Re: [tuning] Digest Number 3537

Dave Keenan wrote:

"To some degree that's true. But in my view that's far better than the
situation you get with the "tuning by ratios" definition. With that
definition I can play you any set of pitches (including a perfect
12-equal) and claim they constitute a JI scale and there is no way you
can prove me wrong, since there is always some ratio that's within the
accuracy of any measurement you can make.

At least with the "tunable by ear" definition, if someone claims to be
a super-tuner and able to hear near-beatlessness in say 12-equal, we
can subject them to randomised blind tests. Of course they may refuse,
but we can draw our own conclusions from that too. :-) "

In my opinion, a more useful definition of Just Intonation identifies it with a _process_ of tuning ever-closer to some rational ideal, rather than the specific result of such a procedure. Such a definition has the (to my ears, at least) musical quality of embedding it in a real time activity, takes into consideration the variables of musical performance and audition, and distances us productively away from the viewpoint that would hold that a given temperament is "the same as" Just Intonation.

Let me be clear that I have a serious interest in approximations of intervals, intervallic tolerance, and in the compositional use of intervallic ambiguity or puns. But when I conceive a score in Just Intonation I want to be able to compose, perform, and listen to it with what might be called a Just Intonation orientation or teleology (without excluding the possiblity of ambiguities and puns in a JI context), and I believe that it is both musically useful and ethically important to distinguish this from other orientations. This may well be, for other musicians, a distinction of negligible dimension, but it has increasingly proved useful, if not central, to my work.

Daniel Wolf

Hi Daniel,

Good to hear from you.

--- In tuning@yahoogroups.com, Daniel Wolf <djwolf@s...> wrote:
> In my opinion, a more useful definition of Just Intonation
identifies it
> with a _process_ of tuning ever-closer

Yes!

> to some rational ideal, rather

Ah! Now we have a problem. Would you count tuning one's major thirds
ever closer to the rational ideal of 504:635 as an example of the
process of just intonation. These are theoretically only 0.0006 of a
cent from the major thirds of 12-equal and in practice utterly
indistinguishable from them.

If not, then wouldn't it be better to say "to some just or pure ideal"
or "to some beatless ideal", rather than "to some rational [number]
ideal"?

> rather
> than the specific result of such a procedure. Such a definition has
the
> (to my ears, at least) musical quality of embedding it in a real time
> activity, takes into consideration the variables of musical performance
> and audition,

This is good. So primarly Just Intonation is something you _do_ when
tuning one pitch against another, to the best of your ability at a
particular time and place. So degrees of sucess must be admitted, and
therefore degrees of justness.

From there we go to describing certain intervals (in the abstract),
that are close to certain sizes, as just intervals or JI intervals,
and from there we can describe certain scales (in the abstract) as JI
scales because their pitches are fully connected, with more than
merely linear connectivity, by just intervals.

> and distances us productively away from the viewpoint that
> would hold that a given temperament is "the same as" Just
> Intonation.

One doesn't need to invoke the purely mathematical concept of rational
number (with its attendant problems of inaudibility and
immeasurability) in order to distance oneself from this viewpoint. I
assume you saw that I did so in a recent discussion with Gene Ward Smith.

> Let me be clear that I have a serious interest in approximations of
> intervals, intervallic tolerance, and in the compositional use of
> intervallic ambiguity or puns. But when I conceive a score in Just
> Intonation I want to be able to compose, perform, and listen to it with
> what might be called a Just Intonation orientation or teleology
(without
> excluding the possiblity of ambiguities and puns in a JI context),
and I
> believe that it is both musically useful and ethically important to
> distinguish this from other orientations. This may well be, for other
> musicians, a distinction of negligible dimension, but it has
> increasingly proved useful, if not central, to my work.

That makes perfect sense. However there is no more reason that a
sufficiently gentle temperament should prevent such a JI orientation
than do the normal small random inaccuracies of actual instrument tunings.

I agree there can be a sharp line between rational and tempered
orientations to composition, but there is no sharp line between
rational and tempered in the actual tuning of most instruments or in
our ability to distinguish them by listening or measurement.

