112-edo

I am recently interested in 112-edo. The meantone fifth goes full cycle and
produces very accurate notation for 4:5:6:7. Principal Rast scale maps to
natural keys. Pythagorean common pitches are finely represented. Each degree
is half a "comma" large. Nicely resonating superpythagorean fifths are
everywhere. The system has fine enough resolution for such superparticular
ratios as 10:9, 11:10, 12:11, 13:12, 14:13, 15:14, 16:15. The worst are 6:5,
11:10 and 16:15 with about 5 cents error. Transpositions are seamless
though.

These were the pros. What may be the cons?

I can come up with the first one already: How are we to implement it to an
"actual" instrument?

The second problem is the size of the limma. Two of them does not make a
minor tone! One needs to alterate between 8 and 9 steps (86 and 96 cents
respectively).

Now, how can I notate (Sagittal or otherwise) the scale by making the
generator the meantone fifth in SCALA?

George? Manuel?

Cordially,
Oz.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:

> Now, how can I notate (Sagittal or otherwise) the scale by making the
> generator the meantone fifth in SCALA?

set notation e112
set fifth 65

Not sure that this is exactly what you want.
For Sagittal it's not that hard either, you must edit sag_et.par to
add a non-native-fifth notation for 112. The file explains itself
hopefully, otherwise email me.

Manuel

--- In tuning@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@...> wrote:
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@> wrote:
>
> > Now, how can I notate (Sagittal or otherwise) the scale by making
the
> > generator the meantone fifth in SCALA?
>
> set notation e112
> set fifth 65
>
> Not sure that this is exactly what you want.
> For Sagittal it's not that hard either, you must edit sag_et.par to
> add a non-native-fifth notation for 112. The file explains itself
> hopefully, otherwise email me.
>
> Manuel

Oz, the only way I would recommend to notate 112-EDO in Sagittal is
as a subset of 224-ET (which is what you get with "set nota sa112").
While you could add native-fifth notation to sag_et.par, it's not
clear what the sequence of accidentals should be; all I can be sure
of at this point is that it would *not* be every other symbol in the
224 set.

Best,

--George

Dear Manuel, the version I use, which is 2.22i, does not contain the "set
fifth" command.

Dear George, I think what Manuel said about non-native fifth notation is
what I wanted. But since then, I decided that 105-tET is more suitable
regarding the accomodation of a decent "limma". Now, I see that you have not
Sagitally notated this temperament yet. How should one go about it?

Cordially,
Oz.

----- Original Message -----
From: "Manuel Op de Coul" <manuel.op.de.coul@eon-benelux.com>
To: <tuning@yahoogroups.com>
Sent: 13 Kas�m 2006 Pazartesi 19:31
Subject: [tuning] Re: 112-edo

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> > Now, how can I notate (Sagittal or otherwise) the scale by making the
> > generator the meantone fifth in SCALA?
>
> set notation e112
> set fifth 65
>
> Not sure that this is exactly what you want.
> For Sagittal it's not that hard either, you must edit sag_et.par to
> add a non-native-fifth notation for 112. The file explains itself
> hopefully, otherwise email me.
>
> Manuel
>

*

> Oz, the only way I would recommend to notate 112-EDO in Sagittal is
> as a subset of 224-ET (which is what you get with "set nota sa112").
> While you could add native-fifth notation to sag_et.par, it's not
> clear what the sequence of accidentals should be; all I can be sure
> of at this point is that it would *not* be every other symbol in the
> 224 set.
>
> Best,
>
> --George
>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> Dear Manuel, the version I use, which is 2.22i, does not contain
the "set
> fifth" command.
>
> Dear George, I think what Manuel said about non-native fifth
notation is
> what I wanted. But since then, I decided that 105-tET is more
suitable
> regarding the accomodation of a decent "limma". Now, I see that you
have not
> Sagitally notated this temperament yet. How should one go about it?
>
> Cordially,
> Oz.

I would notate 105 as a subset of 210, however, there is at present
only a tentative notation for 210.

However, you're not interested in a 105 notation as subset of 210,
but rather a native-5th notation for 105, which ain't easy (nor
pretty). Only for 3 steps of 105 are there any good options: either
|) or /|), which are 100% valid as either 63:64, or as 1024:1053 and
35:36.

For two steps of 105 the only unaccented symbol that's valid is )|),
as 56:57. If this is used, then |) as 3 steps would be ruled out,
since that symbol clearly indicates that |) should be smaller.
Otherwise, if you want to keep |) for 3 steps, then an accented
symbol .~|( would work (as 135:136) for 2 steps.

After that, it gets worse. For a single step of 105 an accented
symbol .~| is valid as 255:256. If you want an unaccented symbol,
then it would have to be either ~|\ (as 16384:16767) or |( in a
secondary role (as 351:352). Since ~|\ is larger than )|), then |(
looks like a better choice.

These are the lines that you would add to sag_et.par:

n 105 61
|( )|) /|) (|\ )/N /N) /N\

The command would be: set nota sa105n

Please bear in mind that this isn't an "officially approved"
notation, just something that will allow you to play around with 105.

I'm not sure how well Scala presently supports accented symbols, but
if you really need the 7-comma symbol for 3 steps of 105, then these
are the lines with the accented symbols that would replace the above.

n 105 61
.~| .~|( |) N) '(N( ')//N /N\

You can also mix and match, e.g., the following sequence, which uses
the lowest primes, consists entirely of athenian-cored symbols:

n 105 61
|( .~|( |) N) '(N( /N) /N\

Best,

--George

Dear George

SNIP

>
> I would notate 105 as a subset of 210, however, there is at present
> only a tentative notation for 210.
>
> However, you're not interested in a 105 notation as subset of 210,
> but rather a native-5th notation for 105, which ain't easy (nor
> pretty). Only for 3 steps of 105 are there any good options: either
> |) or /|), which are 100% valid as either 63:64, or as 1024:1053 and
> 35:36.

I like /|) for 3deg 105-edo.

>
> For two steps of 105 the only unaccented symbol that's valid is )|),
> as 56:57. If this is used, then |) as 3 steps would be ruled out,
> since that symbol clearly indicates that |) should be smaller.
> Otherwise, if you want to keep |) for 3 steps, then an accented
> symbol .~|( would work (as 135:136) for 2 steps.

It is a pity there is no other unaccented symbol that would work here. In
that case, I prefer )|) for 2deg 105-edo.

>
> After that, it gets worse.

Be still my beating heart!

For a single step of 105 an accented
> symbol .~| is valid as 255:256. If you want an unaccented symbol,
> then it would have to be either ~|\ (as 16384:16767) or |( in a
> secondary role (as 351:352). Since ~|\ is larger than )|), then |(
> looks like a better choice.

I concur. |( is suitable for 1deg 105-edo.

>
> These are the lines that you would add to sag_et.par:
>
> n 105 61
> |( )|) /|) (|\ )/N /N) /N\
>
> The command would be: set nota sa105n

I could only make it work if it was defined as "d 105 61", in which case the
last letter n drops from the command text.

>
> Please bear in mind that this isn't an "officially approved"
> notation, just something that will allow you to play around with 105.
>

I appreciate it very much!

> I'm not sure how well Scala presently supports accented symbols, but
> if you really need the 7-comma symbol for 3 steps of 105, then these
> are the lines with the accented symbols that would replace the above.
>
> n 105 61
> .~| .~|( |) N) '(N( ')//N /N\
>
> You can also mix and match, e.g., the following sequence, which uses
> the lowest primes, consists entirely of athenian-cored symbols:
>
> n 105 61
> |( .~|( |) N) '(N( /N) /N\

My version of Scala displays accented notes so long as the lines in
sag_et.par are entered "d 105 61".

I noticed a discrepancy with the unaccented version. You used "Athena's
helmet" for 3deg and its double for 6deg. Shouldn't the same rut apply to
"Aphrodite's perfect curves mirrored" for 2deg and 5deg (possibly
"Persephone's curves" for the latter)?

>
> Best,
>
> --George
>
>

Cordially,
Oz.

P.S. Is 105 actually an ugly choice for a possible meantone-pythagorean
hybrid in your opinion?

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

> Oz, the only way I would recommend to notate 112-EDO in Sagittal is
> as a subset of 224-ET (which is what you get with "set nota sa112").

This strikes me as awfully strange, as 224 (a temperament I've
actually written music in, BTW!) is a very different character than 112.

For 112, it might depend on what you are using it for. As Oz notes, it
is a good meantone system, and if you are using it for that, standard
notation is one means. He also mentioned the sharp fifth; if you use
that for pajara or superpyth you are in a contorted 56-et, and your
notation could reflect that. Other possible uses are shrutar, myna,
and injera, or even miracle if you want to get sloppy. In each of
these cases, notation could reflect the temperament in question.

Of course, switching between temperaments is closer to what Oz has in mind

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:

> Dear George, I think what Manuel said about non-native fifth notation is
> what I wanted. But since then, I decided that 105-tET is more suitable
> regarding the accomodation of a decent "limma".

It's another good meantone, but the sharp fifth is getting pretty
sharp. On the other hand, it's fine if you want a superpyth fifth, and
I think you mentioned superpythagorean recently.

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

> I would notate 105 as a subset of 210, however, there is at present
> only a tentative notation for 210.
>
> However, you're not interested in a 105 notation as subset of 210,
> but rather a native-5th notation for 105, which ain't easy (nor
> pretty).

Surely the most obvious way to do that is meantone??

Indeed I did. Thanks for confirming my suspicions.

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@coolgoose.com>
To: <tuning@yahoogroups.com>
Sent: 15 Kas�m 2006 �ar�amba 0:42
Subject: [tuning] Re: 112-edo

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> > Dear George, I think what Manuel said about non-native fifth notation is
> > what I wanted. But since then, I decided that 105-tET is more suitable
> > regarding the accomodation of a decent "limma".
>
> It's another good meantone, but the sharp fifth is getting pretty
> sharp. On the other hand, it's fine if you want a superpyth fifth, and
> I think you mentioned superpythagorean recently.
>

Eq-xactly my dear Gene. A switch-swatch is what I have in mind. Now that the
question of 105-edo is pretty much resolved thanks for the most part to
George, I wonder how one may notate 112-edo not as a subset of 224-tET, but
as a distinct system all its own with the only perfect fifth that goes full
cycle as the generator.

224 seems most perfect though, albeit a generally unpracticable octave
division.

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@coolgoose.com>
To: <tuning@yahoogroups.com>
Sent: 15 Kas�m 2006 �ar�amba 0:34
Subject: [tuning] Re: 112-edo

> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>
> > Oz, the only way I would recommend to notate 112-EDO in Sagittal is
> > as a subset of 224-ET (which is what you get with "set nota sa112").
>
> This strikes me as awfully strange, as 224 (a temperament I've
> actually written music in, BTW!) is a very different character than 112.
>
> For 112, it might depend on what you are using it for. As Oz notes, it
> is a good meantone system, and if you are using it for that, standard
> notation is one means. He also mentioned the sharp fifth; if you use
> that for pajara or superpyth you are in a contorted 56-et, and your
> notation could reflect that. Other possible uses are shrutar, myna,
> and injera, or even miracle if you want to get sloppy. In each of
> these cases, notation could reflect the temperament in question.
>
> Of course, switching between temperaments is closer to what Oz has in mind
>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> Dear George
>
> SNIP
>
> >
> > I would notate 105 as a subset of 210, however, there is at
present
> > only a tentative notation for 210.
> >
> > However, you're not interested in a 105 notation as subset of 210,
> > but rather a native-5th notation for 105, which ain't easy (nor
> > pretty)...
> >
> > These are the lines that you would add to sag_et.par:
> >
> > n 105 61
> > |( )|) /|) (|\ )/N /N) /N\
> >
> > The command would be: set nota sa105n
>
> I could only make it work if it was defined as "d 105 61", in which
case the
> last letter n drops from the command text.

I guess that's because there's no entry for 105 as a subset-of-210
notation -- I didn't know what the requirements were for entering a
native notation.

> ...
> I noticed a discrepancy with the unaccented version. You
used "Athena's
> helmet" for 3deg and its double for 6deg. Shouldn't the same rut
apply to
> "Aphrodite's perfect curves mirrored" for 2deg and 5deg (possibly
> "Persephone's curves" for the latter)?

No. The symbols in the right half of the sequence are apotome-
complements of those in the first half, taken in reverse order.
Thus, |( and /N) are complements, as are )|) and )/N, and /|) and
(|\ .

> Cordially,
> Oz.
>
> P.S. Is 105 actually an ugly choice for a possible meantone-
pythagorean
> hybrid in your opinion?

To be honest, I think it is rather ugly.

Please look at 129-edo to see whether that's a possibility. Its best
fifth is approximately 1/5-comma narrow. I was thinking that it
could be notated using the accidentals for 258 (which would still
need to be determined). These could include the inconspicuous
schisma symbol (left-accent) for a single degree.

I'll have to experiment with this before getting back to you, because
I'm not sure how well this will work.

Best,

--George

SNIP

> >
> > P.S. Is 105 actually an ugly choice for a possible meantone-
> pythagorean
> > hybrid in your opinion?
>
> To be honest, I think it is rather ugly.
>
> Please look at 129-edo to see whether that's a possibility. Its best
> fifth is approximately 1/5-comma narrow. I was thinking that it
> could be notated using the accidentals for 258 (which would still
> need to be determined). These could include the inconspicuous
> schisma symbol (left-accent) for a single degree.
>
> I'll have to experiment with this before getting back to you, because
> I'm not sure how well this will work.
>
> Best,
>
> --George
>

I would much rather prefer 117 if you don't mind George.

Oz.

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

> Please look at 129-edo to see whether that's a possibility. Its best
> fifth is approximately 1/5-comma narrow.

It's best fifth is the fifth of 43-et; I don't see why that's better
than 105. The latter can be notated as a chain of fifths.

Yes, the sequence is uninterrupted. Such is not the case with 129.

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@coolgoose.com>
To: <tuning@yahoogroups.com>
Sent: 16 Kas�m 2006 Per�embe 1:37
Subject: [tuning] Re: 105-ET native-5th notation (was: 112-edo)

> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>
> > Please look at 129-edo to see whether that's a possibility. Its best
> > fifth is approximately 1/5-comma narrow.
>
> It's best fifth is the fifth of 43-et; I don't see why that's better
> than 105. The latter can be notated as a chain of fifths.
>
>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:

> I would much rather prefer 117 if you don't mind George.

The flat fifth is fine for meantone, and half of it is fine for
semififths, the closely related 31&55 temperament. It has a decent
version of keemun temperament using the minor third. Porcupine is also
supported, and magic more or less also. On the other hand the sharp
fifth generator isn't that great; you get 39-equal from that.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> SNIP
>
> > >
> > > P.S. Is 105 actually an ugly choice for a possible meantone-
pythagorean
> > > hybrid in your opinion?
> >
> > To be honest, I think it is rather ugly.
> >
> > Please look at 129-edo to see whether that's a possibility. ...
>
> I would much rather prefer 117 if you don't mind George.

Before I even look at that one, could we please take another look at
112? I found, much to my surprise, that a narrow-5th notation
actually works quite well. Simply add the following lines to
sag_et.par:

d 112 65
~|( /|) (|) /|\ (|\ (N( /N\

and remove this line:
s 112
which follows "d 224 131"

Note: The lines that I really wanted to add are these:

n 112 65
~|( /|) (|) /|\ (|\ (N( /N\

without removing the "s 112" line, but for some reason the "set nota
sa112n" command doesn't work, because I get an error message that
sa112n isn't a valid notation, even though "set nota sa__n" works for
the other native-5th lines in the file.

