Schoenberg's Theory of Harmony book and 11-limit

I was reading on the Tonalsoft page about the Schoenberg overtone
scales and the 11-limit. What exactly were the scales from the Theory
of Harmony book? That book seems to be mysteriously hard to find
especially in english except for the online version.

thanks

Hi John,

--- In tuning@yahoogroups.com, "johngilbert3x" <preciousatonement@...>
wrote:
>
> I was reading on the Tonalsoft page about the Schoenberg
> overtone scales and the 11-limit. What exactly were the
> scales from the Theory of Harmony book? That book seems
> to be mysteriously hard to find especially in english except
> for the online version.

What i have on my webpage
http://sonic-arts.org/monzo/schoenberg/harm/ch-4.htm#monznote

is all of the mathematical information that can be
derived from the statements Schoenberg provided in
his book. They are *not* scales that Schoenberg used,
but rather his explanation of the rational basis behind
his use of 12-edo.

Every scale and harmonic structure in his book is
a subset of 12-edo, or the entire 12-edo chromatic scale.
In fact, it's this latter that was the revolutionary
new thing that he was propounding.

I'm very surprised that you say that Schoenberg's book
is hard to find ... it seems like i see at least one
copy in the music section of every Borders or Barnes & Noble
that i visit, in the form of Roy E. Carter's 1978 translation
entitled "Theory of Harmony". A quick search on amazon.com
turned it right up:

http://www.amazon.com/Theory-Harmony-California-Library-Reprint/dp/0520049446/ref=pd_bbs_sr_1/104-9607692-2266322?ie=UTF8&s=books&qid=1185487344&sr=8-1

If you want the original 1911 German edition, yes,
that is very hard to find. You'll have to track it
down thru interlibrary loan like i did, or luck out
and find a copy for sale online. According to amazon.com,
Universal Edition published a reprint in 2001, but that
might be a later revised edition and not the 1911 edition.

You also might want to look at his 1934 lecture
"Problems of Harmony", which was published in
_Style and Idea_ (a book which is readily available too).
I have some of it here:

http://sonic-arts.org/monzo/schoenberg/problems/problems.htm

In that paper, he goes beyond the 11-limit basis
in _Harmonielehre_ and extends it to 13-limit.

_Style and Idea_ is well worth reading if you're
interested in Schoenberg and his work -- it gives
a perspective on how wide-ranging Schoenberg's interests
were.

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

Say, Monz, I noticed on this page (thanks for putting up
this groovy Schoenberg stuff btw.) that in discussing alternative
EDO's, there is once again a mysterious lack of 34-EDO, which is as
far as I can tell far and away the best of them all in a real-world
sense. What gives?

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi John,
>
> --- In tuning@yahoogroups.com, "johngilbert3x" <preciousatonement@>
> wrote:
> >
> > I was reading on the Tonalsoft page about the Schoenberg
> > overtone scales and the 11-limit. What exactly were the
> > scales from the Theory of Harmony book? That book seems
> > to be mysteriously hard to find especially in english except
> > for the online version.
>
>
> What i have on my webpage
> http://sonic-arts.org/monzo/schoenberg/harm/ch-4.htm#monznote
>
> is all of the mathematical information that can be
> derived from the statements Schoenberg provided in
> his book. They are *not* scales that Schoenberg used,
> but rather his explanation of the rational basis behind
> his use of 12-edo.
>
> Every scale and harmonic structure in his book is
> a subset of 12-edo, or the entire 12-edo chromatic scale.
> In fact, it's this latter that was the revolutionary
> new thing that he was propounding.
>
> I'm very surprised that you say that Schoenberg's book
> is hard to find ... it seems like i see at least one
> copy in the music section of every Borders or Barnes & Noble
> that i visit, in the form of Roy E. Carter's 1978 translation
> entitled "Theory of Harmony". A quick search on amazon.com
> turned it right up:
>
> http://www.amazon.com/Theory-Harmony-California-Library-
Reprint/dp/0520049446/ref=pd_bbs_sr_1/104-9607692-2266322?
ie=UTF8&s=books&qid=1185487344&sr=8-1
>
>
> If you want the original 1911 German edition, yes,
> that is very hard to find. You'll have to track it
> down thru interlibrary loan like i did, or luck out
> and find a copy for sale online. According to amazon.com,
> Universal Edition published a reprint in 2001, but that
> might be a later revised edition and not the 1911 edition.
>
>
> You also might want to look at his 1934 lecture
> "Problems of Harmony", which was published in
> _Style and Idea_ (a book which is readily available too).
> I have some of it here:
>
> http://sonic-arts.org/monzo/schoenberg/problems/problems.htm
>
> In that paper, he goes beyond the 11-limit basis
> in _Harmonielehre_ and extends it to 13-limit.
>
> _Style and Idea_ is well worth reading if you're
> interested in Schoenberg and his work -- it gives
> a perspective on how wide-ranging Schoenberg's interests
> were.
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> Say, Monz, I noticed on this page (thanks for putting up
> this groovy Schoenberg stuff btw.) that in discussing alternative
> EDO's, there is once again a mysterious lack of 34-EDO, which is as
> far as I can tell far and away the best of them all in a real-world
> sense. What gives?

Maybe another Wikipedia article? I'm not clear why, in a real world
sense, 34 beats out the rest, though.

Hi Cameron and Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@> wrote:
> >
> > Say, Monz, I noticed on this page (thanks for putting up
> > this groovy Schoenberg stuff btw.) that in discussing
> > alternative EDO's, there is once again a mysterious lack
> > of 34-EDO, which is as far as I can tell far and away
> > the best of them all in a real-world sense. What gives?

Back in 1911 *no-one* was thinking about 34-edo.

As you can see, Schoenberg mentions that his student
Robert Neumann (who has yet to be positively identified
in the scholarly research) advocates 53-edo and quotes
Neumann's reasons for doing so ... then promptly states
that "The average musician will laugh at such speculations
and will not be inclined to see their point". And this
from Schoenberg at a time when he actually *was* kindly
disposed towards the possibility of microtonality!

Other than what Neumann told him about 53-edo, the
only other microtonal tunings Schoenberg knew much
about (AFAIK) were 36-edo from Busoni and 24-edo
from Willi Moellendorf and possibly Richard Stein.

See my webpage publication of Moellendorf's book:

http://tonalsoft.com/monzo/moellendorf/book/contents.htm

And my page about an important letter Schoenberg wrote
to Busoni:

http://tonalsoft.com/monzo/schoenberg/to-busoni-1909-8-24.htm

> Maybe another Wikipedia article? I'm not clear why,
> in a real world sense, 34 beats out the rest, though.

For sure i need to create a Tonalsoft Encyclopedia
page about it ... and for that matter, i still need
to make one about 53-edo too (been meaning to do that
one for a *long* time).

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

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@>
wrote:
> >
> > Say, Monz, I noticed on this page (thanks for putting up
> > this groovy Schoenberg stuff btw.) that in discussing
alternative
> > EDO's, there is once again a mysterious lack of 34-EDO, which is
as
> > far as I can tell far and away the best of them all in a real-
world
> > sense. What gives?
>
> Maybe another Wikipedia article?

There's one already- not so good, but not tragically bad. I'd be
happy to quote and link George's upcoming paper for the wiki, for
it will surely be level-headed and fair in nature.

>I'm not clear why, in a real world
> sense, 34 beats out the rest, though.

Wouldn't you agree that as far as "justification" of intervals
found low in the harmonic series that 34 is the best, overall,
below 53 (ergo very doable, even fretable). If you insist on a
pure 7/4, and intervals incorporating the seventh harmonic in
general, it's about what, third place? but 34 offers an elegant
solution as far as 7/4, by having a 23/13, the harmonic mean
of 7/4 and 16/9. The kicker for me is the "median" intervals,
which are superb- 27/22 is only 1.6 cents flat in 34-EDO, which
means you can have some pure in a WT, etc. There's a
strong (and measurable) feeling, to me, that 34 consists of harmonic
means of "simple" intervals along with audibly "simple" intervals.

There isn't and shouldn't be a universal tuning, in my opinion,
but I believe that 34 is ideal for a great deal of music.

-Cameron Bobro

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@>
wrote:
> >
> > --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@>
wrote:
> > >
> > > Say, Monz, I noticed on this page (thanks for putting up
> > > this groovy Schoenberg stuff btw.) that in discussing
alternative
> > > EDO's, there is once again a mysterious lack of 34-EDO, which
is as
> > > far as I can tell far and away the best of them all in a real-
world
> > > sense. What gives?
> >
> > Maybe another Wikipedia article?
>
> There's one already- not so good, but not tragically bad. I'd be
> happy to quote and link George's upcoming paper for the wiki, for
> it will surely be level-headed and fair in nature.

Ouch! I take that as a hint that I should be working on a 34-tone
paper. Unfortunately, at the moment that's not at the top of my list
of microtonal priorities. While I have a general idea of its
content, I haven't yet organized my thoughts into the kind of logical
flow that I achieved in my 17-tone paper. So now you've encouraged
me to think about it, at least.

In the meantime, one thing I've recently completed (and could make
available soon) is a generalized pajara keyboard design that
accommodates 12, 22, 34, 46, 56, and 58. (The drawing needs a few
finishing touches, so it will be at least a few days.) I've found
that generalized keyboard diagrams, like lattice diagrams, are very
helpful for composing, because they help me to see tonal
relationships.

Since I mentioned 46 in that list, might I add that 46 has been
discussed even less often than 34. Barbour covered 34 in his chapter
on multiple (octave) division (in _Tuning and Temperament_), but
nowhere did he even mention 46. Erv Wilson wrote me that Augusto
Novaro was the first to recognize the merits of 46, in 1927.

> >I'm not clear why, in a real world
> > sense, 34 beats out the rest, though.

A 5th, major 3rd, and major 9th, each somewhat wider than just, seem
to have pleasing melodic properties. Also, since the error of 4:5 is
about half as large as that of 2:3, there is something reasonably
close to proportional beating in the major and minor triads.

> Wouldn't you agree that as far as "justification" of intervals
> found low in the harmonic series that 34 is the best, overall,
> below 53 (ergo very doable, even fretable).

If by this you mean at the 5-limit, then I agree.

> If you insist on a
> pure 7/4, and intervals incorporating the seventh harmonic in
> general, it's about what, third place?

If an alternative to 34 with a better 4:7 is desired, then my choice
is 46 (which also has nice melodic properties, BTW).

> but 34 offers an elegant
> solution as far as 7/4, by having a 23/13, the harmonic mean
> of 7/4 and 16/9.

This is the 4:7 that's used for 34-tone pajara.

> The kicker for me is the "median" intervals,
> which are superb- 27/22 is only 1.6 cents flat in 34-EDO, which
> means you can have some pure in a WT, etc.

My 34-WT has 4 of these with less than 1 cent error and 4 more with
~1.9 cents error. In other keys the interval of 10deg34-WT comes
very close to 11/9, 16/13, or 39/32, so in different keys you get
neutral 3rds in different flavors.

> There's a
> strong (and measurable) feeling, to me, that 34 consists of
harmonic
> means of "simple" intervals along with audibly "simple" intervals.
>
> There isn't and shouldn't be a universal tuning, in my opinion,
> but I believe that 34 is ideal for a great deal of music.

My four favorite divisions below 72 are 19, 31, 34, and 46. (Yes, I
like 17, but since I can have it as a subset of 34, I'll go with the
latter.)

--George

George D. Secor wrote:

> In the meantime, one thing I've recently completed (and could make > available soon) is a generalized pajara keyboard design that > accommodates 12, 22, 34, 46, 56, and 58. (The drawing needs a few > finishing touches, so it will be at least a few days.) I've found > that generalized keyboard diagrams, like lattice diagrams, are very > helpful for composing, because they help me to see tonal > relationships.

I've found the same thing. Not only that, but MOS patterns (that can have melodic uses) are also apparent. I've used these sorts of diagrams when planning some of my more recent pieces for about the last year or so (not that I've done much lately, but it's been useful for finding chord progressions in gorgo, bug, and father temperament).

> My four favorite divisions below 72 are 19, 31, 34, and 46. (Yes, I > like 17, but since I can have it as a subset of 34, I'll go with the > latter.)

I actually haven't done much with 34 although it looks nice as a keemun temperament. I guess if I had to pick 4, I'd go with 15, 19, 22, and 26, although most of the smaller ET's are good for something, and 31 does have some very nice qualities.

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

> Since I mentioned 46 in that list, might I add that 46 has been
> discussed even less often than 34. Barbour covered 34 in his
>chapter
> on multiple (octave) division (in _Tuning and Temperament_), but
> nowhere did he even mention 46. Erv Wilson wrote me that Augusto
> Novaro was the first to recognize the merits of 46, in 1927.

Hm, 46 looks very interesting- you've got a 14/11 (my favorite high
3rd), a 9/5, all kinds of things. 704.3 is about the perfect high
3/2 to my ears, too. The minor second is a tad wide, IMO- 25/24
being my ideal.

Speaking of 25/24, I made a nice (ir)rational 34, with 34 25/24's,
by alternating steps of 50/49 and 49/48. This makes the octave 1.43
cents wide, and the result, I think, is an excellent "real-life"
34 "equal".
>
> > >I'm not clear why, in a real world
> > > sense, 34 beats out the rest, though.
>
> A 5th, major 3rd, and major 9th, each somewhat wider than just,
>seem
> to have pleasing melodic properties.

I also prefer them harmonically to pure, and in actual practice
where pitch is statistically accurate but varying within a couple
of cents at any given moment, they can also "be" pure.

>Also, since the error of 4:5 is
> about half as large as that of 2:3, there is something reasonably
> close to proportional beating in the major and minor triads.
>
> > Wouldn't you agree that as far as "justification" of intervals
> > found low in the harmonic series that 34 is the best, overall,
> > below 53 (ergo very doable, even fretable).
>
> If by this you mean at the 5-limit, then I agree.

> > There's a
> > strong (and measurable) feeling, to me, that 34 consists of
> harmonic
> > means of "simple" intervals along with audibly "simple"
intervals.
> >
> > There isn't and shouldn't be a universal tuning, in my opinion,
> > but I believe that 34 is ideal for a great deal of music.
>
> My four favorite divisions below 72 are 19, 31, 34, and 46. (Yes,
>I
> like 17, but since I can have it as a subset of 34, I'll go with
the
> latter.)

To me, 19 and 31 tend to sound soggy. 19 is useless to me, no median
third, but 31 sports the sevens and is audibly more suited to
certain musics than 34 is. 19, 31, 34, 46: milk, beer, wine, and ?

I think we discussed some time ago how 17 feels like a subset of
something else- I wasn't up for 34 at the time because of the
absence of 7/4 but I've changed my mind because intervals literally
incorporating the 7th partial seem loathe to move or mingle outside
the family, just as it's a major color decision in additive
synthesis to create a strong 7th partial.

Also, the theoretical foundation of the way I make my chords
taller is not by stacking thirds, but by adding intervals with an
ear on the physical timbre of the chord, eg adding 11/10 to 14/11.
Example: "G-E-C-D" might have a "G" of 182/121 (makes a difference
from 3/2 in this approach, try it), an "E" of 14/11 and a "D" of
11/10 (voiced 91/121, 14/11, 2/1, 22/10).

34 works admirably for this approach because of its "harmonic-mean"
nature, I believe. The high third is a 23/18 (less than .9 cents off
even in straight EDO), the harmonic mean of 14/11 and 9/7; the dark
minor third is 20/17 (within one cent), the harmonic mean of 13/11
and 7/6, and so on. So these "mean" intervals, though not directly
percieved as "Just", are "softened" by more distantly percieved
literal consonances in the audible range, while at the same
time lending themselves to different interpretations in the lower
harmonics.

Carl calls all this "eye of newt" but I keep getting comments
from listeners which seem to confirm that it all works precisely as
intended, so, whatever. :-).

Anyway as far as other kinds of music, I find 34 eerily reminiscent
of the intonational flavor of pre-war Eastern European orchestral
recordings (Melodiya), and of old recordings of Romantic music in
general. A couple of artists I know have made remarks to similar
effect, "sounds like...1930!", perhaps remarkable considering the
highly synthetic timbres I use.

