Paultone/Twintone as a basis for Jazz harmony

Hi. I'm new here, so I may not have all the terms right, but I'd like
to make some observations about the linear temperament used for Paul
Ehrlich's Decatonic scales. Paul originally conceived this system for
22 tone equal temperament which has 709 cent 5ths, I think I saw a
message suggesting 34 tone equal temperament (706 cent 5ths.) Anyway,
no one seems to have noticed that the two basic unison vectors 64/63,
50/49 are also consistent with the common as dirt 12 tone equal
temperament (700 cent 5ths.) Given this, it would be extremely
surprising if decatonic scales weren't used in Western music.

As it turns out, decatonic scales HAVE been used extensively in Jazz.
One of the most common bebop scales is a major scale with added blue
3rd, 5th and 7th: C D Eb E F F# G A Bb B. Note, 8 short steps and two
long ones. This scale is featured prominently in everything from Blue
Monk to Purple Haze. Not all jazz works with this temperament ("I've
got rhythm" is comma pump city,) but a surprising amount does,
especially the bluesier stuff.

Aside from 2, 3, 5, and 7, this system represents two additional primes
in a meaningful way, 17 and 11 -- by tempering out 85/84 and 99/98. 17
is especially important because it is used in the diminished 7th chord.
Adding another unison vector can result in a closed equal temperament:
34/33 results in 12 equal, 55/54 results in 22 equal, and 243/242
results in 34 equal. That said, I don't like any of these systems
particularly -- 22 equal tunes the minor and neutral third as the same
interval and has jarringly sharp perfect 5ths, and 12 and 34 equal have
annoyingly sharp harmonic 7ths. I did some experimenting with my own
arrangement of Blue Monk in scala, and found the optimal temperament to
be somewhere in the neighborhood of 56 equal (707 cent 5ths.) This has
a number of advantages: Both 11/8 and 11/7 are acceptably tuned, the
major 3rds are very pure (within 1 cent), and the large and small steps
differ from one another by almost exactly the golden ratio, which
results in maximal melodic variety -- the ratio of a long swing 8th to
a short one also approximates the golden mean by the way.

Anyway, my feeling is paultone/twintone is in fact the second most
common linear temperament in western music after meantone, and
therefore warrants more study.

Any thoughts?

Ray Perlner

--- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@...>
wrote:
>
> Hi. I'm new here, so I may not have all the terms right, but I'd
like
> to make some observations about the linear temperament used for
Paul
> Ehrlich's Decatonic scales. Paul originally conceived this system
for
> 22 tone equal temperament which has 709 cent 5ths, I think I saw a
> message suggesting 34 tone equal temperament (706 cent 5ths.)
Anyway,
> no one seems to have noticed that the two basic unison vectors
64/63,
> 50/49 are also consistent with the common as dirt 12 tone equal
> temperament (700 cent 5ths.) Given this, it would be extremely
> surprising if decatonic scales weren't used in Western music.
>
> As it turns out, decatonic scales HAVE been used extensively in
Jazz.
> One of the most common bebop scales is a major scale with added
blue
> 3rd, 5th and 7th: C D Eb E F F# G A Bb B. Note, 8 short steps and
two
> long ones. This scale is featured prominently in everything from
Blue
> Monk to Purple Haze. Not all jazz works with this temperament
("I've
> got rhythm" is comma pump city,) but a surprising amount does,
> especially the bluesier stuff.
>
> Aside from 2, 3, 5, and 7, this system represents two additional
primes
> in a meaningful way, 17 and 11 -- by tempering out 85/84 and 99/98.
17
> is especially important because it is used in the diminished 7th
chord.
> Adding another unison vector can result in a closed equal
temperament:
> 34/33 results in 12 equal, 55/54 results in 22 equal, and 243/242
> results in 34 equal. That said, I don't like any of these systems
> particularly -- 22 equal tunes the minor and neutral third as the
same
> interval and has jarringly sharp perfect 5ths, and 12 and 34 equal
have
> annoyingly sharp harmonic 7ths. I did some experimenting with my
own
> arrangement of Blue Monk in scala, and found the optimal
temperament to
> be somewhere in the neighborhood of 56 equal (707 cent 5ths.) This
has
> a number of advantages: Both 11/8 and 11/7 are acceptably tuned,
the
> major 3rds are very pure (within 1 cent), and the large and small
steps
> differ from one another by almost exactly the golden ratio, which
> results in maximal melodic variety -- the ratio of a long swing 8th
to
> a short one also approximates the golden mean by the way.
>
> Anyway, my feeling is paultone/twintone is in fact the second most
> common linear temperament in western music after meantone, and
> therefore warrants more study.
>
> Any thoughts?
>
> Ray Perlner

There is a lot that could be said about your post. But I just want to
comment on your use of the golden ratio, as a way of unifying harmony
and rhythm. I always felt that rhythm in itself must also
be "tempered", or at least not just plain simple fractions all the
time. It's interesting that 4 and 3 are the main time signatures, and
these multiply out to 12. (Any relation to 12-ET?). Anyway, I didn't
know that about swing-eighths, and I'm a jazz musician myself. Thanks.
(sqrt(5)+1)/2 and its reciprocal kind of mix geometric and arithmetic
means I guess.

It would help if someone would give a listing of paultone/meantone
with period and generators, etc.

Paul Hj

--- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@...>
wrote:
>
> Hi. I'm new here, so I may not have all the terms right, but I'd
like
> to make some observations about the linear temperament used for
Paul
> Ehrlich's Decatonic scales.

We now call it 'pajara'.

> Paul originally conceived this system for
> 22 tone equal temperament which has 709 cent 5ths, I think I saw a
> message suggesting 34 tone equal temperament (706 cent 5ths.)
Anyway,
> no one seems to have noticed that the two basic unison vectors
64/63,
> 50/49 are also consistent with the common as dirt 12 tone equal
> temperament (700 cent 5ths.)

I know for a fact that Paul is well aware of this, inasmuch as he
wrote the following to me (off-list) a couple of years ago:

<< As you know, Pajara is essentially the same system proposed in my
XH17 paper (though there it was tuned essentially in 22-equal).
The 'unfortunate' thing about Pajara from a xenharmonic standpoint is
that its vanishing ratios -- 50:49, 64:63, 225:224, 2048:2025, . . . -
- all vanish in 12-equal. So one can't use this system to create
cyclic chord progressions which would be 'impossible' in 12-equal --
hence it's not really as subversive as many xenharmonic composers
would wish their tuning system to be. >>

> Given this, it would be extremely
> surprising if decatonic scales weren't used in Western music.
>
> As it turns out, decatonic scales HAVE been used extensively in
Jazz.
> One of the most common bebop scales is a major scale with added
blue
> 3rd, 5th and 7th: C D Eb E F F# G A Bb B. Note, 8 short steps and
two
> long ones. This scale is featured prominently in everything from
Blue
> Monk to Purple Haze.

I don't know whether Paul ever happened to make that observation (and
he hasn't been around here lately, so I don't expect he'll be reading
this unless someone contacts him off-list).

> Not all jazz works with this temperament ("I've
> got rhythm" is comma pump city,) but a surprising amount does,
> especially the bluesier stuff.
>
> Aside from 2, 3, 5, and 7, this system represents two additional
primes
> in a meaningful way, 17 and 11 -- by tempering out 85/84 and 99/98.
17
> is especially important because it is used in the diminished 7th
chord.
> Adding another unison vector can result in a closed equal
temperament:
> 34/33 results in 12 equal, 55/54 results in 22 equal, and 243/242
> results in 34 equal. That said, I don't like any of these systems
> particularly -- 22 equal tunes the minor and neutral third as the
same
> interval and has jarringly sharp perfect 5ths, and 12 and 34 equal
have
> annoyingly sharp harmonic 7ths. I did some experimenting with my
own
> arrangement of Blue Monk in scala, and found the optimal
temperament to
> be somewhere in the neighborhood of 56 equal (707 cent 5ths.)

You are in close agreement with the results of a pajara-temperament
evaluation I conducted recently on the main list:
/tuning/topicId_67957.html#68521

One point that was brought up was that it's questionable whether the
complexity of 56 is worth the improvement over 34 (especially since
pajara in the best 12 (11-limit) or 16 (7- and 9-limit) keys of my 34-
tone well-temperament are virtually indistinguishable from 56-ET
pajara.

> This has
> a number of advantages: Both 11/8 and 11/7 are acceptably tuned,
the
> major 3rds are very pure (within 1 cent), and the large and small
steps
> differ from one another by almost exactly the golden ratio, which
> results in maximal melodic variety -- the ratio of a long swing 8th
to
> a short one also approximates the golden mean by the way.
>
> Anyway, my feeling is paultone/twintone is in fact the second most
> common linear temperament in western music after meantone, and
> therefore warrants more study.

Indeed! -- and also more *use* in the alternative tunings that
support it.

--George Secor

!It occurred to me that it would be a good idea to post the midi
!files I've been messing around with, but I can't figure out how to
!do that, so I just posted the SCL file. It does the right thing in
!12, 34, 56, 100, or 22 equal, but I can't vouch for anything else.
!I'm vaugely hoping someone will turn it into a sound file. Bonus
!points for finding less computerized sounding timbres. Many thanks to
!Thelonius Monk for providing the melody and (hopefully) not suing me.