Manuel, please note!!! I suspect that there's a bug in Scala that
doesn't allow native-5th notations for divisions >99, i.e., having
more than 2 digits.

A brief explanation of this 112-edo symbol set is in order. This is
basically an extended meantone notation (as Gene suggested), very
similar to the 31-ET notation for the 11 limit, where the 11M-diesis
(32:33) symbol is used for both the 11M-diesis and 7-comma
alterations. A 4:5:6:7 chord on C may therefore be notated as either
C,E,G,A# or C,E,G,Bb\!/ .

However, note that, because the 11M-diesis (5deg) is not exactly half
as many degrees as the apotome (9deg), there must be two 11-diesis
symbols, so you must take care to use the proper one. If you're
altering upward from a nominal to get an otonal 11 relationship, then
the /|\ symbol is used, but if it's downward, then the (!) symbol is
used. For example, 8:11 on C is C,F/|\, whereas on F it's F,B(!) .
This also affects the way you notate the 7-comma; e.g., 4:5:6:7 on G
will be either G,B,D,F(!) or G,B,D,E#. The utonal 11's work the
opposite way: e.g., 16/11 of C is G\!/, whereas 16/11 of B is F(|) .
This applies to *all tunings* where the 11M diesis is *not exactly
half* the size of the apotome, for both temperaments and (always) JI.

The /|) and (|\ symbols must be interpreted as 13M and 13L dieses
(1024:1053 and 26:27), respectively, not as 35M and 35L (35:36 and
8192:8505, their primary roles), so in order for the lines to be
completely correct, those two symbols should have right accents, like
this:

n 112 65
~|( /|). (|) /|\ (|\' (N( /N\

Examples of instances where "completely correct" interpretation of
the symbols has significance is 1) when extreme pitch precision is
required, or 2) when transferring the symbols for playback in another
temperament (where 35M and 13M could be different numbers of degrees).

But since Dave and I haven't yet worked with Manuel to implement
right-accents adequately in Scala, the unaccented symbols will have
to do for now.

The ~|( symbol is valid as the 17-comma, so this meantone-notation
for 112 is actually 17-limit! For example, a 10:12:14:17 diminished-
7th chord would be notated E,G,Bb\!/,Db~|( .

Oz, please give this a try. If you like it, we might also try a wide-
5th notation (observing that the best 5th of 56-ET allows a nice
implementation of pajara, with 11 thrown in for good measure).

--George

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

> Oz, please give this a try. If you like it, we might also try a wide-
> 5th notation (observing that the best 5th of 56-ET allows a nice
> implementation of pajara, with 11 thrown in for good measure).

I tried to interest Paul Erlich in pajara on 56, but no dice. However,
it is a good alternative. One nice feature is that the 5-limit tunings
in particular are better, and the major third not so flat. 56 = 22+34,
and it represents a compromise between the 34 version of pajara,
strongly weighted in favor of the 5-limit over the 7-limit, and the 22
version.

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>Manuel, please note!!! I suspect that there's a bug in Scala that
>doesn't allow native-5th notations for divisions >99, i.e., having
>more than 2 digits.

Ok, it's fixed now. I'll hear what's needed for the right accents.
Best,

Manuel

All the more reason to return to 112 then. Is there anything 112 does not
support?

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@coolgoose.com>
To: <tuning@yahoogroups.com>
Sent: 16 Kas�m 2006 Per�embe 2:34
Subject: [tuning] Re: 105-ET native-5th notation (was: 112-edo)

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> > I would much rather prefer 117 if you don't mind George.
>
> The flat fifth is fine for meantone, and half of it is fine for
> semififths, the closely related 31&55 temperament. It has a decent
> version of keemun temperament using the minor third. Porcupine is also
> supported, and magic more or less also. On the other hand the sharp
> fifth generator isn't that great; you get 39-equal from that.
>
>

> All the more reason to return to 112 then. Is there anything 112
> does not support?

Hi Ozan - Can you fill in the blanks?

The scale I'm looking for must

. Have fewer than _____ tones per octave.
. Have a large fifth between _____ cents and _____ cents.
. Have a small fifth between _____ cents and _____ cents.
. Approximate primes/ratios __, __, __, and __ within __ cents.

Then I can give you a complete list of all the ETs that meet
your requirements.

-Carl

--- In tuning@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@...> wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
> >Manuel, please note!!! I suspect that there's a bug in Scala that
> >doesn't allow native-5th notations for divisions >99, i.e., having
> >more than 2 digits.
>
> Ok, it's fixed now.

Thanks, Manuel! Support like this can't be beat! :-)

> I'll hear what's needed for the right accents.

That's still in the future.

Best,

--George

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
>
> > Oz, please give this a try. If you like it, we might also try a
wide-
> > 5th notation (observing that the best 5th of 56-ET allows a nice
> > implementation of pajara, with 11 thrown in for good measure).
>
> I tried to interest Paul Erlich in pajara on 56, but no dice.

Probably too many tones, but just as likely that he doesn't consider
it very near to optimal.

> However,
> it is a good alternative. One nice feature is that the 5-limit
tunings
> in particular are better, and the major third not so flat. 56 =
22+34,
> and it represents a compromise between the 34 version of pajara,
> strongly weighted in favor of the 5-limit over the 7-limit, and the
22
> version.

Gene, this is exactly the way my thinking about pajara has gravitated
over the last couple of months. I figured that, since the
considerable error of 5:7 in pajara is a constant (~17.488 cents,
independent of the size of the generator), it's therefore not
possible to avoid its the adverse effect of its contribution to the
total 7-limit error, no matter how you calculate it. So I've
accepted the fact that 7-limit chords won't sound anywhere near as
consonant in pajara as they do in 31-ET and have concluded that the
optimal pajara generator should be the one that minimizes the 5-limit
errors while limiting the absolute error of each 7-limit consonance
to no more than the 5:7 error. This is accomplished with a generator
of ~106.843c, which makes 4:5 just. Now if you're thinking of going
up to the 9- or 11-limit, there's very good news: all 11-limit
consonances are also within the 5:7 error. (In 22-ET, by contrast,
the max error is progressively exceeded at *both* the 9 and 11
limits, and its approximation of 9:11, in particular, leaves much to
be desired.)

Several years ago Paul confronted me with the problem that, because I
wouldn't be able to find a 7-limit minimax generator for pajara on
account of that constant 5:7 error, I would have to use some other
method to find an optimal generator for pajara. I finally have an
answer for him, and it's significantly different from what he
advocates.

The 56-ET pajara generator (~107.143c) is very close to the optimal
figure given above, and it also results in a pajara tuning in which
the consonances do not exceed the 5:7 error at the 11-limit.

However, if 56 is too many tones for you, and if you want a pajara-
capable tuning that offers free modulation, then consider the
following. I was curious to see what I would get when I combined two
circles of my 17-tone well-temperament 600 cents apart. I was
delighted to discover that the 16 best 4:5:6:7 chords of the
resulting 34-WT are produced by a two chains of generators of
~107.220c (very close to the pajara-56-ET figure), and that all
sixteen 11-limit otonal hexads (4:5:6:7:9:11) built on the same
starting tones are such that no consonance exceeds the above-
mentioned 5:7 error. Furthermore, since the 17-WT subsets were
designed to be non-5 *13-limit*, and since *17* is very good in the
34 division, 12 of those 16 keys have *all of the 17-limit
consonances* represented within that 5:7 error (and most of them are
much better than that). Since 34-WT has up-down symmetry, this
should also hold true for utonalities.

Here's a Scala listing:

! secor_34wt.scl
!
George Secor's 34-tone well temperament (with 10 exact 11/7)
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

If C is taken as 1/1, then the 16 best otonal keys will be in the
following two chains of 5ths: from Bb through B, and from E\! through
E#\!; the 12 best 17-limit otonal keys will be in the chains from Bb
through A and from E\! through D#\! -- but the major triads with the
lowest errors are in keys other than these, e.g., B and F#.

Please try it, comparing with pajara-generated chords in 22-ET, 34-
ET, and 56-ET, and see what you think. You can crunch numbers till
doomsday, but the bottom line is, "how does it sound?"

BTW, I wrote Graham off-list the other day that I think that Paul's
total neglect of 34 in his pajara (22-tone) paper was a serious
omission.

(Too bad Paul's not here to defend himself -- perhaps he'll come out
of hiding.)

--George

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> ----- Original Message -----
> From: "Gene Ward Smith" <genewardsmith@...>
> To: <tuning@yahoogroups.com>
> Sent: 16 Kasým 2006 Perþembe 2:34
> Subject: [tuning] Re: 105-ET native-5th notation (was: 112-edo)
>
>
> > --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@> wrote:
> >
> > > I would much rather prefer 117 if you don't mind George.
> >
> > The flat fifth is fine for meantone, and half of it is fine for
> > semififths, the closely related 31&55 temperament. It has a decent
> > version of keemun temperament using the minor third. Porcupine is
also
> > supported, and magic more or less also. On the other hand the
sharp
> > fifth generator isn't that great; you get 39-equal from that.
>
> All the more reason to return to 112 then. Is there anything 112
does not
> support?

There are plenty of things that any given division of the octave
won't support. It's easier to list the things that it *will* support.

What's of primary concern to you is whether 112-edo will do enough of
the things *you require* well enough for *your purposes*.

I've started looking at possible a wide-fifth notation for 112, and I
already see that it'll be more complicated than the narrow-fifth one
(which is rather nice, IMO). I'll let you know what I come up with.

Best,

--George

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@>
> wrote:

> > I tried to interest Paul Erlich in pajara on 56, but no dice.
>
> Probably too many tones, but just as likely that he doesn't consider
> it very near to optimal.

The only way you can argue 22 is better is if you are interested only
in the 7-odd-limit. That is, 7-limit, but not 9-limit. Also, you need
to assume it is just as important to get the 7 right as the 3 and 5.
But if you look at the 5, 9, or 11 limits, 56 is better. It is also
better when you weight 7 less highly. I think if you do a listening
test you are likely to conclude it sounds better.

Your first point, of course, is too many notes. This is indeed a
problem. I wonder, though, about instruments where the range of
generators for a MOS could be set on the fly.

> This is accomplished with a generator
> of ~106.843c, which makes 4:5 just.

I don't know that this is "best" but somewhere in that range is where
I think by listening you will find the best pajara generator; I think
the fifth of 22 is too sharp really.

The size of fifth making 5/4 pure in pajara is 2^(7/4)/sqrt(5), which
is 706.843 cents; this is very close to the 146 equal version of
pajara, but 56-et has a fifth a fraction of a cent sharper which is a
semiconvergent here.

> However, if 56 is too many tones for you, and if you want a pajara-
> capable tuning that offers free modulation, then consider the
> following. I was curious to see what I would get when I combined two
> circles of my 17-tone well-temperament 600 cents apart. I was
> delighted to discover that the 16 best 4:5:6:7 chords of the
> resulting 34-WT are produced by a two chains of generators of
> ~107.220c (very close to the pajara-56-ET figure), and that all
> sixteen 11-limit otonal hexads (4:5:6:7:9:11) built on the same
> starting tones are such that no consonance exceeds the above-
> mentioned 5:7 error.

It might be interesting to look at 34-et pajara well-temperaments as
such. All those exact 11/7s are intriguing, however.

> Please try it, comparing with pajara-generated chords in 22-ET, 34-
> ET, and 56-ET, and see what you think. You can crunch numbers till
> doomsday, but the bottom line is, "how does it sound?"

Absolutely, and it sounds as if you could be on to something.

> (Too bad Paul's not here to defend himself -- perhaps he'll come out
> of hiding.)

I wish he would. Someone was asking how Fokker defined Fokker blocks,
and that's the sort of question we used to be able to leave to Paul.

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

> There are plenty of things that any given division of the octave
> won't support. It's easier to list the things that it *will* support.

That's true of the size of division we are looking at, around 100
notes to the octave. It becomes increasingly less true as you increase
the number of notes, so long as you are unconcerned about supporting
ultra temperaments of the kind whose interest is more theoretical than
practical.

It's been suggested that 152-et is a good Swiss army knife
temperament, but I think really the first plausible candidate is
270-et, which unfortunately has three times as many notes as Oz wants.
However, unless you are a JI purist it does a good JI, and supports a
lot of temperaments. As you increase the number of divisions of the
octave, equal divisions can do more and more, but after a while we
cease considering them to be temperaments, and regard them as a means
of approximating intervals. It also becomes true that alternating two
different sizes of the same interval becomes harder and harder to
notice, and hence you get the flexibility to do just about anything
except make purists happy.

> Someone was asking how Fokker defined Fokker blocks, and
> that's the sort of question we used to be able to leave to Paul.

If that person (Rick Holmes?) got his ears on, I'll take a stab.

I am not aware of Fokker discussing the restriction that the
block's commas be smaller than its 2nds (I haven't read all
of his papers however). I am similarly unaware of him giving
any examples where this was not the case.

Gene came up with something called epimorphism. Gene, can you
remind us what relation epimorphism bears to the above property?
I believe Gene was calling epimorphic blocks "Fokker blocks".
'z that right, Gene?

Epimorphism is defined here:

http://www.tonalsoft.com/enc/e/epimorphic.aspx

which I believe is just saying there's a mapping (specifically
a homomorphism I think) from the rationals to the integers
that, when you plug in a ratio in the scale, gives you the
corresponding scale degree. The deeper implications of this
are not immediately obvious (to me).

-Carl

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

> I am not aware of Fokker discussing the restriction that the
> block's commas be smaller than its 2nds (I haven't read all
> of his papers however). I am similarly unaware of him giving
> any examples where this was not the case.
>
> Gene came up with something called epimorphism. Gene, can you
> remind us what relation epimorphism bears to the above property?

Its related to the question of what you need as conditions to get the
Erlich "hypothesis" to be true, but that seems to have only an
indirect connection to the above (that is, if the commas are larger
than some scale steps, you could more easily be non-epimorphic.)

> I believe Gene was calling epimorphic blocks "Fokker blocks".
> 'z that right, Gene?

I was calling something a Fokker block if the boundries of the octave
class regions which the octave classes are partioned into can be taken
to be paralleograms. I actually like to define and compute these
things in terms of a unimodular matrix of monzos and the corresponding
unimodular matrix of vals, but I am assuming the question is about
what defintions Fokker used.

> > I am not aware of Fokker discussing the restriction that the
> > block's commas be smaller than its 2nds (I haven't read all
> > of his papers however). I am similarly unaware of him giving
> > any examples where this was not the case.
> >
> > Gene came up with something called epimorphism. Gene, can you
> > remind us what relation epimorphism bears to the above property?
>
> Its related to the question of what you need as conditions to get
> the Erlich "hypothesis" to be true, but that seems to have only an
> indirect connection to the above (that is, if the commas are larger
> than some scale steps, you could more easily be non-epimorphic.)
>
> > I believe Gene was calling epimorphic blocks "Fokker blocks".
> > 'z that right, Gene?
>
> I was calling something a Fokker block if the boundries of the
> octave class regions which the octave classes are partioned into
> can be taken to be paralleograms. I actually like to define and
> compute these things in terms of a unimodular matrix of monzos
> and the corresponding unimodular matrix of vals, but I am
> assuming the question is about what defintions Fokker used.

Thanks for clearing this up. I'll endeavor to remember it.

