Why not perfect tuning?

Rather than try to devise a scheme which tries to reduce the average
error of tuning a tiny bit, why not have absolutely perfect tuning?
Why not actually play the notes at frequencies that are the ones that
you want and have the harmonic relations to each other that are desired?

This post may be taken as being a severe criticism of all ET schemes as,
having recognised the problem, they fall short in finding a solution.
Solutions to all the problems raised will be suggested and a grand
structure erected for all to criticise in turn and hopefully for some to
take on board and bring to fruition the full potential of the ideas.

In the days when instruments had to be pre-tuned and each extra note
available was an added expense and complication then compromises had to
be made. However today we can produce by electronic means any waveform
that we want and so there is no need to make compromises at all (unless
of course you happen to play a guitar or other fixed tuned instrument).
There is no additional cost to having an infinite variety of notes
available than the one time effort of programming a computer, a small
price to pay for producing music that speaks the composer's intentions
without any compromise at all.

There are a variety of factors that will come into such an effort.
Some people argue that slightly imperfect tuning adds a tension to a
chord whereas the perfect chord may sound a bit dull. This may be so,
but I would argue that this is a separate factor that should always be
controlled by choice rather than by chance and that there are additional
factors involved which should also be considered.

The most important additional factor is that of the beats between notes.
With any form of ET scale, no matter how many notes in it, the beats
between notes are not related to the key in a meaningful way but rather
are generally quite discordant. However with JI the beats are always
other notes of the scale, and if variations in frequency are to be added
for tension then consideration should be given to how the beats fall
with respect to the scale. This must be done for every case. As an
example, the 7th interval is quite ugly in normal 12-TET but in 8:15 JI
is a quite acceptable chord.

Another factor which should be considered at some point is the key of a
piece in relation to the tempo of the piece. Many great works of music
have a tempo (considered as an oscillation rate) which is "in tune" with
the key of the piece. Of course tempo varies in pieces so this may be
taken to mean the typical tempo. Also, variations in tempo will
themselves therefore tend to be by ratios which are musical ratios.

Again, to achieve the best results the frequencies used should be
related to the natural world, which includes the consequences on the
human body, the relationship to the building in which it is being played
and even to the whole earth resonances. This may seem fanciful, but in
the end the instrument that is being played is the human body and mind
and to play it most successfully requires understanding its origins and
resonances.

How then is such a grand scheme to be implemented? It will certainly
require an intelligent computer program between the keyboard and the
sound production, assuming that a keyboard is used. It might equally
apply to music played from a score as to keyboard input and perhaps it
would be most logical to consider it as a device between a MIDI stream
of notes and the actual sound production.

The general idea of this I call AJI for Automatic Just Intonation and I
discuss it at http://www.kcbbs.gen.nz/users/rtomes/aji-main.htm and
include several examples of the frequencies played for each note in the
example pieces. In essence the idea is to recognise that if the notes
played are, for example, C-E-G then the frequency ratios should be
exactly 4:5:6 and so to do that. I ignore for the moment the question
of deliberate tension requirements by slight variations in frequency
which can be addressed later.

The determination of the ratios of a particular chord is a relatively
easy matter, as see http://www.kcbbs.gen.nz/users/rtomes/aji-a.gif which
is a simple "slide rule" for determining for any combination of notes
what the underlying ratios that should be used are. Actually there are
a couple of complications, but this works in most cases. However a
piece of music is not a series of isolated chords but a whole which
flows along. As such, the previous notes and chords need to be
considered, particularly in some ambiguous cases where two or even more
possible harmonic intentions need to be considered.

I will suggest that there is a "natural tuning" which works best and
that it is not based on A=440 Hz but a slightly higher value of ~450 Hz.
I have come to this conclusion based on three quite separate pieces of
evidence which I will not go into here. There is some discussion at
http://www.kcbbs.gen.nz/users/rtomes/alex-ha.htm that touches on some of
these points. Tempo also will naturally follow to have certain discrete
values that are favoured in use.

The most common frequencies for the notes sould be these:

D A E
600 450 675

Bb F C G
480 360 540 405

The above values are equal to the Earth's Schumann resonance (the
natural earth electromagnetic oscillation) multiplied by the JI ratios
of 24:27:30:32:36:40:45 and are therefore natural strong resonances.

These are arranged so that there are exact ratios of 3 horizontally and
of 5 diagonally up to the right with all notes being reduced by ratios
of 2 to a common octave. Of course if we shift key by modulation then
we arrive at a new scale such as:

A E B
450 675 506.25

F C G D
360 540 405 607.5

And in so doing we discover that D has changed from 600 Hz to 607.5 Hz.
Such changes by a ratio of 80:81 are well known and we need to recognise
that in a single piece of music sometimes D means one of these and
sometimes the other. No compromise can deal with the fact that these
are two separate notes which are only confused because of our inadequate
notation. In Indian music these are clearly recognised as different
notes and played as such.

In playing a piece of music in the key of F say, if we play Bb-D-F then
D must be 600 Hz to harmonise properly while in the chord G-B-D it must
be 607.5 Hz.

I will close by mentioning that when exact harmonics are not played (for
reasons of "interest") then the beats should be designed to fit with the
scale and the rhythm. For example, suppose we wanted to play Bb-F in
the above tuning but wanted some tension then we would want to vary from
the exact 240 Hz and 360 Hz frequencies. In so doing a good variation
would be by 3.75 Hz as that (as the beats that will be generated) is in
tune with the notes themselves and hopefully also in keeping with the
rhythm which might be say 112.5 BPM or 1.875 Hz.

-- Ray Tomes -- http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm --
Cycles email list -- http://www.kcbbs.gen.nz/users/af/cyc.htm
Alexandria eGroup list -- http://www.kcbbs.gen.nz/users/af/alex.htm
Boundaries of Science http://www.kcbbs.gen.nz/users/af/scienceb.htm

The Schuman resonance is constantly shifting so unless you have some
detector hooked to your automatic intonation weegee the supposed resonance
will be more fanciful than actual.

There's no right and wrong in art, just "play", and things that make you go
"oooo" and things that leave you cold. If a composer writes in et then
playing in et is not a compromise. There's no such thing as "perfect
tuning", just various tunings that you like to compose in/play in or not.

dante

Rosati wrote:

> From: "Rosati" <dante@pop.interport.net>
>
> The Schuman resonance is constantly shifting so unless you have some
> detector hooked to your automatic intonation weegee the supposed resonance
> will be more fanciful than actual.

It seems that this resonance should be able to be measured and the performer
could do what Charlemane Palestine did. Start performances by adjusting his
base pitch to the resonance of the room except to the earth. Serialism is
worth exploring and might teach us something but natural resonance is some wee
gee hocus-pocus. Is it your academia that's showing.:)

>
>
> There's no right and wrong in art, just "play", and things that make you go
> "oooo" and things that leave you cold.

Then why have any tuning at all

> If a composer writes in et then
> playing in et is not a compromise.