> > The kicker for me is the "median" intervals,
> > which are superb- 27/22 is only 1.6 cents flat in 34-EDO, which
> > means you can have some pure in a WT, etc.
>
> My 34-WT has 4 of these with less than 1 cent error and 4 more
>with
> ~1.9 cents error. In other keys the interval of 10deg34-WT comes
> very close to 11/9, 16/13, or 39/32, so in different keys you get
> neutral 3rds in different flavors.

That's very nice indeed- is your 34-WT in the Scala archive? I don't
find it.

-Cameron Bobro

You might also try using Large (L) and small (s) intervals to construct your multiple EDO harmonies,

as many of the EDOs also have meantone-type characteristics in which the traditional harmonic patterns are really apparent.

see:

http://www.lucytune.com/tuning/equal_temp.html

(Yes carl, another link, to save me the effort of re-typing and to overcome the limitations of yahoo groups ;-)

BTW Carl, where is you list of references that you claim I have failed to justify?

Charles Lucy lucy@lucytune.com

----- Promoting global harmony through LucyTuning -----

For information on LucyTuning go to: http://www.lucytune.com

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http://www.lullabies.co.uk

Skype user = lucytune

http://www.myspace.com/lucytuning

On 2 Aug 2007, at 04:19, Herman Miller wrote:

> George D. Secor wrote:
>
> > In the meantime, one thing I've recently completed (and could make
> > available soon) is a generalized pajara keyboard design that
> > accommodates 12, 22, 34, 46, 56, and 58. (The drawing needs a few
> > finishing touches, so it will be at least a few days.) I've found
> > that generalized keyboard diagrams, like lattice diagrams, are very
> > helpful for composing, because they help me to see tonal
> > relationships.
>
> I've found the same thing. Not only that, but MOS patterns (that can
> have melodic uses) are also apparent. I've used these sorts of > diagrams
> when planning some of my more recent pieces for about the last year or
> so (not that I've done much lately, but it's been useful for finding
> chord progressions in gorgo, bug, and father temperament).
>
> > My four favorite divisions below 72 are 19, 31, 34, and 46. (Yes, I
> > like 17, but since I can have it as a subset of 34, I'll go with the
> > latter.)
>
> I actually haven't done much with 34 although it looks nice as a > keemun
> temperament. I guess if I had to pick 4, I'd go with 15, 19, 22, > and 26,
> although most of the smaller ET's are good for something, and 31 does
> have some very nice qualities.
>
>
>

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

> In the meantime, one thing I've recently completed (and could
> make available soon) is a generalized pajara keyboard design
> that accommodates 12, 22, 34, 46, 56, and 58. (The drawing
> needs a few finishing touches, so it will be at least a few
> days.) I've found that generalized keyboard diagrams, like
> lattice diagrams, are very helpful for composing, because
> they help me to see tonal relationships.
>
> Since I mentioned 46 in that list, might I add that 46 has
> been discussed even less often than 34. Barbour covered 34
> in his chapter on multiple (octave) division (in _Tuning and
> Temperament_), but nowhere did he even mention 46. Erv Wilson
> wrote me that Augusto Novaro was the first to recognize the
> merits of 46, in 1927.

James Paul White mentioned its bad qualities on page 266-7 in
part 2 of "Is Perfect Intonation Practicable?" Music, vol. 8. Music
Magazine Publishing Co., Chicago, 1895. p.262-275:

"In the other elements of harmony, both these systems approach
considerably nearer to the true chords than does the uncial,
espectially the 46, but, for general music, they would be
unpleasant, the 29 on account of its bad harmony, and both
systems on account of their enharmonic commas, or rather, what
those commas involve - the very thing we are coming at."

Clark

nice find! -C.

> James Paul White mentioned its bad qualities on page 266-7 in
> part 2 of "Is Perfect Intonation Practicable?" Music, vol. 8. Music
> Magazine Publishing Co., Chicago, 1895. p.262-275:
>
> "In the other elements of harmony, both these systems approach
> considerably nearer to the true chords than does the uncial,
> espectially the 46, but, for general music, they would be
> unpleasant, the 29 on account of its bad harmony, and both
> systems on account of their enharmonic commas, or rather, what
> those commas involve - the very thing we are coming at."
>
> Clark

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...>
wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
>
> > Since I mentioned 46 in that list, might I add that 46 has been
> > discussed even less often than 34. Barbour covered 34 in his
chapter
> > on multiple (octave) division (in _Tuning and Temperament_), but
> > nowhere did he even mention 46. Erv Wilson wrote me that Augusto
> > Novaro was the first to recognize the merits of 46, in 1927.
>
> Hm, 46 looks very interesting- you've got a 14/11 (my favorite high
> 3rd), a 9/5, all kinds of things. 704.3 is about the perfect high
> 3/2 to my ears, too.

Yep! You also have most of the above in 41, but I find fault with 41
in that the 5/4 is low (not good melodically) and that both the major
& minor 3rds are tempered too much. Also, it may be argued that 34
and 46 are both 17-limit, while 41 is 15-limit.

> The minor second is a tad wide, IMO- 25/24
> being my ideal.

Yes, that's exactly what I said in my 17-tone paper.

In July 2002 (after I wrote the paper) I had a discussion with Paul
Erlich on the harmonic entropy list. Paul adjusted variables in his
harmonic entropy program to produce a graph that closely corresponded
to observations I had made many years earlier with a retunable
electronic organ, in which I had determined specific interval sizes
as points of local maximum dissonance. As it turned out, the global
maximum harmonic entropy is at 67 cents in this graph (close to 70
cents).

You'll be interested to learn that the local HE maximum between 5/4
and 9/7 is at 422 cents in this graph, which is very close to 23/18
(which you mentioned below). Also, there is a slightly greater local
HE maximum between 6/5 and 5/4 at 356 cents, which is very close to
27/22 (which you also mentioned). (There's a second peak, almost as
great, at 340 cents, so you might want to listen to 28/23 in JI or 46-
ET, and especially to the 18:23:28 triad, which is isoharmonic.)
This supports a point of speculation in my paper that intervals that
are melodically most effective are close to local maximum points of
dissonance, discord, or disorder (i.e., harmonic entropy).

> Speaking of 25/24, I made a nice (ir)rational 34, with 34 25/24's,
> by alternating steps of 50/49 and 49/48. This makes the octave 1.43
> cents wide, and the result, I think, is an excellent "real-life"
> 34 "equal".

Hmm, I'll have to take a closer look at that.

> > > ...
> > > There isn't and shouldn't be a universal tuning, in my opinion,
> > > but I believe that 34 is ideal for a great deal of music.
> >
> > My four favorite divisions below 72 are 19, 31, 34, and 46.
(Yes, I
> > like 17, but since I can have it as a subset of 34, I'll go with
the
> > latter.)
>
> To me, 19 and 31 tend to sound soggy.

Presumably on account of their melodic properties, I would think.
I've observed that these may require a "break-in" period during which
one's ears become accustomed to the smaller-than-optimal major 2nds
and larger minor 2nds.

> 19 is useless to me, no median
> third,

That's one of the reasons I've avoided 19-ET, preferring a multi-
purpose well-temperament with 3 extra tones for the 11's (19+3WT).

I assume that you don't like 22, either. I dislike 22 on account of
the heavy tempering of the minor 3rd (and its small major 3rd doesn't
help, either). IMO, the first wide-fifth EDO that delivers the goods
is 34.

> but 31 sports the sevens and is audibly more suited to
> certain musics than 34 is.

Yes. For me, it's not a matter of preferring one over the other, but
rather recognizing that each has different strengths and weaknesses.

> 19, 31, 34, 46: milk, beer, wine, and ?

??

> I think we discussed some time ago how 17 feels like a subset of
> something else- I wasn't up for 34 at the time because of the
> absence of 7/4

You're not the only one to think that. When I first looked at the 34
division, I thought that the poor representation of 7 was a fatal
flaw. Paul Erlich also eliminated 34 from consideration in his 22-
tone (pajara) paper because of 7-limit inconsistency and concluded
that 22-ET was the best, most practical, implementation of pajara.

However, my recent listening-test survey on this list has proved
otherwise, due in large part to the fact that we seem to accept a
larger positive than negative error for 4:7 (possibly because of our
extensive exposure to 12-ET, with its 600-cent tritone).

I can't help noticing a resemblance between these observations and
the introductory section of my 17-tone paper -- I think I've just
laid out the opening thoughts for a 34-tone paper. :-)

> but I've changed my mind because intervals literally
> incorporating the 7th partial seem loathe to move or mingle outside
> the family, just as it's a major color decision in additive
> synthesis to create a strong 7th partial.
>
> Also, the theoretical foundation of the way I make my chords
> taller is not by stacking thirds, but by adding intervals with an
> ear on the physical timbre of the chord, eg adding 11/10 to
14/11.

The resultant 7/5 looks nice on paper; I'll have to try it.

> Example: "G-E-C-D" might have a "G" of 182/121 (makes a difference
> from 3/2 in this approach, try it), an "E" of 14/11 and a "D" of
> 11/10 (voiced 91/121, 14/11, 2/1, 22/10).

Coincidentally, a few days ago I was wondering what sort of triad I
would get with stacked 3rds of 14/11 and 13/11 (which gives a 5th of
182/121), but I haven't had a chance to try it. Now I'm really
curious!

> 34 works admirably for this approach because of its "harmonic-mean"
> nature, I believe. The high third is a 23/18 (less than .9 cents
off
> even in straight EDO), the harmonic mean of 14/11 and 9/7; the dark
> minor third is 20/17 (within one cent), the harmonic mean of 13/11
> and 7/6, and so on. So these "mean" intervals, though not directly
> percieved as "Just", are "softened" by more distantly percieved
> literal consonances in the audible range, while at the same
> time lending themselves to different interpretations in the lower
> harmonics.
>
> Carl calls all this "eye of newt" but I keep getting comments
> from listeners which seem to confirm that it all works precisely as
> intended, so, whatever. :-).

I'll have to try these out. It's amazing how many different
approaches to tonality can be found when you think outside the 12-box.

> Anyway as far as other kinds of music, I find 34 eerily reminiscent
> of the intonational flavor of pre-war Eastern European orchestral
> recordings (Melodiya), and of old recordings of Romantic music in
> general. A couple of artists I know have made remarks to similar
> effect, "sounds like...1930!", perhaps remarkable considering the
> highly synthetic timbres I use.

That's interesting. Could you direct me to any sound files for this?

> > > The kicker for me is the "median" intervals,
> > > which are superb- 27/22 is only 1.6 cents flat in 34-EDO,
which
> > > means you can have some pure in a WT, etc.
> >
> > My 34-WT has 4 of these with less than 1 cent error and 4 more
with
> > ~1.9 cents error. In other keys the interval of 10deg34-WT comes
> > very close to 11/9, 16/13, or 39/32, so in different keys you get
> > neutral 3rds in different flavors.
>
> That's very nice indeed- is your 34-WT in the Scala archive? I
don't
> find it.

I don't know of any procedure for putting tunings in the Scala
archive, so it's probably not there. However, I've put .scl files
for my best tunings here:
/tuning-math/files/secor/scl/
so you can look at the descriptions and see what else I've been up
to. You need to be a member of tuning-math to access these files; 34-
WT is 6th from the bottom.

You'll also find it here, with a brief explanation:
/tuning/topicId_67957.html#68032

--George

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
> > Hm, 46 looks very interesting- you've got a 14/11 (my favorite
>high
> > 3rd), a 9/5, all kinds of things. 704.3 is about the perfect
>high
> > 3/2 to my ears, too.
>
> Yep! You also have most of the above in 41, but I find fault with
41
> in that the 5/4 is low (not good melodically) and that both the
>major
> & minor 3rds are tempered too much. Also, it may be argued that
34
> and 46 are both 17-limit, while 41 is 15-limit.
>
> > The minor second is a tad wide, IMO- 25/24
> > being my ideal.
>
> Yes, that's exactly what I said in my 17-tone paper.

That's actually how I found this tuning community- I came by
17 simply by repeatedly singing and listening for an ideal semitone,
while tweaking the tuning on a synth. Because I like the "minor
ninth" I was doing it harmonically as well as melodically and it
didn't take long to discover that what I kept arriving at was 25/24.
When I typed this interval and other goodies into google, I found
Margo Schulter and this whole community.

>
> In July 2002 (after I wrote the paper) I had a discussion with
>Paul
> Erlich on the harmonic entropy list. Paul adjusted variables in
his
> harmonic entropy program to produce a graph that closely
corresponded
> to observations I had made many years earlier with a retunable
> electronic organ, in which I had determined specific interval
>sizes
> as points of local maximum dissonance. As it turned out, the
global
> maximum harmonic entropy is at 67 cents in this graph (close to 70
> cents).
>
> You'll be interested to learn that the local HE maximum between
>5/4
> and 9/7 is at 422 cents in this graph, which is very close to
>23/18
> (which you mentioned below). Also, there is a slightly greater
>local
> HE maximum between 6/5 and 5/4 at 356 cents, which is very close
to
> 27/22 (which you also mentioned). (There's a second peak, almost
>as
> great, at 340 cents, so you might want to listen to 28/23 in JI or
>46-
> ET, and especially to the 18:23:28 triad, which is isoharmonic.)
> This supports a point of speculation in my paper that intervals
>that
> are melodically most effective are close to local maximum points
>of
> dissonance, discord, or disorder (i.e., harmonic entropy).

Very interesting indeed, to put it mildly. I'm very familiar with
28/23, using it a great deal (take a look at 24/23, 25/23, 26/23,
27/23, 28/23 in light of a 34 WT). Your point about melodic
effectiveness is very interesting, especially from my point of
hearing, as you're nailing intervals I use heavily, but I have grave
misgivings about "harmonic entropy" as a measure of
consonance- maybe it's a matter of terminology. For you see,
I find these intervals very consonant, and as far as I can tell,
so does pretty much everyone else I come in contact with in everyday
musical life.

If, however, the "dissonance" referred to in HE is a kind of
uncertainty or ambiguity as to an interval's status vis a vis the
first dozen partials, then I'd have to agree that these intervals
are almost maximally "dissonant" in that respect. My take on it is
that these intervals "float" in relationship to the lower partials,
while at the same time being truly consonant "in the corner of the
eye", or the edge of the ear, so to speak.

> > > > ...
> > > > There isn't and shouldn't be a universal tuning, in my
>>opinion,
> > > > but I believe that 34 is ideal for a great deal of music.
> > >
> > > My four favorite divisions below 72 are 19, 31, 34, and 46.
> (Yes, I
> > > like 17, but since I can have it as a subset of 34, I'll go
>with
> the
> > > latter.)
> >
> > To me, 19 and 31 tend to sound soggy.
>
> Presumably on account of their melodic properties, I would think.
> I've observed that these may require a "break-in" period during
which
> one's ears become accustomed to the smaller-than-optimal major
>2nds
> and larger minor 2nds.

Harmonically, too- to my ears, 5/4 tolerates almost no downward
tempering, quickly clamoring to be a median third, and flattened
3/2's are a specific flavor.
>
> > 19 is useless to me, no median
> > third,
>
> That's one of the reasons I've avoided 19-ET, preferring a multi-
> purpose well-temperament with 3 extra tones for the 11's (19+3WT).
>
> I assume that you don't like 22, either. I dislike 22 on account
>of
> the heavy tempering of the minor 3rd (and its small major 3rd
>doesn't
> help, either). IMO, the first wide-fifth EDO that delivers the
>goods
> is 34.

I think 22 just sounds... kind of silly, actually. To each his
own, of course.
>
> > but 31 sports the sevens and is audibly more suited to
> > certain musics than 34 is.
>
> Yes. For me, it's not a matter of preferring one over the other,
>but
> rather recognizing that each has different strengths and
>weaknesses.
>
> > 19, 31, 34, 46: milk, beer, wine, and ?
>
> ??

19 sounds like milk, 31 like beer, 34 like wine, and I don't know
about 46. 22 sounds like some kind of artificially flavored soda-pop.
Different strokes, of course.

