but there isn't any rational fifth around 709 cents which is
going to work like you suggest, it seems to me. Any very small
small-height interval like 3/2 has a sort of zone of repulsion around
it, you might say, keeping the other rational intervals of small
height farther away than average.

> When I get time, I'll finish a little
> Csound program that runs through high fifths and you can hear for
> yourself.

That would be good! Make a believer of me.

> Does "pajara" refer to only one tempering of 3/2?

No; it refers to the mapping. It's a matter of how you use the
tempered fifth more than its numerical value.

> In a well-
> temperament like Secor's there are patently more than one version of
> 3/2, and I see (and hear) nothing wrong with using in the same
> tuning several tempered 3/2's which would usually be associated with
> other octave divisions.

There's nothing wrong with it, but it isn't what theory should first
look at because it's more complex.

--- In MakeMicroMusic@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
> It's common in the sense that 12-edo is common, and it turns up in
>a
> number of the less-accurate 7-limit temperaments. Aside from
>pajara,
> there's dominant, blackwood, augene, porcupine, superpyth and
>beatles.
> Of these, porcupine has garnered the most notice, I think. The Kees
> tuning of dominant, which is simply the Pythagorean tuning, which
>you
> then use with the grotty sharp major thirds and sevenths, is pretty
> familiar in the guise of its 12-edo version, I guess. It's meantone
> where you pretend a V7 is really an otonal tetrad.

I find that sevens seem to scream for more more their own kind. The
partial itself has an unusual power- even the weediest additive
synth patch tends to get stronger if you boost the 7th harmonic. In
a tuning 7/4 for example is very bossy, but it is "the" interval for
a truly tonicizing chord on the dominant, not the whiny leading tone
whatever its size; just take a look at the most prominent difference
tones with the tonic and the fifth.

Very problematic.

> I've suggested making use of sharp wolf intervals precisely tuned
>to rational ones such as 26/17 or 32/21 as a way of slightly taming
>the wolf,

I find that you're correct- those particular intervals are very
simple and strong and fall into a peculiar class, don't know what to
call it. Some kind of metallic fifth (I'd call the meantone fifths
more wooden). You have to go into more complex ratios to maintain
the fifth character.

104/69 is slightly over 710 cents and definitely a fifth. The more
I listen to it the more I think it has a wooden quality like a
meantone fifth...hmmm, I wonder if its rational mirror on the other
side of 3/2 has a similar quality? Have to try it (around 694, hmm).
If it does the next step would be to mirror known good low fifths
and hear what happens.

Just did a blind listening and thought that a 450/299 fifth at 707.7
cents sounded SHARPER than the 104/69 710 cent fifth- I believe the
character of the thing is very important.

>but there isn't any rational fifth around 709 cents which is
> going to work like you suggest, it seems to me. Any very small
> small-height interval like 3/2 has a sort of zone of repulsion
>around
> it, you might say, keeping the other rational intervals of small
> height farther away than average.

I suspect that the zone of repulsion has a number of holes in it,
we'll find out with time. Of course this all raises the question
of tuning accuracy, which is 1/10,000th of a Hz or so in Csound-
just how small are those holes? I believe that they'll actually work
in unison with other fifths- a violin playing a 3/2 and a synth a
104/69 is, I believe, going to work as a thickening effect and
not an out of tune effect, but we'll have to hear.
>
> > When I get time, I'll finish a little
> > Csound program that runs through high fifths and you can hear
>>for
> > yourself.
>
> That would be good! Make a believer of me.

Well we'll see- do you have that list of high fifths? It's pretty
much a matter of cut and paste now that the appropriate sound (I
think) has been programmed.

>
> > In a well-
> > temperament like Secor's there are patently more than one
>>version of
> > 3/2, and I see (and hear) nothing wrong with using in the same
> > tuning several tempered 3/2's which would usually be associated
with
> > other octave divisions.
>
> There's nothing wrong with it, but it isn't what theory should
>first look at because it's more complex.

Not to mention fretting, another problematic issue with different
fifths.

take care,

-Cameron Bobro

The 621/416 approx. 694-cent fifth I mentioned above as the mirror
against 3/2 of the high approx. 710 cent fifth is obviously a meantone
fifth (2/7 comma and generates a tuning 1.3 cents different from an
equal-beating 2/7 meantone in Scala), and does share a remarkable
audible similiarity with its upper reflection. One's high, one's low,
but I hear tham as coming from the same pallette. Hmm!