-- Dave Keenan

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> This is good. So primarly Just Intonation is something you _do_ when
> tuning one pitch against another, to the best of your ability at a
> particular time and place. So degrees of sucess must be admitted, and
> therefore degrees of justness.

The trouble with this definition is that pitchbends in a midi file or
Csound frequency statements are no longer just intonation, because you
don't tune any instrument.

> > Let me be clear that I have a serious interest in approximations of
> > intervals, intervallic tolerance, and in the compositional use of
> > intervallic ambiguity or puns. But when I conceive a score in Just
> > Intonation I want to be able to compose, perform, and listen to it
with
> > what might be called a Just Intonation orientation or teleology
> (without
> > excluding the possiblity of ambiguities and puns in a JI context),
> and I
> > believe that it is both musically useful and ethically important to
> > distinguish this from other orientations. This may well be, for other
> > musicians, a distinction of negligible dimension, but it has
> > increasingly proved useful, if not central, to my work.
>
> That makes perfect sense. However there is no more reason that a
> sufficiently gentle temperament should prevent such a JI orientation
> than do the normal small random inaccuracies of actual instrument
tunings.

That depends on whether just intonation is for the ear or for the
mind. People seem willing to accept 49152 equal as JI, since it is not
chosen for the purpose of approximation as such, but is simply how
close pitch bending allows you to get. On the other hand, 3125 equal
would not by many people be considered JI, since you are deliberately
chosing the numbers to give good approximations. The fact that you may
be unable to tell any difference may not be good enough, because you
feel unsure of your ability to exclude puns. This raises the question
of what you call it when you use pure ratios and then allow puns, of
course. It also raises the question of when you should quit worrying
about puns, because they have become impractical. 2401/2400 is a very
small interval, but it is quite easy to pun with. 3125-et would allow
78125000/78121827 as a pun. Is this complex enough to exclude, thereby
making 3125 effectively pun free?

> I agree there can be a sharp line between rational and tempered
> orientations to composition, but there is no sharp line between
> rational and tempered in the actual tuning of most instruments or in
> our ability to distinguish them by listening or measurement.

I'm not clear the line is always so sharp. It seems to me it got
blurry when someone first noticed that in a pure Pythagoran scale,
C:Fb made an awfully good major third.

hi Dave,

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> I agree there can be a sharp line between rational
> and tempered orientations to composition, but there is
> no sharp line between rational and tempered in the
> actual tuning of most instruments or in our ability
> to distinguish them by listening or measurement.

so in other words, you're proposing that -- excepting
digitally produced music which *can* be quantified
precisely -- we can *never* be totally sure exactly
what we are perceiving mathematically of a musical
composition's tuning?

sounds like a "Keenan Uncertainty Principle" of tuning.

:)

given my ideas on "finity" and "xenharmonic bridges",
it should be easy to see that i agree with this.

-monz

Hi Dave!
The problem I have with your argument Dave is that you are taking a case that might never appear.
It is if i started talking about ET in the billions and claim that there is no difference between it and the continuum, reducing the idea of tuning at all to absurdity.
It seem that the definition of JI is best done by people who are doing it?

Also the assumption that a difference of .0006 is not meaningful, it all depends on how many of these you are using.
Until people actually use such things it is unscientific to assert we know what the results will be. Also as i pointed out in my last post is that 12 ET is not always tuned any more exact than JI so .0006 might not be what one would end up with if one tuned up both.

The tuning world already got bitten on the ass for assuming the 768 was as fine of resolution one needed.
Even those who objected did not know for sure exactly what would happen until it was done.
Until it is done and observed, we do not know

The idea that JI must be tunable by ear. there is no such stipulation about ET. What if i said that ET can only be tuned by machine which is just as reasonable, well throw out all those guitars (which i have to say as an aside, is one the worse instrument to realize a tuning, though i can be done) .