>
> > I think we discussed some time ago how 17 feels like a subset of
> > something else- I wasn't up for 34 at the time because of the
> > absence of 7/4
>
> You're not the only one to think that. When I first looked at the
>34
> division, I thought that the poor representation of 7 was a fatal
> flaw. Paul Erlich also eliminated 34 from consideration in his 22-
> tone (pajara) paper because of 7-limit inconsistency and concluded
> that 22-ET was the best, most practical, implementation of pajara.
>
> However, my recent listening-test survey on this list has proved
> otherwise, due in large part to the fact that we seem to accept a
> larger positive than negative error for 4:7 (possibly because of
>our
> extensive exposure to 12-ET, with its 600-cent tritone).

Just calculating roughly in my head, I would guess that sharpening
the 7/4 would actually create more overall dissonance in the first
half-dozen partials, due to critical bandwidth interactions. To me,
the thing with lowering the 7/4 much is that it simply doesn't sound
like a seventh anymore, but wants to be a 15/9 or something like
that.

>
> I can't help noticing a resemblance between these observations and
> the introductory section of my 17-tone paper -- I think I've just
> laid out the opening thoughts for a 34-tone paper. :-)
>
> > but I've changed my mind because intervals literally
> > incorporating the 7th partial seem loathe to move or mingle
outside
> > the family, just as it's a major color decision in additive
> > synthesis to create a strong 7th partial.
> >
> > Also, the theoretical foundation of the way I make my chords
> > taller is not by stacking thirds, but by adding intervals with an
> > ear on the physical timbre of the chord, eg adding 11/10 to
> 14/11.
>
> The resultant 7/5 looks nice on paper; I'll have to try it.

You don't know how relieved I am to find that you noticed the 7/5
immediately. :-) Because the children and desires of intervals and
chords, whether fulfulled or thwarted, are innate, and that's what
drives my whole approach- this particular chord is designed to do
the Holst/sci-fi "tritone" thingy, you know which one I mean, in the
smoothest and most limpid way.
>
> > Example: "G-E-C-D" might have a "G" of 182/121 (makes a
>difference
> > from 3/2 in this approach, try it), an "E" of 14/11 and a "D" of
> > 11/10 (voiced 91/121, 14/11, 2/1, 22/10).
>
> Coincidentally, a few days ago I was wondering what sort of triad
>I
> would get with stacked 3rds of 14/11 and 13/11 (which gives a 5th
of
> 182/121), but I haven't had a chance to try it. Now I'm really
> curious!

I am finding in this case that tempering the x/lls and their
children (I throw in harmonic means as scale steps) up or down to
taste by 364/363, the difference between 182/121 and 3/2, creates a
more coherent pallette. Actually "impurity" is better for me,
because I'm loading specific frequency regions with energy- if
they're about 4-8 cents, more or less, wide, the whole thing seems
to work better. Absence of phase cancellation may be involved.
>
> > 34 works admirably for this approach because of its "harmonic-
>mean"
> > nature, I believe. The high third is a 23/18 (less than .9 cents
> off
> > even in straight EDO), the harmonic mean of 14/11 and 9/7; the
dark
> > minor third is 20/17 (within one cent), the harmonic mean of
13/11
> > and 7/6, and so on. So these "mean" intervals, though not
directly
> > percieved as "Just", are "softened" by more distantly percieved
> > literal consonances in the audible range, while at the same
> > time lending themselves to different interpretations in the lower
> > harmonics.
> >
> > Carl calls all this "eye of newt" but I keep getting comments
> > from listeners which seem to confirm that it all works precisely
as
> > intended, so, whatever. :-).
>
> I'll have to try these out. It's amazing how many different
> approaches to tonality can be found when you think outside the 12-
>box.
>
> > Anyway as far as other kinds of music, I find 34 eerily
reminiscent
> > of the intonational flavor of pre-war Eastern European
orchestral
> > recordings (Melodiya), and of old recordings of Romantic music
in
> > general. A couple of artists I know have made remarks to similar
> > effect, "sounds like...1930!", perhaps remarkable considering
the
> > highly synthetic timbres I use.
>
> That's interesting. Could you direct me to any sound files for
this?

I'll be posting some music this week- technically my DVD should be
done "any day now" but the video artists are all on the seaside...
>
> > > > The kicker for me is the "median" intervals,
> > > > which are superb- 27/22 is only 1.6 cents flat in 34-EDO,
> which
> > > > means you can have some pure in a WT, etc.
> > >
> > > My 34-WT has 4 of these with less than 1 cent error and 4 more
> with
> > > ~1.9 cents error. In other keys the interval of 10deg34-WT
>comes
> > > very close to 11/9, 16/13, or 39/32, so in different keys you
>get
> > > neutral 3rds in different flavors.
> >
> > That's very nice indeed- is your 34-WT in the Scala archive? I
> don't
> > find it.
>
> I don't know of any procedure for putting tunings in the Scala
> archive, so it's probably not there.

I don't either, I guess you ask Manuel?

>However, I've put .scl files
> for my best tunings here:
> /tuning-math/files/secor/scl/
> so you can look at the descriptions and see what else I've been up
> to. You need to be a member of tuning-math to access these files;
34-
> WT is 6th from the bottom.
>
> You'll also find it here, with a brief explanation:
> /tuning/topicId_67957.html#68032

Thanks, George- I see that it's as you say, two 17's hinged in the
middle so to speak, looking forward to trying it out!

take care,

-Cameron Bobro

Hi Cameron,

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

> Very interesting indeed, to put it mildly. I'm very
> familiar with 28/23, using it a great deal (take a look
> at 24/23, 25/23, 26/23, 27/23, 28/23 in light of a
> 34 WT). Your point about melodic effectiveness is
> very interesting, especially from my point of hearing,
> as you're nailing intervals I use heavily, but I have
> grave misgivings about "harmonic entropy" as a measure
> of consonance- maybe it's a matter of terminology. For
> you see, I find these intervals very consonant, and as
> far as I can tell, so does pretty much everyone else
> I come in contact with in everyday musical life.

Hmm, that's interesting, because on the middle graphic
(made by Paul Erlich) on my Encyclopedia page for
"harmonic entropy":

http://tonalsoft.com/enc/h/harmonic-entropy.aspx

Paul listed the cents values for the local maxima,
i.e., the pitch-sizes of those intervals which are
least likely to be pinpointed by the brain as a
specific ratio.

On that graphic, you can see that 73 cents
(very close to 24/23) and 345 cents (close to 28/23)
are local maxima, thus Paul's theory disagrees
with your statement that these are "very consonant".
The other ratios you list fall on the downward slope
between 73 cents and the 6/5 ratio, so they are
considered relatively concordant according to HE.

(Note that i'm using "accordance" for psycho-acoustic
sensory perception of intervals without a musical
context, restricting "sonance" for cases which
involve musical context. I'm always consistent
about this; there are pages about those terms in
the Encyclopedia. )

> If, however, the "dissonance" referred to in HE is a kind of
> uncertainty or ambiguity as to an interval's status vis a vis the
> first dozen partials, then I'd have to agree that these intervals
> are almost maximally "dissonant" in that respect. My take on it is
> that these intervals "float" in relationship to the lower partials,
> while at the same time being truly consonant "in the corner of the
> eye", or the edge of the ear, so to speak.

HE is basically just a result of the numbers in a ratio.
It doesn't have anything to do with an arbitrary
cutoff point in the harmonic series.

I do have a page with more detail about HE:

http://sonic-arts.org/td/erlich/entropy.htm

however, note that Paul has pretty much deprecated it
at this point ... it was really just me trying to
understand his ideas from discussion with him, back
when he first developed the concept. He set up an
entire Yahoo group to discuss the topic:

/harmonic_entropy/

> Thanks, George- I see that it's as you say, two 17's
> hinged in the middle so to speak,

Around here we've come up with the term "bike chain"
for something like that:

http://tonalsoft.com/enc/b/bike-chain.aspx

The idea has been around for awhile: in his 1906 book
_Sketch for a New Aesthetic of Music_, Busoni advocated
36-edo -- but not as 3 bike-chains of 12-edo, which
we today might expect, but rather as 2 bike-chains
of 18-edo. Busoni was captivated by the idea of
1/3-tones, but he still wanted the semitones too.

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

Cameron Bobro wrote:
> 34 works admirably for this approach because of its "harmonic-mean" > nature, I believe. The high third is a 23/18 (less than .9 cents off > even in straight EDO), the harmonic mean of 14/11 and 9/7; the dark > minor third is 20/17 (within one cent), the harmonic mean of 13/11 > and 7/6, and so on. also, on august 1st, wrote:

> 34 offers an elegant > solution as far as 7/4, by having a 23/13, the harmonic mean
> of 7/4 and 16/9. Not to be a nitpicker, but the correct mathematics term is "mediant"...I had a feeling "harmonic mean" was wrong, so I looked it up, and indeed it was.

The mediant of two rationals is what you are talking about: (n1+n2)/(d1+d2)

I'm surprised Gene didn't already chastise you! ;)

Anyway, that aside, all the things you've been saying (and George, too) have certainly piqued my interest in 34.

Cheers,
Aaron.

Thanks Aaron- I should have realized something was amiss because
I've been referring to "median thirds".

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <aaron@...> wrote:
>
> Cameron Bobro wrote:
> > 34 works admirably for this approach because of its "harmonic-
mean"
> > nature, I believe. The high third is a 23/18 (less than .9 cents
off
> > even in straight EDO), the harmonic mean of 14/11 and 9/7; the
dark
> > minor third is 20/17 (within one cent), the harmonic mean of
13/11
> > and 7/6, and so on.
> also, on august 1st, wrote:
>
> > 34 offers an elegant
> > solution as far as 7/4, by having a 23/13, the harmonic mean
> > of 7/4 and 16/9.
>
> Not to be a nitpicker, but the correct mathematics term
is "mediant"...I
> had a feeling "harmonic mean" was wrong, so I looked it up, and
indeed
> it was.
>
> The mediant of two rationals is what you are talking about:
(n1+n2)/(d1+d2)
>
> I'm surprised Gene didn't already chastise you! ;)
>
> Anyway, that aside, all the things you've been saying (and George,
too)
> have certainly piqued my interest in 34.
>
> Cheers,
> Aaron.
>

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi Cameron,
>
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@>
wrote:
>
> > Very interesting indeed, to put it mildly. I'm very
> > familiar with 28/23, using it a great deal (take a look
> > at 24/23, 25/23, 26/23, 27/23, 28/23 in light of a
> > 34 WT). Your point about melodic effectiveness is
> > very interesting, especially from my point of hearing,
> > as you're nailing intervals I use heavily, but I have
> > grave misgivings about "harmonic entropy" as a measure
> > of consonance- maybe it's a matter of terminology. For
> > you see, I find these intervals very consonant, and as
> > far as I can tell, so does pretty much everyone else
> > I come in contact with in everyday musical life.
>
>
> Hmm, that's interesting, because on the middle graphic
> (made by Paul Erlich) on my Encyclopedia page for
> "harmonic entropy":
>
> http://tonalsoft.com/enc/h/harmonic-entropy.aspx
>
> Paul listed the cents values for the local maxima,
> i.e., the pitch-sizes of those intervals which are
> least likely to be pinpointed by the brain as a
> specific ratio.
>
> On that graphic, you can see that 73 cents
> (very close to 24/23) and 345 cents (close to 28/23)
> are local maxima, thus Paul's theory disagrees
> with your statement that these are "very consonant".

According to the charts on your page, the skeleton of
my music basically consists of the points of
maximum harmonic entropy.

> (Note that i'm using "accordance" for psycho-acoustic
> sensory perception of intervals without a musical
> context, restricting "sonance" for cases which
> involve musical context. I'm always consistent
> about this; there are pages about those terms in
> the Encyclopedia. )

Good idea, for it is demonstratable that one
way of creating a startling musical "dissonance" is to throw a
pure interval into a piece using a highly tempered tuning.

From your page:

"[John Chalmers, Yahoo tuning message]
I recommend we distinguish between "sensory consonance" (aka
roughness, sonance etc.) and "contextual consonance" as Tenney does
in his History of Consonance and Dissonance."

I'm going with this one; I'll try to be consistent.

>
>
> > If, however, the "dissonance" referred to in HE is a kind of
> > uncertainty or ambiguity as to an interval's status vis a vis the
> > first dozen partials, then I'd have to agree that these
intervals
> > are almost maximally "dissonant" in that respect. My take on it
>is
> > that these intervals "float" in relationship to the lower
>partials,
> > while at the same time being truly consonant "in the corner of
>the
> > eye", or the edge of the ear, so to speak.
>
>
> HE is basically just a result of the numbers in a ratio.
> It doesn't have anything to do with an arbitrary
> cutoff point in the harmonic series.

Unfortunately the numbers in a ratio really don't mean jack, except
as (dangerously) convenient reference points:
it's the partials that matter, whatever you call them. I call the
13th integer partial, any frequency indistinguishable from what
frequency it happens to be at the moment, and that narrow region on
either side that lies in the critical band safety zone, "Frank".

And there cannot be any "arbitrary cutoff point" in the harmonic
series- there are regions, graded, subjective and varying according
to timbre and person.

Once again...
> Paul listed the cents values for the local maxima,
> i.e., the pitch-sizes of those intervals which are
> least likely to be pinpointed by the brain as a
> specific ratio.

Does this equate "consonance" with ease of equation with
specific ratios?

>
>
> I do have a page with more detail about HE:
>
> http://sonic-arts.org/td/erlich/entropy.htm
>
> however, note that Paul has pretty much deprecated it
> at this point ... it was really just me trying to
> understand his ideas from discussion with him, back
> when he first developed the concept. He set up an
> entire Yahoo group to discuss the topic:
>
> /harmonic_entropy/

Hmmm... well I'm a member of the group, I peek in once in a
while. Now my curiosity is piqued- unless I'm missing
something here, "harmonic entropy" seems to be violently at
odds with personal experience.

-Cameron Bobro

> > Hmm, that's interesting, because on the middle graphic
> > (made by Paul Erlich) on my Encyclopedia page for
> > "harmonic entropy":
> >
> > http://tonalsoft.com/enc/h/harmonic-entropy.aspx
> >
> > Paul listed the cents values for the local maxima,
> > i.e., the pitch-sizes of those intervals which are
> > least likely to be pinpointed by the brain as a
> > specific ratio.
> >
> > On that graphic, you can see that 73 cents
> > (very close to 24/23) and 345 cents (close to 28/23)
> > are local maxima, thus Paul's theory disagrees
> > with your statement that these are "very consonant".
>
> According to the charts on your page, the skeleton of
> my music basically consists of the points of
> maximum harmonic entropy.

Nothnig wrong with that. But I hardly think it would
lead to concordance. I have two tracks from you on my
hard drive if memory serves. Neither was particularly
concordant but I enjoyed them both. Care to give a
specific example?

> And there cannot be any "arbitrary cutoff point" in the harmonic
> series- there are regions, graded, subjective and varying according
> to timbre and person.

Who's claiming there is?

-Carl

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...>
wrote:
>
> > ...
> > > The minor second is a tad wide, IMO- 25/24
> > > being my ideal.
> >
> > Yes, that's exactly what I said in my 17-tone paper.
>
> That's actually how I found this tuning community- I came by
> 17 simply by repeatedly singing and listening for an ideal
semitone,
> while tweaking the tuning on a synth. Because I like the "minor
> ninth" I was doing it harmonically as well as melodically and it
> didn't take long to discover that what I kept arriving at was 25/24.
> When I typed this interval and other goodies into google, I found
> Margo Schulter and this whole community.

Wow, that's wild! (But not as wild as the ride I'm about to take you
on.)

I guess one should never underestimate the power of a lowly
semitone. I'm still amazed (and amused) that, a few years back,
somebody actually suggested naming one after me, and that the idea
actually caught on (before I learned about it):
/tuning/topicId_27028.html#27062
The double irony in this is: 1) that I've never had any interest in
using Miracle for anything (other than as the basis for a keyboard
layout); and 2) that "somebody" to whom I referred is none other than
Dave Keenan, who has since voiced his objection to labeling tunings
with names that have nothing to do with their actual properties.