The concept of ET was born as a compromise and that success and yes I think It
was a success has maybe given others the notion that ET is the way to go. No
ET can match 12 in it simplicity and its ability to do a lot with so little.

> There's no such thing as "perfect
> tuning", just various tunings that you like to compose in/play in or not.

Sterns arguments that nothing can said about any tuning be apriori might be
the best Philosophical stand in many ways. But there is some inner musical
reality that history (my academia showing:)) and transcultural surveys show
are preferred by human beings.
There is something that make people prefer steps between frequencies, melodies
in general, 5 and 7 tone scales over 6,8,or 9 tone scales. Human beings are
not objective data consuming machine who have no natural preference. To think
so one would have to except Psychology as False if not impossible. All tunings
are not Created Equal!

>
>
> dante
>
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www.anaphoria.com

>From: Kraig Grady <kraiggrady@anaphoria.com>
>
>
>
>Rosati wrote:
>
>> From: "Rosati" <dante@pop.interport.net>
>>
>> The Schuman resonance is constantly shifting so unless you have some
>> detector hooked to your automatic intonation weegee the supposed
resonance
>> will be more fanciful than actual.
>
>It seems that this resonance should be able to be measured and the
performer
>could do what Charlemane Palestine did. Start performances by adjusting his
>base pitch to the resonance of the room except to the earth. Serialism is
>worth exploring and might teach us something but natural resonance is some
wee
>gee hocus-pocus. Is it your academia that's showing.:)
>

I never said serialism might teach us something, you must be mixing me up
with someones else's post. I've got nothing against the idea of resonance
but if you're going to do it you might as well try to make it a real
resonance rather than an imaginary one. I'm not sure how the schumann
resonance is monitored but my guess would be that its not a trival
measurement technologicaly. Since Ray mentioned it, he must have
investigated this (I would hope) and perhaps could enlighten us as to how
its done.

>>
>>
>> There's no right and wrong in art, just "play", and things that make you
go
>> "oooo" and things that leave you cold.
>
>Then why have any tuning at all
>

If having no tuning at all is what floats your boat then go for it.

>> There's no such thing as "perfect
>> tuning", just various tunings that you like to compose in/play in or not.
>
>Sterns arguments that nothing can said about any tuning be apriori might be
>the best Philosophical stand in many ways. But there is some inner musical
>reality that history (my academia showing:)) and transcultural surveys show
>are preferred by human beings.
>There is something that make people prefer steps between frequencies,
melodies
>in general, 5 and 7 tone scales over 6,8,or 9 tone scales. Human beings are
>not objective data consuming machine who have no natural preference. To
think
>so one would have to except Psychology as False if not impossible. All
tunings
>are not Created Equal!

While it is true that there are cross cultural tendencies like the ones you
point out, that does not mean that experiments that go beyond these are
pointless. It might also be observed that there are transcultural
tendencies to violence, war and exploitation of one group by another.
"Natural" does not necessarily equal "better".

I just get a little skeptical when I read claims that one or another tuning
system is "better" or the "one true tuning". This is the same kind of bilge
which, when applied to religion, results in intolerance and the Spanish
Inquisition (I bet you weren't expecting the Spanish Inquisition :-)

dante

[Dante Rosati, TD 175.13]
> It might also be observed that there are transcultural
> tendencies to violence, war and exploitation of one group by
> another. "Natural" does not necessarily equal "better".

That's the best point I've seen here all year.

And many of the tendencies we have that most of us wish would
go away arise from our most primitive instincts, and are
situated in what brain-theorists call the 'reptilian brain',
which is our brain stem and evolutionarily the oldest part
of our nervous system. These tendencies are basically concerned
with territoriality, aggression, and sex.

Most of us like the last one, but the other two seem pretty
pointless in todays incredibly overpopulated world.
(Well, put that way, sex seems pretty pointless too.
But we still like it ...)

The solution to minimizing these tendencies is widely seen as
the encouragement and evolvement of the tendencies which arise
out of the specifically mammalian and human parts of the
brain, the limbic system and cerebral cortex, respectively.
These tendencies involve, respectively, nurturing and thinking.

Just thought I'd throw that 0.02 Semitones in.

And I'm also suspect that a 'perfect' *anything* is possible
in *this* universe. Perfection is in God, if you believe
in it/Him, not in anything created outside of or by it/Him.

Perhaps in other parts of the Polyverse....

(I bet you weren't expecting me to mention God.)

(NOBODY expects the Spanish Inquisition!)

-monz

Joseph L. Monzo monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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"Rosati" <dante@pop.interport.net> wrote:
>... I'm not sure how the schumann
>resonance is monitored but my guess would be that its not a trival
>measurement technologicaly. Since Ray mentioned it, he must have
>investigated this (I would hope) and perhaps could enlighten us as to how
>its done.

The schumann resonance is measured with radio equipment and it does vary
by up to about 30 cents either side of the average value (I think on a
time scale of hours to days). I have thought that ideally music should
follow the natural rhythms in real time, and no doubt top performers do
this because they are sensitive to the subtle vibrations in nature.

>I just get a little skeptical when I read claims that one or another tuning
>system is "better" or the "one true tuning". This is the same kind of bilge
>which, when applied to religion, results in intolerance and the Spanish
>Inquisition (I bet you weren't expecting the Spanish Inquisition :-)

I think that it is better to play the frequencies that are intended
rather than to use any compromise. You may say that ET frequencies are
intended by someone but I don't believe that this is normally so.
Everyone knows that ET is a compromise.

Kraig Grady <kraiggrady@anaphoria.com> answered your other points well.

>[Ray Tomes:]
>>Solutions to all the problems raised will be suggested and a grand structure erected

<stearns@capecod.net> wrote:
>Q: If the grand structure is being erected, what makes it so 'natural?'

Nature is a grand stucture. I want to emulate that.

Actually part of what I am suggesting is dismantling a grand structure.
The experiment with ET and the problem of the ambiguous notation of
music needs to be dismantled. If musical notation actually stated the
frequencies of notes to be played then there would be no need for a
tuning list at all. Tuning is in a blind alley.

It seems that no-one is addressing the key point.
Why have a compromise at all?

-- Ray Tomes -- http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm --
Cycles email list -- http://www.kcbbs.gen.nz/users/af/cyc.htm
Alexandria eGroup list -- http://www.kcbbs.gen.nz/users/af/alex.htm
Boundaries of Science http://www.kcbbs.gen.nz/users/af/scienceb.htm

>The experiment with ET and the problem of the ambiguous notation of
>music needs to be dismantled

See? I told you: the Spanish Inquisition.

dante

Ray you seem to find that alot around here people addressing everything BUT the point you
are interested in,mind that spelling and trim those replies though!
The point being
We should not compromise at all.
P

>
>
> It seems that no-one is addressing the key point.
> Why have a compromise at all?
>
> -- Ray Tomes -- http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm --
> Cycles email list -- http://www.kcbbs.gen.nz/users/af/cyc.htm
> Alexandria eGroup list -- http://www.kcbbs.gen.nz/users/af/alex.htm
> Boundaries of Science http://www.kcbbs.gen.nz/users/af/scienceb.htm
>
> ------------------------------------------------------------------------
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> http://www.onelist.com
> Come join one of the 145,000 email communities at ONElist!
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[Ray Tomes:]
>and no doubt top performers do this because they are sensitive to the
subtle vibrations in nature.