What about all the pieces by La Monte Young holding chords of high harmonics by machine. It is questionable that these could be tuned by ear, at least in a persons or instruments lifetime.
There are all these JI scales that you are generating by omitting commas ( a most untunable method in its very conception) are these not JI?

What if we limited Ets to the number that we can actually realize? Well all of Gene's work would be meaningless.

tuning@yahoogroups.com wrote:

>
> >
>
>Message: 6 > Date: Fri, 03 Jun 2005 23:00:39 -0000
> From: "Dave Keenan" <d.keenan@bigpond.net.au>
>Subject: Definitions of JI (was: Digest Number 3537)
>
>Hi Daniel,
>
>Good to hear from you.
>
>--- In tuning@yahoogroups.com, Daniel Wolf <djwolf@s...> wrote:
> >
>>In my opinion, a more useful definition of Just Intonation
>> >>
>identifies it > >
>>with a _process_ of tuning ever-closer
>> >>
>
>Yes!
>
> >
>>to some rational ideal, rather >> >>
>
>Ah! Now we have a problem. Would you count tuning one's major thirds
>ever closer to the rational ideal of 504:635 as an example of the
>process of just intonation. These are theoretically only 0.0006 of a
>cent from the major thirds of 12-equal and in practice utterly
>indistinguishable from them.
>
>If not, then wouldn't it be better to say "to some just or pure ideal"
>or "to some beatless ideal", rather than "to some rational [number]
>ideal"?
>
> >
>>rather
>>than the specific result of such a procedure. Such a definition has
>> >>
>the > >
>>(to my ears, at least) musical quality of embedding it in a real time >>activity, takes into consideration the variables of musical performance >>and audition,
>> >>
>
>This is good. So primarly Just Intonation is something you _do_ when
>tuning one pitch against another, to the best of your ability at a
>particular time and place. So degrees of sucess must be admitted, and
>therefore degrees of justness.
>
>>From there we go to describing certain intervals (in the abstract),
>that are close to certain sizes, as just intervals or JI intervals,
>and from there we can describe certain scales (in the abstract) as JI
>scales because their pitches are fully connected, with more than
>merely linear connectivity, by just intervals.
>
> >
>>and distances us productively away from the viewpoint that >>would hold that a given temperament is "the same as" Just >>Intonation.
>> >>
>
>One doesn't need to invoke the purely mathematical concept of rational
>number (with its attendant problems of inaudibility and
>immeasurability) in order to distance oneself from this viewpoint. I
>assume you saw that I did so in a recent discussion with Gene Ward Smith.
>
> >
>>Let me be clear that I have a serious interest in approximations of >>intervals, intervallic tolerance, and in the compositional use of >>intervallic ambiguity or puns. But when I conceive a score in Just >>Intonation I want to be able to compose, perform, and listen to it with >>what might be called a Just Intonation orientation or teleology
>> >>
>(without > >
>>excluding the possiblity of ambiguities and puns in a JI context),
>> >>
>and I > >
>>believe that it is both musically useful and ethically important to >>distinguish this from other orientations. This may well be, for other >>musicians, a distinction of negligible dimension, but it has >>increasingly proved useful, if not central, to my work.
>> >>
>
>That makes perfect sense. However there is no more reason that a
>sufficiently gentle temperament should prevent such a JI orientation
>than do the normal small random inaccuracies of actual instrument tunings.
>
>I agree there can be a sharp line between rational and tempered
>orientations to composition, but there is no sharp line between
>rational and tempered in the actual tuning of most instruments or in
>our ability to distinguish them by listening or measurement.
>
>-- Dave Keenan
>
>
>
>
>________________________________________________________________________
>________________________________________________________________________
>
>Message: 7 > Date: Sat, 04 Jun 2005 01:03:39 -0000
> From: "Gene Ward Smith" <gwsmith@svpal.org>
>Subject: Re: Definitions of JI (was: Digest Number 3537)
>
>--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> >
>>This is good. So primarly Just Intonation is something you _do_ when
>>tuning one pitch against another, to the best of your ability at a
>>particular time and place. So degrees of sucess must be admitted, and
>>therefore degrees of justness.
>> >>
>
>The trouble with this definition is that pitchbends in a midi file or
>Csound frequency statements are no longer just intonation, because you
>don't tune any instrument.
>
> >