Listening to the two, then to the approx. 707.7 fifth, the 707.7
sounds just awful.... the 414/275 at about 708 cents sounds great.
Okay, okay, I'll try to get the Csound orc/sco up tonight, if not,
then tomorrow.

--- In MakeMicroMusic@yahoogroups.com, "Cameron Bobro"
<misterbobro@...> wrote:

> Well we'll see- do you have that list of high fifths?

Tell me how high you want to go and how large the numerator can be.

--- In MakeMicroMusic@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:

> Tell me how high you want to go and how large the numerator can be.
>

I think that mirroring meantone and other time tested kinds of
tempered fifths is a good idea, the more I experiment with it the
more sense it makes, so, even though there seem to be some "fliers"
that work out at 720 cents or so, 712 (mirroring lower fifths at 690)
might be a good limit.

Haven't found any "rational sounding" high fifths with numerators
over 1000 (864, to be precise), so that might be a good limit, too.

-Cameron Bobro

--- In MakeMicroMusic@yahoogroups.com, "Cameron Bobro"
<misterbobro@...> wrote:

> Haven't found any "rational sounding" high fifths with numerators
> over 1000 (864, to be precise), so that might be a good limit, too.

I think limiting the denominators to less than 200 more than
sufficies, at least I hope so.

299/199, 296/197, 293/195, 290/193, 287/191, 284/189, 281/187,
278/185, 275/183, 272/181, 269/179, 266/177, 263/175, 260/173,
257/171, 254/169, 251/167, 248/165, 245/163, 242/161, 239/159,
236/157, 233/155, 230/153, 227/151, 224/149, 221/147, 218/145,
215/143, 212/141, 209/139, 206/137, 203/135, 200/133, 197/131,
194/129, 191/127, 188/125, 185/123, 182/121, 179/119, 176/117,
173/115, 170/113, 167/111, 164/109, 161/107, 158/105, 155/103,
152/101, 301/200, 149/99, 295/196, 146/97, 289/192, 143/95, 283/188,
140/93, 277/184, 137/91, 271/180, 134/89, 265/176, 131/87, 259/172,
128/85, 253/168, 125/83, 247/164, 122/81, 241/160, 119/79, 235/156,
116/77, 229/152, 113/75, 223/148, 110/73, 217/144, 107/71, 211/140,
104/69, 205/136, 101/67, 300/199, 199/132, 297/197, 98/65, 291/193,
193/128, 288/191, 95/63, 282/187, 187/124, 279/185, 92/61, 273/181,
181/120, 270/179, 89/59, 264/175, 175/116, 261/173, 86/57

Wow, thanks! That is just great. It's going to take a good while to
first classify them according to how I believe "things actually
work", which will test the classification itself of course (so far
it seems to be working) then do listening in different contexts.

-Cameron Bobro

--- In MakeMicroMusic@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In MakeMicroMusic@yahoogroups.com, "Cameron Bobro"
> <misterbobro@> wrote:
>
> > Haven't found any "rational sounding" high fifths with numerators
> > over 1000 (864, to be precise), so that might be a good limit,
too.
>
> I think limiting the denominators to less than 200 more than
> sufficies, at least I hope so.
>
> 299/199, 296/197, 293/195, 290/193, 287/191, 284/189, 281/187,
> 278/185, 275/183, 272/181, 269/179, 266/177, 263/175, 260/173,
> 257/171, 254/169, 251/167, 248/165, 245/163, 242/161, 239/159,
> 236/157, 233/155, 230/153, 227/151, 224/149, 221/147, 218/145,
> 215/143, 212/141, 209/139, 206/137, 203/135, 200/133, 197/131,
> 194/129, 191/127, 188/125, 185/123, 182/121, 179/119, 176/117,
> 173/115, 170/113, 167/111, 164/109, 161/107, 158/105, 155/103,
> 152/101, 301/200, 149/99, 295/196, 146/97, 289/192, 143/95,
>283/188,
> 140/93, 277/184, 137/91, 271/180, 134/89, 265/176, 131/87,
>259/172,
> 128/85, 253/168, 125/83, 247/164, 122/81, 241/160, 119/79,
235/156,
> 116/77, 229/152, 113/75, 223/148, 110/73, 217/144, 107/71,
>211/140,
> 104/69, 205/136, 101/67, 300/199, 199/132, 297/197, 98/65,
>291/193,
> 193/128, 288/191, 95/63, 282/187, 187/124, 279/185, 92/61,
>273/181,
> 181/120, 270/179, 89/59, 264/175, 175/116, 261/173, 86/57
>

Cameron- looking forward to your results! (On the tuning list,
perhaps?)