No doubt?

>I think that it is better to play the frequencies that are intended

Intended by whom?

>Everyone knows that ET is a compromise.

They do?

>Nature is a grand stucture.

Sounds reasonable enough...

>I want to emulate that.

And what if I also want to "emulate that," but the "that" that your so
unconditionally espousing doesn't seem to agree whatsoever with the "that"
that I seem to understand as "that?" What if I happen to 'see' (...hear has
apparently left the conversation) no problem with the "problem..." no
compromise with the "compromise..." and no absolutely perfect tuning in the
"absolutely perfect tuning..." From what you say (and probably much more to
the point - the way you say it), anyone who might have an 'orientation,' or
point of view somewhat similar to mine, is left with only one option: It's
wrong - so take it or fake it.

Dan

Rosati wrote:

> From: "Rosati" <dante@pop.interport.net>
>
> >From: Kraig Grady <kraiggrady@anaphoria.com>
> >
> >
> >
> >Rosati wrote:
> >
> >> From: "Rosati" <dante@pop.interport.net>
> >>
> >> The Schuman resonance is constantly shifting so unless you have some
> >> detector hooked to your automatic intonation weegee the supposed
> resonance
> >> will be more fanciful than actual.
> >
> >It seems that this resonance should be able to be measured and the
> performer
> >could do what Charlemane Palestine did. Start performances by adjusting his
> >base pitch to the resonance of the room except to the earth. Serialism is
> >worth exploring and might teach us something but natural resonance is some
> wee
> >gee hocus-pocus. Is it your academia that's showing.:)
> >
>
> I never said serialism might teach us something, you must be mixing me up
> with someones else's post.

You at least take it seriously and unbiased

> I've got nothing against the idea of resonance
> but if you're going to do it you might as well try to make it a real
> resonance rather than an imaginary one.

Is ET real or imaginary? Is what Academics teach real or imaginary?

> >> There's no such thing as "perfect
> >> tuning", just various tunings that you like to compose in/play in or not.

Perfect, no. more complete yes!
It's easy to accept any tuning for brief periods of time. Especially if you
switch from one to the other. I would like to see people stick to one for a
couple years and explore it fully and then see where it leads. When I first
started working in tuning I began with 31 ET for many of the same reasons
people work with any ET. After a couple of years I couldn't stand not having a
9/8 or 9/8 like interval. This could only come about by using the tuning for a
period of time exclusively. Paul Erlich at least is taken the time to examine
where his logic has lead him and he might change and he might not. At these it
isn't the flippant jump from one tuning to another. You not going to convince
me that we can absorb a tuning that quickly. So the idea of using "what turns
you on" is not so simple

> >Sterns arguments that nothing can said about any tuning be apriori might be
> >the best Philosophical stand in many ways. But there is some inner musical
> >reality that history (my academia showing:)) and transcultural surveys show
> >are preferred by human beings.
> >There is something that make people prefer steps between frequencies,
> melodies
> >in general, 5 and 7 tone scales over 6,8,or 9 tone scales. Human beings are
> >not objective data consuming machine who have no natural preference. To
> think
> >so one would have to except Psychology as False if not impossible. All
> tunings
> >are not Created Equal!
>
> While it is true that there are cross cultural tendencies like the ones you
> point out, that does not mean that experiments that go beyond these are
> pointless.

I agree. there are cultures where the elements I mentioned are not the
predominant ones.

> It might also be observed that there are transcultural
> tendencies to violence, war and exploitation of one group by another.
> "Natural" does not necessarily equal "better".

It's better to face them head on at least. If they are lets examine it not run
away in some arbitrary manner. The whole western idea of going outside of
what's here (SKY GOD MENTALITIES) might benefit more from going in deeper
(Earth Goddess) into what us here. Psychology will inform you that if these
qualities are natural you are not going to get away from it with reason. The
french Revolution was the most intellectual and also the bloodiest. If anything
Hitler was attempting to "correct " nature by purging the defective gene pool.
Science has once again brought us to the same place. No transcendence of nature
is desirable until you know what nature is! Run away from the harmonic series
and it will come back to bite you on the ass!

>
>
> I just get a little skeptical when I read claims that one or another tuning
> system is "better" or the "one true tuning". This is the same kind of bilge
> which, when applied to religion, results in intolerance and the Spanish
> Inquisition (I bet you weren't expecting the Spanish Inquisition :-)
>
> dante

-- Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com

monz@juno.com wrote:

> From: monz@juno.com
>
> [Dante Rosati, TD 175.13]
> > It might also be observed that there are transcultural
> > tendencies to violence, war and exploitation of one group by
> > another. "Natural" does not necessarily equal "better".
>
> That's the best point I've seen here all year.
>
> And many of the tendencies we have that most of us wish would
> go away arise from our most primitive instincts,

How do we know it is primitive. Maybe war and aggression is a more resent
trait of civilization!

> and are
> situated in what brain-theorists call the 'reptilian brain',
> which is our brain stem and evolutionarily the oldest part
> of our nervous system.

I don't think I ever ran across this one. The lizards in my yard are awful
peaceful while the gangs at the bottom of my driveway are not!

> These tendencies are basically concerned
> with territoriality, aggression, and sex.

the first two are qualities of civilization and grow proportionally with
it. Maybe we need to "devolve" to plants where there is not the survival of
the fittest but that which maximizes their interrelationship with the
environment prospers!

> Most of us like the last one, but the other two seem pretty
> pointless in todays incredibly overpopulated world.
> (Well, put that way, sex seems pretty pointless too.
> But we still like it ...)

According to W. Reich the last will get rid of the former two on your list!

> The solution to minimizing these tendencies is widely seen as
> the encouragement and evolvement of the tendencies which arise
> out of the specifically mammalian and human parts of the
> brain, the limbic system and cerebral cortex, respectively.
> These tendencies involve, respectively, nurturing and thinking.

>
> And I'm also suspect that a 'perfect' *anything* is possible
> in *this* universe. Perfection is in God, if you believe
> in it/Him, not in anything created outside of or by it/Him.
>
> Perhaps in other parts of the Polyverse....
>
> (I bet you weren't expecting me to mention God.)
>
> (NOBODY expects the Spanish Inquisition!)
>
> -monz

-- Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com

Rosati wrote:

> From: "Rosati" <dante@pop.interport.net>
>
> >The experiment with ET and the problem of the ambiguous notation of
> >music needs to be dismantled
>
> See? I told you: the Spanish Inquisition.

Was the call to end tonality also a Spanish Inquisition?

-- Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com

"D. Stearns" wrote:

>
>
> And what if I also want to "emulate that," but the "that" that your so
> unconditionally espousing doesn't seem to agree whatsoever with the "that"
> that I seem to understand as "that?" What if I happen to 'see' (...hear has
> apparently left the conversation) no problem with the "problem..." no
> compromise with the "compromise..." and no absolutely perfect tuning in the
> "absolutely perfect tuning..." From what you say (and probably much more to
> the point - the way you say it), anyone who might have an 'orientation,' or
> point of view somewhat similar to mine, is left with only one option: It's
> wrong - so take it or fake it.

For many gods (and goddesses) sake's, go ahead and follow your ear, there is
more to music that ET/JI anyway, and only those that break with all convention
are the ones that can find it! I don't argue with you on a philosophical level
but an empirical one. I find it extremely difficult to play much less compose
in ETs. The opinion is my own evaluation as to why this is true. The very
principles on these observations might be wrong. My father was a psychic that
despite his ability could not explain it. Nor could find a good explanation
from others. Yet the ability remained despite all reason saying it couldn't.
Just like the world of ET, JI is unlimited in its applications and at a
certain point they are perceptibly indistinguishable. My life would be alot
easier if there was an ET that could do what I want to do and sounded the way
I wanted it to without so many notes. What do you think about the practice of
non fixed pitched instruments who play 12 et adjusting their pitches in
ensemble playing?

Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com

Message: 4
Date: Wed, 12 May 1999 07:20:04 -0400
From: monz@juno.com
Subject: Why not perfect tuning?

[Dante Rosati, TD 175.13]
> It might also be observed that there are transcultural
> tendencies to violence, war and exploitation of one group by
> another. "Natural" does not necessarily equal "better".

That's the best point I've seen here all year.

And many of the tendencies we have that most of us wish would
go away arise from our most primitive instincts, and are
situated in what brain-theorists call the 'reptilian brain',
which is our brain stem and evolutionarily the oldest part
of our nervous system. These tendencies are basically concerned
with territoriality, aggression, and sex.

Most of us like the last one, but the other two seem pretty
pointless in todays incredibly overpopulated world.
(Well, put that way, sex seems pretty pointless too.
But we still like it ...)

The solution to minimizing these tendencies is widely seen as
the encouragement and evolvement of the tendencies which arise
out of the specifically mammalian and human parts of the
brain, the limbic system and cerebral cortex, respectively.
These tendencies involve, respectively, nurturing and thinking.

I believe that a balance between the reptilian impulses and the gray matter is best. The afore mentioned aggression and violence of the dark side are they, to use Yoda's phrase.
I believe in balance... just how Musick is beautiful when there is a dynamic balance between dissonance (tension) and consonance (release), and aggressive, sexual grooves and sweet ambient atmospheres.

IMNSHO ;)

And I'm also suspect that a 'perfect' *anything* is possible
in *this* universe. Perfection is in God, if you believe
in it/Him, not in anything created outside of or by it/Him.

Mathematicks is perfect by definition.
It is the only thing absolute in the universe... well... maybe... ;)

(I bet you weren't expecting me to mention God.)

(NOBODY expects the Spanish Inquisition!)

Surprise is their chief weapon... surprise and a fanatic devotion to the pope...

-monz
Musick, Magick, and Mo'f'ck
Alex J. "Azi Vajravai" Van Wey
http://members.xoom.com/Azi_Vajravai

Ray Tomes wrote,

>It seems that no-one is addressing the key point.
>Why have a compromise at all?

Perhaps you should go back and read the last couple of months of archives,
especially all messages titled "Adaptive JI". We've discussed several
reasons why compromises are necessary -- to prevent wandering tonics, to
tune chords with contradictory consonances like major add 6 add 9 and
augmented, and to balance the conflicting demands of harmonic and melodic
perfection.

Ray Tomes wrote,

>>It seems that no-one is addressing the key point.
>>Why have a compromise at all?

I wrote,

>Perhaps you should go back and read the last couple of months of archives,
especially all messages titled >"Adaptive JI". We've discussed several
reasons why compromises are necessary -- to prevent wandering >tonics, to
tune chords with contradictory consonances like major add 6 add 9 and
augmented, and to balance >the conflicting demands of harmonic and melodic
perfection.

And this is assuming perfect control over a potentially infinite number of
pitches. For acoustic instruments such as keyboards and harps, which are
fixed-pitch, and for other acoustic instruments where small pitch variations
cannot be controlled with the precision of electronic synthesizers (guitars,
winds, brasses), the finite number of pitches presents a more direct reason
why compromises are necessary or desirable.

"Paul H. Erlich" wrote:

> From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>
> Ray Tomes wrote,
>
> >>It seems that no-one is addressing the key point.
> >>Why have a compromise at all?
>
> I wrote,
>
> >Perhaps you should go back and read the last couple of months of archives,
> especially all messages titled >"Adaptive JI". We've discussed several
> reasons why compromises are necessary -- to prevent wandering >tonics, to
> tune chords with contradictory consonances like major add 6 add 9 and
> augmented, and to balance >the conflicting demands of harmonic and melodic
> perfection.

Finally a straight answer. I understand the wandering tonic but not the 6 added
ninth thing please explain. you can do that in JI. Since no one else will
mention some of the other advantages of ET, what about the ability especially
in the odd ones of stretching a melody by making all one unit interval 2 or 3
or 4 or all. Kind of what Bartok did at the end of Music for string ,perc.
celeste! Serialist could really exploit the above technique!

>
>
> And this is assuming perfect control over a potentially infinite number of
> pitches. For acoustic instruments such as keyboards and harps, which are
> fixed-pitch, and for other acoustic instruments where small pitch variations
> cannot be controlled with the precision of electronic synthesizers (guitars,
> winds, brasses), the finite number of pitches presents a more direct reason
> why compromises are necessary or desirable.

-- Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com

[Ray Tomes, TD 176.7]
> If musical notation actually stated the frequencies of notes
> to be played then there would be no need for a tuning list at
> all. Tuning is in a blind alley.
>
> It seems that no-one is addressing the key point.
> Why have a compromise at all?

Because no matter how specifically we actually play or otherwise
present our tuning, there are perceptual and cognitive limitations
on how accurately our bodies can receive and process that
information.

That's what all the Tuning List discussion is about.

Joseph L. Monzo monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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>From: Kraig Grady <kraiggrady@anaphoria.com>
>
>
>
>Rosati wrote:
>
>> From: "Rosati" <dante@pop.interport.net>
>>
>> >The experiment with ET and the problem of the ambiguous notation of
>> >music needs to be dismantled
>>
>> See? I told you: the Spanish Inquisition.
>
>Was the call to end tonality also a Spanish Inquisition?

Who called for an end to tonality? Schoenberg said there was still lots of
great music yet to be written in C major.

dante

Rosati wrote:

>
> >
> >Was the call to end tonality also a Spanish Inquisition?
>
> Who called for an end to tonality? Schoenberg said there was still lots of
> great music yet to be written in C major.

he predicted Terry Riley!
I think Boulez came out and said something close to this effect! Most of the
serial school took this stand as far as I could tell.
I think some great music will probably be written in 12 ET. I really hope
some one does it with one of the others. I'm always ready to hear good music!

-- Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com

[Paul Erlich, TD 177.6]
>
> We've discussed several reasons why compromises are necessary
> -- to prevent wandering tonics, to tune chords with contradictory
> consonances like major add 6 add 9 and augmented, and to balance
> the conflicting demands of harmonic and melodic perfection.

I'm not disagreeing with your point, Paul, just with your use
of the word 'necessary'.

I agree with you that compromise is necessary for the reasons
you state, but don't infer that it's necessary _per se_
- it's not. If you are willing to allowing wandering tonics
etc. in your music, you can use a nice extended JI all the way.

-monz

Joseph L. Monzo monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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Paul H. Erlich:
>... look in archives ... messages titled "Adaptive JI".
Thanks Paul, I will do that.

>And this is assuming perfect control over a potentially infinite number of
>pitches. For acoustic instruments such as keyboards and harps, which are
>fixed-pitch, and for other acoustic instruments where small pitch variations
>cannot be controlled with the precision of electronic synthesizers (guitars,
>winds, brasses), the finite number of pitches presents a more direct reason
>why compromises are necessary or desirable.

Obviously I am mainly considering electronic music where there is total
control over the frequency, waveform etc. However strings and voices
can also do many of these things quite naturally if the minds of the
people have not been polluted by ET to the extent of believing it to be
"in tune". I am considering the prospect of the composer being able to
do anything that is desired. Firstly that means getting a good JI by
default and then looking at when and what variations from that are
desired and specifying them.

And just a note about spelling (for which I got criticised by several
people). I happen to come from a part of the world where we spell words
in English rather than American so I write things like "colour" and
"centre" and I don't intend to be AmericaniZed. Other than that I am a
crappy speller but sometimes I hit the spell check button and save you
all from some of them.

Also, a short note to Alex Vanwey and his 13 limit notation.
Certainly a notation is needed that is definitive. A good one appears
in the book "An Introduction to Indian Music" by E Clements (1912) which
in addition to "b" and "#" uses "+", "*" and a "b" that looks like a
vertical line with a 7 written on it. This last one is brilliant
because he uses it for the 7 ratio notes.

If you are going to use day names for the notes, why not make a meal of
it. The natural can be lunch with sharp and flat being dinner and
breakfast. Then you can have morning and afternoon tea (OK coffee for
Americans), supper and brunch. Those ones that really fall in the
cracks (like 11 and 13) can be midnight snacks. :-)

-- Ray Tomes -- http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm --
Cycles email list -- http://www.kcbbs.gen.nz/users/af/cyc.htm
Alexandria eGroup list -- http://www.kcbbs.gen.nz/users/af/alex.htm
Boundaries of Science http://www.kcbbs.gen.nz/users/af/scienceb.htm

[Kraig Grady, TD 177.15]
> I understand the wandering tonic but not the 6 added ninth
> thing please explain.

There was a discussion on the List between Paul Erlich and
I last year about this:

[Paul Erlich, Mills TD 1400.3, 29 Apr 1998]
> [Example] 3. 6/9 (meaning root, maj. 3rd, perf. 5th. maj. 6th, maj.
9th)
>
> Here, we have a chain of 5ths, but also a major 3rd and two minor
> thirds. If all these intervals are to be consonant, no just
> interpretation will really do. This chord requires a tuning where
> the 80:81 syntonic comma vanishes

My response:

[me, monz, Mills TD 1403.10]
>
> Matrix Graph:
>
> maj 3rd
> 5^1 maj 3rd
> 5/4 6th _ - 3^4
> 3.86 9th _ - 3^3 - 81/64
> \ _ - 3^2- 27/16 4.08
> \ 5th - 9/8 9.06
> root _ - 3^1 2.04
> n^0 - 3/2
> 1/1 7.02
> 0.00
>
> Analysis:
> *********
>
> 12-eq ratio s-t. prop. IDs
> ===== ===== ==== ==== ======
> 9th 9/8 2.04 144 9
> 6th 27/16 9.06 108 27
> 5th 3/2 7.02 96 3
> 3rd (P) 81/64 4.08 81 5
> 3rd (j) 5/4 3.86 80 81
> root 1/1 0.00 64 1
> _____ ___
> n^0 n^0
>
> (I had enough room here to show only one syntonic comma, that
> between the "just" and Pythagorean major 3rds. The 13th and
> possibly even the 9th could also be interpreted as 5-limit.)

In other words, in this type of ET chord, a JI interpretation
is difficult because the '6th' can only be consonant as either
a 5:3 or a 27:16, but not both.

Joseph L. Monzo monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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monz@juno.com wrote:

> From: monz@juno.com
>
> [Kraig Grady, TD 177.15]
> > I understand the wandering tonic but not the 6 added ninth
> > thing please explain.
>
> [Paul Erlich, Mills TD 1400.3, 29 Apr 1998]
> > [Example] 3. 6/9 (meaning root, maj. 3rd, perf. 5th. maj. 6th, maj.
> 9th)
> >
> > Here, we have a chain of 5ths, but also a major 3rd and two minor
> > thirds. If all these intervals are to be consonant, no just
> > interpretation will really do. This chord requires a tuning where
> > the 80:81 syntonic comma vanishes

Why do they all have to be consonant? Who says so!

-- Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com

I (Paul E.) wrote,

>> >Perhaps you should go back and read the last couple of months of
archives,
>> especially all messages titled >"Adaptive JI". We've discussed several
>> reasons why compromises are necessary -- to prevent wandering >tonics, to
>> tune chords with contradictory consonances like major add 6 add 9 and
>> augmented, and to balance >the conflicting demands of harmonic and
melodic
>> perfection.

Kraig Grady wrote,

>Finally a straight answer. I understand the wandering tonic but not the 6
added
>ninth thing please explain. you can do that in JI.

In JI, take a major triad and add a sixth and a ninth to it. Your ninth will
be 9/4, a 3:2 above the 3/2. If your sixth is a 5/3, you get a wolf against
the third, while if your sixth is a 27/16, you get a wolf against the ninth.

>Since no one else will
>mention some of the other advantages of ET,

Here and in general I defend not ET but meantone and other linear
temperaments.

Kraig Grady wrote:
>
>monz@juno.com wrote:
>
>> From: monz@juno.com
>>
>> [Kraig Grady, TD 177.15]
>> > I understand the wandering tonic but not the 6 added ninth
>> > thing please explain.
>>
>> [Paul Erlich, Mills TD 1400.3, 29 Apr 1998]
>> > [Example] 3. 6/9 (meaning root, maj. 3rd, perf. 5th. maj. 6th, maj.
>> 9th)
>> >
>> > Here, we have a chain of 5ths, but also a major 3rd and two minor
>> > thirds. If all these intervals are to be consonant, no just
>> > interpretation will really do. This chord requires a tuning where
>> > the 80:81 syntonic comma vanishes
>
>Why do they all have to be consonant? Who says so!

If you had to perform a pop or jazz tune with a C6/9 chord in it, you
certainly wouldn't want to have any wolves in the chord. It would be highly
contrary to the aesthetic of those styles, which use this chord as a
relative consonance. Of course if you're writing your own music, you are
free to use whatever dissonant chords you want. Interesting how here
JI=dissonance!

