19, 31 etc

One of the most important things I learned early on in my study of tunings was Ivor Darreg's observation that each system has it's own sound, or mood, and that's true. Therefore, I think I could conceivably find something of interest in just about any system. That being said, I think that some systems are probably more versatile than others...if you don't have good 3rds/5ths, it might be hard to write complex chordal music, for example.
If I had more $$$, I would have a warehouse full of guitars in many different tunings, and I would compose in all of them, maybe some more than others of course. I like 19 cause it extends forms from 12 in a real subtle way, but, as in "The Spider" from my "Other Worlds" CD, you can get pretty whacked as well. And 34 is an amazing system, little used by hardly anybody, and it's very difficult, especially because of the comma, but that's also one of it's best features...there's a lot more I am discovering about it. I haven't done anything yet in 31, and that was a decision I made years ago...I got the 31 and 34 guitars about the same time, and lot of folks had already composed in 31, so I wanted to try 34, as it was fairly unexplored territory. But, I have some 31 ideas in my head, more to come soon.
I am composing on fretless guitar right now, a new piece is almost ready to be recorded...the potential of fretless is unlimited, and I see many compositions in the future. I also have a pseudo 13 limit scale I've been working on in 34, and it is way out there, very alien sounding...it has 13 notes, and I'm getting it organized, looking for motifs, and I know a strong piece is in there, maybe several. I'm gonna put a post up soon about the compositional process, cause I think it's fascinating to see a piece emerge from...where DO they come from, anyway? It's a mysterious process, and the most fun part of music, for me...best...Hstick
myspace.com/microstick microstick.net

[Non-text portions of this message have been removed]

--- In MakeMicroMusic@yahoogroups.com, <microstick@...> wrote:
>
> One of the most important things I learned early on in my study
of tunings was Ivor Darreg's observation that each system has it's
own sound, or mood, and that's true. Therefore, I think I could
conceivably find something of interest in just about any system. That
being said, I think that some systems are probably more versatile
than others...if you don't have good 3rds/5ths, it might be hard to
write complex chordal music, for example.

> If I had more $$$, I would have a warehouse full of guitars in
many different tunings, and I would compose in all of them, maybe
some more than others of course. I like 19 cause it extends forms
from 12 in a real subtle way, but, as in "The Spider" from my "Other
Worlds" CD, you can get pretty whacked as well. And 34 is an amazing
system, little used by hardly anybody,

I would say that 34 is one of the most underrated divisions around.
A couple of weeks ago I decided to try two circles of my 17-tone well-
temperament (a 13-limit non-5 tuning) 600 cents apart in order to get
primes 5 and 17, and I was quite surprised with the result -- it
sounded a lot better than I expected.

The chief problem with 34 is prime 7: it's too low in pitch (both
melodically and hermonically), and it keeps 34-ET from being 17-limit
consistent. However, if you use the tone 1 degree higher for 7 (as
found in 17-ET), it not only sounds better (more believable), but it
also makes 34-ET 17-limit "pseudo-consistent."

> and it's very difficult, especially because of the comma, but
that's also one of it's best features...there's a lot more I am
discovering about it. I haven't done anything yet in 31, and that was
a decision I made years ago...I got the 31 and 34 guitars about the
same time, and lot of folks had already composed in 31, so I wanted
to try 34, as it was fairly unexplored territory. But, I have some 31
ideas in my head, more to come soon.

> I am composing on fretless guitar right now, a new piece is
almost ready to be recorded...the potential of fretless is unlimited,
and I see many compositions in the future. I also have a pseudo 13
limit scale I've been working on in 34, and it is way out there, very
alien sounding...it has 13 notes, and I'm getting it organized,
looking for motifs, and I know a strong piece is in there, maybe
several.

Sounds intriguing. What are the intervals in your scale?

> I'm gonna put a post up soon about the compositional process,
cause I think it's fascinating to see a piece emerge from...where DO
they come from, anyway?

The last time I attempted to address a question like that, I was
scolded for being way-OT (by one of my best friends, BTW), which
makes me wonder: Is not the inspiration for one's musical creativity
relevant to this forum?

> It's a mysterious process, and the most fun part of music, for
me...best...Hstick

I'll be looking forward to hearing what you come up with.