0 tempo 650_000
0 frequency 262
0 notation L22
0 outfile bluemonk.mid

!
0 track 1
0 program 33
0 note Ax.-3 100
140 note Ax.-3 100
240 note Gx.-3 240
480 note G.-3 240
720 note Fx.-3 240
960 note Dx.-2 240
1200 note cx.-2 240
1440 note C.-2 240
1680 note Ax.-3 240
1920 note D.-2 240
2160 note Dx.-2 120
2280 note E.-2 120
2400 note F#.-2 240
2640 note G.-2 240
2880 note Ax.-2 240
3120 note A.-2 240
3360 note Gx.-2 240
3600 note D.-2 240
3840 note Dx.-2 240
4080 note Ax.-2 240
4320 note A.-2 240
4560 note Gx.-2 240
4800 note G.-2 140
4940 note F#.-2 100
5040 note E.-2 240
5280 note Dx.-2 140
5420 note D.-2 100
5520 note Cx.-2 240
5760 note Ax.-3 240
6000 note D.-2 240
6240 note F#.-2 240
6480 note G.-2 240
6720 note Ax.-2 380
7100 note G.-2 240
7340 note A.-2 340
7680 note F#.-2 240
7920 note E.-2 240
8160 note Dx.-2 240
8400 note Cx.-2 240
8640 note F#.-3 240
8880 note A.-3 240
9120 note Dx.-2 240
9360 note Cx.-2 240
9600 note D.-2 240
9840 note Dx.-2 240
10080 note E.-2 240
10320 note F#.-2 240
10560 note Ax.-2 240
10800 note G.-2 240
11040 note Gx.-2 240
11280 note A.-2 240
11520 note Ax.-2 250
11770 note Cx.-1 260
12030 note [45/88] 270
12300 note B.-2 280
12580 note Ax.-2 1140
12580 note Ax.-2 1150
12580 note Ax.-2 1160

!
0 track 2
0 program 58
0 note D.-1 140
140 note Cx#.-1 100
240 note Cx.-1 140
380 note [45/88] 340
720 note C.-1 240
960 note Dx.-1 140
1100 note Ax.-1 100
1200 note A.-1 140
1340 note G.-1 580
1920 note F#.-1 240
2160 note Ax.-1 240
2400 note Dx.-1 240
2640 note E.-1 140
2780 note F#.-1 480
3260 note Fx.-1 580
3840 note G.-1 1920
5760 note F#.-1 240
6000 note Ax.-1 240
6240 note Dx.-1 240
6480 note E.-1 140
6620 note F#.-1 480
7100 note G.-1 100
7100 note G.-1 100
7200 note Gx.-1 480
7200 note Gx.-1 480
7680 note A.-1 960
8640 note Dx.-1 480
9120 note F#.-1 240
9360 note E.-1 140
9500 note Dx.-1 100
9600 note D.-1 720
10320 note F#.-1 380
10700 note F#.-1 2960

!
0 track 3
0 program 57
0 note D 140
140 note Dx 100
240 note E 140
380 note F# 580
960 note G 140
1100 note Gx 100
1200 note A 140
1340 note Ax 580
1920 note F# 140
2060 note G 100
2160 note F# 140
2300 note E 100
2400 note Dx 140
2540 note F#.-1 100
2640 note Cx 140
2780 note D 240
3020 note Cx 240
3260 note C 580
3840 note G 140
3980 note Gx 100
4080 note A 140
4220 note Ax 580
4800 note Ax 140
4940 note B 100
5040 note C.1 140
5180 note Cx.1 580
5760 note F# 140
5900 note G 100
6000 note F# 140
6140 note E 100
6240 note Dx 140
6380 note F#.-1 100
6480 note Cx 140
6620 note D 820
7440 note F# 80
7520 note F# 80
7600 note F# 80
7680 note F# 120
7800 note F#.-1 840
8640 note F# 140
8780 note G 100
8880 note F# 140
9020 note E 100
9120 note Dx 140
9260 note F#.-1 100
9360 note Cx 140
9500 note D 340
9840 note F# 140
9980 note G 100
10080 note F# 140
10220 note E 100
10320 note Dx 140
10460 note F#.-1 100
10560 note Cx 140
10700 note D 3000

!
0 track 4
0 program 66
0 note Ax.-1 140
140 note C 100
240 note Cx 140
380 note D 340
720 note Cx# 240
960 note Dx 140
1100 note F# 100
1200 note Fx 140
1340 note G 580
1920 note D 240
2160 note D 140
2300 note Cx 100
2400 note A.-1 480
2880 note Ax.-1 960
3840 note Dx 140
3980 note F# 100
4080 note Fx 140
4220 note G 580
4800 note G 140
4940 note Gx 100
5040 note A 140
5180 note Ax 580
5760 note D 240
6000 note D 140
6140 note Cx 100
6240 note A.-1 380
6620 note Ax.-1 820
8640 note A.-1 960
9600 note Ax.-1 720
10320 note Ax.-1 240
10560 note Gx#.-1 140
10700 note Gx.-1 3000

!
0 track 5
0 program 67
0 note F#.-1 140
140 note Fx.-1 100
240 note G.-1 140
380 note Gx.-1 580
960 note Ax.-1 140
1100 note B.-1 100
1200 note C 140
1340 note Cx 580
1920 note Gx.-1 380
2300 note G.-1 100
2400 note F#.-1 620
3020 note A.-1 240
3260 note Gx.-1 580
3840 note Ax.-1 140
3980 note B.-1 100
4080 note C 140
4220 note Cx 580
4800 note Cx 140
4940 note D 100
5040 note Dx 140
5180 note E 580
5760 note Gx.-1 380
6140 note G.-1 100
6240 note F#.-1 380
6620 note Gx.-1 480
7100 note G.-1 100
7200 note Gx.-1 240
8640 note F.-1 960
9600 note Gx.-1 720
10320 note Gx.-1 240
10560 note Gx#.-1 140
10700 note Ax.-1 2980

>
> There is a lot that could be said about your post. But I just want
to
> comment on your use of the golden ratio, as a way of unifying
harmony
> and rhythm. I always felt that rhythm in itself must also
> be "tempered", or at least not just plain simple fractions all the
> time. It's interesting that 4 and 3 are the main time signatures,
and
> these multiply out to 12. (Any relation to 12-ET?). Anyway, I
didn't
> know that about swing-eighths, and I'm a jazz musician myself.
Thanks.
> (sqrt(5)+1)/2 and its reciprocal kind of mix geometric and
arithmetic
> means I guess.

The most salient feature of the golden mean here is that it has no
good rational approximations at any scale (it's continued fraction is
1:1:1:1:1...) Thus when the scale is made up of two intervals whose
sizes are related by that ratio, it is heard always as two distinct
interval sizes rather than a single interval size with skips.
Likewise, when a rhythm is made up of two note lengths related by the
golden mean, the ear does not tend to subdivide the longer note in
terms of the other. The result is a fluid beat that does not sound
squared off like the ticking of a metronome.

>
> It would help if someone would give a listing of paultone/meantone
> with period and generators, etc.
>
> Paul Hj
>

It would seem the temperament I'm using is actually called Pajara:

Here's the version based on the golden ratio:

period 600c
generator 106.8c

primes : (2 3 5 7 11 17)
periods : (2 3 5 6 8 8)
generators: (0 1 -1 -1 -6 1)

s = 106.8c
T = 600c-106.8c*4 = 172.8c
172.8c/106.8c=1.6179775...

Bebop scale : TsssssTsss

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:
>
> There is a lot that could be said about your post. But I just want
> to comment on your use of the golden ratio, as a way of unifying
> harmony and rhythm. I always felt that rhythm in itself must also
> be "tempered", or at least not just plain simple fractions all the
> time. It's interesting that 4 and 3 are the main time signatures, and
> these multiply out to 12. (Any relation to 12-ET?).

It is interesting that you raise this question - for quite exactly
this question, whether there is a relation between 3 and 4 being the
main time signatures and 12EDO tuning, is actually the reason that my
latest two compositions are in 5EDO-5/8 rhythm and 17EDO-17/8 rhythm.

I had asked it several years ago here, in connection with a theory of
motif classification. (The thread is:
/tuning-math/message/4616 - the
answers were rather negative then.) In case of interest, I have some
details of the underlying theory on
http://home.datacomm.ch/straub/mamuth/5tet_e.html . This would be
another nice case for Polya enumeration - I have had in mind for some
time to try it out in GAP (not found time yet :-().
--
Hans Straub

--- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@...>
wrote:

> The most salient feature of the golden mean here is that it has no
> good rational approximations at any scale (it's continued fraction
is
> 1:1:1:1:1...) Thus when the scale is made up of two intervals whose
> sizes are related by that ratio, it is heard always as two distinct
> interval sizes rather than a single interval size with skips.
> Likewise, when a rhythm is made up of two note lengths related by
the
> golden mean, the ear does not tend to subdivide the longer note in
> terms of the other. The result is a fluid beat that does not sound
> squared off like the ticking of a metronome.
>
> >
> > It would help if someone would give a listing of
paultone/meantone
> > with period and generators, etc.
> >
> > Paul Hj
> >
>
> It would seem the temperament I'm using is actually called Pajara:
>
> Here's the version based on the golden ratio:
>
> period 600c
> generator 106.8c
>
> primes : (2 3 5 7 11 17)
> periods : (2 3 5 6 8 8)
> generators: (0 1 -1 -1 -6 1)
>
> s = 106.8c
> T = 600c-106.8c*4 = 172.8c
> 172.8c/106.8c=1.6179775...
>
> Bebop scale : TsssssTsss

Cool a "golden pajara" Even though I think your generators should
be (0 1 -2 -2 -6 1)I believe the commas are 64/63 & 50/49 plus
some of the others you mentioned
>

>
> Cool a "golden pajara" Even though I think your generators should
> be (0 1 -2 -2 -6 1)I believe the commas are 64/63 & 50/49 plus
> some of the others you mentioned
> >
>
You are quite right. Thanks for the correction. I must have been half
asleep when I did that tuning map. Actually, now that I think about
it, I prefer a tuning map that uses a more consonant generator: 4/3
rather than 17/16.

This gives:
Period: 600c
Generator: 493.2c

(2 3 5 7 11 17)
(2 4 3 4 2 9)
(0 -1 2 2 6 -1)

Also, while I was messing with tuning maps, I observed that
interesting things happen when one tries to extend this tuning to the
13 limit. The most natural way to do this is to note that 85/84 =
170/168 is already tempered out, so in order to make sure that our
mapping harmonic series remains monotonic in the range of the 169th
harmonic we must temper out 170/169 and 169/168 as well. This splits
the temperament, resulting in one with a generator that is a
tridecimal interval of some sort, such as 16/13. E.g.:

Period: 600c
Generator: 353.4c

(2 3 5 7 11 13 17)
(2 2 5 6 14 8 7)
(0 2 -4 -4 -12 -1 2)
"golden double pajara"?