[Example] 3. 6/9 (meaning root, maj. 3rd, perf. 5th. maj. 6th, maj.
9th)

So if I stick the chord C-E-G-A-D (is it meaningful that this contains
egad?) into my AJI calculator it comes up with three possible candidates
for the ratios as being 12:15:18:20:27, 14:18:21:24:32 and
16:20:24:27:36. The first one is probably correct, but depending on
context the second one might be better.
I would definitely look at the preceeding chords to make a decision.
Note that the second one does not treat the C-E-G as a 4:5:6 major
chord.

-- Ray Tomes -- http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm --
Cycles email list -- http://www.kcbbs.gen.nz/users/af/cyc.htm
Alexandria eGroup list -- http://www.kcbbs.gen.nz/users/af/alex.htm
Boundaries of Science http://www.kcbbs.gen.nz/users/af/scienceb.htm

[Ray Tomes, TD 176.7:]
> It seems that no-one is addressing the key point.
> Why have a compromise at all?

I like JI, and don't want to compromise when it's not necessary. But
there are problem chords and sequences of chords. For example, what
would you do when tuning:

. The infamous comma-pump chord sequence, I-vi-ii-V-I, with tied
notes whenever possible.

. The transition from G-B-D-F to G-C-F, with the outer notes tied.

. An augmented triad: C-E-G#.

. A full diminished triad: A-C-Eb-Gb.

. Chords with four or more successive fifths, such as C-D-F-G.

These all require compromises of some sort, don't they?

JdL

>>If you are going to use day names for the notes, why not make a meal of
it.<<

from this, I can tell that you didn't look at the notation itself, but
rather the scala. I used specific symbols for the days of the week, that
derive from astrological planet symbols (you'll recognise Tuesday as "Male"
and Friday as "Female").

Go to the website http://members.xoom.com/Azi_Vajravai/ji.html and look at
the symbols. Then tell me what you think.

If I had symbols for breakfast, tea, lunch, tea, dinner, and tea - I'd use
them. Originally I was going to use the first 7 letters of the Hebrew
alphabet... but the 7-days works much more... magickally.

If you want to hear a bit of my 12 tet musick:
http://members.xoom.com/Azi_Vajravai/music.html

Musick, Magick, and Mo'f'ck
Alex J. "Azi Vajravai" Van Wey
http://members.xoom.com/Azi_Vajravai

_______________________________________________________________
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Brett Barbaro wrote:

> From: "Brett Barbaro" <barbaro@noiselabs.com>
>
> I (Paul E.) wrote,
>
> >> >Perhaps you should go back and read the last couple of months of
> archives,
> >> especially all messages titled >"Adaptive JI". We've discussed several
> >> reasons why compromises are necessary -- to prevent wandering >tonics, to
> >> tune chords with contradictory consonances like major add 6 add 9 and
> >> augmented, and to balance >the conflicting demands of harmonic and
> melodic
> >> perfection.
>
> Kraig Grady wrote,
>
> >Finally a straight answer. I understand the wandering tonic but not the 6
> added
> >ninth thing please explain. you can do that in JI.
>
> In JI, take a major triad and add a sixth and a ninth to it. Your ninth will
> be 9/4, a 3:2 above the 3/2. If your sixth is a 5/3, you get a wolf against
> the third, while if your sixth is a 27/16, you get a wolf against the ninth.

I find both these chords sound fine! Sometimes I prefer a 27/20 or a 40/27 to
there simpler counterparts. Something I share with the Persians. With mean tone
my only real experience outside of a few minutes it with 31 et which after a
few years I found too placid for my taste. Have you ever heard the one of
Keplers found in Barbours!

-- Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com

Brett Barbaro wrote:

> From: "Brett Barbaro" <barbaro@noiselabs.com>
>
> Kraig Grady wrote:
> >
> >monz@juno.com wrote:
> >
> >> From: monz@juno.com
> >>
> >> [Kraig Grady, TD 177.15]
> >> > I understand the wandering tonic but not the 6 added ninth
> >> > thing please explain.
> >>
> >> [Paul Erlich, Mills TD 1400.3, 29 Apr 1998]
> >> > [Example] 3. 6/9 (meaning root, maj. 3rd, perf. 5th. maj. 6th, maj.
> >> 9th)
> >> >
> >> > Here, we have a chain of 5ths, but also a major 3rd and two minor
> >> > thirds. If all these intervals are to be consonant, no just
> >> > interpretation will really do. This chord requires a tuning where
> >> > the 80:81 syntonic comma vanishes
> >
> >Why do they all have to be consonant? Who says so!
>
> If you had to perform a pop or jazz tune with a C6/9 chord in it, you
> certainly wouldn't want to have any wolves in the chord. It would be highly
> contrary to the aesthetic of those styles, which use this chord as a
> relative consonance. Of course if you're writing your own music, you are
> free to use whatever dissonant chords you want. Interesting how here
> JI=dissonance!

I love dissonance! This wolf on my harmonium was never that prevalent. I still
hear it as consonant!

-- Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com

John A. deLaubenfels wrote:

>I like JI, and don't want to compromise when it's not necessary. But
>there are problem chords and sequences of chords. For example, what
>would you do when tuning:

> . The infamous comma-pump chord sequence, I-vi-ii-V-I, with tied
> notes whenever possible.

I would play a different note (*81/80) for the common note in the ii and
V chords and therefore return to the original chord. The reason is that
to do otherwise is to introduce an additional ratio of 5 into one note
which is moving too far from the starting point (i.e to have 5^2 which
in my point scoring system is a big no-no compared to the other option).
Of course it is possible to do a similar thing in even slower motion and
to successfully move the tonic by a ratio of 81/80. If someone tries
that hard then they probably intended it.

> . The transition from G-B-D-F to G-C-F, with the outer notes tied.

It depends on the context. If this was the start of a piece then the
G-B-D-F would be 4:5:6:7 and C would be 16/3 with no changes to G and F.
However I have just conceded that 4:5:6:7 is not the only possibility as
I had previously maintained.

> . An augmented triad: C-E-G#.

Again it depends on context, only more so. Please tell me the key of
the piece when this comes up or the previous few chords and I will
answer.

> . A full diminished triad: A-C-Eb-Gb.

Likewise.

> . Chords with four or more successive fifths, such as C-D-F-G.

Most likely treat all the fifths as 3/2 ratios to get 8:12:18:27, but
context could change that as could the inversion of the chord. In the
arrangement that you give it would most likely be 24:27:32:36 but
16:18:21:24 or 18:20:24:27 or even 27:30:36:40 are possible depending on
what came before (what key in other words). In C use 24:27:32:36.

>These all require compromises of some sort, don't they?

Yes, but in the thinking or understanding rather than in the
frequencies. These chords are difficult to understand the meaning of no
matter what tuning system is used. It is much easier to answer
definitively when the full context is clear as in the first example.
Music is all about the relationships, not just within the chord, but
also between chords and so we must expect that context is going to alter
things. This is an advantage, not a disadvantage.