--George

it seems it is essential detail of any tuning process.
it is often the driving force that makes us choose one over the other, and without,
one wonders why one would bother with tuning at all.
if they insist on it being off list maybe we would have to start a
tuning inspiration list, which i would rather not.

34 also relates to the fibonacci series.
and Neil music in it has proved that it indeed sounds better than we expect, which makes me distrust to a certain degree all the over cataloging of tunings.
one has to hear any or all of them to really judge them

George D. Secor wrote:
>
> >> I'm gonna put a post up soon about the compositional process, >> > cause I think it's fascinating to see a piece emerge from...where DO > they come from, anyway? >
> The last time I attempted to address a question like that, I was > scolded for being way-OT (by one of my best friends, BTW), which > makes me wonder: Is not the inspiration for one's musical creativity > relevant to this forum?
>
> > -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

that's right.....you have to get so0e dirt under your nails, and I'll
tell you something else--real instruments aren't midi jobs, and every
last instrument distorts the shit out of tunings through the prism of
its interface and timbre and player, etc etc etc. This is micro
tunings dirty little secret.....that no tuning is exactly the same
tuning on every instrument. For example, i have a 13edo tenor
ukulele, and i can tune the open strings to a near just tuning and
play and still hear the "13" ,but it's not 13 at all, it's something
entirely different....yet, it's an instrument tuned to 13. Same for
20edo guitar… I can play things on it tuned formally to 20, and still
it will sound NOTHING like the same thing played on a midikeyboard
tuned to exactly the same tuning. context and interface are what it's
all about.....tunings are grossly overrated, underrated and
misinterpreted by inorganic midi analysis, psycho acoustical
fetishism, and mathematics. The truth is out there, but it's a wild,
willlld beast….and while psychoacoustics do a good job of
demystifying for those who need demystification (even if it's just a
nice little fairy tale for smart people), but midi and mathematics
don't.
--- In MakeMicroMusic@yahoogroups.com, Kraig Grady <kraiggrady@...>
wrote:
>
> it seems it is essential detail of any tuning process.
> it is often the driving force that makes us choose one over the
other,
> and without,
> one wonders why one would bother with tuning at all.
> if they insist on it being off list maybe we would have to start a
> tuning inspiration list, which i would rather not.
>
>
> 34 also relates to the fibonacci series.
> and Neil music in it has proved that it indeed sounds better than
we
> expect, which makes me distrust to a certain degree all the over
> cataloging of tunings.
> one has to hear any or all of them to really judge them
>
>
> George D. Secor wrote:
> >
> >
> >> I'm gonna put a post up soon about the compositional process,
> >>
> > cause I think it's fascinating to see a piece emerge from...where
DO
> > they come from, anyway?
> >
> > The last time I attempted to address a question like that, I was
> > scolded for being way-OT (by one of my best friends, BTW), which
> > makes me wonder: Is not the inspiration for one's musical
creativity
> > relevant to this forum?
> >
> >
> >
>
> --
> Kraig Grady
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
> The Wandering Medicine Show
> KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles
>

>34 also relates to the fibonacci series.
> and Neil music in it has proved that it indeed sounds better than we
>expect, which makes me distrust to a certain degree all the over
>cataloging of tunings.

It does? Most predictions say that 34 should sound really good.

-Carl

metal bars stay in tune, except in big weather changes , but who plays outside in the cold