While theoretically very nice, this temperament clearly complicates
things somewhat. In particular, 12ET 22ET and 56ET are no longer
compatible with the temperament -- they need twice as many notes, and
while it's vaguely possible that a 24 note periodicity block might be
useful, odds are you'll want to use scales where 4/3 is still a
generator in order to render tridecimal harmonies. Fortunately, 10
notes is one possible cardinality, so tridecimal notes can be
rendered with a specialized accidental applied to the notes of the
standard decatonic scale which alters their pitch by 33 cents
(exactly half the size of the accidentals usually used for decatonic
key signatures). These subchromatic alterations would be a good
starting point for writing truly xenharmonic decatonic music.

Another choice is to look for a first order tridecimal comma that we
can temper out. We find that 78/77 doesn't get us in all too much
trouble (all 13 limit intervals are within 20c or so of just.)
However, this temperament puts the 13th harmonic quite far afield in
our tuning map -- 9 generators from a half octave -- this means
tridecimal intervals are unlikely to be melodically meaningful in a
78/77 pajara temperament. Tempering out both 78/77 and 169/168
results in 34 note equal temperament. And since the next more
accurate 169/168 temperaments don't close until 112, 146 and 180
steps respectively, I think this offers a very good argument for
using either 34ET, 34WT, or some other 34 note scale as a pitch set.

>the ratio of a long swing 8th to
>a short one also approximates the golden mean by the way.

Do you have a cite for this?

-Carl

>Here's the version based on the golden ratio:
>
>period 600c
>generator 106.8c
>
>primes : (2 3 5 7 11 17)
>periods : (2 3 5 6 8 8)
>generators: (0 1 -1 -1 -6 1)
>
>s = 106.8c
>T = 600c-106.8c*4 = 172.8c
>172.8c/106.8c=1.6179775...
>
>Bebop scale : TsssssTsss

Paul's 7-limit TOP version has

s = 106.57
T = 172.17
T/s = 1.6155

So, pretty close.

-Carl

--- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@...>
wrote:
>
> >
> > Cool a "golden pajara" Even though I think your generators should
> > be (0 1 -2 -2 -6 1)I believe the commas are 64/63 & 50/49 plus
> > some of the others you mentioned
> > >
> >
> You are quite right. Thanks for the correction. I must have been
half
> asleep when I did that tuning map. Actually, now that I think about
> it, I prefer a tuning map that uses a more consonant generator: 4/3
> rather than 17/16.
>
> This gives:
> Period: 600c
> Generator: 493.2c
>
> (2 3 5 7 11 17)
> (2 4 3 4 2 9)
> (0 -1 2 2 6 -1)
>
> Also, while I was messing with tuning maps, I observed that
> interesting things happen when one tries to extend this tuning to
the
> 13 limit. The most natural way to do this is to note that 85/84 =
> 170/168 is already tempered out, so in order to make sure that our
> mapping harmonic series remains monotonic in the range of the 169th
> harmonic we must temper out 170/169 and 169/168 as well. This
splits
> the temperament, resulting in one with a generator that is a
> tridecimal interval of some sort, such as 16/13. E.g.:
>
> Period: 600c
> Generator: 353.4c
>
> (2 3 5 7 11 13 17)
> (2 2 5 6 14 8 7)
> (0 2 -4 -4 -12 -1 2)
> "golden double pajara"?
>
> While theoretically very nice, this temperament clearly complicates
> things somewhat. In particular, 12ET 22ET and 56ET are no longer
> compatible with the temperament -- they need twice as many notes,
and
> while it's vaguely possible that a 24 note periodicity block might
be
> useful, odds are you'll want to use scales where 4/3 is still a
> generator in order to render tridecimal harmonies. Fortunately, 10
> notes is one possible cardinality, so tridecimal notes can be
> rendered with a specialized accidental applied to the notes of the
> standard decatonic scale which alters their pitch by 33 cents
> (exactly half the size of the accidentals usually used for
decatonic
> key signatures). These subchromatic alterations would be a good
> starting point for writing truly xenharmonic decatonic music.
>
> Another choice is to look for a first order tridecimal comma that
we
> can temper out. We find that 78/77 doesn't get us in all too much
> trouble (all 13 limit intervals are within 20c or so of just.)
> However, this temperament puts the 13th harmonic quite far afield
in
> our tuning map -- 9 generators from a half octave -- this means
> tridecimal intervals are unlikely to be melodically meaningful in a
> 78/77 pajara temperament. Tempering out both 78/77 and 169/168
> results in 34 note equal temperament. And since the next more
> accurate 169/168 temperaments don't close until 112, 146 and 180
> steps respectively, I think this offers a very good argument for
> using either 34ET, 34WT, or some other 34 note scale as a pitch set.

Excellent. What software are you using to do all these calculations?
(Also, does 1729 in your username mean anything?)

Paul Hj

--- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@...>
wrote:
> ...
> Also, while I was messing with tuning maps, I observed that
> interesting things happen when one tries to extend this tuning to
the
> 13 limit. ...
>
> ... Tempering out both 78/77 and 169/168
> results in 34 note equal temperament. And since the next more
> accurate 169/168 temperaments don't close until 112, 146 and 180
> steps respectively, I think this offers a very good argument for
> using either 34ET, 34WT, or some other 34 note scale as a pitch set.

Yep! And since prime 17 was already there, we're now at the 17 limit.

It's interesting to observe that, if you take every other note of 34
to get a 17-tone octave, the only primes that drop out are 5 and 17,
which results in a 13-limit non-5 temperament.

Since you've already mentioned pajara and jazz in the same breath,
you might want to listen to a few bars of a 17-tone jazz piece I
started a "few" years ago (but haven't gotten around to finishing --
yet). Please see the 2nd link in this message:

/makemicromusic/topicId_14872.html#14943

A description of the 9-out-of-17-tone MOS scale subset I used is in
my 17-tone paper (via the 1st link).

--George

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
>
> Excellent. What software are you using to do all these calculations?

Windows calculator, a pen, and paper -- hence the mistakes in my first
tuning map. I useful trick for finding the max error in a high limit
tuning is to look at the sharpest and flatest harmonics in that limit
and take the difference in their errors. In double golden pajara,the
first 19 harmonics are between 0.1c and 17.6c sharp, therefore, no
interval is more than 17.6 cents out of tune in the 19 limit.

> (Also, does 1729 in your username mean anything?)

Yes.
http://en.wikipedia.org/wiki/1729_(number)

>
> Paul Hj
>

I just worked out a few more harmonics in the golden double pajara
temperament. Rather than writing a tuning map here, I'll write the
harmonics as decatonic intervals, because I think it gives much better
musical intuition:

Harmonic, Semitones, Tones, Decatonic name, error
2/1: 8 2 P10 0.0c
3/2: 5 1 P6 +4.8c
5/4: 2 1 M3 +0.1c
7/4: 6 2 M8 +17.6c
9/8: 2 0 m2 +9.6c
11/8: 2 2 A4 +7.9c
13/8: 5.5 1.5 N7 +6.1c
15/8: 7 2 M9 +4.9c
17/16: 1 0 m1 +1.8c
19/16: 0.5 1.5 sA2 +13.1c
21/16: 3 1 P4 +22.4c
23/16: 3.5 1.5 sA6 +4.7c
25/16: 4 2 A6 +0.2c
27/16: 7 1 m8 +14.4c
29/16: 8 1 m9 -2.4c

Note that all the intervals representing these harmonics and their
subharmoics can be decomposed into major (172.8c), minor(106.8), and
neutral (139.8c) steps, and they are therefore fairly likely to be
melodically comprehensible. The fact that the 21st and 29th harmonics
are outside of the range +0.1c to +17.6c error means that harmonies in
these limits have more error than harmonies in the lower limits. In
34Tet, The 19th harmonic also increases the temperament error.
Nonetheless, these harmonics are fairly melodically simple and may
still be useful.

Regarding some of the posts about rhythm. I do think scale structure
affects meter, but I don't believe it's as simple as 12 notes=4*3 meter.

First off, while a 12 note periodicity block is used in common practice
music, which is the most dominated by duple and triple meter, the most
structurally important periodicity block by far is the seven note
diatonic scale. Thus, I would be more inclined to believe the 12=4*3
suggestion if classical music was based on diminished and augmented
scales. I did some experimentation playing various scales ranging from
5 to 12 notes on my and I observed the following basic principles:

1) The scales could be grouped into strong and weak beats, and the
strong beats could be subgrouped into weaker and stronger beats. At
each level of organization, the weak beats outnumber the strong beats
by a factor of 1 or 2, or somwhere in between -- playing in compound
meters. Note that this is "in the neighborhood" of the golden section,
using a physicists definition (I was a physics major once upon a time.)

2) The scale usually ended (a full octave above the first note) on a
strong beat.

3) The strong beats generally form a simpler periodicity block than the
scale as a whole, featuring lower limit consonances. The strongest
beats are usually displaced from one another by a perfect 5th.

I think it is reasonable to assume that a 5 note composition might be
compatible with a 3+2 meter, but I think it would be unreasonable to
assume that a 17 note scale would have anything to do with a 17 beat
rhythmic cycle.

More about the Golden Section and swing eigths --
Sadly, I do not have a reference mentioning swing eighths and the
golden section in the same breath. I was speaking out of general
intuition. I've heard swing eighths described as a Q:E:e = 3:2:1 type
rhythm, but I observed while programming my arrangement of blue monk
that that was too much swing. The golden section seemed at that point
to be the intuitively obvious underlying meter.
I have since looked up "swung note" on wikipedia It gives three
approximations:
2:1:1 -- How it's written. Straight Eighths.
3:2:1 -- How it's usually described to classical musicians who haven't
got a clue how to swing a note. Hard Swing.
5:3:2 -- How it's actually played more or less.

These certainly look like convergents to the golden section. I also
read a number of the referenced articles where they actually measured
swing eighths in jazz recordings. It seems the situation is a good deal
more complicated than the simple model presented above. In
particularly, the drummer plays a different swing ratio than the
soloist, and both ratios are dependent upon tempo. Here's what I think
is actually going on:

At mid to slow tempo (In the ballpark of a heartbeat -- again in the
physics sense)
The soloist plays notes in the golden section
The drummer's upbeat subdivides the short eighth note in the golden
section -- closer to the soloist's upbeat than his downbeat.