>Wow, thanks! That is just great. It's going to take a good while to
>first classify them according to how I believe "things actually
>work", which will test the classification itself of course (so far
>it seems to be working) then do listening in different contexts.
>
>-Cameron Bobro
>
>> > Haven't found any "rational sounding" high fifths with numerators
>> > over 1000 (864, to be precise), so that might be a good limit,
>> > too.
>>
>> I think limiting the denominators to less than 200 more than
>> sufficies, at least I hope so.
>>
>> 299/199, 296/197, 293/195, 290/193, 287/191, 284/189, 281/187,
>> 278/185, 275/183, 272/181, 269/179, 266/177, 263/175, 260/173,
>> 257/171, 254/169, 251/167, 248/165, 245/163, 242/161, 239/159,
>> 236/157, 233/155, 230/153, 227/151, 224/149, 221/147, 218/145,
>> 215/143, 212/141, 209/139, 206/137, 203/135, 200/133, 197/131,
>> 194/129, 191/127, 188/125, 185/123, 182/121, 179/119, 176/117,
>> 173/115, 170/113, 167/111, 164/109, 161/107, 158/105, 155/103,
>> 152/101, 301/200, 149/99, 295/196, 146/97, 289/192, 143/95,
>> 283/188, 140/93, 277/184, 137/91, 271/180, 134/89, 265/176,
>> 131/87, 259/172, 128/85, 253/168, 125/83, 247/164, 122/81,
>> 241/160, 119/79, 235/156, 116/77, 229/152, 113/75, 223/148,
>> 110/73, 217/144, 107/71, 211/140, 104/69, 205/136, 101/67,
>> 300/199, 199/132, 297/197, 98/65, 291/193, 193/128, 288/191,
>> 95/63, 282/187, 187/124, 279/185, 92/61, 273/181, 181/120,
>> 270/179, 89/59, 264/175, 175/116, 261/173, 86/57

--- In MakeMicroMusic@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> Cameron- looking forward to your results! (On the tuning list,
> perhaps?)

Yes, the tuning list is more appropriate I agree. You could
synthesize some and check it out, too. When I say "classify",
classification is the byproduct of a generating technique: I octave-
reduce (or expand in generation) the denominator and consider that
interval functionally. 289/192 would then be a high fifth of the
fifth family, 299/199 more in the "minor sixth" family. According to
my theory, both should sound very good, either high or mirrored
downward against 3/2, whereas 284/189, although it is lower as far
as odd numbers, will probably sound more dissonant because 189 is in
some kind of flat-fifth family. The most distant from 3/2 of the
three, and not of the lowest odd-number, 289/192, might even be the
most "fifth" sounding of the three.

Haven't tested these three together yet, but this is the kind of
thing I'm doing.

-Cameron Bobro

>> Cameron- looking forward to your results! (On the tuning list,
>> perhaps?)
>
>Yes, the tuning list is more appropriate I agree. You could
>synthesize some and check it out, too. When I say "classify",
>classification is the byproduct of a generating technique: I octave-
>reduce (or expand in generation) the denominator and consider that
>interval functionally. 289/192 would then be a high fifth of the
>fifth family, 299/199 more in the "minor sixth" family. According to
>my theory, both should sound very good, either high or mirrored
>downward against 3/2, whereas 284/189, although it is lower as far
>as odd numbers, will probably sound more dissonant because 189 is in
>some kind of flat-fifth family. The most distant from 3/2 of the
>three, and not of the lowest odd-number, 289/192, might even be the
>most "fifth" sounding of the three.

Wow, this blows my mind. Unfortunately my synthesis capabilities
are pretty wimpy at the moment (see my post on tuning). But I would
like to explore this with you.

-Carl

--- In MakeMicroMusic@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> Wow, this blows my mind. Unfortunately my synthesis capabilities
> are pretty wimpy at the moment (see my post on tuning). But I would
> like to explore this with you.
>
> -Carl
>

Okay, let's take it over there! I'm delighted.