>>>Let me be clear that I have a serious interest in approximations of >>>intervals, intervallic tolerance, and in the compositional use of >>>intervallic ambiguity or puns. But when I conceive a score in Just >>>Intonation I want to be able to compose, perform, and listen to it
>>> >>>
>with > >
>>>what might be called a Just Intonation orientation or teleology
>>> >>>
>>(without >> >>
>>>excluding the possiblity of ambiguities and puns in a JI context),
>>> >>>
>>and I >> >>
>>>believe that it is both musically useful and ethically important to >>>distinguish this from other orientations. This may well be, for other >>>musicians, a distinction of negligible dimension, but it has >>>increasingly proved useful, if not central, to my work.
>>> >>>
>>That makes perfect sense. However there is no more reason that a
>>sufficiently gentle temperament should prevent such a JI orientation
>>than do the normal small random inaccuracies of actual instrument
>> >>
>tunings.
>
>That depends on whether just intonation is for the ear or for the
>mind. People seem willing to accept 49152 equal as JI, since it is not
>chosen for the purpose of approximation as such, but is simply how
>close pitch bending allows you to get. On the other hand, 3125 equal
>would not by many people be considered JI, since you are deliberately
>chosing the numbers to give good approximations. The fact that you may
>be unable to tell any difference may not be good enough, because you
>feel unsure of your ability to exclude puns. This raises the question
>of what you call it when you use pure ratios and then allow puns, of
>course. It also raises the question of when you should quit worrying
>about puns, because they have become impractical. 2401/2400 is a very
>small interval, but it is quite easy to pun with. 3125-et would allow
>78125000/78121827 as a pun. Is this complex enough to exclude, thereby
>making 3125 effectively pun free? >
> >
>>I agree there can be a sharp line between rational and tempered
>>orientations to composition, but there is no sharp line between
>>rational and tempered in the actual tuning of most instruments or in
>>our ability to distinguish them by listening or measurement.
>> >>
>
>I'm not clear the line is always so sharp. It seems to me it got
>blurry when someone first noticed that in a pure Pythagoran scale, >C:Fb made an awfully good major third.
>
>
>
>
>________________________________________________________________________
>________________________________________________________________________
>
>Message: 8 > Date: Fri, 3 Jun 2005 22:55:11 -0400
> From: Cris Forster <76153.763@compuserve.com>
>Subject: D'Alembert and String Tension Equations
>
>Dear Fellow Builders,
>
>Before I begin, please note that due to email font limitations, in the >following equations, the unique symbol ( _| ) represents the >Square Root Symbol.
>
>A very popular equation for the mode frequencies of flexible >strings
>
>[[ Eq. i(a) ]]: F(n) = ( n / ( 2*L )) * _| ( T / Mul )
>
>states: The frequency of a given mode F(n), equals >the quantity of the mode number ( n ), any integer, divided by 2 >times the string Length, multiplied by the quantity of the square >root of the string Tension divided by the string Mass per unit >length.
>
>Since Mul = Pi * r^2 * rho
>
>we also have:
>
>[[ Eq. i(b) ]]: F(n) = ( n / ( 2*L )) * _| ( T / (Pi * r^2 * rho)) >
>We may solve Equation i(b) for the following three variables:
>
>Acoustically correct: T = 4 * F^2 * L^2 * Pi * r^2 * rho
>Acoustically correct: L = (1 / ( F * D )) * _| ( T / ( Pi * rho ))
>Acoustically correct: D = (1 / ( F * L )) * _| ( T / ( Pi * rho ))
>
>where ( D ) is the diameter of the string.
>
>******************************
>******************************
>
>Only these three solutions --
>which ubiquitously exclude the mode number squared ( n^2 ) -- >are correct.
>
>A given string has only one value for T; a given string has only >one value for L; and a given string has only one value for D.
>In other words, for the upper modes of vibration, T does not and >cannot change, L does not and cannot change, and D also does >not and cannot change.
>
>******************************
>******************************
>
>In direct contrast, refer to Equation i(b), and note that although the >following solution for T is algebraically correct >
>Acoustically Incorrect: T = (4 * F^2 * L^2 * Pi * r^2 * rho) / n^2
>
>it is _acoustically_ incorrect and, therefore, does not constitute a >solution, or solutions, for the variable T of a flexible string.
>
>To understand an acoustically correct solution, consider the >following alternative for the mode frequencies of flexible strings:
>
>[[ Eq. ii ]]: F(n) = c(t) / [ 2*L / (n) ]
>
>where c(t) is the speed of transverse waves in a string, in >meters/second.
>
>The following quote is from my manuscript
>_Musical Mathematics: A Practice in the Mathematics of Tuning >Instruments and Analyzing Scales_, Chapter 10:
>
>******************************
>
>The solution to this problem came from Jean le Rond D'Alembert >(1717-1783), who solved a second order partial differential >equation, known as the _wave equation_, which enabled him to >formulate an equation for the constant speed of transverse >traveling waves in flexible strings. (See Equation 3.11.) >D'Alembert also created a purely mathematical model to show that >standing waves in strings are caused by the superposition of >traveling waves that propagate in opposite directions.
>(See Chapter 3, Figure 6.) This principle applies to all acoustic >sound-producing systems.
>
>******************************
>
>D'Alembert's famous solution of the _wave equation_
>
>c(t) = _| ( T / Mul)
>
>states: The speed of transverse waves in a string equals the >quantity of the square root of the string Tension divided by the >string Mass per unit length.
>
>The following quote is from -- University Physics, 7th ed. Addison-
>Wesley Publishing Company, Reading, Massachusetts, 1988. -- >one of the finest physics textbooks used throughout the US:
>
>******************************
>
>p. 501: "The wave speed ( c ) is the same for all frequencies."
>
>******************************
>
>Since c(t) is the same for all frequencies, the tension ( T ) must be >constant for all frequencies, and, therefore, does not and cannot >change; and the mass per unit length ( Mul ) must also be >constant for all frequencies, and, therefore, does not and cannot >change.
>
>In Equation ii, the quantity [ 2*L / (n) ] describes how -- due to the >superposition of transverse traveling waves propagating in >opposite directions -- the wavelengths of the upper modes of >vibrations _decrease_ in length. Therefore, ( n ) is a variable that >exclusive modifies 2*L. It categorically never modifies: T, L, or D.
>
>For the first mode of vibration, wavelength (lambda) = (2*L) / 1;
>for the second mode of vibration, lambda = (2*L) / 2;
>for the third mode of vibration, lambda = (2*L) / 3; etc.
>
>Given the strict acoustic context of [ 2*L / (n) ],
>only the mode number variable ( n )
>accounts for increases in the mode frequencies of strings.
>
>Given the string properties in my Tuning Group Message # 58811, >solutions for Equation ii are as follows:
>
>For F(1): 392.0 Hz = _| (843.8 N / .00574 kg/m) / ((2 * .489 m) / 1);
>for F(2): 784.0 Hz = _| (843.8 N / .00574 kg/m) / ((2 * .489 m) / 2);
>etc.
>
>Therefore, in my Tuning Group Message # 58798,
>only the following four equations for Tension are correct:
>
>((1)) T = F^2 * L^2 * D^2 * Pi * rho
>
>((2)) T = 4 * F^2 * L^2 * Pi * r^2 * rho
>
>((3)) T = 4 * F^2 * L^2 * M/u.l.
>
>((4)) T = 4 * F^2 * L * M
>
>These four equations exclude the mode number squared ( n^2 ) >because all flexible strings have only one tension, or, equivalently, >have only one transverse wave speed for all modes of vibration.
>
>******************************
>
>With respect to the speed of longitudinal waves in strings, the >following equation
>
>[[ Eq. iii ]]: c(l) = _| (E / rho)
>
>states: The speed of a longitudinal wave equals the >quantity of the square root of the stringing material modulus of >elasticity ( E ) divided by the stringing material mass density ( rho). >Here there exists only one longitudinal wave speed for all strings >made from the same material.
>
>******************************
>
>Sincerely,
>
>
>Cris Forster, Music Director
>www.Chrysalis-Foundation.org
>
>http://www.Chrysalis-Foundation.org/musical_mathematics.htm
>
>
>
>
>
>________________________________________________________________________
>________________________________________________________________________
>
>Message: 9 > Date: Sat, 04 Jun 2005 09:06:51 -0000
> From: "monz" <monz@tonalsoft.com>
>Subject: Re: Definitions of JI (was: Digest Number 3537)
>
>hi Dave,
>
>--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> >
>>I agree there can be a sharp line between rational
>>and tempered orientations to composition, but there is
>>no sharp line between rational and tempered in the
>>actual tuning of most instruments or in our ability
>>to distinguish them by listening or measurement.
>> >>
>
>
>so in other words, you're proposing that -- excepting
>digitally produced music which *can* be quantified
>precisely -- we can *never* be totally sure exactly >what we are perceiving mathematically of a musical
>composition's tuning?
>
>sounds like a "Keenan Uncertainty Principle" of tuning.
>
>:)
>
>
>given my ideas on "finity" and "xenharmonic bridges",
>it should be easy to see that i agree with this.
>
>
>-monz
>
>
>
>
> >
>
>
>
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>
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--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