At faster tempos, the shorter notes grow as a proportion of the whole
sequence, because the human ear cannot distinguish beats that are much
closer together than about 100ms.

--- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@...>
wrote:
>
> Regarding some of the posts about rhythm. I do think scale
structure
> affects meter, but I don't believe it's as simple as 12 notes=4*3
meter.
>
> First off, while a 12 note periodicity block is used in common
practice
> music, which is the most dominated by duple and triple meter, the
most
> structurally important periodicity block by far is the seven note
> diatonic scale. Thus, I would be more inclined to believe the
12=4*3
> suggestion if classical music was based on diminished and augmented
> scales. I did some experimentation playing various scales ranging
from
> 5 to 12 notes on my and I observed the following basic principles:
>
> 1) The scales could be grouped into strong and weak beats, and the
> strong beats could be subgrouped into weaker and stronger beats. At
> each level of organization, the weak beats outnumber the strong
beats
> by a factor of 1 or 2, or somwhere in between -- playing in
compound
> meters. Note that this is "in the neighborhood" of the golden
section,
> using a physicists definition (I was a physics major once upon a
time.)
>
> 2) The scale usually ended (a full octave above the first note) on
a
> strong beat.
>
> 3) The strong beats generally form a simpler periodicity block than
the
> scale as a whole, featuring lower limit consonances. The strongest
> beats are usually displaced from one another by a perfect 5th.
>
> I think it is reasonable to assume that a 5 note composition might
be
> compatible with a 3+2 meter, but I think it would be unreasonable
to
> assume that a 17 note scale would have anything to do with a 17
beat
> rhythmic cycle.
>
> More about the Golden Section and swing eigths --
> Sadly, I do not have a reference mentioning swing eighths and the
> golden section in the same breath. I was speaking out of general
> intuition. I've heard swing eighths described as a Q:E:e = 3:2:1
type
> rhythm, but I observed while programming my arrangement of blue
monk
> that that was too much swing. The golden section seemed at that
point
> to be the intuitively obvious underlying meter.
> I have since looked up "swung note" on wikipedia It gives three
> approximations:
> 2:1:1 -- How it's written. Straight Eighths.
> 3:2:1 -- How it's usually described to classical musicians who
haven't
> got a clue how to swing a note. Hard Swing.
> 5:3:2 -- How it's actually played more or less.
>
> These certainly look like convergents to the golden section. I also
> read a number of the referenced articles where they actually
measured
> swing eighths in jazz recordings. It seems the situation is a good
deal
> more complicated than the simple model presented above. In
> particularly, the drummer plays a different swing ratio than the
> soloist, and both ratios are dependent upon tempo. Here's what I
think
> is actually going on:
>
> At mid to slow tempo (In the ballpark of a heartbeat -- again in
the
> physics sense)
> The soloist plays notes in the golden section
> The drummer's upbeat subdivides the short eighth note in the golden
> section -- closer to the soloist's upbeat than his downbeat.
>
> At faster tempos, the shorter notes grow as a proportion of the
whole
> sequence, because the human ear cannot distinguish beats that are
much
> closer together than about 100ms.

Thanks. Bringing rhythm into all this theory is important I think.
(I would also like to bring in more voice-leading and counterpoint,
or at least more with cadences and simple chord progressions). My
whole thing with 4 X 3 relates to the Kronecker decomposition of
12. Of course 7 and 5 note scales are important, but even they
can be broken into 4 + 3 and 3 + 2 etc. There is a lot of theory
on this newsgroup relating to MOS's and so forth which it looks like
you are already familiar with. 7 and 5 come into play of course
as the steps in a P5 and P4. I think this can be tied into the
number of black and white keys on the piano (diatonic and pentatonic
scales).

If you take trapezoid approximations of the Int_(1,2) 1/x
broken into two pieces you obtain 7/12 for log(2)(3/2) and 5/12
for the log(2)(4/3) and (24/35) for the ln(2) (actually get
2/5 for ln(3/2) and 2/7 for ln(4/3) and divide each of these
by ln(2)=~24/35). It's a little crude but it works. 35 makes an
appearance here, also in Gene's 7-limit lattice with coordinates
of 15/14, 21/20 and 35/24, also there are 35 hexachord types
and 35 pentachord types based on unique interval vectors. (Reduced
for Z-relation, but in the case of hexachords, it is partitions)
Well enough numerology, but I thought I would bring up a few points.

Oh yeah the Ramanujan license plate story. I knew that.

--- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@...> wrote:
>
> !It occurred to me that it would be a good idea to post the midi
> !files I've been messing around with, but I can't figure out how to
> !do that, so I just posted the SCL file. It does the right thing in
> !12, 34, 56, 100, or 22 equal, but I can't vouch for anything else.
> !I'm vaugely hoping someone will turn it into a sound file.

I can do that. Is there any reason not to put the result up on my web
site for downloading? What codec do you prefer?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@> wrote:
> >
> > !It occurred to me that it would be a good idea to post the midi
> > !files I've been messing around with, but I can't figure out how to
> > !do that, so I just posted the SCL file. It does the right thing in
> > !12, 34, 56, 100, or 22 equal, but I can't vouch for anything else.
> > !I'm vaugely hoping someone will turn it into a sound file.
>
> I can do that. Is there any reason not to put the result up on my web
> site for downloading? What codec do you prefer?
>
I certainly don't object to your posting the file. The tune and chords
belong to Thelonius Monk of course, so if you were really paranoid you
might want to ask the Monk estate before posting the file to your site -
- but you probably don't need to be that paranoid.

As far as codec, I have no strong preference. The timbres in the midis
I have are kind of ugly sounding, so if you can get better ones, and
maybe make an mp3, that might be good. I'm not much of a digital audio
geek, so if you can get something that sounds good and will play on my
computer I'll be happy.

--- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@...>
wrote:

> > Cool a "golden pajara" Even though I think your generators should
> > be (0 1 -2 -2 -6 1)I believe the commas are 64/63 & 50/49 plus
> > some of the others you mentioned
> > >
> >
> You are quite right. Thanks for the correction.

There are two basic approaches to extending pajara to the 11-limit,
both supported by 22 edo. Tempering out 50/49, 64/63 and 55/54
("pajaric") gives a wedgie <<2 -4 -4 10 -11 -12 9 2 37 42|| and works
with a sharper fifth, and tempering out 50/49, 64/63 and 99/98 leads
to the wedgie <<2 -4 -4 -12 -11 -12 -26 2 -14 -20|| and a flatter
optimal fifth. Even so this isn't supported by the nearest-integer
("patent val") tuning for 56 or 34, but you can use <56 89 130 158
194| and <34 54 79 96 118| for that instead.

An interesting alternative ("pajarous") is to temper out 45/44 (and
therefore also 56/55) instead, using an even flatter fifth. This
could be described (in terms of patent vals which support it) as
10&12 instead of 12&22. The 56-val for this would be <56 89 130 158
196| and the 34-val <34 54 79 96 119|. This gets way more out of tune
for 11 than I would ever want to consider, but Herman could take a
look and see if he thought it was usable.

The various ways of extending to the 11-limit lead to various
versions of 13-limit pajara, in particular by adding 65/64 as a
comma. We then can get various 17-limit versions by adding in 85/84.

> While theoretically very nice, this temperament clearly complicates
> things somewhat.

I think it's pretty complex for the tuning accuracy here.

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

Aside from general accuracy, I like 99/98 much better than 55/54.
First, minor and neutral thirds sound quite different and should be
rendered as such whenever possible, and second, 100/98 is already
tempered out, so not tempering out 99/98 and 100/99 makes the
harmonic series nonmonotonic earlier than it needs to be, which
really rubs me the wrong way. Finally, the jazz guys call the 11th
harmonic a sharp 11th, not a flat tritone -- this is consistent with
the 99/98 temperament in that the 11th harmonic is treated as an
alteration of the perfect 4th rather than the tritone when it is
played in a melody.

As far as tempering out 169/168 and 247/245, it doesn't actually make
the temperament as complicated as it might seem. If, rather than
thinking of the temperament as an interpretation of equal
temperaments, we think of it as a staight linear temperament, we have
a ten note scale and accidentals (see my other post
/tuning-math/message/15966 -- I
messed up the decatonic note name for the 23rd harmonic it should
read sA5 not sA6.) Anyway, you have a scale with only 10 notes,
within which the first 29 harmonics are all proper intervals, and all
19 limit harmonies are tuned within 17.6c error -- I doubt there is a
19 limit temperament which improves on all of these factors.

>
> --- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@>
> wrote:
>
> > > Cool a "golden pajara" Even though I think your generators
should
> > > be (0 1 -2 -2 -6 1)I believe the commas are 64/63 & 50/49 plus
> > > some of the others you mentioned
> > > >
> > >
> > You are quite right. Thanks for the correction.
>
> There are two basic approaches to extending pajara to the 11-limit,
> both supported by 22 edo. Tempering out 50/49, 64/63 and 55/54
> ("pajaric") gives a wedgie <<2 -4 -4 10 -11 -12 9 2 37 42|| and
works
> with a sharper fifth, and tempering out 50/49, 64/63 and 99/98
leads
> to the wedgie <<2 -4 -4 -12 -11 -12 -26 2 -14 -20|| and a flatter
> optimal fifth. Even so this isn't supported by the nearest-integer
> ("patent val") tuning for 56 or 34, but you can use <56 89 130 158
> 194| and <34 54 79 96 118| for that instead.
>
> An interesting alternative ("pajarous") is to temper out 45/44 (and
> therefore also 56/55) instead, using an even flatter fifth. This
> could be described (in terms of patent vals which support it) as
> 10&12 instead of 12&22. The 56-val for this would be <56 89 130 158
> 196| and the 34-val <34 54 79 96 119|. This gets way more out of
tune
> for 11 than I would ever want to consider, but Herman could take a
> look and see if he thought it was usable.
>
> The various ways of extending to the 11-limit lead to various
> versions of 13-limit pajara, in particular by adding 65/64 as a
> comma. We then can get various 17-limit versions by adding in 85/84.
>
> > While theoretically very nice, this temperament clearly
complicates
> > things somewhat.
>
> I think it's pretty complex for the tuning accuracy here.
>

> Since you've already mentioned pajara and jazz in the same breath,
> you might want to listen to a few bars of a 17-tone jazz piece I
> started a "few" years ago (but haven't gotten around to finishing --
> yet). Please see the 2nd link in this message:
>
> /makemicromusic/topicId_14872.html#14943
>
> A description of the 9-out-of-17-tone MOS scale subset I used is in
> my 17-tone paper (via the 1st link).
>
> --George
>
Cool. Distinctly jazzy but very alien. I think a good bit about what
makes it sound so unusual is not the presence of 13 limit harmonies,
but the absence of 5 limit harmonies. A bit like the way the sound of
the wholetone scale in 12 equal is defined by the absence of 3 limit
harmonies (perfect 5ths) rather than the presence of 5, 7, and 11 limit
harmonies.