daniel_anthony_stearns wrote:
> that's right.....you have to get so0e dirt under your nails, and I'll > tell you something else--real instruments aren't midi jobs, and every > last instrument distorts the shit out of tunings through the prism of > its interface and timbre and player, etc etc etc. This is micro > tunings dirty little secret.....that no tuning is exactly the same > tuning on every instrument. For example, i have a 13edo tenor > ukulele, and i can tune the open strings to a near just tuning and > play and still hear the "13" ,but it's not 13 at all, it's something > entirely different....yet, it's an instrument tuned to 13. Same for > 20edo guitar� I can play things on it tuned formally to 20, and still > it will sound NOTHING like the same thing played on a midikeyboard > tuned to exactly the same tuning. context and interface are what it's > all about.....tunings are grossly overrated, underrated and > misinterpreted by inorganic midi analysis, psycho acoustical > fetishism, and mathematics. The truth is out there, but it's a wild, > willlld beast�.and while psychoacoustics do a good job of > demystifying for those who need demystification (even if it's just a > nice little fairy tale for smart people), but midi and mathematics > don't. > --- In MakeMicroMusic@yahoogroups.com, Kraig Grady <kraiggrady@...> > wrote:
> >> it seems it is essential detail of any tuning process.
>> it is often the driving force that makes us choose one over the >> > other, > >> and without,
>> one wonders why one would bother with tuning at all.
>> if they insist on it being off list maybe we would have to start a
>> tuning inspiration list, which i would rather not.
>>
>>
>> 34 also relates to the fibonacci series.
>> and Neil music in it has proved that it indeed sounds better than >> > we > >> expect, which makes me distrust to a certain degree all the over >> cataloging of tunings.
>> one has to hear any or all of them to really judge them
>>
>>
>> George D. Secor wrote:
>> >>> >>> >>>> I'm gonna put a post up soon about the compositional process, >>>> >>>> >>> cause I think it's fascinating to see a piece emerge from...where >>> > DO > >>> they come from, anyway? >>>
>>> The last time I attempted to address a question like that, I was >>> scolded for being way-OT (by one of my best friends, BTW), which >>> makes me wonder: Is not the inspiration for one's musical >>> > creativity > >>> relevant to this forum?
>>>
>>> >>> >>> >> -- >> Kraig Grady
>> North American Embassy of Anaphoria Island <http://anaphoria.com/>
>> The Wandering Medicine Show
>> KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles
>>
>> >
>
>
>
>
>
>
> > Yahoo! Groups Links
>
>
>
> >
>
>
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

On 5/15/06, Kraig Grady <kraiggrady@anaphoria.com> wrote:
[...]
> 34 also relates to the fibonacci series.
> and Neil music in it has proved that it indeed sounds better than we
> expect, which makes me distrust to a certain degree all the over
> cataloging of tunings.
> one has to hear any or all of them to really judge them

The fact that 34 is a fibonacci number doesn't really mean anything.
The linear temperament that goes through the fibonacci numbered EDOs
is frankly no good. In fact, it's one of the worst, because the
interval that's the golden ratio part of an octave just doesn't sound
that great.

I do agree with you about the need to deeply explore individual
temperaments though. I've been exploring Hanson/Keemun for months now
and I think I almost know it well enough to write pieces in it.

Keenan

in what sense does it not sound good, phi does not sound good or the approximation is too far off.

and what is the Hanson/Keemun temperment and did the two connect at all on this, or is like that one that someone added my name on to recently that has nothing to do with me

Keenan Pepper wrote:
>
> In fact, it's one of the worst, because the
> interval that's the golden ratio part of an octave just doesn't sound
> that great.
>
> I do agree with you about the need to deeply explore individual
> temperaments though. I've been exploring Hanson/Keemun for months now
> and I think I almost know it well enough to write pieces in it.
>
> Keenan
>
> > -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

On 5/16/06, Kraig Grady <kraiggrady@anaphoria.com> wrote:
> in what sense does it not sound good, phi does not sound good or the
> approximation is too far off.

I wasn't talking about the interval of phi itself, but the interval
that's the phi part of an octave, 741.64 cents. It sounds bad to me
because it's too wide to be a fifth but too narrow to be a 14/9, but
that's completely subjective of course.

> and what is the Hanson/Keemun temperment and did the two connect at all on
> this, or is like that one that someone added my name on to recently that has
> nothing to do with me

It's the temperament produced by a chain of 6/5s where 5 of them
approximate 5/2. Hanson is the 5-limit version and Keemun is the
7-limit version where 3 6/5s approximate 7/4. I'm not sure what
"Hanson" refers to, but Keemun is a kind of tea.