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

> What if we limited Ets to the number that we can actually realize? Well
> all of Gene's work would be meaningless.

If you'd ever bothered to learn anything about my work you'd know how
utterly false this is. If you don't want to learn anything of recent
developments, please at least do not mischaracterize what people have
been doing or the music people have been writing. If you are under the
impression you already know it all when it comes to tuning or at least
JI, I don't think so.

-Gene- If you looked at what i said, it is a defence of your work.
that you exceed the limits of ETs beyond what is easily accessable to
most.
It is something that we or at least you should do.
Yasser is an example of a brilliant mind who limited the extent of his
vision by too much practicality at the time.
We had just come out of the depression which i imagine also had some
bering on this.
I am sure he might have looked at those higher in his own series if
he had lived at a different time.
I admit that is quite allot of your work i do not understand.
That is because i am not familiar with certain mathematical jargon.
While we are on the subject,
Often i wish you would show an example or two for more often than not
i can see things and poatterns easier than grasp them in such a languange

-- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
>
> > What if we limited Ets to the number that we can actually realize?
Well
> > all of Gene's work would be meaningless.
>
> If you'd ever bothered to learn anything about my work you'd know how
> utterly false this is. If you don't want to learn anything of recent
> developments, please at least do not mischaracterize what people have
> been doing or the music people have been writing. If you are under the
> impression you already know it all when it comes to tuning or at least
> JI, I don't think so.

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> hi Dave,
>
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > I agree there can be a sharp line between rational
> > and tempered orientations to composition, but there is
> > no sharp line between rational and tempered in the
> > actual tuning of most instruments or in our ability
> > to distinguish them by listening or measurement.
>
>
> so in other words, you're proposing that -- excepting
> digitally produced music which *can* be quantified
> precisely -- we can *never* be totally sure exactly
> what we are perceiving mathematically of a musical
> composition's tuning?
>
> sounds like a "Keenan Uncertainty Principle" of tuning.
>
> :)

It isn't mine. Maybe all you need is Cantor's theorem or something.
The laws of Thermodynamics just make things worse.

What is it Gene? You know what we're talking about here don't you? It
might be characterised as "the immeasurability of (numerical)
rationality", or "there's always one more decimal place", or "no
matter how irrational your number and no matter how precise your
tolerance there's always a ratio closer to it than your tolerance".

It just seems self-evident to me, but maybe there's some way to make
it clearer to people.

But then we have people who understand this but then say that it
doesn't matter, because what determines whether an interval is just or
not is someone's _intention_. If someone _intended_ there to be a
strict ratio between frequencies, no matter how complex that ratio,
then the interval is just.