I assume you've also heard paul erlich's decatonic swing, but just in
case you haven't, here it is. This would tend to indicate that he is
aware that pajara temperaments are compatible with jazz.
http://lumma.org/tuning/erlich/decatonic-swing.mp3

--- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@...> wrote:
>
> !It occurred to me that it would be a good idea to post the midi
> !files I've been messing around with, but I can't figure out how to
> !do that, so I just posted the SCL file. It does the right thing in
> !12, 34, 56, 100, or 22 equal, but I can't vouch for anything else.
> !I'm vaugely hoping someone will turn it into a sound file. Bonus
> !points for finding less computerized sounding timbres. Many thanks to
> !Thelonius Monk for providing the melody and (hopefully) not suing me.

Here you are:

http://www.xenharmony.org/mp3/misc/blue22.mp3

http://www.xenharmony.org/mp3/misc/bluemonk.mp3

For reasons unknown to me, in a few places a C raised by a quartertone
(single step of 22) is replaced by a C raised by 45/44. Hence I give
two versions, one with the JI quartertones, and one in 22-et. I didn't
produce a 56-et version because I was unsure of the claim that this
gives the correct values. Ideally, one would like a pajara-based
notation with assignable values for the generators.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@>
wrote:
> >
> > !It occurred to me that it would be a good idea to post the midi
> > !files I've been messing around with, but I can't figure out how
to
> > !do that, so I just posted the SCL file. It does the right thing
in
> > !12, 34, 56, 100, or 22 equal, but I can't vouch for anything
else.
> > !I'm vaugely hoping someone will turn it into a sound file. Bonus
> > !points for finding less computerized sounding timbres. Many
thanks to
> > !Thelonius Monk for providing the melody and (hopefully) not
suing me.
>
> Here you are:
>
> http://www.xenharmony.org/mp3/misc/blue22.mp3
>
> http://www.xenharmony.org/mp3/misc/bluemonk.mp3
>
> For reasons unknown to me, in a few places a C raised by a
quartertone
> (single step of 22) is replaced by a C raised by 45/44. Hence I
give
> two versions, one with the JI quartertones, and one in 22-et. I
didn't
> produce a 56-et version because I was unsure of the claim that this
> gives the correct values. Ideally, one would like a pajara-based
> notation with assignable values for the generators.
>

My impression was that the square bracket notation chooses the scale
tone closest to the just interval and not the just interval itself.
The note I was intending to write was a lowered C# rather than a
raised C. The problem is, the notation was rendering notes in 22
equal and then finding the closest step of 34 or 56 equal. Since the
56ET and 34ET semitones are narrower than the 22ET semitones, when I
typed "equal 56" "example bluemonk.scl" it was choosing the note 3/56
above C rather than the one 2/56 above C. This was (rather audibly)
the wrong note. What I did was kludgy but if the program performs as
advertised, it should render all the notes correctly in either 34 or
56 tone equal temperament (or 12 for that matter). I agree it would
be better to have a generalized pajara notation to do this, but it
was faster to do what I did rather than figure out how to code one up.

Sorry for the confusion, and thanks for posting the mp3s.

--- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@...> wrote:

> My impression was that the square bracket notation chooses the scale
> tone closest to the just interval and not the just interval itself.

Ah. I didn't know that, I thought it was interpreting it as
parenthesis. Well, now I don't feel bad I couldn't tell the difference.

> The note I was intending to write was a lowered C# rather than a
> raised C. The problem is, the notation was rendering notes in 22
> equal and then finding the closest step of 34 or 56 equal. Since the
> 56ET and 34ET semitones are narrower than the 22ET semitones, when I
> typed "equal 56" "example bluemonk.scl" it was choosing the note 3/56
> above C rather than the one 2/56 above C.

How does it even decide what a semitone is? I've never understood
Manuel's notation scheme, and don't trust it to give the result I want
in cases like this, therefore.

[ Attachment content not displayed ]

--- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@...> wrote:
>
> !It occurred to me that it would be a good idea to post the midi
> !files I've been messing around with, but I can't figure out how to
> !do that,

I'd like to listen to these. If you click on the 'Files' section of
this group, you can create a folder and save them there (in this path):

/tuning-math/files/

There's not much room left there, but probably enough for your midi
files.

--George

[ Attachment content not displayed ]

At 03:09 PM 1/24/2007, you wrote:
>!It occurred to me that it would be a good idea to post the midi
>!files I've been messing around with, but I can't figure out how to
>!do that, so I just posted the SCL file.

One nitpick: this is in fact a .seq (Scala sequence) file.

-Carl

On 28/01/07, Gene Ward Smith <genewardsmith@coolgoose.com> wrote:

> There are two basic approaches to extending pajara to the 11-limit,
> both supported by 22 edo. Tempering out 50/49, 64/63 and 55/54
> ("pajaric") gives a wedgie <<2 -4 -4 10 -11 -12 9 2 37 42|| and works
> with a sharper fifth, and tempering out 50/49, 64/63 and 99/98 leads
> to the wedgie <<2 -4 -4 -12 -11 -12 -26 2 -14 -20|| and a flatter
> optimal fifth. Even so this isn't supported by the nearest-integer
> ("patent val") tuning for 56 or 34, but you can use <56 89 130 158
> 194| and <34 54 79 96 118| for that instead.
>
> An interesting alternative ("pajarous") is to temper out 45/44 (and
> therefore also 56/55) instead, using an even flatter fifth. This
> could be described (in terms of patent vals which support it) as
> 10&12 instead of 12&22. The 56-val for this would be <56 89 130 158
> 196| and the 34-val <34 54 79 96 119|. This gets way more out of tune
> for 11 than I would ever want to consider, but Herman could take a
> look and see if he thought it was usable.
>
> The various ways of extending to the 11-limit lead to various
> versions of 13-limit pajara, in particular by adding 65/64 as a
> comma. We then can get various 17-limit versions by adding in 85/84.
>
> > While theoretically very nice, this temperament clearly complicates
> > things somewhat.
>
> I think it's pretty complex for the tuning accuracy here.

I was browsing my temperament-finder output for stupidly high prime
limits some time ago, and I'm sure I saw a lot of Pajaras getting in
there. But as this net bar won't let me access my own website I can't
check it now. There's certainly a diaschismic temperament with the
58&46 mapping of 7 (but not all primes, I think 17 or 19 is different)
that's a stand-out in the 31 limit.

In practice, maybe the errors would be too high for all intervals to
be recognized, but only practice can demonstrate that.

Graham

--- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@...> wrote:
>
> !It occurred to me that it would be a good idea to post the midi
> !files I've been messing around with, but I can't figure out how to
> !do that, ...

I answered this yesterday, but it didn't show up (yet).

I'd love to listen to your midi files. There's only a limited amount
of space available in the "Files" section of this group, but since
midi's are quite small that shouldn't be a problem. Why don't you
create a folder there and upload them.

--George

I wrote:

> I was browsing my temperament-finder output for stupidly high prime
> limits some time ago, and I'm sure I saw a lot of Pajaras getting in
> there. But as this net bar won't let me access my own website I can't
> check it now. There's certainly a diaschismic
...

Oops. I've got a new website now, haven't I? So I apologize to the
owners of this net bar. The relevant script is at

http://x31eq.com/temper/regular.html

I found this extension of Pajara

34&12
598.586 cents period
105.765 cents generator
106.015 cents generator for pure octaves

Period and Generator Mappings
2 3 5 7 11 13 17 19 23 29 31
< 2, 3, 5, 6, 8, 9, 8, 8, 11, 11, 11 ]
< 0, 1, -2, -2, -6, -9, 1, 3, -11, -7, -6 ]

Constituent Equal Temperaments
2 3 5 7 11 13 17 19 23 29 31
< 34, 54, 79, 96, 118, 126, 139, 145, 154, 166, 169 ]
< 12, 19, 28, 34, 42, 45, 49, 51, 55, 59, 60 ]

Complexity 6.277
RMS Weighted Error 2.214 cents/octave

To find it, search for

31
7
25
50
50
0.07

I spotted one more Pajara in the output. The specific results are
very fragile at such a high limit so it does depend on what you search
for.

Graham

--- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@...>
wrote:
> [GS:]
> > Since you've already mentioned pajara and jazz in the same
breath,
> > you might want to listen to a few bars of a 17-tone jazz piece I
> > started a "few" years ago (but haven't gotten around to
finishing --
> > yet). Please see the 2nd link in this message:
> >
> > /makemicromusic/topicId_14872.html#14943
> >
> > A description of the 9-out-of-17-tone MOS scale subset I used is
in
> > my 17-tone paper (via the 1st link).
> >
> > --George
> >
> Cool. Distinctly jazzy but very alien.

Hmmm, alien jazz, huh? I like that label. It's too bad that John
Williams didn't know about this tuning. He could have used it in the
cantina scene near the beginning of the first (Episode IV) Star Wars
film. ;-)

> I think a good bit about what
> makes it sound so unusual is not the presence of 13 limit
harmonies,
> but the absence of 5 limit harmonies. A bit like the way the sound
of
> the wholetone scale in 12 equal is defined by the absence of 3
limit
> harmonies (perfect 5ths) rather than the presence of 5, 7, and 11
limit
> harmonies.