-Carl

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
> >
> > --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@>
wrote:
>
> > > I tried to interest Paul Erlich in pajara on 56, but no dice.
> >
> > Probably too many tones, but just as likely that he doesn't
consider
> > it very near to optimal.
>
> The only way you can argue 22 is better is if you are interested
only
> in the 7-odd-limit. That is, 7-limit, but not 9-limit.

In Paul's 22-tone paper, that turns out to be the case.

> Also, you need
> to assume it is just as important to get the 7 right as the 3 and 5.

A careful reading of the paper leads me to the conclusion that Paul
made that assumption.

Whatever the case, the main purpose of the paper is to uncover scales
that exhibit properties similar to the diatonic scale of the
conventional major-minor harmonic system, and Paul's presentation is
absolutely brilliant. Whether or not 22-ET is the best practical
implementation of those scales is a secondary issue that in no way
detracts from the main point.

> But if you look at the 5, 9, or 11 limits, 56 is better.

Yes, indeed! Given that the error of 5:7 (and 7:10) in pajara is
independent of the generator, you can pretty much write off 7 and
instead take advantage of the fact that 3, 5, 9, and 11 can all be
dramatically improved in one fell swoop.

> It is also
> better when you weight 7 less highly. I think if you do a listening
> test you are likely to conclude it sounds better.

Yep.

> Your first point, of course, is too many notes. This is indeed a
> problem. I wonder, though, about instruments where the range of
> generators for a MOS could be set on the fly.

Do you mean, set on the fly to favor different keys while you're
performing music in a WT, or do you just want to sustain chords while
you're changing the generator, to find the one that sounds best?
Those are things I can already do (and have done) to a limited extent
with my Scalatron; you can also do the latter in Scala by sustaining
a chord on the clavier while you change the tuning (must be same # of
notes/octave).

> > This is accomplished with a generator
> > of ~106.843c, which makes 4:5 just.
>
> I don't know that this is "best" but somewhere in that range is
where
> I think by listening you will find the best pajara generator;

That generator is mathematically best for the 5 limit, but after
listening to it I would agree that the just 4:5 may not be the best
thing if everything else is tempered -- something like the proverbial
sore thumb, only in reverse. Making the fifth a little wider (as in
56-ET) would change that, and would also improve 7 a little.

> I think
> the fifth of 22 is too sharp really.

Yes, and noticeably flatter-than-just major 3rds are melodically not
very desirable.

> The size of fifth making 5/4 pure in pajara is 2^(7/4)/sqrt(5),
which
> is 706.843 cents; this is very close to the 146 equal version of
> pajara, but 56-et has a fifth a fraction of a cent sharper which is
a
> semiconvergent here.
>
> > However, if 56 is too many tones for you, and if you want a
pajara-
> > capable tuning that offers free modulation, then consider the
> > following. I was curious to see what I would get when I combined
two
> > circles of my 17-tone well-temperament 600 cents apart. I was
> > delighted to discover that the 16 best 4:5:6:7 chords of the
> > resulting 34-WT are produced by a two chains of generators of
> > ~107.220c (very close to the pajara-56-ET figure), and that all
> > sixteen 11-limit otonal hexads (4:5:6:7:9:11) built on the same
> > starting tones are such that no consonance exceeds the above-
> > mentioned 5:7 error.
>
> It might be interesting to look at 34-et pajara well-temperaments as
> such. All those exact 11/7s are intriguing, however.

That description is one I copied & edited my 17-WT .scl file, which
reads, "George Secor's well temperament with 5 pure 11/7 and 3 near
just 11/6". The just 7:11's are mostly in less-likely-to-be-used
keys, but there are also 6 near-just 6:11's (with 0.09c error) in 34-
WT, and they're all in keys that are among the best in 11-limit
pajara. In brief, the following holds true in every key of both 17-
WT and 34-WT:
3-error < 11-error <= 7-error

Something you may find even more intriguing is that 34-WT has both a
very good 13 (better than 56-ET) and 17, so it actually takes pajara
to the 17 limit. This is a consequence of the varied 5ths in the 17-
tone circles coming into play (13 being almost directly opposite 1/1
in the circle).

> > Please try it, comparing with pajara-generated chords in 22-ET,
34-
> > ET, and 56-ET, and see what you think. You can crunch numbers
till
> > doomsday, but the bottom line is, "how does it sound?"
>
> Absolutely, and it sounds as if you could be on to something.

I hope so. Hudson & Margo both gave my 17-WT enthusiastic reviews on
MMM, so now I'll be very disappointed if I don't see messages from at
least 4 different people saying that my 34-WT is twice as good. ;-)

--George

I'm reading this thread but there's no traction. Would
anyone care to synthesize the tetrads of 22 and 56 for
comparison? (In a past life I would just do this instead
of requesting it...)

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> I'm reading this thread but there's no traction. Would
> anyone care to synthesize the tetrads of 22 and 56 for
> comparison? (In a past life I would just do this instead
> of requesting it...)

I could but if someone has a pajara midi file I'd rather retune that.

> > I'm reading this thread but there's no traction. Would
> > anyone care to synthesize the tetrads of 22 and 56 for
> > comparison? (In a past life I would just do this instead
> > of requesting it...)
>
> I could but if someone has a pajara midi file I'd rather
> retune that.

I'm not aware of any such MIDI files incidentally, but I don't
think full-music transcriptions are good first tests. It's
good to not have to control for all the elements of music. If
you wanted to get something less static than single chords, it
would be trivial to write a cycle of decatonic tetrads in a
Scala seq file.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> I'm reading this thread but there's no traction. Would
> anyone care to synthesize the tetrads of 22 and 56 for
> comparison? (In a past life I would just do this instead
> of requesting it...)
>
> -Carl

"Oh, bother!", said Pooh.
"So sorry you asked.
There's so much to do
As life rushes past."

With time so precious, I try not to do things microtonal that could
be done just as well (if not better) by others. (You'd much rather
have me finishing a generalized keyboard design for 22, 34, 46, 56,
and 58, wouldn't you?) And if I made sound files of a progression of
chords, then you (or somebody else) would probably want a score.

Why don't you listen to the following pajara chords with the Scala
chromatic clavier (in 22, 34, and 56-ET)? If you set the following
commands first, then I think you'll see the notes with the spellings
I've indicated below in each of these tunings:

set nota saNN (where NN = 22, 34, 56)
set sagi mixed
set sagi short

4:5:6 - C E\ G
4:5:6:7 - C E\ G Bb
10:12:15 - E\ G B\
10:12:15:17 - E\ G B\ C#\ or E\ G B\ Db/
That last chord is equivalent to the first inversion of
1/4:1/5:1/6:1/7; I gave two spellings, because I'm not sure which one
you'll see.

Note that you're using 28deg34 and 46deg56 for 7, which is two
fourths upward from 4.

Also try these for the 9- and 11-limit:
6:7:9 - G Bb D
6:7:9:11 - G Bb D Gb

Gb is equivalent to F^ in 34 and 56 and to F/ in 22.

I've also put my 34-tone well-temperament here, if you want to try it:
/tuning-math/files/secor/scl/
The filename is secor34wt.scl; in Scala, set notation sa34.

Just before posting this, I noticed that you're discussing .seq files
with Gene; that would be good.

--George

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@> wrote:
> >
> > ----- Original Message -----
> > From: "Gene Ward Smith" <genewardsmith@>
> > To: <tuning@yahoogroups.com>
> > Sent: 16 Kasým 2006 Perþembe 2:34
> > Subject: [tuning] Re: 105-ET native-5th notation (was: 112-edo)
> >
> > > --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@>
wrote:
> > >
> > > > I would much rather prefer 117 if you don't mind George.
> > >
> > > The flat fifth is fine for meantone, and half of it is fine for
> > > semififths, the closely related 31&55 temperament. It has a
decent
> > > version of keemun temperament using the minor third. Porcupine
is also
> > > supported, and magic more or less also. On the other hand the
sharp
> > > fifth generator isn't that great; you get 39-equal from that.
> >
> > All the more reason to return to 112 then. Is there anything 112
does not
> > support?
>
> There are plenty of things that any given division of the octave
> won't support. It's easier to list the things that it *will*
support.
>
> What's of primary concern to you is whether 112-edo will do enough
of
> the things *you require* well enough for *your purposes*.

Among the best things that 112 will support are:

1) 1/4-Didymus-comma meantone, 9-limit
2) 1/5-Archytas-comma superpythagorean
3) 1/5-Archytas-comma pajara, 11-limit, via the 56-edo subset

There's not really a Pythagorean option here; you'd need 224-ET for
that.

> I've started looking at possible a wide-fifth notation for 112, and
I
> already see that it'll be more complicated than the narrow-fifth
one
> (which is rather nice, IMO). I'll let you know what I come up with.

Oz, assuming that you've downloaded the latest version of Scala and
have put back the "s 112" line after "d 224 131", try this in
sag_et.par for a wide-5th (superpythagorean) 112 notation (you'll
have to replace the narrow-5th one I previously gave you):

n 112 66
|) |\ )|) /| /|) /|\ (| (|) (|\ N\ )/N /N N) /N\

The /|) and (|\ symbols represent the 35M and 35L dieses,
respectively, and are *not* valid for prime 13; for 13 the (| symbol
replaces both /|). And (|\' .

Because 112-edo is highly inconsistent, many of these degree
assignments were judgment calls: 7C, 11M, and 13L are all about
midway between degrees of 112, so I could have "rounded" them either
up or down. For all of these I chose up (which is not the closest
for any of them), because the fifth (3rd harmonic) and ninth (9th
harmonic) are also altered upward in this (wide-fifth) notation.
Consequently, you'll get much better approximations of 6:7, 7:9,
6:11, and 9:11 than you would otherwise.

If you're using this for pajara (through the 11-limit), then only one
difference needs to be observed: the |) symbol will be omitted for
prime 7, since the 7-comma vanishes in that scale system.

--George

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

Here's a simple pajara cadence, 1-IV-V7-I.

http://bahamas.eshockhost.com/~xenharmo/midi/examples/cadence/

--- In tuning@yahoogroups.com, "George D. Secor" wrote:
>
[snip]
> > > However, if 56 is too many tones for you, and if you want a
pajara-capable tuning that offers free modulation, then consider the
following. I was curious to see what I would get when I combined two
circles of my 17-tone well-temperament 600 cents apart. I was
delighted to discover that the 16 best 4:5:6:7 chords of the
resulting 34-WT are produced by a two chains of generators of
~107.220c (very close to the pajara-56-ET figure), and that all
sixteen 11-limit otonal hexads (4:5:6:7:9:11) built on the same
starting tones are such that no consonance exceeds the above-
mentioned 5:7 error.
> >
> > It might be interesting to look at 34-et pajara well-temperaments
as such. All those exact 11/7s are intriguing, however.
>
> That description is one I copied & edited my 17-WT .scl file, which
reads, "George Secor's well temperament with 5 pure 11/7 and 3 near
just 11/6". The just 7:11's are mostly in less-likely-to-be-used
keys, but there are also 6 near-just 6:11's (with 0.09c error) in 34-
WT, and they're all in keys that are among the best in 11-limit
pajara. In brief, the following holds true in every key of both 17-
WT and 34-WT:
> 3-error < 11-error <= 7-error
>
> Something you may find even more intriguing is that 34-WT has both
a very good 13 (better than 56-ET) and 17, so it actually takes
pajara to the 17 limit. This is a consequence of the varied 5ths in
the 17-tone circles coming into play (13 being almost directly
opposite 1/1 in the circle).

This sounds very juicy ...

> > > Please try it, comparing with pajara-generated chords in 22-ET,
34-ET, and 56-ET, and see what you think. You can crunch numbers
till doomsday, but the bottom line is, "how does it sound?"
> >
> > Absolutely, and it sounds as if you could be on to something.
>
> I hope so. Hudson & Margo both gave my 17-WT enthusiastic reviews
on MMM, so now I'll be very disappointed if I don't see messages from
at least 4 different people saying that my 34-WT is twice as good. ;-)

Hi George,

Haven't been able to follow MMM lately, but would hate to miss out on
something so capable as this seems: could you please point me to a
specific message, or tell me where to download a sample (I'm assuming
you've made some)?

Regards,
Yahya

I loathe the superpythagorean notation George. The only reason I would
consider wide fifths are for transitions involving alterations of fifths.
Now, how about 119? Is it more consistent?

----- Original Message -----
From: "George D. Secor" <gdsecor@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 22 Kas�m 2006 �ar�amba 0:22
Subject: [tuning] 112-edo (was: 105-ET native-5th notation)

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@> wrote:
> >
> > ----- Original Message -----
> > From: "Gene Ward Smith" <genewardsmith@>
> > To: <tuning@yahoogroups.com>
> > Sent: 16 Kas�m 2006 Per�embe 2:34
> > Subject: [tuning] Re: 105-ET native-5th notation (was: 112-edo)
> >
> > > --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@>
wrote:
> > >
> > > > I would much rather prefer 117 if you don't mind George.
> > >
> > > The flat fifth is fine for meantone, and half of it is fine for
> > > semififths, the closely related 31&55 temperament. It has a
decent
> > > version of keemun temperament using the minor third. Porcupine
is also
> > > supported, and magic more or less also. On the other hand the
sharp
> > > fifth generator isn't that great; you get 39-equal from that.
> >
> > All the more reason to return to 112 then. Is there anything 112
does not
> > support?
>
> There are plenty of things that any given division of the octave
> won't support. It's easier to list the things that it *will*
support.
>
> What's of primary concern to you is whether 112-edo will do enough
of
> the things *you require* well enough for *your purposes*.

Among the best things that 112 will support are:

1) 1/4-Didymus-comma meantone, 9-limit
2) 1/5-Archytas-comma superpythagorean
3) 1/5-Archytas-comma pajara, 11-limit, via the 56-edo subset

There's not really a Pythagorean option here; you'd need 224-ET for
that.

> I've started looking at possible a wide-fifth notation for 112, and
I
> already see that it'll be more complicated than the narrow-fifth
one
> (which is rather nice, IMO). I'll let you know what I come up with.

Oz, assuming that you've downloaded the latest version of Scala and
have put back the "s 112" line after "d 224 131", try this in
sag_et.par for a wide-5th (superpythagorean) 112 notation (you'll
have to replace the narrow-5th one I previously gave you):

n 112 66
|) |\ )|) /| /|) /|\ (| (|) (|\ N\ )/N /N N) /N\

The /|) and (|\ symbols represent the 35M and 35L dieses,
respectively, and are *not* valid for prime 13; for 13 the (| symbol
replaces both /|). And (|\' .

Because 112-edo is highly inconsistent, many of these degree
assignments were judgment calls: 7C, 11M, and 13L are all about
midway between degrees of 112, so I could have "rounded" them either
up or down. For all of these I chose up (which is not the closest
for any of them), because the fifth (3rd harmonic) and ninth (9th
harmonic) are also altered upward in this (wide-fifth) notation.
Consequently, you'll get much better approximations of 6:7, 7:9,
6:11, and 9:11 than you would otherwise.

If you're using this for pajara (through the 11-limit), then only one
difference needs to be observed: the |) symbol will be omitted for
prime 7, since the 7-comma vanishes in that scale system.

--George

George, I very much like the notation for 112-tET. My only complaint is the
existence of two kinds of limmas. There ought to be an option to make the
apotome even as in 105-tET without sacrificing the consistency of 112.
Perhaps 119? Notice that Petr's latest extended-octave meantone can be
accomodated by an extended-octave 119.

What is pajara by the way?