Keenan

a

Keenan Pepper wrote:
>
> I wasn't talking about the interval of phi itself, but the interval
> that's the phi part of an octave, 741.64 cents. It sounds bad to me
> because it's too wide to be a fifth but too narrow to be a 14/9, but
> that's completely subjective of course.
> thank for clearing that up.
it can be used in its own right having nothing to do with say diatonic harmony.
> It's the temperament produced by a chain of 6/5s where 5 of them
> approximate 5/2. Hanson is the 5-limit version and Keemun is the
> 7-limit version where 3 6/5s approximate 7/4. I'm not sure what
> "Hanson" refers to, but Keemun is a kind of tea.
> Hanson was Larry Hanson who Erv worked with quite a bit and still has a file cabinet at his house of musical notes, we need to go through at some point.
yes it was the chain of 6/5 that Hanson had come up with, but because so , it lead to his own generalized keyboard, that allowed all type of tunings that Bosonquet could not handle as well. 72 and the hebdomekontany being one of these. He also worked with Neil Haverstick
> Keenan
>
>
>
> > Yahoo! Groups Links
>
>
>
> > -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

--- In MakeMicroMusic@yahoogroups.com, <microstick@...> wrote:

> And 34 is an amazing system, little used by hardly anybody, and it's
> very difficult, especially because of the comma, but that's also one
> of it's best features...there's a lot more I am discovering about it.

One interesting feature of 34 is that there are two reasonable
approaches to 7-limit chords using it, the keemun system and the
pajara system. In keemun, you approach 7-limit chords from a kleismic
point of view, when you stack six 6/5s to get a 3, and 5 6/5s to get a
5/2, you also stack 3 6/5s to get a 7/4 (which is also a 12/7 in this
system.)
Pajara is Paul Erlich's favorite on 22-et, but it should sound a lot
different in 34, with its improved octaves, and a sharp 7/4 of
988 cents. This amounts to using a dominant seventh chord as your
7-limit otonal tetrad, and while it is quite sharp, it isn't nearly as
sharp as the 1000 cent 12-limit version. Do either of these or maybe
both correspond at all to how you approach playing in 34-et?

--- In MakeMicroMusic@yahoogroups.com, "George D. Secor" <gdsecor@...>
wrote:

> The chief problem with 34 is prime 7: it's too low in pitch (both
> melodically and hermonically), and it keeps 34-ET from being 17-limit
> consistent. However, if you use the tone 1 degree higher for 7 (as
> found in 17-ET), it not only sounds better (more believable), but it
> also makes 34-ET 17-limit "pseudo-consistent."

The pajara method. This may sound better in part because it is so
familiar.

> The last time I attempted to address a question like that, I was
> scolded for being way-OT (by one of my best friends, BTW), which
> makes me wonder: Is not the inspiration for one's musical creativity
> relevant to this forum?

The dirty little secret around here is that theory of one kind or
another is immensely inspirational for a lot of the music which gets done.

--- In MakeMicroMusic@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In MakeMicroMusic@yahoogroups.com, <microstick@> wrote:
>
> > And 34 is an amazing system, little used by hardly anybody, and
it's
> > very difficult, especially because of the comma, but that's also
one
> > of it's best features...there's a lot more I am discovering about
it.
>
> One interesting feature of 34 is that there are two reasonable
> approaches to 7-limit chords using it, the keemun system and the
> pajara system. In keemun, you approach 7-limit chords from a
kleismic
> point of view, when you stack six 6/5s to get a 3, and 5 6/5s to
get a
> 5/2, you also stack 3 6/5s to get a 7/4 (which is also a 12/7 in
this
> system.)
>
> Pajara is Paul Erlich's favorite on 22-et, but it should sound a lot
> different in 34, with its improved octaves, and a sharp 7/4 of
> 988 cents. This amounts to using a dominant seventh chord as your
> 7-limit otonal tetrad, and while it is quite sharp, it isn't nearly
as
> sharp as the 1000 cent 12-limit version. ...

The wide 4:7 of 988 cents sounds much better, IMO, than the 953-cent
interval that occurs in 34-tone Keemun. I would also note that 34's
best approximations of 5:7, 6:7, and 7:9 are three good reasons for
preferring its second-best 4:7 over its 'best'.

Another nicety of 34 is that 4:5 is slightly wide, which I think most
will prefer melodically to the narrow 4:5 of 22. (Need I compare
2:3?) So you get some good things in exchange for making 7 a bit
worse.

I assume that you're including the pentachordal form of Paul's
decatonic scale in your mention of 'pajara'. It's unfortunate that
Paul didn't mention anywhere in his 22-tone paper that it's possible
to use his decatonic scales in 34. The interval sequence ssLsssLsss
(the pentachordal major form) has L=3 and s=2 degrees in 22, but in
34 it would be L=5 and s=3.