In other words, justness is inaudible. I find this to be an extreme
departure from the original meaning of the word "just" in the context
of musical harmony.

It's also unclear to me who needs to do the intending.

Does this mean that if I merely _intend_ my acoustic piano to be tuned
to 5-limit JI then it is? Even though the tuner misunderstood me and
actually tuned it to 12-equal. No, it will become clear that it is not
5-limit-JI the moment I start playing it.

So there must be some criteria for success or otherwise in _realising_
the intention. Maybe in the case of a piano you would consider the
intention to have been realised if it was within say +-2 cents, or
maybe you're incredibly strict and insist on +- 0.2 cents (and so must
have it retuned immediately before every performance), but whatever
the case you must accept some tolerance. Since you are unwilling to
allow infinite _time_ for frequency measurements to be made, then you
must accept finite accuracy (this is the classical uncertainty principle).

But what if instead of 5-limit JI I _intend_ the piano (which a moment
ago everyone agreed was tuned to an exceedingly accurate 12-equal) to
be tuned to certain complex rational ratios which differ from
theoretical 12-equal only in the hundredth decimal place of cents?
Does it then become a JI piano?

Or is it the intention of the composer of the piece being played that
determines whether it is JI or not? Or the intention of the designer
of the rational scale that the composer chose to use (without
understanding anything about ratios)? Or the intention of the
performer, or the intentions of each listener individually?

Or is it the instrument tuner's intention that matters? Must I call
the tuner back and give her the list of ratios and ask her to retune
it to those ratios, even though when she gets out her calculator and
does the necessary calculations she is going to give me some funny
looks and tell me that the piano is already tuned to those ratios.

Must I insist that she first mistune it and then retune it with the
_intention_ that it be tuned to those ratios, and if she says that she
somehow just can't seem to muster up that intention because she
_knows_ she will actually be tuning it to 12-equal, should I find
another tuner (who might lie about this so as to get the job).

Doesn't it seem like there is something just a little bit wrong with
basing determinations of justness solely on _intention_ to tune by
rational ratios (no matter how complex).

Intention is important in a court of law regarding the difference
between murder and manslaughter, but the intended victim still
actually has to be measurably dead for it to be called murder. Whereas
"a measurably rational number" is an oxymoron.

As monz mentioned, although one can't tell by measurement, it is
possible to know that two frequencies are related by a strictly
rational ratio if one know that they are produced by a mechanism that
logically guarantees this, such as digital counters with a common
clock, or even an acoustic instrument that uses resonators to extract
different harmonics from a single continuously-driven oscillator such
as bowed string or wind.

But if we were to insist that henceforth only such instruments could
be called JI we'd be flying in the face of an awful lot of historical
usage of the term. And there's also the point that when one combines
notes which are completely phase-locked like this, it can sound more
like one is just creating different timbres rather than intervals or
chords. It appears that it is the slow beat or drift in phase of one
note against the other that allows them to retain some degree of
independent existence when combined.

-- Dave Keenan

--- In tuning@yahoogroups.com, "banaphshu" <kraiggrady@a...> wrote:

> -Gene- If you looked at what i said, it is a defence of your work.
> that you exceed the limits of ETs beyond what is easily accessable to
> most.

Thanks for wanting to defend my work, but using ets of large
dimensions is hardly everything I do or consider.

> I admit that is quite allot of your work i do not understand.

Personally, I think if you can figure out Erv you can figure out the
tuning-math stuff if you want to put your mind to it.

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> What is it Gene? You know what we're talking about here don't you? It
> might be characterised as "the immeasurability of (numerical)
> rationality", or "there's always one more decimal place", or "no
> matter how irrational your number and no matter how precise your
> tolerance there's always a ratio closer to it than your tolerance".

Density of the rationals. An ordered set is said to be "dense" if
for every x < y there is a z such that x < z < y. This entails the
density of the rationals in the topological sense, which is closely
related, so you can say "density" meaning either and come up winners.