----- Original Message -----
From: "George D. Secor" <gdsecor@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 16 Kas�m 2006 Per�embe 19:05
Subject: [tuning] 112-edo (was: 105-ET native-5th notation)

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > SNIP
> >
> > > >
> > > > P.S. Is 105 actually an ugly choice for a possible meantone-
> pythagorean
> > > > hybrid in your opinion?
> > >
> > > To be honest, I think it is rather ugly.
> > >
> > > Please look at 129-edo to see whether that's a possibility. ...
> >
> > I would much rather prefer 117 if you don't mind George.
>
> Before I even look at that one, could we please take another look at
> 112? I found, much to my surprise, that a narrow-5th notation
> actually works quite well. Simply add the following lines to
> sag_et.par:
>
> d 112 65
> ~|( /|) (|) /|\ (|\ (N( /N\
>
> and remove this line:
> s 112
> which follows "d 224 131"
>
> Note: The lines that I really wanted to add are these:
>
> n 112 65
> ~|( /|) (|) /|\ (|\ (N( /N\
>
> without removing the "s 112" line, but for some reason the "set nota
> sa112n" command doesn't work, because I get an error message that
> sa112n isn't a valid notation, even though "set nota sa__n" works for
> the other native-5th lines in the file.
>
> Manuel, please note!!! I suspect that there's a bug in Scala that
> doesn't allow native-5th notations for divisions >99, i.e., having
> more than 2 digits.
>
> A brief explanation of this 112-edo symbol set is in order. This is
> basically an extended meantone notation (as Gene suggested), very
> similar to the 31-ET notation for the 11 limit, where the 11M-diesis
> (32:33) symbol is used for both the 11M-diesis and 7-comma
> alterations. A 4:5:6:7 chord on C may therefore be notated as either
> C,E,G,A# or C,E,G,Bb\!/ .
>
> However, note that, because the 11M-diesis (5deg) is not exactly half
> as many degrees as the apotome (9deg), there must be two 11-diesis
> symbols, so you must take care to use the proper one. If you're
> altering upward from a nominal to get an otonal 11 relationship, then
> the /|\ symbol is used, but if it's downward, then the (!) symbol is
> used. For example, 8:11 on C is C,F/|\, whereas on F it's F,B(!) .
> This also affects the way you notate the 7-comma; e.g., 4:5:6:7 on G
> will be either G,B,D,F(!) or G,B,D,E#. The utonal 11's work the
> opposite way: e.g., 16/11 of C is G\!/, whereas 16/11 of B is F(|) .
> This applies to *all tunings* where the 11M diesis is *not exactly
> half* the size of the apotome, for both temperaments and (always) JI.
>
> The /|) and (|\ symbols must be interpreted as 13M and 13L dieses
> (1024:1053 and 26:27), respectively, not as 35M and 35L (35:36 and
> 8192:8505, their primary roles), so in order for the lines to be
> completely correct, those two symbols should have right accents, like
> this:
>
> n 112 65
> ~|( /|). (|) /|\ (|\' (N( /N\
>
> Examples of instances where "completely correct" interpretation of
> the symbols has significance is 1) when extreme pitch precision is
> required, or 2) when transferring the symbols for playback in another
> temperament (where 35M and 13M could be different numbers of degrees).
>
> But since Dave and I haven't yet worked with Manuel to implement
> right-accents adequately in Scala, the unaccented symbols will have
> to do for now.
>
> The ~|( symbol is valid as the 17-comma, so this meantone-notation
> for 112 is actually 17-limit! For example, a 10:12:14:17 diminished-
> 7th chord would be notated E,G,Bb\!/,Db~|( .
>
> Oz, please give this a try. If you like it, we might also try a wide-
> 5th notation (observing that the best 5th of 56-ET allows a nice
> implementation of pajara, with 11 thrown in for good measure).
>
> --George
>
>

--- In tuning@yahoogroups.com, "yahya_melb" <yahya@...> wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" wrote:
> > ...
> > Something you may find even more intriguing is that 34-WT has
both
> a very good 13 (better than 56-ET) and 17, so it actually takes
> pajara to the 17 limit. This is a consequence of the varied 5ths
in
> the 17-tone circles coming into play (13 being almost directly
> opposite 1/1 in the circle).
>
> This sounds very juicy ...
>
> > > > Please try it, comparing with pajara-generated chords in 22-
ET,
> 34-ET, and 56-ET, and see what you think. You can crunch numbers
> till doomsday, but the bottom line is, "how does it sound?"
> > >
> > > [GW Smith:]
> > > Absolutely, and it sounds as if you could be on to something.
> >
> > I hope so. Hudson & Margo both gave my 17-WT enthusiastic
reviews
> on MMM, so now I'll be very disappointed if I don't see messages
from
> at least 4 different people saying that my 34-WT is twice as
good. ;-)
>
> Hi George,
>
> Haven't been able to follow MMM lately, but would hate to miss out
on
> something so capable as this seems: could you please point me to a
> specific message, or tell me where to download a sample (I'm
assuming
> you've made some)?
>
> Regards,
> Yahya

Hi Yahya,

I wasn't sure whether you were asking about 34-WT, 17-WT, or both, so
I checked to see when you last posted to MMM and saw that it was well
before my 17-tone paper (recently published in Xenharmonikon 18) was
announced there, coincidentally on the heels of the 17-tone piano
project -- so I guess you have a lot of catching up to do.

Here's my initial announcement about the paper (the link is no longer
good):
/makemicromusic/topicId_14872.html#14889

Here are links to both my 17-WT paper and an mp3 file of part of a
jazz composition in 17-WT I was working on a while back:
/makemicromusic/topicId_14872.html#14993

This is Hudson Lacerda's very kind review of the 17-WT tuning:
/makemicromusic/topicId_14872.html#14993
and another message with a link to an improvisation he did in 17-WT:
/makemicromusic/topicId_14872.html#15062
plus my very favorable comments about it:
/makemicromusic/topicId_14872.html#15094

Margo also made some nice observations:
/makemicromusic/topicId_15142.html#15142

There's not very much I can direct you to yet about 34-WT, except
that this is the message that started the present discussion:
/tuning/topicId_67957.html#68032
I have only one sound file in 34-WT, but it would require so much
explanation that I think it would be best if I didn't share it at
this point. You should get a general idea of 7-limit chords in the
best keys of 34-WT from the 56-ET files Gene prepared (I haven't had
a chance to listen to them yet):
/tuning/topicId_67957.html#68107
After you've listened to those, we would appreciate your opinion
about whether the 22- or 56-ET (pajara) examples are better,
particularly for the V7 chord.

Best,

--George

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> I loathe the superpythagorean notation George. The only reason I would
> consider wide fifths are for transitions involving alterations of
fifths.
> Now, how about 119? Is it more consistent?

Not really.

I guess I don't understand exactly what you're trying to do. If you
want a tuning that supports both a meantone and a pythagorean notation,
then you'll need a tuning with both meantone and just (or nearly just)
fifths. Since that would involve tones ~5 cents apart, you would then
be then looking at something on the order of 224-ET (which is 15-limit
consistent, and which would also allow you to keep the 112-edo meantone
subset notation).

Have you looked at 224?

--George

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> George, I very much like the notation for 112-tET. My only
complaint is the
> existence of two kinds of limmas. There ought to be an option to
make the
> apotome even as in 105-tET without sacrificing the consistency of
112.

112 really isn't really consistent -- the 112 meantone notation
doesn't use its best fifth, yet it's possible to overcome that with a
meaningful set of accidentals. Your complaint about two kinds of
limmas, however, points up a lingering inconsistency.

> Perhaps 119? Notice that Petr's latest extended-octave meantone can
be
> accomodated by an extended-octave 119.

Nope, 119 won't do it.

I believe the limma inconsistency can be remedied by using a division
that has a fifth ~1/5-comma narrow, i.e., somewhere in the
neighborhood of 697.65 cents. IIRC, you said that you also want a
division in which a chain of meantone-like fifths takes you through
all of the tones.

Perhaps Gene or Carl will have some suggestions.

> What is pajara by the way?

It's a scale system with a period of 600 cents and a generator of
~109 cents. For a good understanding of it, you need to read Paul
Erlich's 22-tone paper. The link that I have is no longer good, so I
hope someone else can provide a current one.

Best,

--George

I would like to find a not-so-voluminous equal tuning with meantone fifths
around 696.5 cents, yielding 4:5:6:7 as c:e:g:a#, limmas at about 90 cents,
minor tones at about 180 cents (double-limma), decent apotomes, Pythagorean
and unidecimal major thirds deviating not as much as 3-4 cents, almost just
11:10, 12:11, 13:12, 14:13... (one or two cents octave tempering is also
possible) I guess 112-edo is the best we get at this conjuncture.

Yes, I looked at 224. I have an aversion with divisions greater than 199.

Oz.

----- Original Message -----
From: "George D. Secor" <gdsecor@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 22 Kas�m 2006 �ar�amba 22:17
Subject: [tuning] Re: 112-edo (was: 105-ET native-5th notation)

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > I loathe the superpythagorean notation George. The only reason I would
> > consider wide fifths are for transitions involving alterations of
> fifths.
> > Now, how about 119? Is it more consistent?
>
> Not really.
>
> I guess I don't understand exactly what you're trying to do. If you
> want a tuning that supports both a meantone and a pythagorean notation,
> then you'll need a tuning with both meantone and just (or nearly just)
> fifths. Since that would involve tones ~5 cents apart, you would then
> be then looking at something on the order of 224-ET (which is 15-limit
> consistent, and which would also allow you to keep the 112-edo meantone
> subset notation).
>
> Have you looked at 224?
>
> --George
>
>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:

> Oz, assuming that you've downloaded the latest version of Scala and
> have put back the "s 112" line after "d 224 131", try this in
> sag_et.par for a wide-5th (superpythagorean) 112 notation (you'll
> have to replace the narrow-5th one I previously gave you):

I've been wanting to try this since Manuel told me about it, but where
is it explained?

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> George, I very much like the notation for 112-tET. My only complaint
is the
> existence of two kinds of limmas.

Various intervals are called a limma; the Hughens-Fokker list has
27/25, 135/128, and 256/243. Depending on the approximations, a
temperament may make this into a smaller list. In 12-equal, and
12-equal *only*, these are all the same. In any other temperament
there are at least two "limmas". In any meantone system, 27/25 and
256/243 are the same, since each is an adjustment (up or down) of
16/15 by 81/80.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> I would like to find a not-so-voluminous equal tuning with meantone
fifths
> around 696.5 cents, yielding 4:5:6:7 as c:e:g:a#, limmas at about 90
cents,

How do you define "limma"?

> minor tones at about 180 cents (double-limma), decent apotomes,

How do you define "decent apotomes"?

Pythagorean
> and unidecimal major thirds deviating not as much as 3-4 cents,

You want *both* Pythagorean and unidecimal thirds?

almost just
> 11:10, 12:11, 13:12, 14:13... (one or two cents octave tempering is also
> possible) I guess 112-edo is the best we get at this conjuncture.
>
> Yes, I looked at 224. I have an aversion with divisions greater than
199.

It looks to me your requirements entail a large equal division. That
is, I think they contradict not-so-voluminous. 224 or 270 look good.

I was rather thinking among the lines of 135:128 and 256:243.

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@coolgoose.com>
To: <tuning@yahoogroups.com>
Sent: 23 Kas�m 2006 Per�embe 0:13
Subject: [tuning] Re: 112-edo (was: 105-ET native-5th notation)

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > George, I very much like the notation for 112-tET. My only complaint
> is the
> > existence of two kinds of limmas.
>
> Various intervals are called a limma; the Hughens-Fokker list has
> 27/25, 135/128, and 256/243. Depending on the approximations, a
> temperament may make this into a smaller list. In 12-equal, and
> 12-equal *only*, these are all the same. In any other temperament
> there are at least two "limmas". In any meantone system, 27/25 and
> 256/243 are the same, since each is an adjustment (up or down) of
> 16/15 by 81/80.
>
>

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

> I believe the limma inconsistency can be remedied by using a division
> that has a fifth ~1/5-comma narrow, i.e., somewhere in the
> neighborhood of 697.65 cents. IIRC, you said that you also want a
> division in which a chain of meantone-like fifths takes you through
> all of the tones.
>
> Perhaps Gene or Carl will have some suggestions.

That's the vicinity of 43, so maybe 43, 74, 98, or 55.

If I knew what this alleged limma problem was it would help.

> > What is pajara by the way?
>
> It's a scale system with a period of 600 cents and a generator of
> ~109 cents. For a good understanding of it, you need to read Paul
> Erlich's 22-tone paper. The link that I have is no longer good, so I
> hope someone else can provide a current one.

http://www.xenharmony.org/text/forms-tonality.pdf

The limma problem is the situation with 53-edo: In terms of "commas", 4+4=8,
hence the minor tone, 4+5=9, hence the wholetone.

Oz.

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@coolgoose.com>
To: <tuning@yahoogroups.com>
Sent: 23 Kas�m 2006 Per�embe 0:27
Subject: [tuning] Re: 112-edo (was: 105-ET native-5th notation)

> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>
> > I believe the limma inconsistency can be remedied by using a division
> > that has a fifth ~1/5-comma narrow, i.e., somewhere in the
> > neighborhood of 697.65 cents. IIRC, you said that you also want a
> > division in which a chain of meantone-like fifths takes you through
> > all of the tones.
> >
> > Perhaps Gene or Carl will have some suggestions.
>
> That's the vicinity of 43, so maybe 43, 74, 98, or 55.
>
> If I knew what this alleged limma problem was it would help.
>
> > > What is pajara by the way?
> >
> > It's a scale system with a period of 600 cents and a generator of
> > ~109 cents. For a good understanding of it, you need to read Paul
> > Erlich's 22-tone paper. The link that I have is no longer good, so I
> > hope someone else can provide a current one.
>
> http://www.xenharmony.org/text/forms-tonality.pdf
>
>
>

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > I would like to find a not-so-voluminous equal tuning with meantone
> fifths
> > around 696.5 cents, yielding 4:5:6:7 as c:e:g:a#, limmas at about 90
> cents,
>
> How do you define "limma"?

Pythagorean.

>
> > minor tones at about 180 cents (double-limma), decent apotomes,
>
> How do you define "decent apotomes"?
>

Pythagorean.

> Pythagorean
> > and unidecimal major thirds deviating not as much as 3-4 cents,
>
> You want *both* Pythagorean and unidecimal thirds?

Yes.

>
> almost just
> > 11:10, 12:11, 13:12, 14:13... (one or two cents octave tempering is also
> > possible) I guess 112-edo is the best we get at this conjuncture.
> >
> > Yes, I looked at 224. I have an aversion with divisions greater than
> 199.
>
> It looks to me your requirements entail a large equal division. That
> is, I think they contradict not-so-voluminous. 224 or 270 look good.
>
>

I would be most pleased if I can avoid large divisions such as those you
describe.

If I get it right, this pajara has exactly 5.5 generators within the period,
with the smallest interval at 54.545 cents?