--George

--- In MakeMicroMusic@yahoogroups.com, "George D. Secor" <gdsecor@...>
wrote:

> I assume that you're including the pentachordal form of Paul's
> decatonic scale in your mention of 'pajara'.

Any of the ten or twelve note scales Paul discussed can be played in
34, because they are all pajara scales. Just as a lot of meatone music
can be swapped around to different equal temperaments, you could take
anything in 22 which sticks to the pajara system and play it in 34
instead. You have 22+12=34, so they are all in it together. You could
even try 34+12=46, but now you are using the "wrong" tuning of the
7th. It's really the dominant seventh in 46, not the otonal tetrad.

It's unfortunate that
> Paul didn't mention anywhere in his 22-tone paper that it's possible
> to use his decatonic scales in 34.

I think he wanted to concentrate on 22, but I can't recall him
mentioning porcupine or superpyth either, which makes it really a
pajara paper, not a 22 paper, which certainly could have mentioned 34.

>The wide 4:7 of 988 cents sounds much better, IMO, than the 953-cent
>interval that occurs in 34-tone Keemun.

And wide 5:4s tend to sound better than 5:4s flattened by the same
amount. So while cents error is a quick and dirty measure of
discordance, harmonic entropy is ultimately needed. It tells us
that there's a local maximum of discordance around 950 cents, and
that discordance increases more rapidly South of 969 than North.

-Carl

>I think he wanted to concentrate on 22, but I can't recall him
>mentioning porcupine or superpyth either, which makes it really a
>pajara paper, not a 22 paper, which certainly could have mentioned 34.

He was insisting on consistency.

-Carl

--- In MakeMicroMusic@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >The wide 4:7 of 988 cents sounds much better, IMO, than the 953-
cent
> >interval that occurs in 34-tone Keemun.
>
> And wide 5:4s tend to sound better than 5:4s flattened by the same
> amount.

I agree.

> So while cents error is a quick and dirty measure of
> discordance, harmonic entropy is ultimately needed. It tells us
> that there's a local maximum of discordance around 950 cents, and
> that discordance increases more rapidly South of 969 than North.

I think that melodic considerations and habituation (a preference for
intervals one is most accustomed to hearing) also play a part. It's
not difficult to accept the notion that harmonic preference could be
related to harmonic entropy, but melodic preference would seem to be
much more subjective.

I once thought that melodic preference might be almost completely
determined by habituation, but I quickly discarded that notion the
first time I tried playing major scales in Pythagorean tuning and 17-
ET, both of which I immediately preferred to 12-ET and 1/4-comma
meantone (the tunings to which, at that time, I was most accustomed).

More recently, I've come to the conclusion that the best melodic
tuning for a diatonic scale is one in which the semitone is close to
the point of maximum harmonic entropy (around 67 cents), which brings
us back to your statement. :-)

--George

--- In MakeMicroMusic@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >I think he wanted to concentrate on 22, but I can't recall him
> >mentioning porcupine or superpyth either, which makes it really a
> >pajara paper, not a 22 paper, which certainly could have mentioned
34.
>
> He was insisting on consistency.

I don't think that Paul was insisting on consistency in the strictest
sense: that one is obligated to use the best approximation of *all*
of the 7-limit consonances of an octave division. If that were the
case, then why did he mention 76 and other (larger) divisions of the
octave as possible tunings for decatonic scales? Because he was able
to identify subsets of those tunings that produce decatonic scales
that are error-competitive with 22-ET, in spite of the fact that
decatonic scales in 76-ET, for example, would contain neither its
best 2:3 nor its best 4:7.

If you relax the requirement that decatonic scales in 34-ET use its
best 4:7 (and 7:8), then it's possible to achieve a tonal mapping of
decatonic scales in 34 that's internally consistent (a slightly
different use of the term).

So I still maintain that 34 deserved a mention in Paul's paper.

--George

--- In MakeMicroMusic@yahoogroups.com, "George D. Secor" <gdsecor@...>
wrote:

> More recently, I've come to the conclusion that the best melodic
> tuning for a diatonic scale is one in which the semitone is close to
> the point of maximum harmonic entropy (around 67 cents), which brings
> us back to your statement. :-)

This sounds good for 34-et, but would you harmonize by going outside
of the scale?