Even if some of these chords do sound a bit weird, they would sound
equally weird in ET. Anyone who writes chords like that clearly wants
to get that sort of effect. The vast majority of chords would be
improved and I don't think that any would be made worse, but I am
prepared to be proved wrong on that.

-- Ray Tomes -- http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm --
Cycles email list -- http://www.kcbbs.gen.nz/users/af/cyc.htm
Alexandria eGroup list -- http://www.kcbbs.gen.nz/users/af/alex.htm
Boundaries of Science http://www.kcbbs.gen.nz/users/af/scienceb.htm

Kraig Grady wrote,

>Why do they all have to be consonant? Who says so!

I just received and listened to Kraig Grady's CD "From the Interiors of
Anaphoria". Beautiful stuff! Anyone who equates Just Intonation with
beatless harmony is in for a real shock if they listen to this one!!! I
would like to learn more about the instruments you used, Kraig, there are no
indications in the liner notes as to what they are -- in particular, what's
that organ-like sound?

"Paul H. Erlich" wrote:

> I would like to learn more about the instruments you used, Kraig, there are
> no
> indications in the liner notes as to what they are -- in particular, what's
> that organ-like sound?

That was a return Estey Reed organ. I have 4 working out of the 6 I have
acquired which i use as reed sources. (If anyone knows reed sources let me
know!) Since its really doesn't look any different than any other organ I
didn't include it amongst the photos on my web site. The first cut is Lake
aloe, the second-Mt.Meru, the third a retuned hammer dulcimer. fifth mesa is
used on cuts 5 and 6. The last cut is a Cheng retuned. Yes the beat are very
deliberate and musically viable. Thanks!

-- Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com

[I wrote:]
>>I like JI, and don't want to compromise when it's not necessary. But
>>there are problem chords and sequences of chords. For example, what
>>would you do when tuning:
>>
>> . The infamous comma-pump chord sequence, I-vi-ii-V-I, with tied
>> notes whenever possible.

[Ray Tomes wrote:]
> I would play a different note (*81/80) for the common note in the ii
> and V chords and therefore return to the original chord.

OK. Would you slide the note in transition, or have it jump by a comma?

>> . The transition from G-B-D-F to G-C-F, with the outer notes tied.

> It depends on the context.

OK, fair enough.

> If this was the start of a piece then the G-B-D-F would be 4:5:6:7 and
> C would be 16/3 with no changes to G and F. However I have just
> conceded that 4:5:6:7 is not the only possibility as I had previously
> maintained.

And I, ironically, jumped to the defense of 4:5:6:7. To my ear, it's
got the most, no contest!

>> . An augmented triad: C-E-G#.

> Again it depends on context, only more so.

Fair enough.

>> . A full diminished triad: A-C-Eb-Gb.

> Likewise.

>> . Chords with four or more successive fifths, such as C-D-F-G.

> Most likely treat all the fifths as 3/2 ratios to get 8:12:18:27, but
> context could change that as could the inversion of the chord.

Here I definitely prefer 12-tET, that much-maligned (rightly so most
of the time!) tuning.

>>These all require compromises of some sort, don't they?

> Yes, but in the thinking or understanding rather than in the
> frequencies.

I agree.

> These chords are difficult to understand the meaning of no matter what
> tuning system is used.

I agree.

> Even if some of these chords do sound a bit weird, they would sound
> equally weird in ET. Anyone who writes chords like that clearly wants
> to get that sort of effect. The vast majority of chords would be
> improved and I don't think that any would be made worse, but I am
> prepared to be proved wrong on that.

Right now I like 12-tET for my last 3 examples, but, as you say, an
individual composer might want a different effect.

It seems very clear that many chords, and many sequences of notes,
cannot satisfy all desired tuning "requirements", and so by my use of
the word, there MUST be "compromise". But our differences seem merely
semantic, which is to say, nothing of substance.

I absolutely agree that we can and should take charge of tuning.

JdL

Regarding contradictory consonances (CC):

I gave a possible CC interpretation of the dominant 7th chord where 12-tET
was the near-optimum temperament. But it wasn't terribly convincing. Paul's
example of the 6-9 chord is better (smaller errors), and as Kraig Grady and
Joe Monzo pointed out "who says they have to be consonances".

But assuming in the case of the 6-9 chord, that they are, here's how the
errors distribute in two different tunings.

6-9 chord Errors when in Errors when in
C E G A D Pythagorean 1/4 comma Meantone
4 5 6 9 +21.5c in 4:5 -5.4c in 4:6, 5:6, 6:9
3 5 +21.5c) +5.4c
3 4 +5.4c
8 9 -10.8c
3 4 +5.4c

The errors in 12-tET will be somewhere in between, but closer to those in
Pythagorean. Clearly meantone is better for this chord than 12-tET, but
31-tET or some meantone a little closer to 1/5 comma (and hence closer to
12-tET) would be kinder to the 8:9 (with something like 8c max errors).

Here's what might be an even better example (lower errors when
distributed), a CC interpretation of the diminished 7th chord.

dim 7th Errors when Errors when in
C EbF#A 5:6:7 is just 1/4 comma Meantone
5 6 7 -5.4c in 5:6, +2.3c in 6:7, -3.1c in 5:7
3 5 +13.8c +5.4c
5 6 -5.4c
5 7 -3.1c

Something a little closer to 1/3 comma would get the max error down to 4.6c.

Of course Ray, you have said that the 6/9 chord should be tuned
12:15:18:20:27 and will say the dim7 should be 10:12:14:17, but whether you
agree that the above CC's are _the_ 6-9 or dim7 chords, the point is that
chords do exist which obtain their greatest consonance in an irrational
temperament, not in any kind of Just scale.

Or putting it another way: These chords may have a dissonance minimum in a
harmonic series subset, but they have another dissonance minimum (which may
be even lower) in some (irrational) temperament.

Can someone easily compare these chords in meantone CC versus harmonic
series and give us an opinion on relative consonance, or whether they sound
like completely different chords?

Now that I think about it, the 6-9 chord seems a better counterexample to
your thesis, than the dim7, because of its higher numbers when given as a
harmonic series subset.

Note that your slide-rule predicted 10:13:15:17:22or23 and
11:14:17:19:25or26 before the more reasonable 12:15:18:20:27 for the 6-9
chord, unless we agree to leave off the primes above 7 as you suggest in
your article on your website.

Do you see that leaving off those primes is just another way of saying that
sometimes, having simple ratios between _pairs_ of notes in the chord is
more important than having small numbers (or even whole numbers) for the
chord as a whole.

Paul also mentioned the augmented chord which has a CC interpretation as:

Aug
C E G#
4 5
4 5
5 8

This is optimally tuned in 12-tET with 13.7c errors all round.

Your slide rule predicts successively 7:9:11, 8:10:13, 10:13:16 11:14:18,
12:15:19, 13:16:20. Leaving off primes above 7 doesn't help here. We end up
at 14:18:22 which reduces to 7:9:11. I think the above CC is more consonant
than a 7:9:11, but I also think they are quite different chords.