SNIP

>
> > > What is pajara by the way?
> >
> > It's a scale system with a period of 600 cents and a generator of
> > ~109 cents. For a good understanding of it, you need to read Paul
> > Erlich's 22-tone paper. The link that I have is no longer good, so I
> > hope someone else can provide a current one.
>
> http://www.xenharmony.org/text/forms-tonality.pdf
>
>

Hi George,

--- In tuning@yahoogroups.com, "George D. Secor" wrote:
>
> --- In tuning@yahoogroups.com, "yahya_melb" wrote:
> >
> > --- In tuning@yahoogroups.com, "George D. Secor" wrote:
> > > ...
> > > Something you may find even more intriguing is that 34-WT has
both a very good 13 (better than 56-ET) and 17, so it actually takes
pajara to the 17 limit. This is a consequence of the varied 5ths in
the 17-tone circles coming into play (13 being almost directly
opposite 1/1 in the circle).
> >
> > This sounds very juicy ...
> >
> > > > > Please try it, comparing with pajara-generated chords in 22-
ET, 34-ET, and 56-ET, and see what you think. You can crunch numbers
till doomsday, but the bottom line is, "how does it sound?"
> > > >
> > > > [GW Smith:]
> > > > Absolutely, and it sounds as if you could be on to something.
> > >
> > > I hope so. Hudson & Margo both gave my 17-WT enthusiastic
reviews on MMM, so now I'll be very disappointed if I don't see
messages from at least 4 different people saying that my 34-WT is
twice as good. ;-)
> >
> > Hi George,
> >
> > Haven't been able to follow MMM lately, but would hate to miss
out on something so capable as this seems: could you please point me
to a specific message, or tell me where to download a sample (I'm
assuming you've made some)?
> >
> > Regards,
> > Yahya
>
> Hi Yahya,
>
> I wasn't sure whether you were asking about 34-WT, 17-WT, or both,
so I checked to see when you last posted to MMM and saw that it was
well before my 17-tone paper (recently published in Xenharmonikon 18)
was announced there, coincidentally on the heels of the 17-tone piano
project -- so I guess you have a lot of catching up to do.

;-) Guess so!

> Here's my initial announcement about the paper (the link is no
longer good):
> /makemicromusic/topicId_14872.html#14889
>
> Here are links to both my 17-WT paper and an mp3 file of part of a
jazz composition in 17-WT I was working on a while back:
> /makemicromusic/topicId_14872.html#14993

Hmmm ... I found this message at #14943 ! I've downloaded and read
your paper, and also played your jazzy sample; sounds pretty good
(except for those prominent parallel chords).

Your paper was so well written that I've read it in the wee hours
without difficulty! Summarising, for my own use, the practical
effects of what I read:
a) the most effective melodic semitone is around 70 cents;
b) the most effective harmonic semitone is around 70 cents;
c) cadences with higher contrasts of consonance are more effective,
thus favouring high-limit (or irrational) chords as preparatory
dissonances before final low-limit chords as consonances;
d) there are plenty of scales and progressions left to discover
within 17WT;
e) 17WT approximates ratios of all primes to 13 (except 5) very well
indeed, giving us plenty of 7-limit, 11-limit and 13-limit intervals
to play with;
f) 17WT approximates ratios of 5 rather (optimally) badly, so we
can't expect to use the idioms arising from the 5 limit thirds and
sixths;
g) as a work-around to the "no 5s" nature of 17WT, we could add 5
extra tones, much as you did with 19+3.

If I understood your recent post correctly, 34WT gives us all the
benefits of 17WT, but with 5s thrown in.

> This is Hudson Lacerda's very kind review of the 17-WT tuning:
> /makemicromusic/topicId_14872.html#14993

I read that. I'll have a look at the rest of your links at the next
opportunity. Thanks for the info!

Regards,
Yahya

> and another message with a link to an improvisation he did in 17-WT:
> /makemicromusic/topicId_14872.html#15062
> plus my very favorable comments about it:
> /makemicromusic/topicId_14872.html#15094
>
> Margo also made some nice observations:
> /makemicromusic/topicId_15142.html#15142
>
> There's not very much I can direct you to yet about 34-WT, except
that this is the message that started the present discussion:
> /tuning/topicId_67957.html#68032
> I have only one sound file in 34-WT, but it would require so much
explanation that I think it would be best if I didn't share it at
this point. You should get a general idea of 7-limit chords in the
best keys of 34-WT from the 56-ET files Gene prepared (I haven't had
a chance to listen to them yet):
> /tuning/topicId_67957.html#68107
> After you've listened to those, we would appreciate your opinion
about whether the 22- or 56-ET (pajara) examples are better,
particularly for the V7 chord.
>
> Best,
>
> --George
>

> Here's a simple pajara cadence, 1-IV-V7-I.
>
> http://bahamas.eshockhost.com/~xenharmo/midi/examples/cadence/

Thanks!

I hear very little difference between cadence22.mid and
cadence56.mid.

But I would not call this a good test of 7-limit consonance,
since it only has one 7-limit chord.

-Carl

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:

> > How do you define "limma"?

> Pythagorean.

In which case, why in the world are you complaining about having two
of them? You can hardly want two different fifths, but not two
different Pythagorean limmas.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> If I get it right, this pajara has exactly 5.5 generators within the
period,
> with the smallest interval at 54.545 cents?

That's pajara in 22-equal tuning. However, it can be tuned in various
ways; 5.5 (22 equal) to 5.6 (56 equal) generators per period being a
reasonable range. George may even lobby for 5.67 (34 equal.)

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > Here's a simple pajara cadence, 1-IV-V7-I.
> >
> > http://bahamas.eshockhost.com/~xenharmo/midi/examples/cadence/
>
> Thanks!
>
> I hear very little difference between cadence22.mid and
> cadence56.mid.
>
> But I would not call this a good test of 7-limit consonance,
> since it only has one 7-limit chord.

You might listen to the kirkwood examples again.

--- In tuning@yahoogroups.com, "yahya_melb" <yahya@...> wrote:
>
> Hi George,
>
> --- In tuning@yahoogroups.com, "George D. Secor" wrote:
> > ...
> > Here are links to both my 17-WT paper and an mp3 file of part of
a
> jazz composition in 17-WT I was working on a while back:
> > /makemicromusic/topicId_14872.html#14993
>
> Hmmm ... I found this message at #14943 !

Sorry about the typo!

> I've downloaded and read
> your paper, and also played your jazzy sample; sounds pretty good
> (except for those prominent parallel chords).

I picked a style in which parallel 5ths are not forbidden -- but I
agree that they leave something to be desired, and I would make it a
point not to have them through the entire piece (if I ever get around
to finishing it).

> Your paper was so well written that I've read it in the wee hours
> without difficulty!

Perhaps not: if it's really that good, then perhaps you had
difficulty putting it down and going to bed at a more reasonable
hour. ;-)

> Summarising, for my own use, the practical
> effects of what I read:
> a) the most effective melodic semitone is around 70 cents;
> b) the most effective harmonic semitone is around 70 cents;

Some time after I wrote the paper, I was talking to Paul Erlich (on
the harmonic entropy list), and we seemed to be homing in on a more
precise value (of ~67 cents, i.e., something near 51:53).

> c) cadences with higher contrasts of consonance are more
effective,
> thus favouring high-limit (or irrational) chords as preparatory
> dissonances before final low-limit chords as consonances;
> d) there are plenty of scales and progressions left to discover
> within 17WT;
> e) 17WT approximates ratios of all primes to 13 (except 5) very
well
> indeed, giving us plenty of 7-limit, 11-limit and 13-limit
intervals
> to play with;
> f) 17WT approximates ratios of 5 rather (optimally) badly, so we
> can't expect to use the idioms arising from the 5 limit thirds and
> sixths;
> g) as a work-around to the "no 5s" nature of 17WT, we could add 5
> extra tones, much as you did with 19+3.

Yep, apart from the historical connections, you've covered most of
the main points.

> If I understood your recent post correctly, 34WT gives us all the
> benefits of 17WT, but with 5s thrown in.

Yes, and prime 17 also. (Anything above that will likely suffer the
effects of harmonic entropy.) I was a bit surprised that having
twice as many tones results in only two more primes being represented.

> > This is Hudson Lacerda's very kind review of the 17-WT tuning:
> > /makemicromusic/topicId_14872.html#14993
>
> I read that. I'll have a look at the rest of your links at the
next
> opportunity. Thanks for the info!

And thanks for the feedback.

Best,

--George

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > Here's a simple pajara cadence, 1-IV-V7-I.
> >
> > http://bahamas.eshockhost.com/~xenharmo/midi/examples/cadence/
>
> Thanks!
>
> I hear very little difference between cadence22.mid and
> cadence56.mid.
>
> But I would not call this a good test of 7-limit consonance,
> since it only has one 7-limit chord.
>
> -Carl

I listened to Gene's midi examples and concluded that midi files
aren't very good for this purpose, inasmuch as the selection of
timbre is critical: you need a suitable harmonic content, and no
vibrato or other transients that might obscure the beating (and other
disturbances) produced by the tuning itself.

Since I eventually expect to be writing a paper on 34-WT and pajara,
I decided that it would be appropriate to prepare some mp3 examples
to accompany it (which I did over the long weekend). I'll make these
available for your evaluation very soon.

--George

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@> wrote:
> >
> > If I get it right, this pajara has exactly 5.5 generators within the
> period,
> > with the smallest interval at 54.545 cents?
>
> That's pajara in 22-equal tuning. However, it can be tuned in various
> ways; 5.5 (22 equal) to 5.6 (56 equal) generators per period being a
> reasonable range. George may even lobby for 5.67 (34 equal.)

You may even lobby for it, because it's not bad at all.

--George

> > > Here's a simple pajara cadence, 1-IV-V7-I.
> > >
> > > http://bahamas.eshockhost.com/~xenharmo/midi/examples/cadence/
> >
> > Thanks!
> >
> > I hear very little difference between cadence22.mid and
> > cadence56.mid.
> >
> > But I would not call this a good test of 7-limit consonance,
> > since it only has one 7-limit chord.
> >
> > -Carl
>
> I listened to Gene's midi examples and concluded that midi files
> aren't very good for this purpose, inasmuch as the selection of
> timbre is critical: you need a suitable harmonic content, and no
> vibrato or other transients that might obscure the beating (and
> other disturbances) produced by the tuning itself.
>
> Since I eventually expect to be writing a paper on 34-WT and
> pajara, I decided that it would be appropriate to prepare some
> mp3 examples to accompany it (which I did over the long weekend).
> I'll make these available for your evaluation very soon.
>
> --George

Great! Thanks, G.S.

-Carl

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

> You may even lobby for it, because it's not bad at all.

I think the 7-limit sounds kind of harsh if you compare it to 56.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
>
> > You may even lobby for it, because it's not bad at all.
>
> I think the 7-limit sounds kind of harsh if you compare it to 56.

I find the improvement at the 5-limit of 34 over 56 quite dramatic,
and I was wondering whether that might upstage the more subtle (yet
very significant) improvement at the 7-, 9-, and 11-limits in a
listening test. I've already indicated that I prefer the best keys
of 17-WT (which is very close to a 56-subset pajara) over 17-ET, so I
would agree with you in choosing 56 over 34-ET.

I think it'll be interesting to see how the voting will turn out for
22-ET vs. 34-ET. I've made mp3 files of 3 different chord
progressions in each of 22-ET, 34-ET, 34-WT, 56-ET, JI (for
reference), and 12-ET (for fun), and I expect to have these up
sometime tomorrow.

--George

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>. I've already indicated that I prefer the best keys
> of 17-WT (which is very close to a 56-subset pajara) over 17-ET, so
>I
> would agree with you in choosing 56 over 34-ET.

Working with 17 every day, I also consider it a (powerful and
practical) subset of something else, just a feeling I get from
listening. As I search for logical ways to expand the tuning (17 being
my central tuning), I find 51 divisions as the lowest reasonable
division, and 153 as a likely path for marking the centerpoints of
equal "regions".
>
> I think it'll be interesting to see how the voting will turn out for
> 22-ET vs. 34-ET. I've made mp3 files of 3 different chord
> progressions in each of 22-ET, 34-ET, 34-WT, 56-ET, JI (for
> reference), and 12-ET (for fun), and I expect to have these up
> sometime tomorrow.

Looking forward to it!

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

> Working with 17 every day, I also consider it a (powerful and
> practical) subset of something else, just a feeling I get from
> listening. As I search for logical ways to expand the tuning (17 being
> my central tuning), I find 51 divisions as the lowest reasonable
> division, and 153 as a likely path for marking the centerpoints of
> equal "regions".

Not from listening but just from the math, 34 and 68 seem to be
clearly the two most logical ways of extending 17. I don't see what
the interest is in 51 or 153 particularly.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@> wrote:
>
> > Working with 17 every day, I also consider it a (powerful and
> > practical) subset of something else, just a feeling I get from
> > listening. As I search for logical ways to expand the tuning (17
>>being
> > my central tuning), I find 51 divisions as the lowest reasonable
> > division, and 153 as a likely path for marking the centerpoints of
> > equal "regions".
>
> Not from listening but just from the math, 34 and 68 seem to be
> clearly the two most logical ways of extending 17. I don't see what
> the interest is in 51 or 153 particularly.
>

51 is the lowest *17 with a good 7/4, isn't it? I'm also thinking of
resultant tones from my own versions of 17, which I'd like to use
for a kind of phantom-common-tone modulation from one tuning to the
next.

-Cameron Bobro

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@> wrote:
>
> > [Gene wrote:]
> > > Here's a simple pajara cadence, 1-IV-V7-I.
> > > ...
> > > http://bahamas.eshockhost.com/~xenharmo/midi/examples/cadence/
> >
> > Thanks!
> >
> > ... I would not call this a good test of 7-limit consonance,
> > since it only has one 7-limit chord.
> >
> > -Carl
>
> I listened to Gene's midi examples and concluded that midi files
> aren't very good for this purpose, inasmuch as the selection of
> timbre is critical: you need a suitable harmonic content, and no
> vibrato or other transients that might obscure the beating (and
other
> disturbances) produced by the tuning itself.
>
> Since I eventually expect to be writing a paper on 34-WT and
pajara,
> I decided that it would be appropriate to prepare some mp3 examples
> to accompany it (which I did over the long weekend). I'll make
these
> available for your evaluation very soon.
>
> --George

Here are my comparative examples of pajara tunings, for your
scrutiny. The files are arranged in 3 groups.

I've used notation that's valid for all of the tunings in the
examples:

down up amount of alteration
--------------------------------
\ / 5-comma (80:81)
v ^ 11M-diesis (32:33)
v ^ 13L-diesis (26:27)
\ / 17-comma (2176:2187)
vanishes 7-comma (63:64)

GROUP A
-------

In pajara the principal consonances (in closed position) are the
otonal tetrad, 4:5:6:7, and the utonal tetrad, 1/7:1/6:1/5:1/4. The
examples in this group consist of a progression of 4 consonant pajara
(7-limit) chords:

G B\ D G - 4:5:6:8 - otonal (major) triad
C A Eb/ G - 1/12:1/7:1/5:1/4 - utonal tetrad, open position
D A C F#\ - 4:6:7:10 - otonal (harmonic 7th) tetrad, open position
G B\ D G - 2:5:6:8 - otonal (major) triad

First listen to these in JI, so you have some idea what we're trying
to approximate. So in 7-limit JI we have:
http://xenharmony.wikispaces.com/space/showimage/Pajara-JI7a.mp3

Paul Erlich suggested that, since the utonal tetrad isn't as
consonant as the otonal, one might prefer an alternate JI tuning, of
which the most consonant voicing is 10:12:15:17 (in the first
inversion). In the following 17-limit JI version of the previous
example, the 2nd chord becomes 10:17:24:30 (in open position):
http://xenharmony.wikispaces.com/space/showimage/Pajara-JI17a.mp3

Would you agree that this is more consonant that the 7-limit example?