Mention of a keyboard layout reminded me of something. Here's my
Pajara keyboard design, with the keys labeled for 34 & 46:
/tuning-
math/files/secor/kbds/KbPaj34-46.gif
Since it's not clear what color arrangement would be most meaningful
in the all of the divisions that this keyboard will handle, the key
colors in the outer rows are only tentative.

I just thought of something funny about yet another kind of semitone:
Why not name the pajara generator using the term "erlich" (lower case
e) after its discoverer Paul Erlich?

Hang on; there's more. I was also thinking, to make the triangle
complete, that it was fitting for Paul Erlich to propose naming a
family of tunings with a minor 3rd generator and a low 7 mapping
(which, in Gene's words, would be "34 trying to act like 19, not 22")
after Dave Keenan -- except that Dave objected and spoiled all the
fun (so now it's called "keemun".

Anyway, to get back to Margo and the 17-tone revolution, she and I
had briefly discussed the possibility that eastern Europe and western
Asia might have been more fertile ground than western Europe for wide-
fifth tunings, particularly 17 and the neutral (median) 3rds it
contains, but apart from the mention of Zalzal's 11th-harmonic
fretting (by both Helmholtz and Partch), we had very little to go
on. I'm happy to see that our suspicions were correct.

> > In July 2002 (after I wrote the paper) I had a discussion with
Paul
> > Erlich on the harmonic entropy list. Paul adjusted variables in
his
> > harmonic entropy program to produce a graph that closely
corresponded
> > to observations I had made many years earlier with a retunable
> > electronic organ, in which I had determined specific interval
sizes
> > as points of local maximum dissonance. As it turned out, the
global
> > maximum harmonic entropy is at 67 cents in this graph (close to
70
> > cents).
> >
> > You'll be interested to learn that the local HE maximum between
5/4
> > and 9/7 is at 422 cents in this graph, which is very close to
23/18
> > (which you mentioned below). Also, there is a slightly greater
local
> > HE maximum between 6/5 and 5/4 at 356 cents, which is very close
to
> > 27/22 (which you also mentioned). (There's a second peak, almost
as
> > great, at 340 cents, so you might want to listen to 28/23 in JI
or 46-
> > ET, and especially to the 18:23:28 triad, which is isoharmonic.)
> > This supports a point of speculation in my paper that intervals
that
> > are melodically most effective are close to local maximum points
of
> > dissonance, discord, or disorder (i.e., harmonic entropy).
>
> Very interesting indeed, to put it mildly. I'm very familiar with
> 28/23, using it a great deal (take a look at 24/23, 25/23, 26/23,
> 27/23, 28/23 in light of a 34 WT).

Ah, yes! For example, I see that the 18:23:28 triad I suggested has
a maximum error <7 cents with C taken as 16 (or 1/1).

> Your point about melodic
> effectiveness is very interesting, especially from my point of
> hearing, as you're nailing intervals I use heavily, but I have grave
> misgivings about "harmonic entropy" as a measure of
> consonance- maybe it's a matter of terminology. For you see,
> I find these intervals very consonant, and as far as I can tell,
> so does pretty much everyone else I come in contact with in
everyday
> musical life.

So far harmonic entropy has been correlated to consonance only when
considering intervals, and only in isolation. As soon as you hear
these intervals in a musical context, then one's perception of
consonance and dissonance is also affected by other factors. In
classical harmony a 3:4 is often considered dissonant, yet
acoustically it's near the top of the list of consonant intervals.

> If, however, the "dissonance" referred to in HE is a kind of
> uncertainty or ambiguity as to an interval's status vis a vis the
> first dozen partials, then I'd have to agree that these intervals
> are almost maximally "dissonant" in that respect. My take on it is
> that these intervals "float" in relationship to the lower partials,
> while at the same time being truly consonant "in the corner of the
> eye", or the edge of the ear, so to speak.
>
> > > > > ...
> > > > > There isn't and shouldn't be a universal tuning, in my
opinion,
> > > > > but I believe that 34 is ideal for a great deal of music.
> > > >
> > > > My four favorite divisions below 72 are 19, 31, 34, and 46.
(Yes, I
> > > > like 17, but since I can have it as a subset of 34, I'll go
with the
> > > > latter.)
> > >
> > > To me, 19 and 31 tend to sound soggy.
> >
> > Presumably on account of their melodic properties, I would
think.
> > I've observed that these may require a "break-in" period during
which
> > one's ears become accustomed to the smaller-than-optimal major
2nds
> > and larger minor 2nds.
>
> Harmonically, too- to my ears, 5/4 tolerates almost no downward
> tempering, quickly clamoring to be a median third, and flattened
> 3/2's are a specific flavor.

Yep, that's "soggy" all right! :-)

> > > 19 is useless to me, no median
> > > third,
> >
> > That's one of the reasons I've avoided 19-ET, preferring a multi-
> > purpose well-temperament with 3 extra tones for the 11's (19+3WT).
> >
> > I assume that you don't like 22, either. I dislike 22 on account
of
> > the heavy tempering of the minor 3rd (and its small major 3rd
doesn't
> > help, either). IMO, the first wide-fifth EDO that delivers the
goods
> > is 34.
>
> I think 22 just sounds... kind of silly, actually. To each his
> own, of course.

Just so others don't get the wrong impression, I wouldn't say that 19
or 22 are worthless, only that it takes some time to get used to
them, whereas some other divisions sound good "right out of the box."

> ... To me,
> the thing with lowering the 7/4 much is that it simply doesn't
sound
> like a seventh anymore, but wants to be a 15/9 or something like
> that.

Perhaps a 12/7. In 19 you have a single interval trying to be both
7:12 and 4:7; this works much better if the 19 tones are well-
tempered.

> > ...
> > > Also, the theoretical foundation of the way I make my chords
> > > taller is not by stacking thirds, but by adding intervals with
an
> > > ear on the physical timbre of the chord, eg adding 11/10 to
> > 14/11.
> >
> > The resultant 7/5 looks nice on paper; I'll have to try it.

(I did, and it is nice!)

> You don't know how relieved I am to find that you noticed the 7/5
> immediately. :-) Because the children and desires of intervals and
> chords, whether fulfulled or thwarted, are innate, and that's what
> drives my whole approach- this particular chord is designed to do
> the Holst/sci-fi "tritone" thingy, you know which one I mean, in
the
> smoothest and most limpid way.

Yes.

> > ...
> > > Anyway as far as other kinds of music, I find 34 eerily
reminiscent
> > > of the intonational flavor of pre-war Eastern European
orchestral
> > > recordings (Melodiya), and of old recordings of Romantic music
in
> > > general. A couple of artists I know have made remarks to
similar
> > > effect, "sounds like...1930!", perhaps remarkable considering
the
> > > highly synthetic timbres I use.
> >
> > That's interesting. Could you direct me to any sound files for
this?
>
> I'll be posting some music this week-

Okay, thanks. I'll watch for it.

Best,

--George

Hi Cameron,

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

> Hmmm... well I'm a member of the group, I peek in once in a
> while. Now my curiosity is piqued- unless I'm missing
> something here, "harmonic entropy" seems to be violently at
> odds with personal experience.

To use one example, the basic idea of HE is this:

The 3/2 ratio is readily perceivable because it
is such a strong concordance, thus it has a low dip
on the chart (a "minimum").

There are a lot of ratios in the pitch-size vicinity
of the 3/2 ratio which have larger integers in their
terms (numerator/denominator) which the human listener
will still perceive as a 3/2. That is why the low dip
for 3/2 has such a wide bell-curve on either side of it.

As the pitch-size of the ratios under consideration
get farther away from exactly 3/2 in either direction
(i.e., larger or smaller), the odds of a listener
perceiving it as a 3/2 will decrease up to the point
of maximum HE (the highest points on the chart next
to the 3/2 dip) -- on the chart on the Encyclopedia
page, Paul labeled these 639 and 765 cents. These
are the points of least precision in the perception
of which ratio the interval is.

Then beyond that, there are slopes on either side
which eventually dip down to the minima at the
7/5 and 8/5 ratios.

The one variable parameter involved, which Paul
calls "s", is how accurate a listener's perception is.
I suppose that's what a lot of discussion on the
HE list is or could be about.

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

Howdy Partners,

During the current references to 34 ED's of the O, I
have noticed a lack of any reference whatsoever to the
fact that 34-ET always generates two separate cycles
of 17 fifths each, a situation that Siemen Terpstra
once critiqued as being musically schizophrenic, or
some such.

Johnny, on the other hand, might find it a
compositional challenge! Dichotemic Multi Micro
Xenotonality, or some such.

In the early 1990's Siem and I once had the honor of
playing Los Angeles microtonal theorist Larry A.
Hanson's brand new 34-TPO electric guitar.

We found that the multitude of frets made us both
prefer to tune the instrument to an open tuning. 31
to 34 TPO seems to be the outer playability limit for
equal tempered guitar frettings.

A colleague of Erv Wilson and John Chalmers, Larry
Hanson's work led him to a scale, or scales, based on
cycles of Thirds. He passed away a few months later.

Mark, or some such.

--- Cameron Bobro <misterbobro@yahoo.com> wrote:

> Thanks Aaron- I should have realized something was
> amiss because
> I've been referring to "median thirds".
>
> --- In tuning@yahoogroups.com, "Aaron K. Johnson"
> <aaron@...> wrote:
> >
> > Cameron Bobro wrote:
> > > 34 works admirably for this approach because of
> its "harmonic-
> mean"
> > > nature, I believe. The high third is a 23/18
> (less than .9 cents
> off
> > > even in straight EDO), the harmonic mean of
> 14/11 and 9/7; the
> dark
> > > minor third is 20/17 (within one cent), the
> harmonic mean of
> 13/11
> > > and 7/6, and so on.
> > also, on august 1st, wrote:
> >
> > > 34 offers an elegant
> > > solution as far as 7/4, by having a 23/13, the
> harmonic mean
> > > of 7/4 and 16/9.
> >
> > Not to be a nitpicker, but the correct mathematics
> term
> is "mediant"...I
> > had a feeling "harmonic mean" was wrong, so I
> looked it up, and
> indeed
> > it was.
> >
> > The mediant of two rationals is what you are
> talking about:
> (n1+n2)/(d1+d2)
> >
> > I'm surprised Gene didn't already chastise you! ;)
> >
> > Anyway, that aside, all the things you've been
> saying (and George,
> too)
> > have certainly piqued my interest in 34.
> >
> > Cheers,
> > Aaron.
> >
>
>
>

____________________________________________________________________________________
Looking for a deal? Find great prices on flights and hotels with Yahoo! FareChase.
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George D. Secor wrote:

> Mention of a keyboard layout reminded me of something. Here's my > Pajara keyboard design, with the keys labeled for 34 & 46:
> /tuning-
> math/files/secor/kbds/KbPaj34-46.gif
> Since it's not clear what color arrangement would be most meaningful > in the all of the divisions that this keyboard will handle, the key > colors in the outer rows are only tentative.

This looks good. Any particular reason for using D (|) - F (|) - G (|) and so on rather than D )||( - F )||( - G )||( ? Do you have a particular 11-limit version of pajara in mind, or is there a 7-limit interpretation of (|) that makes sense in this context? I haven't been keeping up with all the latest Sagittal developments.

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> George D. Secor wrote:
>
> > Mention of a keyboard layout reminded me of something. Here's my
> > Pajara keyboard design, with the keys labeled for 34 & 46:
> > /tuning-
> > math/files/secor/kbds/KbPaj34-46.gif
> > Since it's not clear what color arrangement would be most
meaningful
> > in the all of the divisions that this keyboard will handle, the
key
> > colors in the outer rows are only tentative.
>
> This looks good. Any particular reason for using D (|) - F (|) - G
(|)
> and so on rather than D )||( - F )||( - G )||( ? Do you have a
> particular 11-limit version of pajara in mind, or is there a 7-
limit
> interpretation of (|) that makes sense in this context? I haven't
been
> keeping up with all the latest Sagittal developments.

Hi, Herman. Thanks for bringing this up.

Yes, I did have an 11-limit version of pajara in mind, but it doesn't
play out in this diagram. The standard symbol sets for 34 and 56
both contain /|\:
34: /| /|\ ||\ /||\
56: |) /| /|\ (|) ||\ ||) /||\
The position of the /|\ symbol on the diagram does not correspond to
a generalized 11-limit pajara mapping, since it's valid for 34, but
not for 56. For a generalized pajara notation, the /|\ and (|)
symbols should be switched, and I should be showing degrees of 56
instead of 46 (from which you can deduce degrees of 22 as the
difference between the two).

The fundamental problem here is that I've labeled this as a "Pajara
Keyboard". While it is indeed intended for pajara tunings (most
notably 22, 34, and 56), it's also excellent for some other divisions
that aren't as well suited to pajara, particularly 46 and 58. So
perhaps I should call this a pajara-plus keyboard and point out that
the notation for 46 and 58 don't agree completely with pajara.

Since Cameron and I were interested primarily in 34 and 46, I used
Sagittal spellings from the standard 34 and 46 symbol sets that are
valid for both 34 and 46. The standard symbol set for 46 is:
46: /| /|\ (|) ||\ /||\
Although (|) doesn't occur in the 34 set, it's equivalent to /|\ in
34, so the spellings using (|), though not standard, are valid for
34, so you can make a mental substitution of /|\ for (|) when viewing
this for 34.

To clear this up, I'd better make a separate diagram for pajara
showing 34 and 56, and perhaps that will help me decide on colors for
the remote keys.

--George

George D. Secor wrote:
> The fundamental problem here is that I've labeled this as a "Pajara > Keyboard". While it is indeed intended for pajara tunings (most > notably 22, 34, and 56), it's also excellent for some other divisions > that aren't as well suited to pajara, particularly 46 and 58. So > perhaps I should call this a pajara-plus keyboard and point out that > the notation for 46 and 58 don't agree completely with pajara.

Makes sense. Like the way the Bosanquet keyboard can be used for other fifth-based tunings even though it was originally for a schismatic temperament (53-ET). Probably Larry Hanson's keyboard has other uses as well.

> Since Cameron and I were interested primarily in 34 and 46, I used > Sagittal spellings from the standard 34 and 46 symbol sets that are > valid for both 34 and 46. The standard symbol set for 46 is:
> 46: /| /|\ (|) ||\ /||\
> Although (|) doesn't occur in the 34 set, it's equivalent to /|\ in > 34, so the spellings using (|), though not standard, are valid for > 34, so you can make a mental substitution of /|\ for (|) when viewing > this for 34.

I guess that explains where the (|) came from. It seems like )||( would be good for 2 steps of 34, but I don't know what that works out to be in 46-ET.

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> George D. Secor wrote:
> > The fundamental problem here is that I've labeled this as
> > a "Pajara Keyboard". While it is indeed intended for pajara
> > tunings (most notably 22, 34, and 56), it's also excellent
> > for some other divisions that aren't as well suited to pajara,
> > particularly 46 and 58. So perhaps I should call this a
> > pajara-plus keyboard and point out that the notation for 46
> > and 58 don't agree completely with pajara.
>
> Makes sense.

Not to me. What's the difference between keyboards? They're
all just two-dimensional arrays of digitals. What's meant
by "keyboard geometry"? The shapes of the digitals, the choice
of mapping vectors??