I think it's both, but I have to insist that the presence of 11
and/or 13 is essential to its strangeness. If you eliminate prime 5
from the 5-limit, then you're left with 3-limit (or Pythagorean)
harmony, which occurs in my sample in septimal form (because of the
wide 5ths) as subminor triads in the opening bars. Those 7-limit
triads, in and of themselves, are not what makes it alien-sounding.
Rather, it's that their root tones are in a chain of neutral 2nds
(involving primes 11 and/or 13), so *initially* it's the *melodic*
rather than the harmonic element that's unusual.

It's not until after the first 8 bars that you begin to hear 11 & 13
*harmony*, and only after the 16th bar does this occur as block
chords. I originally had 6:7:9:11 chords (tempered) in the opening
bars, but I quickly dropped the 11, because it would have been too
weird too soon. It was better to introduce the 11's and 13's
gradually, first melodically, then harmonically.

> I assume you've also heard paul erlich's decatonic swing,

Yes.

> but just in
> case you haven't, here it is. This would tend to indicate that he
is
> aware that pajara temperaments are compatible with jazz.
> http://lumma.org/tuning/erlich/decatonic-swing.mp3

Yes, and he also offered a brief explanation why; see
/tuning/topicId_4604.html#4604
specifically, the following paragraph:

<< Here's where the cultural/historical aspect of my theory comes in.
In medieval music, only fourths and fifths were consonances. Ever
since thirds and sixths took over that role in the Renaissance, and
triadic texture became the norm for stability, a bare fourth or fifth
would be an interruption that texture, a sonority with lower tension
than that associated with the point of repose. "Too plain" is an apt
description. In jazz, triads are "too plain." In my decatonic system,
7-limit tetrads become the norm, the point of lowest tension. But I
don't think "plainness" takes away from "consonance". What could be
more consonant that plain old octaves and unisons? >>

Also note:
/tuning/topicId_11494.html#11494
which contains the following paragraph:

<< _However_, I part with both you an Blackwood in that I see some
fluidity in what constitutes consonance, depending on musical style.
In Gothic music, for example, a triad was felt to be dissonant, and
had to resolve to a 3-limit dyad. In blues and rock the dominant
seventh can be used for all the chords in the I-IV-I-V-IV-I
progression, and it has no need to resolve to a triad. Similarly,
much latin jazz in a mixed-minor mode ends on a minor chord with
added major sixth. In this spirit, my paper (http://www-
math.cudenver.edu/~jstarret/22ALL.pdf) proposes a new type of "tonal"
music where approximations to 4:5:6:7 and 1/7:1/6:1/5:1/4 tetrads
can, with enough exposure, become recognized as the
canonical "consonances", to which other chords would resolve, and
compared to which, triads would sound incomplete. >>

I need to listen to your bluemonk files before making any more
comments.

--George

To test the 13 and 19 limit mappings of my tuning, I added a new
arrangemt of blue monk with some slight alterations featuring the 13
and 19 limits.

At 02:11 AM 1/30/2007, you wrote:
>To test the 13 and 19 limit mappings of my tuning, I added a new
>arrangemt of blue monk with some slight alterations featuring the 13
>and 19 limits.

Where? I liked the previous one quite a bit.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> At 02:11 AM 1/30/2007, you wrote:
> >To test the 13 and 19 limit mappings of my tuning, I added a new
> >arrangemt of blue monk with some slight alterations featuring the
13
> >and 19 limits.
>
> Where? I liked the previous one quite a bit.
>
> -Carl
>
Thanks. Same folder. It's called bluemonk19lim or something -- not too
different, I just added a couple licks in the 1st, 2nd, and 6th
measure to see how the 19s and 13s sounded, and I took a out a
dissonance that was in the 9th measure, because it ended up sounding
unnecessary in the new version.

>> >To test the 13 and 19 limit mappings of my tuning, I added a new
>> >arrangemt of blue monk with some slight alterations featuring the
>> >13 and 19 limits.
>>
>> Where? I liked the previous one quite a bit.
>
>Thanks. Same folder. It's called bluemonk19lim or something -- not too
>different, I just added a couple licks in the 1st, 2nd, and 6th
>measure to see how the 19s and 13s sounded, and I took a out a
>dissonance that was in the 9th measure, because it ended up sounding
>unnecessary in the new version.

Cool. I think I liked the first version better, but the
difference wasn't huge.

-Carl

Here's a summary of the scales I've been using, described using the
more systematic variant of my E blues notation.
(see /tuning-math/message/15979)

A couple important features of the 34 note mode:

1)In the 34 note scale, the notes of the decatonic scale that are
chromatically altered in both directions are the blue 3rd and 7th
traditionally associated with the blues style.

2)The 34 note scale contains the first 27 harmonics.

George. Does this resemble any of the modes of your 34WT?

34 note scale:
E E^ E+

Gb- Gbv Gb
G- Gv G G^ G+
Ab Ab^ Ab+
A A^ A+
Bb Bb^ Bb+
B B^ B+

Db- Dbv Db
D- Dv D D^ D+
Eb Eb^ Eb+

12 note scale:
E Gb- Gb G Ab A Bb B B+ Db D Eb

10 note scale:
E Gb G Ab A Bb B Db D Eb

5 note mode:
E G A B D

Harmonic series:
E B Ab D G- A+ Dbv Eb Gb- G^ A Bb^ B+ D-

--- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@...>
wrote:
>
> Here's a summary of the scales I've been using, described using the
> more systematic variant of my E blues notation.
> (see /tuning-math/message/15979)
>
> A couple important features of the 34 note mode:
>
> 1)In the 34 note scale, the notes of the decatonic scale that are
> chromatically altered in both directions are the blue 3rd and 7th
> traditionally associated with the blues style.
>
> 2)The 34 note scale contains the first 27 harmonics.
>
> George. Does this resemble any of the modes of your 34WT?

I'm not sure that I understand your question. 34-WT is a tuning, not
a scale. However, it does contain many different scales, each of
which may have various modes (or starting tones). Anyway, I haven't
used 34 for very long, so I've hardly begun to scratch the surface.

> 34 note scale:
> E E^ E+

I must apologize that I've gotten very confused trying follow your
notation, for a couple of reasons:

1) You gave these symbol definitions (in message #15979):

> Start with the E minor pentatonic scale tuned in pajara fourths: B
E A D G
> Define the modifier, b, as lowering the note by a half octave minus
a pajara fourth.
> Define # as raising a note by the same.
>
> Define the modifier ^ as raising a note by two and a half pajara
fourths and
> lowering by an octave.
> Define the modifier v as lowering by the same.
> Define the modifier + as ^^.
> Define the modifier - as vv.

This does not produce a generalized pajara notation. It's impossible
to take 2-1/2 pajara fourths in *all pajara tunings*, because the
fourth is not necessarily an even number of degrees, e.g., 9 degrees
in 22-ET. (Or have I misunderstood what you mean by a "pajara
fourth".)

So I had to assume that you're defining these symbols in terms of 34-
ET, which leads to my next problem:

2) Your up-down symbol pairs are nice, but they're completely
opposite to how I've been accustomed to using them over the past
several years (so I can't help but be confused). This is how I've
been notating 34:

34-division sequence:
Eb Eb/ Ev E\ E E/ E^ (Sagittal ASCII shorthand), or
Eb Eb+ Ev E- E E+ E^ (Sagittal-Wilson ASCII shorthand)

For the past 5 years, Dave Keenan & I have been working on a
comprehensive generalized microtonal notation system that's capable
of notating practically any tuning (either just or tempered). It's
generalized in the sense that the symbols have constant meanings
(i.e., they're defined in such a way that they have the same harmonic
meanings, regardless of the tunings in which they're used). So you
don't have to learn (or even invent) a new set of symbols for a new
tuning, and there will be no conflict in the meaning of any
particular symbol from any one tuning to any another.

In Sagittal, these symbols have the following meanings:

down up pitch alteration
---- -- ----------------
b # apotome (2048:2187)
\ / 5-comma (80:81)
- + 5-comma (80:81) Sagittal-Wilson option
v ^ 11-diesis (32:33)

(The Sagittal-Wilson 5-comma symbol pair has the same meaning as the
regular Sagittal 5-comma pair; the difference is merely cosmetic, for
reasons explained in our documentation.)

In the 34 division, the apotome equates to 4 degrees, the 5-comma 1
degree, and the 11-diesis 2 degrees. In 22-ET, these are 3deg, 1deg,
& 1deg, respectively. In 12-ET, these are 2deg, 0deg, and not
applicable.

If you're interested in further details, you can read our formal
introductory paper (the Xenharmonikon 18 article):
http://dkeenan.com/sagittal/Sagittal.pdf
or browse through the other goodies on the Sagittal website:
http://dkeenan.com/sagittal/

There's a scalable symbol font (available free of charge), and the
introduction to the mythology contains a brief rationale for the
existence of alternative tunings.

Anyway, I looked at some of what you wrote, but I'm not sure that I'm
interpreting it correctly:

> ...
> 12 note scale:
> E Gb- Gb G Ab A Bb B B+ Db D Eb
>
> 10 note scale:
> E Gb G Ab A Bb B Db D Eb

The 10-note scale doesn't seem to be a decatonic pajara scale.

> 5 note mode:
> E G A B D

I thought that this next one might be a sequence of odd harmonics:

> Harmonic series:
> E B Ab D G- A+ Dbv Eb Gb- G^ A Bb^ B+ D-

I thought that the 5th one in the sequence (the 9th harmonic) should
be F#, but it isn't, so I don't follow this. Sorry!

--George

Sorry for the confusion.

First off, this notation is not designed specifically with 34-ET or
any ET in mind for that matter. It works for 34,78,112,146,180...
I don't recommend 34 equal, because the 34 note scale is meant to have
two distinct step sizes. I was wondering if the pattern of long and
short steps in my 34 note scale corresponded in any way to your 34
note well temperament.

The notation I described has a number of redundancies, so a lot of
notes have various enharmonically equivalent names. To get a unique
name, make sure the note has no sharps, no more than one flat, no more
than one of ^ or v, and shares a letter name with a note in the
E-minor pentatonic scale. F#, for example, would be written as G-.