Here's the above progression in 4 different pajara temperaments:
22-ET:
http://xenharmony.wikispaces.com/space/showimage/Pajara-22a.mp3
34-ET (using 28deg34 for 4:7):
http://xenharmony.wikispaces.com/space/showimage/Pajara-34a.mp3
56-ET (using 46deg56 for 4:7):
http://xenharmony.wikispaces.com/space/showimage/Pajara-56a.mp3
34-WT (using 28deg34 for 4:7):
http://xenharmony.wikispaces.com/space/showimage/Pajara-34WTa.mp3

Note that the 2nd and 3rd chords have two common tones (C and A),
which do *not* change in pitch (except that the C's are an octave
apart).

And just for fun, here's a 12-ET example. It's a little lower in
pitch (C=261.63) than all of the others (C=264), and it's here simply
as a reminder of why we've been looking for alternative tunings.

12-ET:
http://xenharmony.wikispaces.com/space/showimage/Pajara-12ab.mp3

For the 2nd chord the first JI example has 1/12:1/7:1/5:1/4, while
the second example replaces that with 10:17:24:30 (changing only the
tenor voice in the 2nd chord), which sounds more consonant, IMO. The
JI examples have a couple of shifts in pitch that are tempered out in
pajara, and they're here to give a clearer idea of what we're trying
to approximate with the temperaments.

GROUP B
-------

Note that the 2nd chord in the Group A examples is *not* an otonal
dominant 9th chord with the root omitted, although it has a similar
sound. Should you be interested in comparing that particular chord
in the above tunings, the Group B examples have that otonal chord in
the progression instead. Warning: this involves a comma shift in the
tenor voice, from A\ to A, which isn't a good pajara progression --
it's here only for comparison with the above.

The examples in this group consist of a progression of 4 consonant 9-
limit chords:

G B\ D G - 4:5:6:8 - otonal (major) triad
C A\ Eb G - 6:10:14:18 - otonal 9th w/o root, open position
D A C F#\ - 4:6:7:10 - otonal (harmonic 7th) tetrad, open position
G B\ D G - 2:5:6:8 - otonal (major) triad

First in 9-limit JI:
http://xenharmony.wikispaces.com/space/showimage/Pajara-JI9b.mp3

And also in the following pajara temperaments:
22-ET:
http://xenharmony.wikispaces.com/space/showimage/Pajara-22b.mp3
34-ET (using 28deg34 for 4:7):
http://xenharmony.wikispaces.com/space/showimage/Pajara-34b.mp3
56-ET (using 46deg56 for 4:7):
http://xenharmony.wikispaces.com/space/showimage/Pajara-56b.mp3
34-WT (using 28deg34 for 4:7):
http://xenharmony.wikispaces.com/space/showimage/Pajara-34WTb.mp3

Notice that the pitch shifts in the tempered examples are larger than
the comma shifts in the JI example -- evidence that this version of
the progression isn't well suited for pajara (where it's more
appropriate to stick with the utonal tetrads in Group A).

If you couldn't hear any difference between the 56-ET and 34-WT
examples within groups A and B, it's because there isn't any
significant difference between the two at either the 7 or 9 limit.
Oh, there is very big difference between the two: 56-ET allows you
to play these otonal and utonal chords in 56 different keys (all in
the same intonation), while 34-WT has 16 "best" keys and 18 that are
not as good (except that they're better at the 5-limit).

For 12-ET nothing changes, so the example from Group A is repeated
here:
http://xenharmony.wikispaces.com/space/showimage/Pajara-12ab.mp3

GROUP C
-------

By including tones from extended pajara chains, we can add primes 11
and 13 to the mix, which allows subtle melodic movement, as well as
more exotic harmony.

The examples in this group consist of a progression of 8 consonant 17-
limit chords:

G B\ D G - 4:5:6:8 - otonal (major) triad
G A C Eb/ - 15:17:20:24 (alias 1/8:1/7:1/6:1/5) - utonal tetrad
G A C^ Ev - 8:9:11:13 - isoharmonic 13-limit tetrad
G A C#\ E - 7:8:10:12 - otonal tetrad
D A D F - 6:9:12:14 - subminor triad
D A C^ F - 6:9:11:14 - subminor triad w/ neutral 7th, open position
D A C F#\ - 4:5:7:10 - otonal tetrad, open position
G B\ D G - 2:5:6:8 - otonal (major) triad

First listen to this in 17-limit JI:
http://xenharmony.wikispaces.com/space/showimage/Pajara-JI17c.mp3

And also in the following temperaments:
22-ET:
http://xenharmony.wikispaces.com/space/showimage/Pajara-22c.mp3
34-ET (using 28deg34 for 4:7):
http://xenharmony.wikispaces.com/space/showimage/Pajara-34c.mp3
56-ET (using 46deg56 for 4:7):
http://xenharmony.wikispaces.com/space/showimage/Pajara-56c.mp3
34-WT (using 28deg34 for 4:7):
http://xenharmony.wikispaces.com/space/showimage/Pajara-34WTc.mp3

If you listen closely, you may hear a couple of differences between
the 56-ET and 34-WT examples in Group C (in the 3rd and 6th chords).

We could already tell from the numbers that 13 isn't well represented
in 22-ET (3rd chord), but would you agree that 11 (in the 6th chord)
doesn't come off very well either?

For a broader perspective, compare 12-ET:
Oops, sorry! -- this progression isn't possible in 12-ET! :-(

Please vote for which of the above 4 pajara temperaments you think is
best.

--George

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

> > Not from listening but just from the math, 34 and 68 seem to be
> > clearly the two most logical ways of extending 17. I don't see what
> > the interest is in 51 or 153 particularly.
> >
>
> 51 is the lowest *17 with a good 7/4, isn't it?

It's got a very flat major third, and an even worse sharp minor third;
admittedly not as bad as 12 in the other direction but with that many
notes I would want something better. 68, on the other hand, does much,
much better in both the 5 and 7 limits.

There's also the question of how to use the 7/4. Use the debased form
of miracle it supports? It also does an uncataloged version of
porcupine where the 7/4 has complexity 16; that doesn't seem promising
either. But I find nothing better than these two. With the same
complexity 68 will get you hemiwuer or octacot (the 88 cent
temperament for those fans of 88 cents.)

> G B\ D G - 4:5:6:8 - otonal (major) triad
> C A Eb/ G - 1/12:1/7:1/5:1/4 - utonal tetrad, open position
> D A C F#\ - 4:6:7:10 - otonal (harmonic 7th) tetrad, open position
> G B\ D G - 2:5:6:8 - otonal (major) triad
>
> First listen to these in JI, so you have some idea what we're
> trying to approximate. So in 7-limit JI we have:
> http://xenharmony.wikispaces.com/space/showimage/Pajara-JI7a.mp3
>
> Paul Erlich suggested that, since the utonal tetrad isn't as
> consonant as the otonal, one might prefer an alternate JI tuning,
> of which the most consonant voicing is 10:12:15:17 (in the first
> inversion). In the following 17-limit JI version of the previous
> example, the 2nd chord becomes 10:17:24:30 (in open position):
> http://xenharmony.wikispaces.com/space/showimage/Pajara-JI17a.mp3
>
> Would you agree that this is more consonant that the 7-limit
> example?

In this example not especially, and I think the 7-limit version
has nicer bite.

> Here's the above progression in 4 different pajara temperaments:
> 22-ET:
> http://xenharmony.wikispaces.com/space/showimage/Pajara-22a.mp3
> 34-ET (using 28deg34 for 4:7):
> http://xenharmony.wikispaces.com/space/showimage/Pajara-34a.mp3
> 56-ET (using 46deg56 for 4:7):
> http://xenharmony.wikispaces.com/space/showimage/Pajara-56a.mp3
> 34-WT (using 28deg34 for 4:7):
> http://xenharmony.wikispaces.com/space/showimage/Pajara-34WTa.mp3
>
> Note that the 2nd and 3rd chords have two common tones (C and A),
> which do *not* change in pitch (except that the C's are an octave
> apart).

The better 5-limit parts of 34 and 56 were immediately
appreciated. 7-limit still hard to tell because there's only
one otonal tetrad in this example. The 22 version had what
sounded like chorusing here, which I hated.

> And just for fun, here's a 12-ET example. It's a little lower in
> pitch (C=261.63) than all of the others (C=264), and it's here
> simply as a reminder of why we've been looking for alternative
> tunings.
>
> 12-ET:
> http://xenharmony.wikispaces.com/space/showimage/Pajara-12ab.mp3

Couldn't get this to download.

> For the 2nd chord the first JI example has 1/12:1/7:1/5:1/4,
> while the second example replaces that with 10:17:24:30 (changing
> only the tenor voice in the 2nd chord), which sounds more
> consonant, IMO. The JI examples have a couple of shifts in
> pitch that are tempered out in pajara, and they're here to
> give a clearer idea of what we're trying to approximate with
> the temperaments.
>
> GROUP B
> -------
>
> Warning: this involves a comma shift in the tenor voice, from
> A\ to A, which isn't a good pajara progression -- it's here
> only for comparison with the above.
>
> The examples in this group consist of a progression of 4
> consonant 9-limit chords:
>
> G B\ D G - 4:5:6:8 - otonal (major) triad
> C A\ Eb G - 6:10:14:18 - otonal 9th w/o root, open position
> D A C F#\ - 4:6:7:10 - otonal (harmonic 7th) tetrad, open position
> G B\ D G - 2:5:6:8 - otonal (major) triad
>
> First in 9-limit JI:
> http://xenharmony.wikispaces.com/space/showimage/Pajara-JI9b.mp3

Ditto. I'm getting a "redirection cycle detected" error.

> Please vote for which of the above 4 pajara temperaments you
> think is best.

I would say 34 based on this.

-Carl

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

> It's got a very flat major third, and an even worse sharp minor
>third;

Does everyone always need a 5/4? 153 can support an excellent 5/4
region if that's what I need. And I'd call 305.882 a flat minor
third, not a sharp one, as well as very close to the 309 cent region
of the 55/46 third, which is directly related to my 11- and 23-
heavy 17 tuning. I'm looking at equally-spaced narrow regions for
RI, not literal ET.

> ...admittedly not as bad as 12 in the other direction but with
that >many
> notes I would want something better.
> There's also the question of how to use the 7/4. Use the debased
>form of miracle it supports?

I'm not going to use that many notes. The trunk is 17, I just need
branches and leaves, which can come and go and vary in color with
the seasons. With a 27/23 Sm3 in my main tuning, for example, 55/46
as a chromatic movement is simply a 55/54 up and 6/5, a 46/45
movement; a lot of elegant superparticular and resultant-tone
related possibilities, seems to me. 153 provides a framework for all
the possibilities I've discoverd so far.

Of course I agree that 34 and 68 are better- for other approaches.

-Cameron Bobro

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
> > First listen to these in JI, so you have some idea what we're
> > trying to approximate. So in 7-limit JI we have:
> > http://xenharmony.wikispaces.com/space/showimage/Pajara-JI7a.mp3

I don't completely buy into "approximation"- the 34s and 56 both sound
better than either JI "original" to my ears.

> > Please vote for which of the above 4 pajara temperaments you
> > think is best.
>
> I would say 34 based on this.

Another vote for 34, specicially the WT.

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

> Would you agree that this is more consonant that the 7-limit example?

I actually liked 7-lim a little better, but much better than either
was the 9-limit version, and to hell with the comma shift.

> If you couldn't hear any difference between the 56-ET and 34-WT
> examples within groups A and B, it's because there isn't any
> significant difference between the two at either the 7 or 9 limit.

Ah. I slightly preferred 34-WT for some reason.

> First listen to this in 17-limit JI:
> http://xenharmony.wikispaces.com/space/showimage/Pajara-JI17c.mp3

This sounds much better in JI than in any of the temperaments.

> Please vote for which of the above 4 pajara temperaments you think is
> best.

34-WT and 56 are close, trailed by 34 and 22.

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@>
> wrote:
>
> > It's got a very flat major third, and an even worse sharp minor
> >third;
>
> Does everyone always need a 5/4? 153 can support an excellent 5/4
> region if that's what I need.

That's great, but I'm talking about 51. What is a "region"?

> And I'd call 305.882 a flat minor
> third, not a sharp one...

In that case you are using the 400 cent major third, which does make
sense. Even so, the result is that your triads are more out of whack
than in 12-et, because the major third is the same, the fifth is
sharper, and only the minor third is improved. You also again need to
think of how to organize the notes, and nothing very interesting in
the rank-2 temperament department seems to be available.

> Of course I agree that 34 and 68 are better- for other approaches.

So far you haven't explained what you are doing in a way I can understand.

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@> wrote:
> > > First listen to these in JI, so you have some idea what we're
> > > trying to approximate. So in 7-limit JI we have:
> > > http://xenharmony.wikispaces.com/space/showimage/Pajara-JI7a.mp3
>
> I don't completely buy into "approximation"- the 34s and 56 both sound
> better than either JI "original" to my ears.

A lot of people like some detuning of JI, but you can avoid the pure
JI sound with a lot less harshness than this, though not using pajara.
I also thought JI17c was a lot better than any of the temperings of it.

> > I would say 34 based on this.
>
>
> Another vote for 34, specicially the WT.

34-wt and 34-et are by no means the same, of course. It's a lot more
like 56-et, which of the equal temperaments presented I think is the
clear winner.

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

> > Please vote for which of the above 4 pajara temperaments you
> > think is best.
>
> I would say 34 based on this.

34-wt or 34-et?

> 34-WT and 56 are close, trailed by 34 and 22.

I would tend to be very critical of comparing a WT with
an ET like this, where the example doesn't use all the
keys of the WT.

-Carl

> > > Please vote for which of the above 4 pajara temperaments you
> > > think is best.
> >
> > I would say 34 based on this.
>
> 34-wt or 34-et?

I didn't listen to the WT version because George's site started
giving me an error. I'll try to download these again tonight,
but see my previous message on the wisdom of this comparison.

I'm surprised that you said 56 was clearly better than 34.
I thought they were pretty close, but 34 sounded better to me.
I'll try again and see if the same thing happens.

-Carl

George D. Secor wrote:

> Please vote for which of the above 4 pajara temperaments you think is > best.

Based on these samples the 34-WT and 56-ET didn't seem notably better than 34-ET to me -- not enough at any rate to justify the extra size of 56-ET. Still, I like the sound of 34-ET in these samples better than the 22-ET version, and 34-ET is still of a more or less manageable size. I'll vote for the 34-ET version.

Hi George and all,

I got to listen again just now, through headphones under good
conditions, and I was able to download all of the examples.
I hadn't heard group C earlier. Ingenious progression, George!
Neil, can you can pick this progression out on your 34 axe?

I'm with Gene (if I read right) that even the *melodic* aspects
of the JI version of group C sound best to me.

Overall, I'd say 34 perhaps earns its keep, but 56 does not.
Paul's psychoacoustics seem to be nicely validated here, as
22 does indeed sound just about (or slightly more) in tune
than 12 only a limit higher (i.e. 22 in 7 sounds slightly
better than 12 in 5).

-Carl

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

> I'm with Gene (if I read right) that even the *melodic* aspects
> of the JI version of group C sound best to me.

But the 56 version was decidedly better than the 34.

> Overall, I'd say 34 perhaps earns its keep, but 56 does not.
> Paul's psychoacoustics seem to be nicely validated here, as
> 22 does indeed sound just about (or slightly more) in tune
> than 12 only a limit higher (i.e. 22 in 7 sounds slightly
> better than 12 in 5).