-Carl

(I am coming in late to this subject having just returned from Australia). One of the Hanson Keyboard works quite well for 34. Which was one of the reason E. Wilson was so pleased with Hanson's design. This is discussed in his Xenharmonikon Article on page 15 and 16.
http://anaphoria.com/hanson.PDF and
http://anaphoria.com/starr.PDF
page 2 and 3 specifically

Posted by: "Herman Miller" Makes sense. Like the way the Bosanquet keyboard can be used for other
fifth-based tunings even though it was originally for a schismatic
temperament (53-ET). Probably Larry Hanson's keyboard has other uses as
well.
--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > > Hmm, that's interesting, because on the middle graphic
> > > (made by Paul Erlich) on my Encyclopedia page for
> > > "harmonic entropy":
> > >
> > > http://tonalsoft.com/enc/h/harmonic-entropy.aspx
> > >
> > > Paul listed the cents values for the local maxima,
> > > i.e., the pitch-sizes of those intervals which are
> > > least likely to be pinpointed by the brain as a
> > > specific ratio.
> > >
> > > On that graphic, you can see that 73 cents
> > > (very close to 24/23) and 345 cents (close to 28/23)
> > > are local maxima, thus Paul's theory disagrees
> > > with your statement that these are "very consonant".
> >
> > According to the charts on your page, the skeleton of
> > my music basically consists of the points of
> > maximum harmonic entropy.
>
> Nothnig wrong with that. But I hardly think it would
> lead to concordance. I have two tracks from you on my
> hard drive if memory serves. Neither was particularly
> concordant but I enjoyed them both.

It seems to me that it's a far cry from not particularly
concordant to maximally dissonant. Of course there's can be
a big difference between an isolated diad and an interval in
context, as George and Joe mentioned, but I'm downright
belaboring a number of these maximally entropic intervals-
shouldn't the result be hopelessly dissonant?

>Care to give a
> specific example?

Even better, , I wrote a piece last night,
typical but deliberately heavy on the HE maxima.

Because the nature of 34 and collections of
intervals with similar properties basically alternates
or interlaces minima and maxima of HE it is entirely
possible that my use of the HE maxima is really a matter
of drawn-out suspensions. (That doesn't explain my
perception of the median third as consonant though.)

By the way I do agree on the literal consonance of
the classic JI intervals, it's downright... tautological?
Even without historical evidence, to deny their literal
consonance would reflect ignorance of the most basic acoustics.
I just believe that there are other kinds of cohesive,
coherent and non-voodoo consonances, "distant consonances",
that are found in places that are decidely NOT the
simple JI ratios.

>
> > And there cannot be any "arbitrary cutoff point" in the harmonic
> > series- there are regions, graded, subjective and varying
according
> > to timbre and person.
>
> Who's claiming there is?

I was under the impression that Joe was under the impression
that I was claiming that there is, which I'm certainly not.

take care,
-Cameron Bobro

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi Cameron,
>
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@>
wrote:
>
> > Hmmm... well I'm a member of the group, I peek in once in a
> > while. Now my curiosity is piqued- unless I'm missing
> > something here, "harmonic entropy" seems to be violently at
> > odds with personal experience.
>
>
> To use one example, the basic idea of HE is this:
>
> The 3/2 ratio is readily perceivable because it
> is such a strong concordance, thus it has a low dip
> on the chart (a "minimum").

The 600 cent half-octave is readily percievable
because it is such a strong dissonance, does it
have a low dip in the chart? Wouldn't it be a
fundamental mistake to equate ease of perception
with consonance?

>
> There are a lot of ratios in the pitch-size vicinity
> of the 3/2 ratio which have larger integers in their
> terms (numerator/denominator) which the human listener
> will still perceive as a 3/2. That is why the low dip
> for 3/2 has such a wide bell-curve on either side of it.

The vast majority of my experience with human listeners
indicates that they percieve things in terms of brightness,
boldness, dullness, fearfulness, languidity, sweetness,
sourness...etc.

>
> As the pitch-size of the ratios under consideration
> get farther away from exactly 3/2 in either direction
> (i.e., larger or smaller), the odds of a listener
> perceiving it as a 3/2 will decrease up to the point
> of maximum HE (the highest points on the chart next
> to the 3/2 dip) -- on the chart on the Encyclopedia
> page, Paul labeled these 639 and 765 cents. These
> are the points of least precision in the perception
> of which ratio the interval is.

36/25 and 36/23 are awfully close to those points, and
it can't be coindicence that they're my most heavily used
dim/aug 5ths ( I use others within a few cents as well).

> The one variable parameter involved, which Paul
> calls "s", is how accurate a listener's perception is.
> I suppose that's what a lot of discussion on the
> HE list is or could be about.

Accurate by what gauge? I don't consider hearing things
how they're "supposed to be", rather than what they
"are", accurate, rather, a sad example of nurture damaging
nature.

36/25 and 36/23 are excellent ratios to use in conjunction
with 3/2, as distinct scale steps.

Hmm...as I started to say to Carl, it may very well be
that it's maximally sastifying to resolve to 3/2 and other
JI classics from those
points and that explains my attraction to this particular
"family" of intervals I've been using and refining over the last
year, which I now find corresponds to HE maxima. Still doesn't
explain why I hear them as "distant consonances".

Anyway it is obvious that I must accept, from parallel personal
experience, that there is a rhyme and reason to
harmonic entropy, for it has precisely listed one of the
"character families" I've been jammering about for the last year or
so, the one I think of as "in the distance".

take care,

-Cameron Bobro

PS. George if you're reading I'll get back to your messages
in a day or two, gotta run and pick up my kid.

Hi Cameron,

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

> > > And there cannot be any "arbitrary cutoff point"
> > > in the harmonic series- there are regions, graded,
> > > subjective and varying according to timbre and person.
> >
> > Who's claiming there is?
>
> I was under the impression that Joe was under the impression
> that I was claiming that there is, which I'm certainly not.

I was simply responding to an assumption you suggested
about the concordance measure being based on "the first
12 partials". Nothing more than that.

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

Hi Cameron,

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
> >
> > Hi Cameron,
> >
> >
> > --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@>
> wrote:
> >
> > [monz:]
> > The 3/2 ratio is readily perceivable because it
> > is such a strong concordance, thus it has a low dip
> > on the chart (a "minimum").
>
> The 600 cent half-octave is readily percievable
> because it is such a strong dissonance, does it
> have a low dip in the chart? Wouldn't it be a
> fundamental mistake to equate ease of perception
> with consonance?

As Paul wrote on the Harmonic Entropy group homepage,
harmonic entropy is about:

>> "... mathematical modeling of ... the component
>> of consonance that is not due to the critical-band
>> roughness discussed by Plomp and Sethares, but an
>> additional component related to the salience of
>> the perception of a chord as a segment of a
>> harmonic series"

He's not proposing HE as a be-all-and-end-all theory,
but only as a model of one component of accordance.
I think this is important, because the Plomp/Sethares
model has been widely accepted by many music-theorists
without considering any other component.

Anyway, according to HE the 600-cent half-octave
is not especially perceptible -- it's on the up-slope
between the minimum at the 7/5 ratio and the maximum
at 651 cents, on the default HE chart:

/tuning/files/dyadic/default.gif

(you have to be a member of the group to see the Files)

The 7/5 ratio is the most perceptible concordance
between 4/3 and 3/2, and 600 cents is close enough
to 7/5 that there's a good chance that a listener
will perceive as a 7/5 and thus as relatively concordant.

HE is really a statistics and probability measure.

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

> It seems to me that it's a far cry from not particularly
> concordant to maximally dissonant. Of course there's can be
> a big difference between an isolated diad and an interval in
> context, as George and Joe mentioned, but I'm downright
> belaboring a number of these maximally entropic intervals-
> shouldn't the result be hopelessly dissonant?

Discordant, yes. Except the highest-entropy intervals are
near-semitones, which your tracks didn't have a lot of.

> I was under the impression that Joe was under the impression
> that I was claiming that there is, which I'm certainly not.

Oh.

-Carl

> > The 3/2 ratio is readily perceivable because it
> > is such a strong concordance, thus it has a low dip
> > on the chart (a "minimum").
>
> The 600 cent half-octave is readily percievable
> because it is such a strong dissonance, does it
> have a low dip in the chart? Wouldn't it be a
> fundamental mistake to equate ease of perception
> with consonance?

That's not what HE does. It equates the ease of
fitting to a harmonic series to concordance.

-Carl

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> George D. Secor wrote:
> > The fundamental problem here is that I've labeled this as
a "Pajara
> > Keyboard". While it is indeed intended for pajara tunings (most
> > notably 22, 34, and 56), it's also excellent for some other
divisions
> > that aren't as well suited to pajara, particularly 46 and 58. So
> > perhaps I should call this a pajara-plus keyboard and point out
that
> > the notation for 46 and 58 don't agree completely with pajara.
>
> Makes sense. Like the way the Bosanquet keyboard can be used for
other
> fifth-based tunings even though it was originally for a schismatic
> temperament (53-ET).

Bosanquet's generalized keyboard was in fact originally for both the
schismatic and extended meantone temperaments. Both Helmholtz and
Bosanquet (in his 1875 treatise) indicate that Bosanquet had an organ
with 2 ranks of pipes, one in a schismatic temperament (probably 53-
equal, but I'd have to check) and the other in extended 1/4-comma
meantone. The tuning could be changed instantly by moving a stop-
lever to switch from one rank to the other.

The term "generalized" refers not only to the property of
transpositional invariance but also the capability of mapping
multiple tunings onto a single keyboard. In this case it
accommodates not only multiple tunings within a family (e.g., 12, 19,
31, 43, etc. in meantone, or 29, 41, 53, etc. in schismatic) but also
multiple families of tunings. Being fully aware of all of this,
Bosanquet classified regular-fifth tunings as negative or positive
according to whether their fifths were narrow or wider than 700 cents.

As you point out, pajara-plus also accommodates multiple families of
tunings. I noticed that 72-ET can be mapped onto it using a 7deg
generator (secor) and 36deg period; for performance this isn't nearly
as good as my decimal keyboard, see filename KbDec72.gif:
/tuning-math/files/secor/kbds/
but at least it would be possible to explore 72 with a pajara-plus
keyboard.

> Probably Larry Hanson's keyboard has other uses as
> well.

Yes, you can map any tuning on it if all of its tones can be arranged
in a single chain of ~5:6's.

> > Since Cameron and I were interested primarily in 34 and 46, I
used
> > Sagittal spellings from the standard 34 and 46 symbol sets that
are
> > valid for both 34 and 46. The standard symbol set for 46 is:
> > 46: /| /|\ (|) ||\ /||\
> > Although (|) doesn't occur in the 34 set, it's equivalent to /|\
in
> > 34, so the spellings using (|), though not standard, are valid
for
> > 34, so you can make a mental substitution of /|\ for (|) when
viewing
> > this for 34.
>
> I guess that explains where the (|) came from. It seems like )||(
would
> be good for 2 steps of 34, but I don't know what that works out to
be in
> 46-ET.

It's a good idea for the 34-ET notation to agree with the 17-ET
notation:
17: /|\ /||\
34: /| /|\ ||\ /||\

One rule we have in Sagittal for assigning standard symbols to EDO's
is: if /||\ (an apotome) is an even number of degrees and /|\ (32:33)
is valid for half that number, then the /|\ symbol assignment (as a
semisharp) is automatic.

But if you're asking whether )||( is valid for 2deg34, the answer is
yes, as is //|, its apotome complement, which is simpler. In 46, //|
is 2deg and )||( is 3deg.

--George

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> --- In tuning@yahoogroups.com, Herman Miller <hmiller@> wrote:
> >
> > George D. Secor wrote:
> > > The fundamental problem here is that I've labeled this as
> > > a "Pajara Keyboard". While it is indeed intended for pajara
> > > tunings (most notably 22, 34, and 56), it's also excellent
> > > for some other divisions that aren't as well suited to pajara,
> > > particularly 46 and 58. So perhaps I should call this a
> > > pajara-plus keyboard and point out that the notation for 46
> > > and 58 don't agree completely with pajara.
> >
> > Makes sense.
>
> Not to me. What's the difference between keyboards? They're
> all just two-dimensional arrays of digitals.

Yes, that's true in the abstract. However, when that array is
translated into concrete columns and rows of keys, various parameters
must be set, which determine the physical properties of that array in
order to make it practical and convenient for the player.

This is how I go about designing a keyboard.

First I set the most basic parameters:
1) The angle between rows and columns;
2) The physical distance between the centers of adjacent keys in a
row;
3) The physical distance between the centers of adjacent keys in a
column.

Once the above parameters are set, then the following decisions are
made:
4) Rotate the array along the z axis by an angle that will make the
keyboard most convenient for the player (it's assumed that lower
pitches will be to the left; I believe that octaves of the same tone
should be laterally aligned);
5) Flip over the entire array, i.e., rotate along the lateral (x)
axis by 180 degrees, to see if the arrangement of pitches is more
convenient, and repeat if not;
6) Determine a practical physical shape for the keys, i.e., one that
will use the space efficiently, be aesthetically pleasing, and be
most convenient for the hand (i.e., will preferably have the y-
dimension larger than the x, or lateral, dimension; if necessary, go
back and redo all of the previous steps to arrive at a better key
shape);
7) Determine a meaningful pattern of key colors that will facilitate
navigation for multiple tunings;
8) Limit the keys to a finite number, according to the desired pitch
range (number of octaves) and number of pitches per octave.

> What's meant
> by "keyboard geometry"? The shapes of the digitals,

No.

> the choice
> of mapping vectors??

Yes and no.

The keyboard geometry depends on the period and (approximate)
generator shared by the tunings that the keyboard will accommodate.
For the Bosanquet geometry the period is 2 and the generator is ~3/2
(or ~4/3, flipped over), so the keyboard geometry depends only on the
mapping of primes 2 and 3; the mapping of primes >3 may vary,
depending on the exact (tempered) size of the period and generator.

Getting back to the difference between "pajara" vs. "pajara-plus" as
a name for the keyboard, the latter term is preferable, because the
combination of a 1/2-octave period and ~105-cent generator will
include tunings that are not considered to be good examples of pajara.

--George

George wrote...

> This is how I go about designing a keyboard.
>
> First I set the most basic parameters:
> 1) The angle between rows and columns;
> 2) The physical distance between the centers of adjacent keys in a
> row;
> 3) The physical distance between the centers of adjacent keys in a
> column.
>
> Once the above parameters are set, then the following decisions are
> made:
> 4) Rotate the array along the z axis by an angle that will make the
> keyboard most convenient for the player (it's assumed that lower
> pitches will be to the left; I believe that octaves of the same
> tone should be laterally aligned);
> 5) Flip over the entire array, i.e., rotate along the lateral (x)
> axis by 180 degrees, to see if the arrangement of pitches is more
> convenient, and repeat if not;
> 6) Determine a practical physical shape for the keys, i.e., one
> that will use the space efficiently, be aesthetically pleasing,
> and be most convenient for the hand (i.e., will preferably have
> the y-dimension larger than the x, or lateral, dimension; if
> necessary, go back and redo all of the previous steps to arrive
> at a better key shape);
> 7) Determine a meaningful pattern of key colors that will
> facilitate navigation for multiple tunings;
> 8) Limit the keys to a finite number, according to the desired
> pitch range (number of octaves) and number of pitches per octave.

Thanks. Those are all good steps.

> > What's meant
> > by "keyboard geometry"?
//
> > the choice
> > of mapping vectors??
>
> Yes and no.
>
> The keyboard geometry depends on the period and (approximate)
> generator shared by the tunings that the keyboard will
> accommodate.

Can you define "accommodate"?

> For the Bosanquet geometry the period is 2 and the generator is
> ~3/2 (or ~4/3, flipped over), so the keyboard geometry depends
> only on the mapping of primes 2 and 3; the mapping of primes >3
> may vary, depending on the exact (tempered) size of the period
> and generator.

Sure.

-Carl

Carl Lumma wrote:
> --- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>> George D. Secor wrote:
>>> The fundamental problem here is that I've labeled this as
>>> a "Pajara Keyboard". While it is indeed intended for pajara
>>> tunings (most notably 22, 34, and 56), it's also excellent
>>> for some other divisions that aren't as well suited to pajara,
>>> particularly 46 and 58. So perhaps I should call this a
>>> pajara-plus keyboard and point out that the notation for 46
>>> and 58 don't agree completely with pajara.
>> Makes sense.
> > Not to me. What's the difference between keyboards? They're
> all just two-dimensional arrays of digitals. What's meant
> by "keyboard geometry"? The shapes of the digitals, the choice
> of mapping vectors??