To avoid confusion, I'll just write the 34-note scale in L and s notation:

ssL ssL ssssL ssL ssL ssL ssL ssL ssssL ssL.

The short step represents 40/39 = 65/64. It's around 33 cents.
The long step represents 36/35 = 81/80. It's around 40.8 cents.

I tried to come up with a notation that showed the structure of the
various scales while corresponding to classical notation as much as
possible. I started with the pentatonic scale, because that is the
simplest, most recognizable feature of the tuning.
Hence, I only use 5 letters unless I'm being lazy when trying to
transcribe something from 12-tet. The next modifier I added was the
flat. It pretty much just toggles between the two cycles of fifths,
although it is reduced by a fourth from the half octave, both to
correspond more closely with classical notation, and to make the
modification small and manageable. You can use sharps too, but they're
not really necessary. Jazz tends to use flats more often so I just
stuck with the flats. The additional modifiers were chosen so that
the modified note and the unmodified note are enharmonically
equivalent in 10TET (just like A and A# are equivelant in 7tet). A
syntonic comma is a bad choice for a modifier, for example, because
5/4 is represented by 3 steps in a decatonic scale, while 81/64 is
represented by 4. It's really a very elegant notation once you get
used to it. I feel strongly about the definition of # and b, but feel
free to use some other set of symbols for ^v+- if you feel like it.

Hi George and Ray,

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@>
> wrote:
> >
> > Hi. I'm new here, so I may not have all the terms right,
> > but I'd like to make some observations about the linear
> > temperament used for Paul Ehrlich's Decatonic scales.

His surname is "Erlich" ... this is a rather common error.

> We now call it 'pajara'.
>
> > Paul originally conceived this system for 22 tone
> > equal temperament which has 709 cent 5ths, I think
> > I saw a message suggesting 34 tone equal temperament
> > (706 cent 5ths.) Anyway, no one seems to have noticed
> > that the two basic unison vectors 64/63, 50/49 are
> > also consistent with the common as dirt 12 tone equal
> > temperament (700 cent 5ths.)
>
> I know for a fact that Paul is well aware of this,
> inasmuch as he wrote the following to me (off-list)
> a couple of years ago:
>
> << As you know, Pajara is essentially the same system
> proposed in my XH17 paper (though there it was tuned
> essentially in 22-equal). The 'unfortunate' thing about
> Pajara from a xenharmonic standpoint is that its vanishing
> ratios -- 50:49, 64:63, 225:224, 2048:2025, . . . -- all
> vanish in 12-equal. So one can't use this system to create
> cyclic chord progressions which would be 'impossible' in
> 12-equal -- hence it's not really as subversive as many
> xenharmonic composers would wish their tuning system
> to be. >>

I find this an interesting line of thought, because
i'm fascinated with exactly the opposite idea: the
way certain temperaments share unison-vectors, and
are thus members of the same family.

In other words, the way that the use of pajara in 12-edo
could serve as a stepping-stone to the use of pajara
in 22-edo, which in turn opens the door to working with
temperament families of which 22-edo is a member but
12-edo is *not*.

One of my favorite papers which explores this idea,
and probably one of the earliest, is Erv Wilson's
"On the Development of Intonational Systems by
Extended Linear Mapping", originally published in
Xenharmonikon 3 (1975):

http://www.anaphoria.com/xen3b.PDF

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@...>
wrote:
>
> Sorry for the confusion.

No apology necessary -- the confusion is all mine. ;-)

> First off, this notation is not designed specifically with 34-ET or
> any ET in mind for that matter. It works for 34,78,112,146,180...

So it's not a pajara notation, then.

> I don't recommend 34 equal, because the 34 note scale is meant to
have
> two distinct step sizes. I was wondering if the pattern of long and
> short steps in my 34 note scale corresponded in any way to your 34
> note well temperament.

Okay, now I understand your question. I doubt that this would be the
case, but you can examine the cents values in this Scala listing (but
first see my quick test, below, which shows that it doesn't):

! secor_34wt.scl
!
George Secor's 34-tone well temperament with 10 pure 11/7 and 6 near
just 11/6
34
!
40.47925
66.74120
107.22045
144.85624
171.11819
214.44090
249.23324
278.33864
321.66136
353.61023
385.55910
428.88181
457.98722
492.77955
536.10226
562.36421
600.00000
640.47925
666.74120
707.22045
744.85624
771.11819
814.44090
849.23324
878.33864
921.66136
953.61023
985.55910
1028.88181
1057.98722
1092.77955
1136.10226
1162.36421
2/1

> The notation I described has a number of redundancies, so a lot of
> notes have various enharmonically equivalent names.

Yes, that can be very useful.

> To get a unique
> name, make sure the note has no sharps, no more than one flat, no
more
> than one of ^ or v, and shares a letter name with a note in the
> E-minor pentatonic scale. F#, for example, would be written as G-.
>
> To avoid confusion, I'll just write the 34-note scale in L and s
notation:
>
> ssL ssL ssssL ssL ssL ssL ssL ssL ssssL ssL.

Since my 34-WT consists of two 17-tote circles of fifths 600 cents
apart, then a transposition of the tuning by 600 cents (either up or
down) will be identical to the original (assuming octave equivalence).

This is not true of two modes of your 34-note scale, starting 17
tones apart:

starting on tone 1: ssLssLssssLssLssLssLssLssLssssLssL
starting on tone 18: ssLssLssLssssLssLssLssLssssLssLssL

> The short step represents 40/39 = 65/64. It's around 33 cents.
> The long step represents 36/35 = 81/80. It's around 40.8 cents.

Since you have a factor of 13 in there, and since pajara isn't
generally supportive of 13, then I would have to conclude that, at
best, you're describing only a special case of pajara.

> I tried to come up with a notation that showed the structure of the
> various scales while corresponding to classical notation as much as
> possible. I started with the pentatonic scale, because that is the
> simplest, most recognizable feature of the tuning.
> Hence, I only use 5 letters unless I'm being lazy when trying to
> transcribe something from 12-tet. The next modifier I added was the
> flat. It pretty much just toggles between the two cycles of fifths,
> although it is reduced by a fourth from the half octave, both to
> correspond more closely with classical notation, and to make the
> modification small and manageable. You can use sharps too, but
they're
> not really necessary. Jazz tends to use flats more often so I just
> stuck with the flats.

I (and on this point I speak for many others in this group, including
Paul Erlich) strongly discourage the use of flat or sharp symbols to
mean anything other than an apotome, i.e., an alteration
corresponding to a 7-position displacement along a chain of fifths.
What you're advocating goes contrary to centuries of common practice,
and to do something like that virtually guarantees confusion.

Since you're dealing with two chains of fifths, I would therefore
advise using some other symbol pair. Sagittal shorthand notation
does not have a single character in this particular instance, but it
does offer the combination pairs b/ and #\, which *do* in fact
represent single symbols in the pure Sagittal font, !!/ and ||\ (but,
again, I require multiple ASCII characters to represent them here).

> The additional modifiers were chosen so that
> the modified note and the unmodified note are enharmonically
> equivalent in 10TET (just like A and A# are equivelant in 7tet). A
> syntonic comma is a bad choice for a modifier, for example, because
> 5/4 is represented by 3 steps in a decatonic scale, while 81/64 is
> represented by 4.

It seems to me that the syntonic comma would be an excellent choice.
If 5/4 and 81/64 are represented by different tones, then there
*should* be a modifier to distinguish them. This would be
particularly true if you're notating a decatonic scale, where you're
dealing with nothing above the 7 (prime) limit. Inasmuch as you
can't use a 7-comma (63:64) symbol here (since it vanishes), then a 5-
comma symbol is clearly the most appropriate. So if you used the
combination pairs b/ and #\ instead of b and #, then you could easily
determine which chain a tone is in by the presence or absence of the
5-comma modifier. (You also get ready-made equivalent spellings such
as F#\ and Gb/ in the 2nd chain that are valid in *all* pajara-
compatible divisions of the octave.) Furthermore, as you extend that
2nd chain of fifths, the b and # accidentals drop out, and you're
left with easily understood / and \ accidentals.

If the first chain is extended, then equivalent spellings require a
new pair of symbols. For this the most logical is the 11-diesis
(32:33), v and ^, for which Gb will be equivalent to F^ (as 11/8 of
C) in the most desirable range of pajara generators (which includes
34, 56, and 22-ET).

But, OTOH, if you insist on using the flat or sharp symbols to
represent something other than the apotome (which already invites
confusion), then I would agree that a 5-comma (80:81) symbol used in
combination with those would be very ill-advised -- something like
confusion squared.

> It's really a very elegant notation once you get
> used to it. I feel strongly about the definition of # and b, but
feel
> free to use some other set of symbols for ^v+- if you feel like it.

If it's any consolation to you, in a 1975 Xenharmonikon article I
advocated using # and b to symbolize 128:135 (which is essentially
what you're doing). I pointed out that, for near-just and wide-fifth
tunings, it required fewer microtonal accidentals and resulted in
some nice parallels between tunings having fifths of different
sizes. In a word, to me it seemed quite elegant.

Since then not one person has indicated any agreement with my
proposal. The only direct feedback I can remember was a letter from
Erv Wilson that contained a firm, but diplomatic scolding for
requiring that Pythagorean tuning be notated differently from the way
everyone has done it for a thousand years. It took me many years,
but I eventually abandoned the idea. With our Sagittal project, Dave
Keenan and I have gone to great lengths to incorporate existing
symbols (and symbol features) into a notation system that is backward-
compatible, logical, elegant, and comprehensive.

Elegance is nice, but, in and of itself, it's not enough. If we're
going to communicate about new tunings, we need to be speaking the
same language (and in this case, conversant with the same notation).
If you're going to redefine symbols to mean something other than what
others already understand them to be, then you do so at your own
peril or discretion (as the case may be).

I apologize if any of this seems a bit harsh -- really, I'd rather be
discussing the music.

--George

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

> So it's not a pajara notation, then.