Does harmonic entropy have a way of comparing the dissonance of a
triad in 12 to a tetrad in 22?

> > Overall, I'd say 34 perhaps earns its keep, but 56 does not.
> > Paul's psychoacoustics seem to be nicely validated here, as
> > 22 does indeed sound just about (or slightly more) in tune
> > than 12 only a limit higher (i.e. 22 in 7 sounds slightly
> > better than 12 in 5).
>
> Does harmonic entropy have a way of comparing the dissonance of
> a triad in 12 to a tetrad in 22?

He didn't have/use harmonic entropy in that paper. He used
unweighted RMS error, I believe. Actually he looks at both
weighted and unweighted, and chooses unweighted.

Harmonic entropy can compare irrational dyads, but as the
public consensus stuff stands, AFAIK not triads. I've been
hoping that will change for some time.

Actually you can sum the dyadic entropies in the triads.
I believe Paul has done some stuff in that direction.

In 2002, Paul showed that voronoi cells around triads in a
Chalmers-style plot

/tuning/files/trivoro.gif

agrees with the generalized Tenney height

/tuning/files/triadic.gif

. . .

-Carl

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
> That's great, but I'm talking about 51.

34, 51, 153... however far out I have to go (as little as possible
of course) to get my various chroma. The tuning is first andforemost
17. I specifically said that 51 is the "least", not the best or the
only.

>What is a "region"?

A region is just that, a narrow region in the pitch continuum
centered on an ET point. Rational intervals fall within that region
so I get transposition and modulation- this is how I work with RI.
Any WT within reason can be described this way, seems to me. For
example, in the 17 tuning I'm using at the moment, the 1/3 tones all
fall within + - 6 cents of 25/24, which is both an ideal "semitone"
in my opinion, and luckily almost identical to one 17-ET degree.

> In that case you are using the 400 cent major third, which does
>make
> sense.

Never going to use a 400 cent major third as a "major third" per se-
there's a resultant tone right above 400 cents that happens
somewhere in the tuning I use, so that might pop up as a color
harmony in the right place.

>Even so, the result is that your triads are more out of whack
> than in 12-et, because the major third is the same, the fifth is
> sharper, and only the minor third is improved. You also again need
>to
> think of how to organize the notes, and nothing very interesting in
> the rank-2 temperament department seems to be available.

What triads? Triads will certainly happen a lot in the course of
events but I don't care if they do or don't.

> So far you haven't explained what you are doing in a way I can
>understand.

For example: there is a melody. It is harmonized by another melody
in the same tuning (17) in counterpoint. The original melody is
repeated, but this time it is harmonized by the "phantom" tones
which appeared in relation to the counter-melody, for example
summation tones.If the phantom tones are to be more than purely
coloristic, ie. function as counter-melodies in their own right,
they need to be tied together in some way, and I'm finding that
supersets of 17 include the phantom tones I'm getting within the 17
I use. It is probably obvious that I intend to use these as
modulation points for modulating to other tunings within a piece of
music- still working on that. :-)

-Cameron Bobro

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > 34-WT and 56 are close, trailed by 34 and 22.
>
> I would tend to be very critical of comparing a WT with
> an ET like this, where the example doesn't use all the
> keys of the WT.

Good point. No need to listen to 33 more sets of files (one for each
key of 34-WT) -- I doubt we would have the time and/or patience for
that! Besides, it's not necessary, because (as I noted in my
commentary) 34-WT has 16 "best" keys and 18 others that are not as
good (except that they're better at the 5-limit). There's virtually
no difference in the amount of tempering at the 9-limit among
those "best pajara" keys, so there would be no point in replicating
examples "a" and "b" for any of those. Instead, I've prepared an
example typical of one of those 18 other keys, one degree lower in
pitch than the 34-WT example:

http://xenharmony.wikispaces.com/space/showimage/Pajara-34ITa.mp3
http://xenharmony.wikispaces.com/space/showimage/Pajara-34ITb.mp3
http://xenharmony.wikispaces.com/space/showimage/Pajara-34ITc.mp3

(I didn't know exactly what to call this intonation, so I used "IT" --
for ill-tempered. ;-)

So you now have the "best" and "worst" extremes to evaluate.

I'll leave off making any other comments till you've had a chance to
listen.

--George

Hi George,

--- In tuning@yahoogroups.com, "George D. Secor" wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" wrote:
> >
> > > 34-WT and 56 are close, trailed by 34 and 22.
> >
> > I would tend to be very critical of comparing a WT with
> > an ET like this, where the example doesn't use all the
> > keys of the WT.
>
> Good point. No need to listen to 33 more sets of files (one for
each key of 34-WT) -- I doubt we would have the time and/or patience
for that! Besides, it's not necessary, because (as I noted in my
commentary) 34-WT has 16 "best" keys and 18 others that are not as
good (except that they're better at the 5-limit). There's virtually
no difference in the amount of tempering at the 9-limit among
those "best pajara" keys, so there would be no point in replicating
examples "a" and "b" for any of those. Instead, I've prepared an
example typical of one of those 18 other keys, one degree lower in
pitch than the 34-WT example:
>
> http://xenharmony.wikispaces.com/space/showimage/Pajara-34ITa.mp3
> http://xenharmony.wikispaces.com/space/showimage/Pajara-34ITb.mp3
> http://xenharmony.wikispaces.com/space/showimage/Pajara-34ITc.mp3
>
> (I didn't know exactly what to call this intonation, so I
used "IT" -- for ill-tempered. ;-)

At last! Just what this list has always needed:
an ill-temperament! How this extends the emotional
palette: now we can all properly express anger and
deep, burning rage! ;-)

Ahem. I thought your third example quite juicy; in
comparison, the first two were rather bland. Yet
(I think it's been said before) I'm not a great fan
of the timbre you're using for the comparisons - it's
rather "buzzy". Might I suggest using two or three
standard sustaining orchestral temperaments (eg viola,
flute, and clarinet) and one or two non-sustaining
ones (eg from amongst piano, harpsichord, guitar,
pizzicato)? That way we'd get a better idea how the
tuning would perform with actual instruments.

But - returning to your examples - no fair! c gets
to sound off some chords not used by a or b. What
gives?

> So you now have the "best" and "worst" extremes to evaluate.
>
> I'll leave off making any other comments till you've had a chance
to listen.
>
> --George

Regards,
Yahya

--- In tuning@yahoogroups.com, "yahya_melb" <yahya@...> wrote:
>
>
> Hi George,
>
> --- In tuning@yahoogroups.com, "George D. Secor" wrote:
> >
> > --- In tuning@yahoogroups.com, "Carl Lumma" wrote:
> > >
> > > > 34-WT and 56 are close, trailed by 34 and 22.
> > >
> > > I would tend to be very critical of comparing a WT with
> > > an ET like this, where the example doesn't use all the
> > > keys of the WT.
> >
> > Good point. No need to listen to 33 more sets of files (one for
> each key of 34-WT) -- I doubt we would have the time and/or
patience
> for that! Besides, it's not necessary, because (as I noted in my
> commentary) 34-WT has 16 "best" keys and 18 others that are not as
> good (except that they're better at the 5-limit). There's
virtually
> no difference in the amount of tempering at the 9-limit among
> those "best pajara" keys, so there would be no point in replicating
> examples "a" and "b" for any of those. Instead, I've prepared an
> example typical of one of those 18 other keys, one degree lower in
> pitch than the 34-WT example:
> >
> > http://xenharmony.wikispaces.com/space/showimage/Pajara-34ITa.mp3
> > http://xenharmony.wikispaces.com/space/showimage/Pajara-34ITb.mp3
> > http://xenharmony.wikispaces.com/space/showimage/Pajara-34ITc.mp3
> >
> > (I didn't know exactly what to call this intonation, so I
> used "IT" -- for ill-tempered. ;-)
>
> At last! Just what this list has always needed:
> an ill-temperament! How this extends the emotional
> palette: now we can all properly express anger and
> deep, burning rage! ;-)

Indeed! I first heard the term "ill-tempered scale" in the 1960's;
it was used to characterize 12-equal.

> Ahem. I thought your third example quite juicy;

Thanks. Carl L. also liked the progression (thanks, Carl!); perhaps
I'll use it in a composition sometime -- some day ...

> in
> comparison, the first two were rather bland.

These were just simple examples of 7- and 9-limit chords.

> Yet
> (I think it's been said before) I'm not a great fan
> of the timbre you're using for the comparisons - it's
> rather "buzzy".

This particular patch was selected in order to best reveal the faults
and virtues of each tuning. This is best accomplished with a rich
harmonic content, without any vibrato or other transients that might
obscure beating or other disturbances (such as the interaction of
combinational tones).

> Might I suggest using two or three
> standard sustaining orchestral temperaments (eg viola,
> flute, and clarinet) and one or two non-sustaining
> ones (eg from amongst piano, harpsichord, guitar,
> pizzicato)? That way we'd get a better idea how the
> tuning would perform with actual instruments.
>
> But - returning to your examples - no fair! c gets
> to sound off some chords not used by a or b. What
> gives?

All 3 are examples of the same tuning, namely one of the keys
(a "worst case") in the "far side" of my 34-tone well-temperament.
You're supposed to compare these to the examples (a, b, and c) in the
other tunings in the comparison, which I already gave here:

/tuning/topicId_67957.html#68285

--George

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@> wrote:
> >
> > > 34-WT and 56 are close, trailed by 34 and 22.
> >
> > I would tend to be very critical of comparing a WT with
> > an ET like this, where the example doesn't use all the
> > keys of the WT.
>
> Good point. // I've prepared an
> example typical of one of those 18 other keys, one degree lower in
> pitch than the 34-WT example:
>
> http://xenharmony.wikispaces.com/space/showimage/Pajara-34ITa.mp3
> http://xenharmony.wikispaces.com/space/showimage/Pajara-34ITb.mp3
> http://xenharmony.wikispaces.com/space/showimage/Pajara-34ITc.mp3
>
> (I didn't know exactly what to call this intonation, so I
> used "IT" -- for ill-tempered. ;-)

I don't want to be an ungrateful ninny, but why did you go for
a half-step down? It does make comparison harder. In any case,
we seem to be dealing with a very small order of difference
between the 'good' and 'bad' keys of this temperament, and
34-tET.

-Carl

Hi George,

--- In tuning@yahoogroups.com, "George D. Secor" wrote:
>
> --- In tuning@yahoogroups.com, "yahya_melb" wrote:
[snip]
> > Ahem. I thought your third example quite juicy;
>
> Thanks. Carl L. also liked the progression (thanks, Carl!);
perhaps I'll use it in a composition sometime -- some day ...
>
> > in comparison, the first two were rather bland.
>
> These were just simple examples of 7- and 9-limit chords.
>
> > Yet (I think it's been said before) I'm not a great fan of the
timbre you're using for the comparisons - it's rather "buzzy".
>
> This particular patch was selected in order to best reveal the
faults and virtues of each tuning. This is best accomplished with a
rich harmonic content, without any vibrato or other transients that
might obscure beating or other disturbances (such as the interaction
of combinational tones).

I understand your rationale. However, for me
to extrapolate from those sounds to those of
common acoustic instruments with the same
tuning would be more than an act of imagination;
it would simply be guessing. So I must withhold
judgment of how well these tunings would work for
me until I can hear them on such instruments.

> > Might I suggest using two or three standard sustaining orchestral
temperaments (eg viola, flute, and clarinet) and one or two non-
sustaining ones (eg from amongst piano, harpsichord, guitar,
pizzicato)? That way we'd get a better idea how the tuning would
perform with actual instruments.
> >
> > But - returning to your examples - no fair! c gets to sound off
some chords not used by a or b. What gives?
>
> All 3 are examples of the same tuning, namely one of the keys
(a "worst case") in the "far side" of my 34-tone well-temperament.
You're supposed to compare these to the examples (a, b, and c) in the
other tunings in the comparison, which I already gave here:
>
> /tuning/topicId_67957.html#68285

'Deed, so you did, but I was unable to listen at
the time. That deficiency now being rectified,
my verdict is:

1) All sound better in your 34WT than any other
approximations to the JI originals.

2) The 22-EDO versions are (at least equal)
worst of all.

3) The other two versions are mostly better than
22-EDO and worse than 34-WT.

4) With that timbre, I *detest* the 'harmony'
8:9:11:13! It makes my skin crawl and my stomach
flip-flop; which could be a good thing; music IS
supposed to move us ...

Regards,
Yahya

--- In tuning@yahoogroups.com, "yahya_melb" <yahya@...> wrote:

> I understand your rationale. However, for me
> to extrapolate from those sounds to those of
> common acoustic instruments with the same
> tuning would be more than an act of imagination;
> it would simply be guessing. So I must withhold
> judgment of how well these tunings would work for
> me until I can hear them on such instruments.

I've done enough recording to probably make a guess,
but in general, I agree. You can't seperate tuning
from timbre.

> > > Might I suggest using two or three standard sustaining
>orchestral
> temperaments (eg viola, flute, and clarinet) and one or two non-
> sustaining ones (eg from amongst piano, harpsichord, guitar,
> pizzicato)? That way we'd get a better idea how the tuning would
> perform with actual instruments.

Unfortunately sampled instruments are usually horrible for
alternate tunings, not least because of formants.

> 1) All sound better in your 34WT than any other
> approximations to the JI originals.

I agree- and the 34WT sounds better than the JI, IMO.
>
> 2) The 22-EDO versions are (at least equal)
> worst of all.

Agree here also. It's sad that 22 sounds so cheesey to me,
at least what I've heard so far.

> 3) The other two versions are mostly better than
> 22-EDO and worse than 34-WT.

Agree here also.

-Cameron Bobro

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
> >
> > --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@> wrote:
> > >
> > > > 34-WT and 56 are close, trailed by 34 and 22.
> > >
> > > I would tend to be very critical of comparing a WT with
> > > an ET like this, where the example doesn't use all the
> > > keys of the WT.
> >
> > Good point. // I've prepared an
> > example typical of one of those 18 other keys, one degree lower
in
> > pitch than the 34-WT example:
> >
> > http://xenharmony.wikispaces.com/space/showimage/Pajara-34ITa.mp3
> > http://xenharmony.wikispaces.com/space/showimage/Pajara-34ITb.mp3
> > http://xenharmony.wikispaces.com/space/showimage/Pajara-34ITc.mp3
> >
> > (I didn't know exactly what to call this intonation, so I
> > used "IT" -- for ill-tempered. ;-)
>
> I don't want to be an ungrateful ninny, but why did you go for
> a half-step down?

In a well-temperament, you have to go into a different key to get
different intonation. I used a key (only a comma lower, which is
less than half of a half-step) that would make the comparison least
difficult -- the keys most nearly opposite the original would have
been an augmented 2nd higher or lower.

> It does make comparison harder. In any case,
> we seem to be dealing with a very small order of difference
> between the 'good' and 'bad' keys of this temperament, and
> 34-tET.

I gather, then, that you think that the "worst" keys of the well-
temperament are acceptable, yes?

--George

SNIP

> 4) With that timbre, I *detest* the 'harmony'
> 8:9:11:13! It makes my skin crawl and my stomach
> flip-flop; which could be a good thing; music IS
> supposed to move us ...
>
> Regards,
> Yahya
>

LOL! Your picturesque allegory surely IS moving. ;)

> > I don't want to be an ungrateful ninny, but why did you go for
> > a half-step down?
>
> In a well-temperament, you have to go into a different key to get
> different intonation. I used a key (only a comma lower, which is
> less than half of a half-step) that would make the comparison least
> difficult -- the keys most nearly opposite the original would have
> been an augmented 2nd higher or lower.