Well, my comment should be seen in the context of my previous question (why an 11-limit sagittal would be used on a pajara keyboard). But there are certainly differences between generalized keyboards; the Bosanquet and Wicki keyboards are based on intervals of fourths (or fifths), while the pajara-plus keyboard has a semitone generator with a period that divides the octave in half. You could transform a Wicki into a Fokker or a Bosanquet keyboard by sliding each row relative to the neighboring rows, but you couldn't get a Hanson keyboard, a lemba keyboard, or a pajara-plus keyboard that way without reassigning pitches.

More specifically, generalized keyboards usually repeat their pattern at the octave. So the choice of which pair of keys to use for the octave determines the overall design of the keyboard. You could retune a Bosanquet-style keyboard to use the pajara-plus mapping, but it would have a different set of notes in each octave.

Kraig Grady wrote:
> (I am coming in late to this subject having just returned from > Australia). One of the Hanson Keyboard works quite well for 34. Which > was one of the reason E. Wilson was so pleased with Hanson's design. > This is discussed in his Xenharmonikon Article on page 15 and 16.
> http://anaphoria.com/hanson.PDF and
> http://anaphoria.com/starr.PDF
> page 2 and 3 specifically
> Thanks, I wasn't familiar with the second link. With a different mapping (so-called "parakleismic"), the Hanson Keyboard could also be used for 42, 80, and 99-ET; it might even be adapted to myna temperament (formerly called "nonkleismic"), which gives you mappings for 27, 31, 35, 58, and 89. Those are the two most obvious ones, but I'm sure there are others.

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

> Discordant, yes. Except the highest-entropy intervals are
> near-semitones, which your tracks didn't have a lot of.

Yes you're right of course. However, I still find an
overly large discrepancy between the sheer bulk of
theoretical discordance and the actual results:

this tuning is pretty high in HE, if I'm not mistaken,

!
Majko, 12 tones from moderately lumpy 34
12
!
25/24
25/23
28/23
23/18
161/121
175/121
182/121
189/121
18/11
42/23
44/23
2/1

but the third song here, "Majko"

http://www.zebox.com/bobro

simply does not sound "discordant" to me, rather, "concordant at a
distance, and with itself". Others may disagree, of course.

I've been reading about HE on Joe Monzo's pages, and perhaps thanks
to Joe's annotations (no offence intended, it's a matter of a
jarring discrepancy between what Ehrlich seems to me at first read
to be saying and Joe's comments) I now can confidently say that I
don't understand what the hell it's all driving at. :-D

-Cameron Bobro

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > > The 3/2 ratio is readily perceivable because it
> > > is such a strong concordance, thus it has a low dip
> > > on the chart (a "minimum").
> >
> > The 600 cent half-octave is readily percievable
> > because it is such a strong dissonance, does it
> > have a low dip in the chart? Wouldn't it be a
> > fundamental mistake to equate ease of perception
> > with consonance?
>
> That's not what HE does. It equates the ease of
> fitting to a harmonic series to concordance.

Hmmmm.... if that's the fundamental idea, it's a damn good one,
as long as it doesn't claim that this "only" source of concordance,
something Joe Monzo says H.E. does not claim. Groovy.

This might mean, it seems to me, that I'm actually reducing HE in
my tunings, for they're full of intervals which are high in
HE vis a vis the "first family", but I deliberately try to
create cohesion by using intervals which are part of a
harmonic series on a kind of phantom tonic (the x/23s for
example), and the idea of an "overall timbre" of a tuning
really just means that certain partials are emphasized, which
may also lead, in context, to the feeling the intervals belonging
to the harmonic series of a phantom tonic.

-Cameron Bobro

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> George wrote...
> ...
> > The keyboard geometry depends on the period and (approximate)
> > generator shared by the tunings that the keyboard will
> > accommodate.
>
> Can you define "accommodate"?

Perhaps I should have said, "The keyboard geometry is defined by the
period and (approximate) generator of a set of tones." A keyboard will
accommodate a tuning if that tuning is capable of being mapped on that
keyboard. In other words, if all of the tones of a tuning may be
arranged with that period and approximate generator, then that tuning
is capable of being mapped on the keyboard using that geometry. (Kinda
redundant, huh?)

--George

Herman Miller wrote...
> But there
> are certainly differences between generalized keyboards; the
> Bosanquet and Wicki keyboards are based on intervals of
> fourths (or fifths), while the pajara-plus keyboard has a
> semitone generator with a period that divides the octave
> in half.

To me, these are mappings, not keyboards.

> More specifically, generalized keyboards usually repeat

This use of the term throws me. IIRC, Bosanquet still
called his 22-ET layout 'the generalized keyboard'. I
guess I should check that.

-Carl

> Kraig Grady wrote:
> > (I am coming in late to this subject having just returned from
> > Australia). One of the Hanson Keyboard works quite well for 34.
> > Which was one of the reason E. Wilson was so pleased with
> > Hanson's design. This is discussed in his Xenharmonikon Article
> > on page 15 and 16.
> > http://anaphoria.com/hanson.PDF and
> > http://anaphoria.com/starr.PDF
> > page 2 and 3 specifically
> >
>
> Thanks, I wasn't familiar with the second link. With a
> different mapping (so-called "parakleismic"), the Hanson ...

I suppose my use of the word "mapping" runs afoul of this
one (the 'regular temperaments' sense).

-Carl

> > > > The 3/2 ratio is readily perceivable because it
> > > > is such a strong concordance, thus it has a low dip
> > > > on the chart (a "minimum").
> > >
> > > The 600 cent half-octave is readily percievable
> > > because it is such a strong dissonance, does it
> > > have a low dip in the chart? Wouldn't it be a
> > > fundamental mistake to equate ease of perception
> > > with consonance?
> >
> > That's not what HE does. It equates the ease of
> > fitting to a harmonic series to concordance.
>
> Hmmmm.... if that's the fundamental idea, it's a damn good
> one, as long as it doesn't claim that this "only" source of
> concordance, something Joe Monzo says H.E. does not claim.
> Groovy.

Paul Erlich likes to say it's the simplest possible model
of concordance. So maybe it's more like a view of many
sources than a single source itself.

-Carl

George wrote...
> > > The keyboard geometry depends on the period and (approximate)
> > > generator shared by the tunings that the keyboard will
> > > accommodate.
> >
> > Can you define "accommodate"?
>
> Perhaps I should have said, "The keyboard geometry is defined
> by the period and (approximate) generator of a set of tones."
> A keyboard will accommodate a tuning if that tuning is capable
> of being mapped on that keyboard. In other words, if all of
> the tones of a tuning may be arranged with that period and
> approximate generator, then that tuning is capable of being
> mapped on the keyboard using that geometry. (Kinda
> redundant, huh?)

Since generalized keyboards are regular tilings, as long as
an interval isn't a period of the scale, it can 'generate'
the keyboard layout. So what special flag marks the generator
as the generator?

I think it would be more fruitful to classify keyboard
layouts in terms of how the first n primes are mapped
relative to 1. Say, a list showing the order of primes
as a ray goes from 1 to 3 and then clockwise 360deg.
Distances (straight-line from 1) to each prime could also
be given under the assumption that, say, the keyboard is
a Cartesian grid (perfectly square). [I'm sure this could
be refined...]

-Carl

Carl Lumma wrote:
> Herman Miller wrote...
>> But there >> are certainly differences between generalized keyboards; the
>> Bosanquet and Wicki keyboards are based on intervals of
>> fourths (or fifths), while the pajara-plus keyboard has a
>> semitone generator with a period that divides the octave
>> in half.
> > To me, these are mappings, not keyboards.

You could consider it a partial mapping (e.g., a 3-limit mapping in the case of Bosanquet-type layouts). Where the 5 and 7 are mapped vary from one temperament to another, but the keyboard layout is still in a sense the same.

The big difference between keyboard layouts, or mappings, or whatever you want to call them, is where the octave gets mapped; this determines which temperaments can be comfortably accommodated by the layout, and which ones end up being awkward. In some ways the main difference between keyboard layouts is how many periods are in an octave. So you could, in theory, use a Bosanquet, Fokker, Wicki, or Hanson keyboard for any temperament with a single period to the octave, but if you do that with a pajara or lemba keyboard, you'll end up not using half the keys.

Once you've set the period, the generator is one of the two keys closest to the line connecting the two ends of the period (any other position will leave keys unused). So in theory you could map any temperament to any keyboard layout having the right number of periods per octave. But try tuning George Secor's pajara keyboard to lemba temperament, and you'll find it's an awkward fit. 5/4 ends up to the right of 3/2, and 4/3 is about midway between 5/4 and 2/1. The 10-note lemba MOS requires you to skip back and forth between keys labeled C E\! C||\ F D F||\ B\!!/ G B\! A!!/ C.

So there's definitely a sense in which, even though one keyboard layout is mathematically as good as any other, the physical arrangement of the keys is more suitable for some temperaments than others.

> > Discordant, yes. Except the highest-entropy intervals are
> > near-semitones, which your tracks didn't have a lot of.
>
> Yes you're right of course. However, I still find an
> overly large discrepancy between the sheer bulk of
> theoretical discordance and the actual results:
>
> this tuning is pretty high in HE, if I'm not mistaken,
>
> !
> Majko, 12 tones from moderately lumpy 34
> 12
> !
> 25/24
> 25/23
> 28/23
> 23/18
> 161/121
> 175/121
> 182/121
> 189/121
> 18/11
> 42/23
> 44/23
> 2/1
>
> but the third song here, "Majko"
>
> http://www.zebox.com/bobro
>
> simply does not sound "discordant" to me, rather, "concordant at a
> distance, and with itself". Others may disagree, of course.

It's a nice piece, by the way. The synth patch has a slow
attack and tremolo, both of which contribute to making the
pitches harder to hear accurately. The writing is mostly
two-part, and is not especially concordant. The vocals do
not sound like they're singing in the same scale as the
synth, but this isn't offensive per se.

> I've been reading about HE on Joe Monzo's pages, and perhaps
> thanks to Joe's annotations (no offence intended, it's a matter
> of a jarring discrepancy between what Ehrlich seems to me at
> first read to be saying and Joe's comments)

Erlich himself noticed that too, I believe.

> I now can confidently say that I don't understand what the
> hell it's all driving at. :-D

Have you read:

http://www.soundofindia.com/showarticle.asp?in_article_id=1905806937

?

-Carl

Herman wrote...
> Once you've set the period, the generator is one of the two
> keys closest to the line connecting the two ends of the
> period (any other position will leave keys unused).

I know it takes time, but could you illustrate this?

> So in theory you could map any temperament to any keyboard
> layout having the right number of periods per octave. But
> try tuning George Secor's pajara keyboard to lemba temperament,
> and you'll find it's an awkward fit. 5/4 ends up to the right
> of 3/2, and 4/3 is about midway between 5/4 and 2/1. The
> 10-note lemba MOS requires you to skip back and forth between
> keys labeled C E\! C||\ F D F||\ B\!!/ G B\! A!!/ C.

Here's the link again:
http://tinyurl.com/26gll4

Can you give that in ET degrees? I'm not good at
translating ASCII to regular Sagittal.

Alternating to get a scale ain't necessarily bad. Guitarists
do it.

> So there's definitely a sense in which, even though one
> keyboard layout is mathematically as good as any other, the
> physical arrangement of the keys is more suitable for some
> temperaments than others.

Under some assumption about what things you're trying to
make easy to play, sure. If the method used to find the
temperament's period and generator is any good, I believe,
sans any such assumption, the best keyboard layout would
simply minimize those distances.

-Carl

Posted by: "Carl Lumma"

Erlich himself noticed that too, I believe.

> I now can confidently say that I don't understand what the
> hell it's all driving at. :-D

Have you read:

http://www.soundofindia.com/showarticle.asp?in_article_id=1905806937 <http://www.soundofindia.com/showarticle.asp?in_article_id=1905806937>

Helmholtz has a similar chart on page 193 of the dover edition with different results. Based on the concept of difference tones.
--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

Cameron~
The interval you mentioned before are found on many folk instruments around the world including Europe. I agree that they are consonant in context. Javanese music is one example that would fall dead without such jewels. Anyway roughness of intervals will keep a music going. One only need consonance if one is concerned with strong cadences.
Ambiguous intervals can be useful if say you have only one pipe with only 7 notes that you want to express varied emotions. Art is always finding ways to use things science is throwing out, on the other hand it often excepts everything and has to be nudged back. a funny game between the two.
--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

On keyboard mappings one might want to check out
page 13 of http://anaphoria.com/tres.PDF
it is the grid for mapping scale of any interval of equivalence (usually but need not be limited to the octave) with any generator in terms of steps. Also good for remapping the same scale over different layouts.
It is this pattern that lead Erv to fixate on the Co-prime pattern which he attributes to Novaro.
--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > > Discordant, yes. Except the highest-entropy intervals are
> > > near-semitones, which your tracks didn't have a lot of.
> >
> > Yes you're right of course. However, I still find an
> > overly large discrepancy between the sheer bulk of
> > theoretical discordance and the actual results:
> >
> > this tuning is pretty high in HE, if I'm not mistaken,
> >
> > !
> > Majko, 12 tones from moderately lumpy 34
> > 12
> > !
> > 25/24
> > 25/23
> > 28/23
> > 23/18
> > 161/121
> > 175/121
> > 182/121
> > 189/121
> > 18/11
> > 42/23
> > 44/23
> > 2/1
> >
> > but the third song here, "Majko"
> >
> > http://www.zebox.com/bobro
> >
> > simply does not sound "discordant" to me, rather, "concordant at
a
> > distance, and with itself". Others may disagree, of course.
>
> It's a nice piece, by the way.

Thanks- I'm taking a little local informal survey as to how
concordant it subjectively sounds.

>The synth patch has a slow
> attack and tremolo, both of which contribute to making the
> pitches harder to hear accurately.

Yes- measurably not nearly as smeared and wobbly as your typical
string section, though.

>The writing is mostly
> two-part, and is not especially concordant.

Expcet for 4 seconds of diads, it's three-part throughout,
deliberately shying away from octaves, with some four-voice chords.
Maybe you're not hearing the bass if you're listening at a computer.
I must admit that although I have great confidence in the
cohesiveness of tuning as I do, I'm surprised and unable to explain
the amount of JI-stylee "disappearing" in the chords, for upon
listening now, even with the bass loud and clear the overall effect
is definitely 2-part as you say.

> The vocals do
> not sound like they're singing in the same scale as the
> synth, but this isn't offensive per se.

You're right, they're in the same tuning but not the same scale,
what I call "true" harmony. A friend of mine can do
this effortlessly, from singing Balkan folk music all her life,
I'll get her to sing on my next project.

>
> > I've been reading about HE on Joe Monzo's pages, and perhaps
> > thanks to Joe's annotations (no offence intended, it's a matter
> > of a jarring discrepancy between what Ehrlich seems to me at
> > first read to be saying and Joe's comments)
>
> Erlich himself noticed that too, I believe.

I'm sure Joe can take some good-natured ribbing, and of course
once someone puts a theory out, it's open to interpretation
by anybody, and interpretations have to be tested against the
theory itself, not against the author's own interpretation.

> Have you read:
>
> http://www.soundofindia.com/showarticle.asp?
>in_article_id=1905806937

That's the clearest thing I've read yet, and seems very reasonable.
One thing I believe is true, from personal experience: once you've
established a harmonic series, even if the tonic is a phantom, you
can add to it, going up out of the range we could reasonably claim
to be audible, and have intervals which are far more sonant than
they "should" be in terms of naked concordance. I believe this
happens in Warren Burt's and Kraig Grady's tunings.

-Cameron Bobro

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> Cameron~
> The interval you mentioned before are found on many folk
>instruments
> around the world including Europe.

Yes, they're familiar to me from earliest memory ("no hippy music in
the house!" :-) )

>I agree that they are consonant in
> context.
>Javanese music is one example that would fall dead >without such
> jewels.

Maybe their character is exceptionally timbre and harmony dependant-
I noticed with amusment,checking out the material at maqam-world,
that I use a great deal of scalar material and intervals that fit
right in but not once has anyone ever noticed, because of timbre and
harmony I assume.