In fact 7:9:11 is better approximated in 12-tET by a major triad than an
augmented (but not very well by either). This just shows up the fact that
12-tET is inconsistent at the 11-limit. Which is why your slide rule, as it
stands, wont work in general for mutiples of 11 or higher primes. An 11
corresponding to F will sometimes give a better result than F#, likewise 13
being either Ab or A.

The harmonic slide-rule is a great educational and reference tool, but what
you need to do is put the harmonics at the correct positions, not at the
nearest degrees of 12-tET. Then you can see not only _what_ matches, but
_how_well_ it matches. And you can use it against other temperaments too.
I'd show the harmonics as thin lines (not the same width as keys). This
also eliminates any dilemmas over 22 vs 23 etc. and you could take it up to
the 32nd harmonic or further.

Of course this doesn't help your table-scanning adaptive JI algorithm since
it can't do the equivalent of a continuous slide. However it might be
improved by using 72-tET as the template. i.e. expand your table by a
factor of six in size, then look not only for matches to the 72-tET chord
that exactly corresponds to the chord played in 12-tET but also to those
72-tET chords which are slightly different but are closer to the played
12-tET chord than to any other 12-tET chord. Somewhat of a combinatorial
explosion but still tractable. Of course this still doesn't address CCs or
tonic drift.

I'm glad you understand tonic drift now.

[Ray Tomes:]
>... there is another way of preventing too much
>tonic wandering and that is simply to keep pulling the error back by say
>10% per chord played. That would be almost imperceptible and prevent
>the tonic getting very far from where it started.

This is exactly what happens in meantone (with no special effort required).
Tonic drift can still happen in adaptive meantone, but not via the common
progressions.

The augmented and dominant 7th (when considered as contradictory
consonances) are not treated well by meantone, but the 6-9 and dim 7 are.
However I notice Paul Erlich's comment that in meantone the augmented _can_
be tuned to a different CC with very low errors.

Aug Errors when in
C E Ab 1/4 comma meantone
4 5
7 9 -7.7c
5 8

Reduces to 2.6c all round, in something a little closer to 1/3 comma.

Regards,
-- Dave Keenan
http://dkeenan.com

Dave Keenan <d.keenan@uq.net.au> wrote
>... the point is that chords do exist which obtain their greatest
>consonance in an irrational temperament, not in any kind of Just scale.

That is a marvellously expressed idea which I have never considered
before. I will have to think about that -- a real compromise.

>Note that your slide-rule predicted 10:13:15:17:22or23 and
>11:14:17:19:25or26 before the more reasonable 12:15:18:20:27 for the 6-9
>chord, unless we agree to leave off the primes above 7 as you suggest in
>your article on your website.

The slide rule is really an oversimplification because it cannot
recognise that ratios like 12 and 16 are far better than 7, 11 and 13.
So the rule about missing 7, 11 and 13 and only using them if they come
up as 14, 22 and 26 is there for that reason. The full and proper
treatment is to consider the range of prime indices for each prime and
used a weighted sum with much bigger weights on the higher primes and to
use the best solution.

>Do you see that leaving off those primes is just another way of saying that
>sometimes, having simple ratios between _pairs_ of notes in the chord is
>more important than having small numbers (or even whole numbers) for the
>chord as a whole.

I would express it as above, but what you say may have a good
correlation with that.

...
>The harmonic slide-rule is a great educational and reference tool, but what
>you need to do is put the harmonics at the correct positions, not at the
>nearest degrees of 12-tET. Then you can see not only _what_ matches, but
>_how_well_ it matches.

That would be true if you are using it as a tool for finding chords in
ET and there is no reason why you shouldn't. But I am using it as a
tool for finding ratios in JI which is not only a different scale but a
different direction of translation. The point is that I don't care if
there is a 7 ratio which is 30 cents out because my idea is that the
instrument will play the exact correct 7 ratio and so the errors will
always be 0 cents. I am trying to allow a keyboard player (or composer)
to make lots of good JI chords and to get the ones that are the most
likely ones intended almost every time.

>And you can use it against other temperaments too.
>I'd show the harmonics as thin lines (not the same width as keys). This
>also eliminates any dilemmas over 22 vs 23 etc. and you could take it up to
>the 32nd harmonic or further.

Yes you can, and the thin lines is a good idea.

I have tables going up further (~288) and you certainly need them.
Quite right that it should include numbers like 48, 54, 60, 63 and 64.
The problem was that the picture really needs to be bigger than a piece
Of paper so maybe someone will volounteer to make the extension graphic.

>Of course this doesn't help your table-scanning adaptive JI algorithm since
>it can't do the equivalent of a continuous slide.

Computers are so fast now that it can get the answer easily fast enough
by considering each unique position of the slide and looking at all the
notes and doing a weighted score on each fit to get the best answer,
possibly incorporating information about the previous centre of gravity
also to resolve ambiguous cases. No problem!

I have to say that the last few days have been by far the best
communication that I have ever had as regards my AJI and related ideas.
I want to thank everyone for the fun that I have had. Thanks!

-- Ray Tomes -- http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm --
Cycles email list -- http://www.kcbbs.gen.nz/users/af/cyc.htm
Alexandria eGroup list -- http://www.kcbbs.gen.nz/users/af/alex.htm
Boundaries of Science http://www.kcbbs.gen.nz/users/af/scienceb.htm

John A. deLaubenfels wrote:

>OK. Would you slide the note in transition, or have it jump by a comma?

I guess that is a matter of personal preference but I would just jump.
The fact is that it takes a while for the brain to get enough cycles to
register the change anyway.

>And I, ironically, jumped to the defense of 4:5:6:7. To my ear, it's
>got the most, no contest!

I have to confess to why I admitted the other chord as a possibility
which surprised me even more than it did you! I wrote a computer
program to calculate all the notes that are reasonably strong ones based
on the thing that I call the centre of gravity being moved around
slightly in the prime index space. This allows making scales that are
very gradually altered from C to G or from a major to a minor etc.

One of the things that happened was that I got a scale out that had as
its four strongest notes 1 4/3 3/2 16/9 and 5/4 was next strongest.
This rather seemed to be telling me that 1 5/4 3/2 16/9 was a viable
chord or in other words 36:45:54:64. I think that the problem is in
recognising that quite a bit of fiddling can be done with indices of 2
and 3 for the price of a single change in the 7 index. This does indeed
seem to depend on the various other agreements between the notes.
Ah! I am getting my rough edges knocked off which is a good thing.

I still think that 4:5:6:7 is normally the best option though.

>I absolutely agree that we can and should take charge of tuning.

I think this is the most important thing but it seems I accidentally
upset some ET supporters with my arrogant attitude.

-- Ray Tomes -- http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm --
Cycles email list -- http://www.kcbbs.gen.nz/users/af/cyc.htm
Alexandria eGroup list -- http://www.kcbbs.gen.nz/users/af/alex.htm
Boundaries of Science http://www.kcbbs.gen.nz/users/af/scienceb.htm