I think for the purposes of comparison I would have adjusted
concert pitch as necessary.

> > It does make comparison harder. In any case,
> > we seem to be dealing with a very small order of difference
> > between the 'good' and 'bad' keys of this temperament, and
> > 34-tET.
>
> I gather, then, that you think that the "worst" keys of the well-
> temperament are acceptable, yes?

Yes.

-Carl

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

To everyone who responded to my request, I apologize that it's taken
so long to reply -- it's been an especially busy time for me this
month (much more so than I expected). Also, there's a very important
pending microtonal commitment that I've let go for too long, so I'll
be going into lurk mode for a while after posting this.

First, I want to say "thank you" for your very helpful comments. As
a follow-up to my 17-tone paper (for which I had to draw heavily on
my own observations and opinions), I hope to write a 34-tone paper in
the near future. This will be a new chapter in my speculative wide-
fifth alternate tuning history, in which the 17-tone octave is
subdivided into sixth-tones to arrive at a 34-tone system that
introduces primes 5 and 17 into the mix. Since the pajara scale
plays a prominent role in the tonal organization of 34, it will be
necessary to provide evidence for my observation that 34-ET and/or 34-
WT are not only "better" tunings for pajara than 22-ET, but that
they're good enough to justify the additional number of tones (at
both the 7 and 9 limits, and especially at the 11 limit).

Since this is in conflict with a conclusion that Paul Erlich made in
his 22-tone paper (which, especially now, would more accurately be
called his 'pajara paper'), it would be presumptuous of me to
challenge that conclusion solely on the basis of my own subjective
observations and opinion. It's very good, therefore, that I'm now
able to say that my tuning evaluations are supported by a half-dozen
fellow microtonalists, who are virtually unanimous in putting 22-ET
in last place (offering such descriptions as "really bad", "which I
hated", "worst of all", and "cheesey", to take a few choice words out
of context ;-). Fortunately, this does not invalidate the main
content of Paul's brilliant paper, which I will continue to regard as
a major theoretical development in the quest for new organizing
principles for tonality.

Getting back to the survey: your opinions are summarized by the
following comments:

Magnus Jonsson (MMM #15265): I'd take the JI versions if I could,
but I know that's against the rules here so I'll pick 34WT as next
best.

Carl Lumma (#62899): I would say 34 based on this [sets A and B].
(#68328): Overall, I'd say 34 perhaps earns its keep, but 56 does
not.

Cameron Bobro (#68301): Another vote for 34, specicially the WT.

Gene Ward Smith (#68312): 34-WT and 56 are close, trailed by 34 and
22.

Herman Miller (#68325): I'll vote for the 34-ET version.

Yahya Abdal-Aziz (#68382): All sound better in your 34WT than any
other approximations to the JI originals. The 22-EDO versions are
(at least equal) worst of all. The other two versions are mostly
better than 22-EDO and worse than 34-WT.

And here's how I voted: 34-WT (best keys), then 56-ET, then 34-ET,
then 34-IT (worst keys of 34-WT), then 22-ET.

Now for some numbers: Since the pajara period is fixed at 600 cents,
pajara tunings may be defined and compared solely in terms of the
specific generator, from which the errors of the various odd harmonic
factors may be calculated (in descending order of generator size):

Generator Error (cents)
Tuning cents 3 5 7 9 11 13 17
------ ----- ---- ---- ----- ----- ----- ---- ----
22-ET 109.1 +7.1 –4.5 +13.0 +14.3 -5.9 -- +4.1
34-WT 107.2 +5.3 –0.8 +16.7 +10.5 varies* +2.3
56-ET 107.1 +5.2 –0.6 +16.9 +10.4 +5.8 –4.8 +2.2
34-ET 105.9 +3.9 +1.9 +19.4 +7.9 +13.3 +6.5 +0.9
34-IT 104.4 +2.4 +4.9 +22.4 +4.8 varies* -0.6

*In 34-WT the error of 11 varies from 5.4 to 16.7 cents in the best
keys (34-WT), and from 11.0 to 22.4 cents in the worst keys (34-IT);
for any given key it will always be within the range of the 3 and 7
errors. The error of 13 varies from 0.2 to 14.4 cents in the best
keys (34-WT), and from -5.5 to 17.2 cents in the worst keys (34-IT).

Observe that the error of 5:7 (+17.5c) is constant across all of the
above tunings. This is due to the 600-cent period, which differs
from 5:7 by that amount. Notice also that in 34-ET, and also in the
worst keys of 34-WT (34-IT), the error of 7 exceeds that of 5:7,
which is an indication that their generators are outside of the
optimal range. The specific advantage of 34-ET, therefore, is its
excellent 5-limit properties.

I evaluated 56-ET as an open-ended pajara tuning, without regard to
the total number of tones in the ET, much as one might evaluate 1/4-
comma meantone as being preferable to 12-ET and 19-ET (and I believe
that this is also how Gene was thinking of it). Carl, I infer that
your opinion that 56 does not earn its keep was due much more to the
number of tones in the ET than to the sound of the tuning, yes?
Graham Breed mentioned to me off-list (prior to my preparing these
tuning examples) that the optimal Tenney-weighted RMS pajara
generator is close to that of 56-ET, so there are some good numbers
to support 56-ET.

Since some will think that 56-ET is too many tones (i.e., not enough
of an improvement over 34-ET to justify the increase), then 34-WT (in
its 16 best keys) may be a more practical alternative that also
offers a very decent representation of prime 13 (inherited from 17-
WT). I constructed 34-WT earlier this year by placing two circles of
my 17-WT (designed in 1978, before anyone had ever heard of pajara)
600 cents apart, and it was only by chance that this resulted in a
pajara tuning that's virtually the same as 56-ET. Both Graham and
Gene thought that I may be onto something, which convinced me that
this listening test should be done.

At this point I'll reply to a few miscellaneous comments:

--- In tuning@yahoogroups.com, "yahya_melb" <yahya@...> wrote:

> I'm not a great fan
> of the timbre you're using for the comparisons - it's
> rather "buzzy". Might I suggest using two or three
> standard sustaining orchestral temperaments (eg viola,
> flute, and clarinet) and one or two non-sustaining
> ones (eg from amongst piano, harpsichord, guitar,
> pizzicato)? That way we'd get a better idea how the
> tuning would perform with actual instruments.

I already said that "this particular patch was selected in order to
best reveal the faults and virtues of each tuning." In other words,
it tends to accentuate the differences between tunings, making
the "best" tunings sound cleaner and the "worst" tunings sound more
disturbing than what you could expect to hear in a reasonably
accurate performance with acoustic instruments.

--- In MakeMicroMusic@yahoogroups.com, Magnus Jonsson <magnus@...>
wrote:
>
> I find it incredibly hard to compare the JI and the ET
> versions. They sound really really different and good in their own
ways. I
> wonder what the JI would sound like if you deliberately introduced
some
> random tuning errors so that some slow beating would be introduced.

They would sound more like a live performance. Notice that, even
with the 8:9:11:13 chord (which contains no highly consonant
intervals), it's very easy to tell which example is in JI. Often
chords that contain 11's and/or 13's tend not to sound very consonant
(and therefore not very different if they're mistuned), but with
isoharmonic chords such as this one the difference is crystal clear.
Please see my 17-tone paper for a definition of 'isoharmonic' and
discussion of a method for building consonant chords above the 9-
limit:

http://xenharmony.wikispaces.com/space/showimage/17puzzle.pdf

> Group C
> ...
> 56c -- bad. inharmonic, like a gong/bell
> 34WTc -- same problem as 34c, also has gong/bell sound

You're probably referring to the effect of either the 8:9:11:13 or
6:9:11:14 chords, which tend to sound inharmonic because of those
strange-sounding primes, 11 and 13.

--- In tuning@yahoogroups.com, "yahya_melb" <yahya@...> wrote:
>
> 4) With that timbre, I *detest* the 'harmony'
> 8:9:11:13! It makes my skin crawl and my stomach
> flip-flop; which could be a good thing; music IS
> supposed to move us ...

Hmmm, sorry about that -- you might be allergic to 13. ;-)

That happens to be one of my favorite chords, which occurs as the 7th
chord (in the 3rd inversion, its most consonant voicing) on the
dominant degree of a "basic scale" that I described in my 17-tone
paper (page 69):

1/1 13/12 7/6 4/3 3/2 13/8 11/6 2/1

Oz identified this (in msg. #60813) as "nothing other than the
fundamental scale for Maqam Huseini." Small world, eh?

--- In MakeMicroMusic@yahoogroups.com, Magnus Jonsson <magnus@...>
wrote:
>
> I'd be interested to hear some
> meantone versions if the progressions are not impossible in
meantone.

I'd love to make a lot more comparisons that we could all listen to,
but I'm really pressed for time right now due to other things that
I've let go for much too long.

I did try some of these chords in Scala using 31-ET. 5-limit chords
(i.e., major & minor triads) sound very similar to those in the 34-WT
examples, due to the almost equal-but-opposite error of the intervals
(see following table).

Error (cents)
Tuning 3 5 7 9
------ ---- ---- ----- -----
31-ET -5.2 +0.8 -1.1 –10.4
34-WT +5.3 –0.8 +16.7 +10.5

The big difference is with prime 7, where it's impossible *in any
pajara tuning* to avoid an error on the order of 17 cents in at least
one pair of 7-limit consonances.

Also, chord progressions will make it evident that there are
significant melodic differences between 31-ET and 34-WT. Pajara
progressions will readily show that Archytas' comma (63:64) doesn't
vanish in 31, while conventional harmonic progressions (which are
generally suitable for a meantone-type temperament) show that
Didymus' comma (80:81) doesn't vanish in 34. Apart from this, I find
34 (both ET and WT) melodically very pleasing and invigorating,
particularly the slightly wide 8:9 (to echo a recent comment by Kraig
Grady).

While I'm on the subject of melodic differences, might I add that
another reason I find 34 (both ET and WT) preferable to 22 for pajara
is that it's better not to have a 4:5 that's noticeably narrower than
just (something that's often mentioned in comparisons of 31 vs. 19).

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > [GS:]
> > C A Eb/ G - 1/12:1/7:1/5:1/4 - utonal tetrad, open position
> > ...
> > Paul Erlich suggested that, since the utonal tetrad isn't as
> > consonant as the otonal, one might prefer an alternate JI tuning,
> > of which the most consonant voicing is 10:12:15:17 (in the first
> > inversion). In the following 17-limit JI version of the previous
> > example, the 2nd chord becomes 10:17:24:30 (in open position):
> > http://xenharmony.wikispaces.com/space/showimage/Pajara-JI17a.mp3
> >
> > Would you agree that this is more consonant that the 7-limit
> > example?
>
> In this example not especially, and I think the 7-limit version
> has nicer bite.

So you didn't agree with me -- well neither did MJ:

--- In MakeMicroMusic@yahoogroups.com, Magnus Jonsson <magnus@...>
wrote:
>
> Group A
>
> JI7a -- the best in group A, the utonal chord feels just right.
> JI17a -- something sounds wrong, like losing pitch height

This is the only place where your impressions were different from
mine. Anyway, this had little or no bearing on deciding which was
the best temperament.

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> I'm with Gene (if I read right) that even the *melodic* aspects
> of the JI version of group C sound best to me.

In retrospect, it's a good thing that I put the JI examples out there
for reference. When it comes to evaluating and comparing
temperaments, numbers in a table give you only a part of the picture -
- mostly harmonic, at that. The melodic effect of a tuning is also
important, and you have to listen carefully before drawing
conclusions.

Again, thanks to everyone for participating in this evaluation. I'll
be taking some time off from here over the next several weeks, but
will peek in from time to time.

Best wishes to all for a joyous holiday season,

--George

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

> Graham Breed mentioned to me off-list (prior to my preparing these
> tuning examples) that the optimal Tenney-weighted RMS pajara
> generator is close to that of 56-ET, so there are some good numbers
> to support 56-ET.

The Kees pajara tuning is similar--it is the pajara tuning which makes
the major thirds pure, and hence is recommendable to fans of 1/4 comma
meantone. The fifth here is (128/25)^(1/4), or 706.843 cents. This is
between 56 and 34, but closer to 56.

SNIP

--- In tuning@yahoogroups.com, "yahya_melb" <yahya@...> wrote:
>
> 4) With that timbre, I *detest* the 'harmony'
> 8:9:11:13! It makes my skin crawl and my stomach
> flip-flop; which could be a good thing; music IS
> supposed to move us ...

Hmmm, sorry about that -- you might be allergic to 13. ;-)

That happens to be one of my favorite chords, which occurs as the 7th
chord (in the 3rd inversion, its most consonant voicing) on the
dominant degree of a "basic scale" that I described in my 17-tone
paper (page 69):

1/1 13/12 7/6 4/3 3/2 13/8 11/6 2/1

Oz identified this (in msg. #60813) as "nothing other than the
fundamental scale for Maqam Huseini." Small world, eh?

SNIP

I would like to make a correction. The first triad resembles Saba until one
hears 4/3 instead of 14/11. In that form, it sounds more of an Arazbar than
Huseini. The last tetrachord is a weird Hijaz, and could readily reminisce a
segment of Huzzam if one extends it to a pentachord down to 4/3, replaces
11/6 with 15/8, and makes the tonic 5/4. But it is a tedious modification as
it is.

Overall, the scale gives the impression of Ushshaq (I confused this with
Huseini), except that the seventh degree should have been a pure fourth away
from the pure fourth.

Oz.

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

> First, I want to say "thank you" for your very helpful comments. As
> a follow-up to my 17-tone paper (for which I had to draw heavily on
> my own observations and opinions), I hope to write a 34-tone paper in
> the near future.

I think it would be very useful if this paper not only included pajara
but also discussed keemun, where the seventh is flat. This is 34 trying
to act like 19, not 22. The Kees tunimg here sets the fifth, not the
third, to be exact and is pretty much the same as 53-et with a weird
choice of mapping for 7. Hence it is not innately a wide-fifth system,
but of course it will be tuned to 34, and it seems like a system worth
exploring to anyone wanting to delve into 34 and with an instrument
available to do so.

The point of it is, the 7 is not very well in tune, but it's not very
complex either. The 7 and 11 note MOS make for natural scales.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
>
> > First, I want to say "thank you" for your very helpful comments.
As
> > a follow-up to my 17-tone paper (for which I had to draw heavily
on
> > my own observations and opinions), I hope to write a 34-tone
paper in
> > the near future.
>
> I think it would be very useful if this paper not only included
pajara
> but also discussed keemun, where the seventh is flat. This is 34
trying
> to act like 19, not 22. The Kees tunimg here sets the fifth, not
the
> third, to be exact and is pretty much the same as 53-et with a
weird
> choice of mapping for 7. Hence it is not innately a wide-fifth
system,
> but of course it will be tuned to 34, and it seems like a system
worth
> exploring to anyone wanting to delve into 34 and with an instrument
> available to do so.
>
> The point of it is, the 7 is not very well in tune, but it's not
very
> complex either. The 7 and 11 note MOS make for natural scales.

Thanks for your suggestion, Gene. Since I'm thinking of this as a 34-
tone (rather than a pajara) paper, I expect to describe several
useful scales in 34 that may be created with a generating interval,
and Keemun will definitely be among them.

--George