> Anyway roughness of intervals will keep a music going. One only
> need consonance if one is concerned with strong cadences.
> Ambiguous intervals can be useful if say you have only one pipe
>with
> only 7 notes that you want to express varied emotions.

I find major and minor thirds incapable of emodying certain things
like a median third does, for example a kind of far-away softness.
The finest example of this I've heard was in a folksong from
Azerbajzhan, which I've not been able to track down since.

>Art is always
> finding ways to use things science is throwing out, on the other
>hand it
> often excepts everything and has to be nudged back. a funny game
>between
> the two.

A lot of the "science" in music and other arts is, in my opinion, a
lot like a cargo cult. Which can be a good thing.

-Cameron Bobro

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:

> Helmholtz has a similar chart on page 193 of the dover edition with
> different results. Based on the concept of difference tones.

I'll have to check that out. As far as difference tones, some of
these "HE maxima" seem very elegant to me- 24/23 and 28/23 have
normalized difference tones of 32/23 and 40/23, which are above 28/23
by 8/7 and seperated by 5/4.

-Cameron Bobro

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> --- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@> wrote:
>
> > Helmholtz has a similar chart on page 193 of the dover edition
with
> > different results. Based on the concept of difference tones.
>
> I'll have to check that out. As far as difference tones, some of
> these "HE maxima" seem very elegant to me- 24/23 and 28/23 have
> normalized difference tones of 32/23 and 40/23, which are above
28/23
> by 8/7 and seperated by 5/4.

Oh- I forgot to note that George's 116.7 cent second, which makes a
really nice m9 and -M7, generates a normalized difference tone exactly
24/23 away from itself. :-)

Hi Cameron and Carl,

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

> > [Cameron:]
> > I've been reading about HE on Joe Monzo's pages, and perhaps
> > thanks to Joe's annotations (no offence intended, it's a matter
> > of a jarring discrepancy between what Ehrlich seems to me at
> > first read to be saying and Joe's comments)
>
> Erlich himself noticed that too, I believe.

Guys, please note that when i first gave the link
to my HE webpage in this thread, i wrote this about it:

/tuning/topicId_72554.html#72607

>> "however, note that Paul has pretty much deprecated it
>> at this point ... it was really just me trying to
>> understand his ideas from discussion with him, back
>> when he first developed the concept."

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

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>
> Mention of a keyboard layout reminded me of something. Here's my
> Pajara keyboard design, with the keys labeled for 34 & 46:
> /tuning-
> math/files/secor/kbds/KbPaj34-46.gif
> Since it's not clear what color arrangement would be most
meaningful
> in the all of the divisions that this keyboard will handle, the key
> colors in the outer rows are only tentative.

I've finalized the colors, and 3 new files are here:
/tuning-math/files/secor/kbds/

KbPaj34-56.gif is labeled for the 34 & 56 divisions, and the notation
is valid for pajara in general.

KbPaj34-46.gif and KbPaj46-58.gif are labeled with symbols valid for
both divisions in each filename.

The outer rows are colored in order to facilitate identification of
keys assigned to the same pitches as the central rows.

For pajara divisions (34 & 56), the duplicates are identified
according to light-dark patterns: red is dark, pink is light, and
gray is either light or dark according to context (i.e., lighter than
red, but darker than pink). For 22, this also applies, although it's
much simpler to observe that duplicate keys are aligned along the y-
axis.

For non-pajara divisions (46 & 58), the duplicates are identified
according to color: almost all of the red keys are duplicates of pink
keys, and almost all of the gray keys are duplicates of other gray
keys.

--George

>> > I now can confidently say that I don't understand what the
>> > hell it's all driving at. :-D
>>
>> Have you read: [the Sounds of India article ?]
>
> Helmholtz has a similar chart on page 193 of the dover edition with
> different results. Based on the concept of difference tones.

Psychoacoustics is repleat with such charts, since they
can be produced by considering beating of harmonic partials,
difference tones, or harmonic entropy. However, I believe
harmonic entropy is the best single-stop for a chart of
this kind.

-Carl

> > The vocals do
> > not sound like they're singing in the same scale as the
> > synth, but this isn't offensive per se.
>
> You're right, they're in the same tuning but not the same scale,

Can you explain the distinction?

-Carl

> I've finalized the colors, and 3 new files are here:
> /tuning-math/files/secor/kbds/
>
> KbPaj34-56.gif is labeled for the 34 & 56 divisions, and the
> notation is valid for pajara in general.
>
> KbPaj34-46.gif and KbPaj46-58.gif are labeled with symbols
> valid for both divisions in each filename.

What, no 22?

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > I've finalized the colors, and 3 new files are here:
> > /tuning-math/files/secor/kbds/
> >
> > KbPaj34-56.gif is labeled for the 34 & 56 divisions, and the
> > notation is valid for pajara in general.
> >
> > KbPaj34-46.gif and KbPaj46-58.gif are labeled with symbols
> > valid for both divisions in each filename.
>
> What, no 22?

Using KbPaj34-56.gif, simply calculate the difference between deg56 (at
the very bottom of each key) and deg34 (at the very top) to get deg22.

--George

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> Carl Lumma wrote:
> > Herman Miller wrote...
> >> But there
> >> are certainly differences between generalized keyboards; the
> >> Bosanquet and Wicki keyboards are based on intervals of
> >> fourths (or fifths), while the pajara-plus keyboard has a
> >> semitone generator with a period that divides the octave
> >> in half.
> >
> > To me, these are mappings, not keyboards.
>
> You could consider it a partial mapping (e.g., a 3-limit mapping in
the
> case of Bosanquet-type layouts). Where the 5 and 7 are mapped vary
from
> one temperament to another, but the keyboard layout is still in a
sense
> the same.

This is because the period is the same and the generator is in the
same ballpark. I've programmed 9-ET on the GK Scalatron, which has
very narrow fifths. It worked okay, although the layout was rather
weird: one of the descending intervals occurring between adjacent
rows of keys had to be played by going to the right.

> The big difference between keyboard layouts, or mappings, or
whatever
> you want to call them, is where the octave gets mapped; this
determines
> which temperaments can be comfortably accommodated by the layout,
and
> which ones end up being awkward. In some ways the main difference
> between keyboard layouts is how many periods are in an octave. So
you
> could, in theory, use a Bosanquet, Fokker, Wicki, or Hanson
keyboard for
> any temperament with a single period to the octave, but if you do
that
> with a pajara or lemba keyboard, you'll end up not using half the
keys.
>
> Once you've set the period, the generator is one of the two keys
closest
> to the line connecting the two ends of the period (any other
position
> will leave keys unused). So in theory you could map any temperament
to
> any keyboard layout having the right number of periods per octave.
But
> try tuning George Secor's pajara keyboard to lemba temperament, and
> you'll find it's an awkward fit. 5/4 ends up to the right of 3/2,
and
> 4/3 is about midway between 5/4 and 2/1. The 10-note lemba MOS
requires
> you to skip back and forth between keys labeled C E\! C||\ F D F||\
> B\!!/ G B\! A!!/ C.

Another example of an awkward fit is that if you attempt to map
Miracle onto the keys of a Bosanquet keyboard, it will not only
require a greater octave distance, but the octaves will go off at an
angle and start to disappear off the edge of the keyboard. Even if
you have a huge number of keys/octave, having to play along an angle
as one goes up and down the range of the keyboard is going to be
uncomfortable enough that I expect you'd wind up with a backache.

> So there's definitely a sense in which, even though one keyboard
layout
> is mathematically as good as any other, the physical arrangement of
the
> keys is more suitable for some temperaments than others.

Yes.

--George

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> George wrote...
> > > > The keyboard geometry depends on the period and (approximate)
> > > > generator shared by the tunings that the keyboard will
> > > > accommodate.
> > >
> > > Can you define "accommodate"?
> >
> > Perhaps I should have said, "The keyboard geometry is defined
> > by the period and (approximate) generator of a set of tones."
> > A keyboard will accommodate a tuning if that tuning is capable
> > of being mapped on that keyboard. In other words, if all of
> > the tones of a tuning may be arranged with that period and
> > approximate generator, then that tuning is capable of being
> > mapped on the keyboard using that geometry. (Kinda
> > redundant, huh?)
>
> Since generalized keyboards are regular tilings, as long as
> an interval isn't a period of the scale, it can 'generate'
> the keyboard layout. So what special flag marks the generator
> as the generator?

Great question! I can think of two ways.

1) To give you the most "generalized" answer possible, let's say that
you're starting out with a two-dimensional (square) grid of
indefinite size, having adjacent tones separated by one particular
interval along one direction of the grid and another interval in the
perpendicular direction. Your choice of intervals, along with the
period, is going to determine the outcome. These intervals don't
have to be consonances; they could be small intervals that occur as
scale steps in some scale subset of a tuning, e.g., a major 2nd and
minor 2nd. (You would want small intervals between adjacent keys of
a keyboard, wouldn't you?) Now, to restate your question, is either
of these intervals, or some other interval, the generator for these
tones? Let's find out.

Extend the grid sufficiently so that you'll find two tones separated
by the interval of a period (octave, 1/2-octave, etc.), and if
there's more than one grid vector having this property, then choose
one. Draw a line connecting the two points in the grid representing
a period. Choose one of these points as the origin or "tonic".

Imagine that lines, parallel to your original line, are drawn through
each of the other points on the grid. Identify those points for
which these imaginary lines are nearest the original line. The
interval between one of these points (choose one) and the origin is
your generator. Which of these points to choose, you ask? It
doesn't matter; it's your choice, and that's mostly a matter of how
you want to conceptualize it. For example, is the meantone generator
a 5th or a 4th? Is the pajara generator ~105c, or ~490c, or ~710c,
or ~1095c? When you perform period-reduction, these generators will
generate the same tones.

2) Or, if you like, you can do it this way. Choose two different
octave divisions (one not a subset of the other) that you would like
to have on a single keyboard. Determine whether they both contain
one or more tones 1/n-octave, to see whether the period will be a
fraction of an octave. Set tone # zero (the origin) to the same
pitch in both divisions and find the smallest difference (absolute
value >0) in pitch between the two divisions. The interval between
these tones and the origin is the generator.

> I think it would be more fruitful to classify keyboard
> layouts in terms of how the first n primes are mapped
> relative to 1. Say, a list showing the order of primes
> as a ray goes from 1 to 3 and then clockwise 360deg.
> Distances (straight-line from 1) to each prime could also
> be given under the assumption that, say, the keyboard is
> a Cartesian grid (perfectly square). [I'm sure this could
> be refined...]

I think not, so all I can say is good luck! :-)

--George

> > George wrote...
> > > A keyboard will accommodate a tuning if that tuning is capable
> > > of being mapped on that keyboard. In other words, if all of
> > > the tones of a tuning may be arranged with that period and
> > > approximate generator, then that tuning is capable of being
> > > mapped on the keyboard using that geometry.

> > Since generalized keyboards are regular tilings, as long as
> > an interval isn't a period of the scale, it can 'generate'
> > the keyboard layout. So what special flag marks the generator
> > as the generator?

> Great question! I can think of two ways.
>
> 1) To give you the most "generalized" answer possible, let's say
> that you're starting out with a two-dimensional (square) grid of
> indefinite size, having adjacent tones separated by one
> particular interval along one direction of the grid and another
> interval in the perpendicular direction. Your choice of
> intervals, along with the period, is going to determine the
> outcome. These intervals don't have to be consonances; they
> could be small intervals that occur as scale steps in some scale
> subset of a tuning, e.g., a major 2nd and minor 2nd. (You would
> want small intervals between adjacent keys of a keyboard,
> wouldn't you?) Now, to restate your question, is either of
> these intervals, or some other interval, the generator for these
> tones? Let's find out.
>
> Extend the grid sufficiently so that you'll find two tones
> separated by the interval of a period (octave, 1/2-octave,
> etc.), and if there's more than one grid vector having this
> property, then choose one. Draw a line connecting the two
> points in the grid representing a period. Choose one of these
> points as the origin or "tonic".
>
> Imagine that lines, parallel to your original line, are drawn
> through each of the other points on the grid. Identify those
> points for which these imaginary lines are nearest the original
> line. The interval between one of these points (choose one)
> and the origin is your generator. Which of these points to
> choose, you ask? It doesn't matter; it's your choice, and
> that's mostly a matter of how you want to conceptualize it.
> For example, is the meantone generator a 5th or a 4th?
> Is the pajara generator ~105c, or ~490c, or ~710c, or ~1095c?
> When you perform period-reduction, these generators will
> generate the same tones.

Fine, but I don't see how this gets us any closer to a
definition of "keyboard". What test can I perform to tell
of two layouts in front of me are the same, or are two
different, keyboards?

By the way, I would not want to assume monotonically
ascending pitches.

-Carl

Carl Lumma wrote:
> Herman wrote...
>> Once you've set the period, the generator is one of the two
>> keys closest to the line connecting the two ends of the
>> period (any other position will leave keys unused).
> > I know it takes time, but could you illustrate this?

On the pajara-plus keyboard, the period is the interval from C to the black key between F and G (17 of 34-ET, or 23 or 46-ET). If you draw a line between them, the black key to the right of C and the F key are the ones that come closest to the line. So either of these could be taken as the generator.

I also have a lemba keyboard illustration, which I've labeled with some of the just intervals that are approximated by lemba temperament (of course, any of these can be adjusted by one or more of the different commas tempered out by lemba).

http://www.prismnet.com/~hmiller/png/lemba-keyboard.png

The period is the interval from 1/1 to 10/7 (which could also have been labeled 7/5). If you draw a line through these keys, the 8/7 key and the 5/4 key are closest. If you were to use this same layout with the 5/3 key identified as the period (relative to 1/1), then the 5/4 or the 4/3 key would be the generator.

If you had a keyboard with this lemba pattern repeated over several octaves, you could (for instance) use the 7/4 key as an octave above 1/1, and have a pretty good layout for meantone with the 5/4 key as the generator. But each successive octave takes you closer to the edge of the keyboard.

>> So in theory you could map any temperament to any keyboard
>> layout having the right number of periods per octave. But
>> try tuning George Secor's pajara keyboard to lemba temperament,
>> and you'll find it's an awkward fit. 5/4 ends up to the right
>> of 3/2, and 4/3 is about midway between 5/4 and 2/1. The
>> 10-note lemba MOS requires you to skip back and forth between
>> keys labeled C E\! C||\ F D F||\ B\!!/ G B\! A!!/ C.
> > Here's the link again:
> http://tinyurl.com/26gll4
> > Can you give that in ET degrees? I'm not good at
> translating ASCII to regular Sagittal.

In degrees of 34-et, that's 0 11 3 14 6 17 28 20 31 23 34. (The scale in 26-ET is 0 3 5 8 10 13 16 18 21 23 26.)

> Alternating to get a scale ain't necessarily bad. Guitarists
> do it.

Within limits, alternating can be acceptable, but that's a bit of a stretch to get lemba onto a pajara keyboard. Actually, if the octaves are mapped vertically, alternating left and right can end up with some nice fingering patterns. It depends on the circumstances of the temperament and the music.

>> So there's definitely a sense in which, even though one
>> keyboard layout is mathematically as good as any other, the
>> physical arrangement of the keys is more suitable for some
>> temperaments than others.
> > Under some assumption about what things you're trying to
> make easy to play, sure. If the method used to find the
> temperament's period and generator is any good, I believe,
> sans any such assumption, the best keyboard layout would
> simply minimize those distances.

There could always be conflicting requirements (especially if you're trying to represent more than one temperament). Generally, a good layout is one that allows you to easily pick out consonant chords (however you define those) and commonly used melodic steps. E.g., major triads are more convenient on the Bosanquet keyboard than the Wicki keyboard if you're using a schismatic temperament (reaching over from D to Gb is a bit of a leap in the dark if you're not familiar with the exact distance).

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > > The vocals do
> > > not sound like they're singing in the same scale as the
> > > synth, but this isn't offensive per se.
> >
> > You're right, they're in the same tuning but not the same scale,
>
> Can you explain the distinction?