>> >I think he wanted to concentrate on 22, but I can't recall him
>> >mentioning porcupine or superpyth either, which makes it really a
>> >pajara paper, not a 22 paper, which certainly could have mentioned
>34.
>>
>> He was insisting on consistency.
>
>I don't think that Paul was insisting on consistency in the strictest
>sense: that one is obligated to use the best approximation of *all*
>of the 7-limit consonances of an octave division. If that were the
>case, then why did he mention 76 and other (larger) divisions of the
>octave as possible tunings for decatonic scales? Because he was able
>to identify subsets of those tunings that produce decatonic scales
>that are error-competitive with 22-ET, in spite of the fact that
>decatonic scales in 76-ET, for example, would contain neither its
>best 2:3 nor its best 4:7.
>
>If you relax the requirement that decatonic scales in 34-ET use its
>best 4:7 (and 7:8), then it's possible to achieve a tonal mapping of
>decatonic scales in 34 that's internally consistent (a slightly
>different use of the term).
>
>So I still maintain that 34 deserved a mention in Paul's paper.

Paul lists all ETs < 35 with 7-limit accuracy at least as good
as 12-tET's 5-limit accuracy, but doesn't consider non-patent
vals. He then looks for scales in the resulting ETs (22, 26, 27,
and 31). He mentions 76 later, in a different context.

These days, I suspect Paul would search not ETs, but linear
temperaments, and see that pajara wins on 7-limit badness in
many formulations. I suspect he'd find 34-tET a valid pajara
tuning. Whether it's actually better than 22 is a question.
The mad error of 34 is slightly better, and the rms slightly
worse. However, according to his calculation on page 21,
22 is closer to the optimal decatonic tuning than 34.

-Carl

--- In MakeMicroMusic@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In MakeMicroMusic@yahoogroups.com, "George D. Secor" <gdsecor@>
> wrote:
>
> > More recently, I've come to the conclusion that the best melodic
> > tuning for a diatonic scale is one in which the semitone is close
to
> > the point of maximum harmonic entropy (around 67 cents), which
brings
> > us back to your statement. :-)
>
> This sounds good for 34-et, but would you harmonize by going outside
> of the scale?

In some instances, yes, but in others not necessarily. For example, if
I were writing in 17 in a minor mode, then I would interpret the minor
triad as a tempered 6:7:9 (i.e., a superpythagorean temperament) and
treat the dominant (supermajor) triad as a dissonance.

--George

--- In MakeMicroMusic@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >> >I think he wanted to concentrate on 22, but I can't recall him
> >> >mentioning porcupine or superpyth either, which makes it really
a
> >> >pajara paper, not a 22 paper, which certainly could have
mentioned
> >>34.
> >>
> > <SNIP>
> >So I still maintain that 34 deserved a mention in Paul's paper.
>
> Paul lists all ETs < 35 with 7-limit accuracy at least as good
> as 12-tET's 5-limit accuracy, but doesn't consider non-patent
> vals. He then looks for scales in the resulting ETs (22, 26, 27,
> and 31). He mentions 76 later, in a different context.
>
> These days, I suspect Paul would search not ETs, but linear
> temperaments, and see that pajara wins on 7-limit badness in
> many formulations. I suspect he'd find 34-tET a valid pajara
> tuning. Whether it's actually better than 22 is a question.
> The mad error of 34 is slightly better, and the rms slightly
> worse. However, according to his calculation on page 21,
> 22 is closer to the optimal decatonic tuning than 34.

Whether 34 is better than 22 for pajara is beside the point I've been
trying to make: that if Neil is looking for ways to use 34, then he
should give pajara serious consideration. (I think you'll agree that
other divisions, such as 12 and 19, are not as good as 31 for
meantone, but that shouldn't stop anyone from using them for that
purpose.) Neil mentioned the problem of dealing with the comma in
34, and pajara provides a way to get around that problem by
approaching tonality non-diatonically.

But regarding how well 34 stacks up against 22 for pajara, I see that
Graham has responded on tuning-math. :-)

--George

>Whether 34 is better than 22 for pajara is beside the point I've been
>trying to make: that if Neil is looking for ways to use 34, then he
>should give pajara serious consideration.

Totally!

>Neil mentioned the problem of dealing with the comma in
>34, and pajara provides a way to get around that problem by
>approaching tonality non-diatonically.

Yup!

-Carl