The notation is pajara in the sense that it assumes that 64/63 and
50/49 are tempered out. It is not pajara in the sense that not all
notes can be generated by chaining fourths. I would say it relates to
pajara in (roughly) the same way that quartertone notation relates to
meantone. An even closer analog, if I understand correctly, is what's
currently being discussed in the other thread regarding 31-equal. More
on that later.

> Since my 34-WT consists of two 17-tote circles of fifths 600 cents
> apart, then a transposition of the tuning by 600 cents (either up or
> down) will be identical to the original (assuming octave equivalence).

Indeed. I often find that a pattern of short and long steps that
corresponds at the perfect fifth is more effective than one
which repeats at the tritone. Erlich's pentachordal decatonic scales
are an example of this phenomenon. That said, a well temperament has
different implications regarding use than a scale -- In a well
temperament, on some level, the short and long intervals are still
thought of as the same interval. Also, my 12 tone scale repeats at the
tritone -- this allows all three diminished sevenths to be tuned
properly.

> > The short step represents 40/39 = 65/64. It's around 33 cents.
> > The long step represents 36/35 = 81/80. It's around 40.8 cents.
>
> Since you have a factor of 13 in there, and since pajara isn't
> generally supportive of 13, then I would have to conclude that, at
> best, you're describing only a special case of pajara.

I consider the basic pajara unison vectors to be 50/49, 64/63, 85/84.
The other unison vectors I'm using, I consider to be strongly implied
by these:

100>99>98 ; 100/98=50/49=1 --> 99/98=1
170>169>168 ; 170/168=85/84=1 --> 169/168=1
250>247>245 ; 250/245=50/49=1 --> 247/245=1
300>299>294 ; 300/294=50/49=1 --> 300/299=1

Thus in order to maintain the monotonicity of the mapping of the first
couple hundred harmonics, we need to temper out these unison vectors.
Since 169/168 is a second order unison vector, the result is we now
have four circles of fifths (or two circles of some generator around
246.1 cents if you prefer -- I, for one, don't.)

> I (and on this point I speak for many others in this group, including
> Paul Erlich) strongly discourage the use of flat or sharp symbols to
> mean anything other than an apotome, i.e., an alteration
> corresponding to a 7-position displacement along a chain of fifths.
> What you're advocating goes contrary to centuries of common practice,
> and to do something like that virtually guarantees confusion.

I think it depends if I'm trying to talk to microtonalists or jazz
musicians. I know from experience that jazz musicians use flats and
sharps pretty much interchangably (i.e. Eb = D#.) Thus, a chromatic
semitone is literally half of a major wholetone in jazz terminology.
So I think a jazz musician would have an easier time with my notation
-- but then 9 times out of 10, a good jazz musician will play the
right thing even if it's written in an ambiguous 12-equal based notation.

> Since you're dealing with two chains of fifths, I would therefore
> advise using some other symbol pair. Sagittal shorthand notation
> does not have a single character in this particular instance, but it
> does offer the combination pairs b/ and #\, which *do* in fact
> represent single symbols in the pure Sagittal font, !!/ and ||\ (but,
> again, I require multiple ASCII characters to represent them here).

If I understand you correctly then, the correct sagital notation for
my # and b is #\ and b/, and the correct notation for my +, - is #\\,
b//, and my ^ and v are whatever half of those is.

> > The additional modifiers were chosen so that
> > the modified note and the unmodified note are enharmonically
> > equivalent in 10TET (just like A and A# are equivelant in 7tet). A
> > syntonic comma is a bad choice for a modifier, for example, because
> > 5/4 is represented by 3 steps in a decatonic scale, while 81/64 is
> > represented by 4.
>
> It seems to me that the syntonic comma would be an excellent choice.
> If 5/4 and 81/64 are represented by different tones, then there
> *should* be a modifier to distinguish them. This would be
> particularly true if you're notating a decatonic scale, where you're
> dealing with nothing above the 7 (prime) limit. Inasmuch as you
> can't use a 7-comma (63:64) symbol here (since it vanishes), then a 5-
> comma symbol is clearly the most appropriate. So if you used the
> combination pairs b/ and #\ instead of b and #, then you could easily
> determine which chain a tone is in by the presence or absence of the
> 5-comma modifier. (You also get ready-made equivalent spellings such
> as F#\ and Gb/ in the 2nd chain that are valid in *all* pajara-
> compatible divisions of the octave.) Furthermore, as you extend that
> 2nd chain of fifths, the b and # accidentals drop out, and you're
> left with easily understood / and \ accidentals.
>
> If the first chain is extended, then equivalent spellings require a
> new pair of symbols. For this the most logical is the 11-diesis
> (32:33), v and ^, for which Gb will be equivalent to F^ (as 11/8 of
> C) in the most desirable range of pajara generators (which includes
> 34, 56, and 22-ET).
>
> But, OTOH, if you insist on using the flat or sharp symbols to
> represent something other than the apotome (which already invites
> confusion), then I would agree that a 5-comma (80:81) symbol used in
> combination with those would be very ill-advised -- something like
> confusion squared.

Here's my problem with using sagittal in this case. It seems like the
commas that are used as modifiers in sagittal are commas that are
tempered out in seven equal, but not in whatever system you're trying
to notate. Hence it's great for music that uses heptatonic scales, and
not so great for music that's based on five and ten note scales.
Consider the case of accidentals in standard notation, they are
modifiers in the sense that they change a note to one which can be
used as a substitute for the original note. They alter the note in the
meantone mapping, while leaving the note enharmonically equivalent in
7-equal. Insofar as possible, I would like to use accidentals for
decatonic music that alter the note, but leave it enharmonically
equivalent in 10 equal.

> > It's really a very elegant notation once you get
> > used to it. I feel strongly about the definition of # and b, but
> feel
> > free to use some other set of symbols for ^v+- if you feel like it.
>
> If it's any consolation to you, in a 1975 Xenharmonikon article I
> advocated using # and b to symbolize 128:135 (which is essentially
> what you're doing). I pointed out that, for near-just and wide-fifth
> tunings, it required fewer microtonal accidentals and resulted in
> some nice parallels between tunings having fifths of different
> sizes. In a word, to me it seemed quite elegant.
>
> Since then not one person has indicated any agreement with my
> proposal. The only direct feedback I can remember was a letter from
> Erv Wilson that contained a firm, but diplomatic scolding for
> requiring that Pythagorean tuning be notated differently from the way
> everyone has done it for a thousand years. It took me many years,
> but I eventually abandoned the idea.

I think I agree with your original proposal, so that's one. Also, in
the past hundred years, I would say the only thing that can be
universally said about the usage of #s and bs in the past 100 years is
that they represent 100c when the tuning is 12-TET. In the modern era
neoclassical composers like John Williams use #s and bs in the
Pythagorean sense, but rock, pop, and jazz musicians do not, nor do 12
tone serialists and the like.

> I apologize if any of this seems a bit harsh -- really, I'd rather be
> discussing the music.

I think to a large degree, the notation affects whether or not you can
think in the system. Having a universal system may be useful for
switching between tunings, but I feel like studying a single tuning in
depth usually requires coming up with notation specific to that system.

In that light, I'll restate the 34-note scale construction briefly in
a notation more similar to yours, not sure what to do about the 33c
intervals, so I'll just switch +- and ^v so that ^ = ++. You can tell
me if my notation has become more in line with what you're used to.
Just remember that + and - are not expressible below the 13 limit,
except as half of the ^ and v intervals.

Start with the (septimal) minor pentatonic scale:

E G A B D

Add a second pentatonic scale transposed down by half a major tone:

E Gb/ G Ab/ A Bb/ B Db/ D.

Extend each spiral of fifths by one step to create the 12 tone scale:

E Gb/v Gb G Ab A Bb B B^ Db/ D Eb/
or by six to produce a twenty two note scale:

E E^
Gb/v Gb/
Gv G G^
Ab/ Ab/^
A A^
Bb/ Bb/^
B B^
Db/v Db/
Dv D D^
Eb/ Eb/^

Split the intervals of the form X to X^ and Xv to X in half to produce
the full 34 note scale, adding 13, 19 and 23 limit intervals to our
harmonic language:

E E+ E^
Gb/v Gb/- Gb/
Gv G- G G+ G^
Ab/ Ab/+ Ab/^
A A+ A^
Bb/ Bb/+ Bb/^
B B+ B^
Db/v Db/- Db/
Dv D- D D+ D^
Eb/ Eb/+ Eb/^

Again, observe that the first 14 odd harmonics of E are:
E B Ab/ D Gv A^ Db/- Eb/ Gb/v G+ A Bb/+ B^ Dv

I believe that E is the only fundamental for which all of the first 14
harmonics are scale tones. This makes it a natural tonic pitch. Also
note that the reputation for flexibility that the blue notes have is
not entirely unearned -- the blue 3rd and 7th are the only notes which
have 5 rather than 3 chromatic variants included in the 34 note scale.

To summarize my beliefs about the specific pajara variant tuning
system I have specified here, it provides a workable tuning when the
pajara fourth used is between about 491c and 494c. From my
experimentation, it sounds to my ears like authentic blues intonation.
I suspect at least some blues musicians approximate this tuning a
goodly portion of the time. Regardless, I feel it can be used as a
guide to create more extstensively microtonal music in the blues idiom.

On a side note, rhythm changes might actually work in this system even
though it looks like a comma pump.

CM7 A-7 D-7 G7 CM7 works fine in pajara if the minor sevenths are
septimal in nature (12:14:18:21), and septimal minor sevenths aren't
really any less consonant than 5-limit minor sevenths.

--- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@>
wrote:
>
> > So it's not a pajara notation, then.
>
> The notation is pajara in the sense that it assumes that 64/63 and
> 50/49 are tempered out. It is not pajara in the sense that not all
> notes can be generated by chaining fourths. ...

Ray, I apologize that I haven't answered this by now, but I've been
trying to pursue 3 different microtonal discussions at once (each one
complicated in its own way), with not enough spare time (and not
enough sleep, lately) to give all of these the timely attention they
deserve.

I believe that it would helpful for me to take at least a few days to
reflect further on several things we discussed, so I expect that I
won't be replying before the first half of next week. Some of this
is relatively new territory for me, and I need to reread some of the
things in your messages to make sure I don't misunderstand what
your're saying.

Best,

--George