12-tET would be a tuning, C major a scale. If you harmonize C Major
with say Eb major, you'd have one voice in the same tuning but not
the same scale. In a rational tuning, or in this tuning with its
couple of tonics, phantom and concrete, "bitonality",
where "tonality" is a function of acoustics not of a specific
tradition, would be a matter of debate, IMO. In this case, scales
abstracted from the synth voices would be different from each other
and different from the vocal part, but all tones would be in the
tuning (and all 12 tones of the tuning are used in the piece).

-Cameron Bobro

> >> Once you've set the period, the generator is one of the two
> >> keys closest to the line connecting the two ends of the
> >> period (any other position will leave keys unused).
> >
> > I know it takes time, but could you illustrate this?
>
> On the pajara-plus keyboard, the period is the interval from
> C to the black key between F and G (17 of 34-ET, or 23 or 46-ET).

OK.

> If you draw a line between them, the black key to the right
> of C and the F key are the ones that come closest to the line.
> So either of these could be taken as
> the generator.

Using degrees of 34 and drawing the line between the plus
signs on the keys, it looks like the closest keys to the
line look like 3 & 14 to me. The black key to the right of
F is the key I'm drawing the line to!

Are you saying if I use a generator of 7 degrees (2 forward,
1 up) and snap back to the origin every time I pass 34, I
won't be able to complete the keyboard?

> >> try tuning George Secor's pajara keyboard to lemba temperament,
> >> and you'll find it's an awkward fit. 5/4 ends up to the right
> >> of 3/2, and 4/3 is about midway between 5/4 and 2/1. The
> >> 10-note lemba MOS requires you to skip back and forth between
> >> keys labeled C E\! C||\ F D F||\ B\!!/ G B\! A!!/ C.
> >
> > Here's the link again:
> > http://tinyurl.com/26gll4
> >
> > Can you give that in ET degrees? I'm not good at
> > translating ASCII to regular Sagittal.
>
> In degrees of 34-et, that's 0 11 3 14 6 17 28 20 31 23 34.

Maybe I'm being dense, but I don't get it. I assume this
is ascending in lemba. How are you mapping it to 34?

> major triads are more
> convenient on the Bosanquet keyboard than the Wicki keyboard if
> you're using a schismatic temperament

So how would one decode this... by "Bosanquet keyboard", you
mean something with best 3:2s at Forward 4, Down 1? I'm not
sure how to describe Wicki.

-Carl

Carl Lumma wrote:
>> If you draw a line between them, the black key to the right
>> of C and the F key are the ones that come closest to the line.
>> So either of these could be taken as >> the generator.
> > Using degrees of 34 and drawing the line between the plus
> signs on the keys, it looks like the closest keys to the
> line look like 3 & 14 to me. The black key to the right of
> F is the key I'm drawing the line to!

"The black key to the right of C" (3 of 34-ET) is one key; "the F key" (14 of 34-ET) is the other.

> Are you saying if I use a generator of 7 degrees (2 forward,
> 1 up) and snap back to the origin every time I pass 34, I
> won't be able to complete the keyboard?

If you're snapping back to the origin, then it's not a uniform mapping. But the thing I was trying to point out is just that if you use 7 degrees as your generator (or anything other than 3 or 14, within that period), many of the notes of the keyboard will be unused. On the other hand, if you want a generator of 7 degrees using this keyboard layout, you could take the key that currently has 3 and map it to 7 instead.

>> In degrees of 34-et, that's 0 11 3 14 6 17 28 20 31 23 34.
> > Maybe I'm being dense, but I don't get it. I assume this
> is ascending in lemba. How are you mapping it to 34?

I'm mapping it to the pajara-plus keyboard; 34 is only used as a reference so you can find which keys I'm talking about. As I mentioned, the actual scale this represents is 0 3 5 8 10 13 16 18 21 23 26 in 26-ET.

I've added degrees of 26-ET to the lemba-keyboard chart so you can see how this comes out on the lemba keyboard.
http://www.io.com/~hmiller/png/lemba-keyboard.png

>> major triads are more
>> convenient on the Bosanquet keyboard than the Wicki keyboard if >> you're using a schismatic temperament
> > So how would one decode this... by "Bosanquet keyboard", you
> mean something with best 3:2s at Forward 4, Down 1? I'm not
> sure how to describe Wicki.

If you're describing 3/2 on the Bosanquet keyboard as forward 4, down 1, the 5/4 is at forward 3, down 2, a convenient reach. The Wicki layout puts the 3/2 in a more convenient location (diagonally up and to the right), leaving the meantone 5/4 at two keys over to the right, but the schismatic 5/4 is two rows straight up and four keys over to the left.

Bosanquet Wicki
|*C | D | E | F#| G#| A#| |*Fb| Gb| Ab| Bb| C | D | E | F# |
| | Db| Eb| F |*G | A | B | | Cb| Db| Eb| F |*G | A | B |
| | | |*Fb| Gb| Ab| Bb| C | | Fb| Gb| Ab| Bb|*C | D | E | F# |

Other intermediate keyboard layouts are also possible, e.g.
| |*Fb| Gb| Ab| Bb| C | D | E |
| Cb| Db| Eb| F |*G | A | B |
| | Ab| Bb|*C | D | E | F# |

In a sense you could say that these three are all the same class of keyboard, and that the differences between them are trivial. But trivial differences add up to larger differences, and so I don't see a problem with identifying these as different keyboards (simply based on how the octaves are mapped, if for no other reason than the fact that practical keyboards are finite in size). On the other hand, the Fokker keyboard is similar enough to the Bosanquet that it could be considered as a minor variation (depending on your point of view). Possibly the Janko keyboard could be included as a member of the same general class.

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > > George wrote...
> > > > A keyboard will accommodate a tuning if that tuning is capable
> > > > of being mapped on that keyboard. In other words, if all of
> > > > the tones of a tuning may be arranged with that period and
> > > > approximate generator, then that tuning is capable of being
> > > > mapped on the keyboard using that geometry.
>
> > > Since generalized keyboards are regular tilings, as long as
> > > an interval isn't a period of the scale, it can 'generate'
> > > the keyboard layout. So what special flag marks the generator
> > > as the generator?
>
> > Great question! I can think of two ways. ...
>
> Fine, but I don't see how this gets us any closer to a
> definition of "keyboard". What test can I perform to tell
> of two layouts in front of me are the same, or are two
> different, keyboards?

I'll answer this as best I can, hoping that I can avoid circularity
in the definition.

The two keyboards are essentially the same if they're constructed
using both the same period and approximate generator.

So how can you determine the intended period and generator merely by
looking at a keyboard, or whether there even is any intended period
or generator?

I grant that this would be impossible if the keys were laid out in a
triangular or checkerboard lattice, with exactly lateral rows and all
keys the same color. This most generalized keyboard of all, however,
would probably not be the most convenient layout for most of the
tunings we're interested in, because, among other things, the octave
distance and direction would probably not be the most comfortable for
real-time performance.

My assumption is that the layout would enable one to deduce the
intended period or octave-vector and also the vector for at least one
other interval (e.g., a fifth), by observing such clues as the
pattern of key colors, size and shape of the keys, and occurrence of
laterally aligned keys.

I think that a keyboard design would not be very good if it didn't
provide some visual clues to help you locate the pitches.

--George

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:

> >> If you draw a line between them, the black key to the right
> >> of C and the F key are the ones that come closest to the line.
> >> So either of these could be taken as
> >> the generator.
> >
> > Using degrees of 34 and drawing the line between the plus
> > signs on the keys, it looks like the closest keys to the
> > line look like 3 & 14 to me. The black key to the right of
> > F is the key I'm drawing the line to!
>
> "The black key to the right of C" (3 of 34-ET) is one key;
> "the F key" (14 of 34-ET) is the other.

Gotcha. It looks like the original context has been trimmed,
but were you saying that any other generator (in terms of a
vector on the keyborad) will leave gaps?

> > Are you saying if I use a generator of 7 degrees (2 forward,
> > 1 up) and snap back to the origin every time I pass 34, I
> > won't be able to complete the keyboard?
>
> If you're snapping back to the origin, then it's not a uniform
> mapping.

It virtully a requirement of uniform periodic mappings. For
example, start at C and go to F, then repeating that '4 up,
1 down' motion to 28 (of 34), and then to 8/34, then to 22...
No?

> But the thing I was trying to point out is just that if you
> use 7 degrees as your generator (or anything other than 3 or
> 14, within that period), many of the notes of the keyboard
> will be unused. On the other hand, if you want a generator
> of 7 degrees using this keyboard layout, you could take the
> key that currently has 3 and map it to 7 instead.

I printed out pajara plus and started marking off the
'up 2, up 1' motion (to which 7deg is currently mapped), while
respecting period boundaries (of 34-ET), and it looks like I
get vertical bands that should cover the keyboard. No?

> >> In degrees of 34-et, that's 0 11 3 14 6 17 28 20 31 23 34.
> >
> > Maybe I'm being dense, but I don't get it. I assume this
> > is ascending in lemba. How are you mapping it to 34?
>
> I'm mapping it to the pajara-plus keyboard; 34 is only used
> as a reference so you can find which keys I'm talking about.
> As I mentioned, the actual scale this represents is
> 0 3 5 8 10 13 16 18 21 23 26 in 26-ET.
>
> I've added degrees of 26-ET to the lemba-keyboard chart so
> you can see how this comes out on the lemba keyboard.
> http://www.io.com/~hmiller/png/lemba-keyboard.png

I forget the context again here, but that's a cool
keyboard / scale.

> >> major triads are more
> >> convenient on the Bosanquet keyboard than the Wicki keyboard if
> >> you're using a schismatic temperament
> >
> > So how would one decode this... by "Bosanquet keyboard", you
> > mean something with best 3:2s at Forward 4, Down 1? I'm not
> > sure how to describe Wicki.
>
> If you're describing 3/2 on the Bosanquet keyboard as forward 4,
> down 1, the 5/4 is at forward 3, down 2, a convenient reach.

The 6/5 considerably less so.

> The Wicki layout puts the 3/2 in a more convenient location
> (diagonally up and to the right), leaving the meantone 5/4 at
> two keys over to the right, but the schismatic 5/4 is two rows
> straight up and four keys over to the left.
>
> Bosanquet Wicki
> |*C | D | E | F#| G#| A#| |*Fb| Gb| Ab| Bb| C | D | E | F# |
> | | Db| Eb| F |*G | A | B | | Cb| Db| Eb| F |*G | A | B |
> | | | |*Fb| Gb| Ab| Bb| C | | Fb| Gb| Ab| Bb|*C | D | E | F# |

I hope that's still diplaying corretly (I had to adjust
because of the quote characters).

Both triads have the same basic shape. I don't really
see the difference. It looks like 5-1-2 or even just 5-1
in the left hand. Slightly harder for the right, but
you could always mirror the keyborads (isn't Wicki meant
to be mirrored?).

> Other intermediate keyboard layouts are also possible, e.g.
> | |*Fb| Gb| Ab| Bb| C | D | E |
> | Cb| Db| Eb| F |*G | A | B |
> | | Ab| Bb|*C | D | E | F# |
>
> In a sense you could say that these three are all the same
> class of keyboard, and that the differences between them are
> trivial. But trivial differences add up to larger differences,
> and so I don't see a problem with identifying these as
> different keyboards (simply based on how the octaves are
> mapped, if for no other reason than the fact that practical
> keyboards are finite in size). On the other hand, the Fokker
> keyboard is similar enough to the Bosanquet that it could be
> considered as a minor variation (depending on your point
> of view). Possibly the Janko keyboard could be included as
> a member of the same general class.

I've always thought of the Bosanquet, Janko, and Fokker
keyboards as the same. Wicki I would probably call
different. It brings up an interesting qusetion, that I
was trying to raise, about how to classify these things.

Gene has a great page on this you've probably seen:

http://66.98.148.43/~xenharmo/bosanquet.html

Erv's 'Keyboard Schemata from the Scale Tree' is another
neat approach (see the Anaphoria site).

Then you've got the thing I suggested last time, with the
rays going around clockwise from 1/1-3/2, identifying
harmonics...

It would be great to have some discussion on this if
anyone has any ideas.

-Carl

Carl Lumma wrote:

> Gotcha. It looks like the original context has been trimmed,
> but were you saying that any other generator (in terms of a
> vector on the keyborad) will leave gaps?

Right, once you've decided where the period goes, the generator has to be one of the two keys closest to the line. Well, I haven't tried to figure it out mathematically, but that's the idea I get from looking at lots of keyboard charts (like the ones in the Wilson archives).

>>> Are you saying if I use a generator of 7 degrees (2 forward,
>>> 1 up) and snap back to the origin every time I pass 34, I
>>> won't be able to complete the keyboard?
>> If you're snapping back to the origin, then it's not a uniform
>> mapping.
> > It virtully a requirement of uniform periodic mappings. For
> example, start at C and go to F, then repeating that '4 up,
> 1 down' motion to 28 (of 34), and then to 8/34, then to 22...
> No?

It's like wrapping around the circle of fifths in 12-ET so that it ends up back at C instead of B#; it depends on the generator being a rational fraction of the period. If the generator is irrational (as in whatever the pajara equivalent of "golden meantone" would be), you won't ever get back to the origin. If you look at just the "upward" component of the generator (noting that the period can only get you horizontal motion), you can see that you'll keep going farther from the origin.

>> But the thing I was trying to point out is just that if you
>> use 7 degrees as your generator (or anything other than 3 or
>> 14, within that period), many of the notes of the keyboard
>> will be unused. On the other hand, if you want a generator
>> of 7 degrees using this keyboard layout, you could take the
>> key that currently has 3 and map it to 7 instead.
> > I printed out pajara plus and started marking off the
> 'up 2, up 1' motion (to which 7deg is currently mapped), while
> respecting period boundaries (of 34-ET), and it looks like I
> get vertical bands that should cover the keyboard. No?

There are other keys off the edge of the keyboard that would be labeled "34". I think what you're doing is probably the same as taking one of those keys above the chart as your period (in which case it would be possible for the 7 key to be the generator).

>> Bosanquet Wicki
>> |*C | D | E | F#| G#| A#| |*Fb| Gb| Ab| Bb| C | D | E | F# |
>> | | Db| Eb| F |*G | A | B | | Cb| Db| Eb| F |*G | A | B |
>> | | | |*Fb| Gb| Ab| Bb| C | | Fb| Gb| Ab| Bb|*C | D | E | F# |
> > I hope that's still diplaying corretly (I had to adjust
> because of the quote characters).
> > Both triads have the same basic shape. I don't really
> see the difference. It looks like 5-1-2 or even just 5-1
> in the left hand. Slightly harder for the right, but
> you could always mirror the keyborads (isn't Wicki meant
> to be mirrored?).

Well, on a QWERTY keyboard one would be TNK, and the other one MRK. Both thirds (MR, RK) would be a large leap on the Wicki (if you had it tuned to a schismatic temperament), while only the fifth (TK) is a good-sized leap on the Bosanquet. It makes a bigger difference if you're trying to play a melody.

> I've always thought of the Bosanquet, Janko, and Fokker
> keyboards as the same. Wicki I would probably call
> different. It brings up an interesting qusetion, that I
> was trying to raise, about how to classify these things.

Bosanquet and Fokker are both 3/7 keyboards in Wilson's terminology; Wicki is a 1/2. The Janko keyboard is an interesting case because it's designed for a specific temperament and essentially only has two rows. You could call it a 0/1 keyboard with a period of 1/6 octave and a generator of 1/12 octave, although it also supports the same fingering patterns as the 3/7 keyboards.

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
> I guess one should never underestimate the power of a lowly
> semitone. I'm still amazed (and amused) that, a few years back,
> somebody actually suggested naming one after me, and that the idea
> actually caught on (before I learned about it):
> /tuning/topicId_27028.html#27062
> The double irony in this is: 1) that I've never had any interest in
> using Miracle for anything (other than as the basis for a keyboard
> layout); and 2) that "somebody" to whom I referred is none other than
> Dave Keenan, who has since voiced his objection to labeling tunings
> with names that have nothing to do with their actual properties.

Hee hee. Don't you remember I weaseled on that by pointing out that
"secor" was a contraction of "SECond, minOR". ;-)

-- Dave Keemun