> In other words, justness is inaudible. I find this to be an extreme
> departure from the original meaning of the word "just" in the context
> of musical harmony.

Well, mathematicians have a saying that it isn't the object, it's the
morphism. In this case it isn't the actual note, it is the mapping
from some group to the note, and a certain mathematical fact about the
nature of that mapping.

> But what if instead of 5-limit JI I _intend_ the piano (which a moment
> ago everyone agreed was tuned to an exceedingly accurate 12-equal) to
> be tuned to certain complex rational ratios which differ from
> theoretical 12-equal only in the hundredth decimal place of cents?
> Does it then become a JI piano?

It becomes a Kirnberger tuning, probably, with fifths tuned to
16384/10935. How in the world do you propose to do this, though? An
electronic piano?

> But if we were to insist that henceforth only such instruments could
> be called JI we'd be flying in the face of an awful lot of historical
> usage of the term. And there's also the point that when one combines
> notes which are completely phase-locked like this, it can sound more
> like one is just creating different timbres rather than intervals or
> chords. It appears that it is the slow beat or drift in phase of one
> note against the other that allows them to retain some degree of
> independent existence when combined.

Maybe the absence of any perceptible slow beat or drift *is* JI.

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> Hi Dave!
> The problem I have with your argument Dave is that you are taking a
case
> that might never appear.
> It is if i started talking about ET in the billions and claim that
> there is no difference between it and the continuum,

There is a mathematical difference but do you really think there is
any difference that matters for musical purposes? Surely you don't
think there is any human on the planet who could _hear_ the difference
between an ascending or descending scale in 1-billion-ET and a
continuous sweep? (Assuming there are no first-order discontinuities
(vertical jumps in the waveform) as you go from one pitch to the next).

> reducing the idea
> of tuning at all to absurdity.

Why would recognising that 1-billion-ET is indistinguishable from the
continuum for all musical purposes render the idea of all tuning absurd?

> It seem that the definition of JI is best done by people who are
doing it?
>

Shouldn't people who are listening to it have some say? Particularly
if they are paying customers (I know, they are all too few :) - paying
specifically to hear that special (audible) quality they call just
iontonation - that special buzz that a just interval has.

> Also the assumption that a difference of .0006 is not meaningful, it
> all depends on how many of these you are using.
> Until people actually use such things it is unscientific to assert we
> know what the results will be.

I think enough scientific tests have been done on enough people to say
that if you think 0.0006 of a cent can be musically significant then
the onus is on you to prove it. But even if we were to find that 0.006
of a cent _can_ be significant, I think it would be unscientific to
suggest that there is _no_ number small enough that it might not be
found to be significant. In principle one would have to test every
human being on the planet to see if they can detect it, and maybe
you'll say that with new humans being born all the time you never know
when one will be born that can detect it, but surely this is all
irrelevant because the great mass of any audience will have
discimination within 3 standard deviations of the average, and we
already have a good-enough idea what that discrimination is like.

Science might currently be unduly pessimistic and be putting the
figure too high, but that doesn't mean there is no such figure. Pick a
number for the smallest significant mistuning as low as you like,
except zero, and my argument still stands.

> Also as i pointed out in my last post is
> that 12 ET is not always tuned any more exact than JI so .0006 might
not
> be what one would end up with if one tuned up both.

I totally agree. But if they are both likely to end up a cent or two
away from theoretical, doesn't that make such a distinction (between a
12-equal major third and a supposedly JI 504:635) seem even less
meaningful?

> The tuning world already got bitten on the ass for assuming the 768
was
> as fine of resolution one needed.

Sure. That was bad, and a measure of the cost/benefit tradeoff at the
time. But that doesn't mean that no sufficiently fine resolution exists.

> Even those who objected did not know for sure exactly what would
happen
> until it was done.
> Until it is done and observed, we do not know

True, but by going a factor of a thousand (or a million or whatever)
beyond anything anyone can imagine mattering, we can be pretty damn sure.

> The idea that JI must be tunable by ear. there is no such stipulation
> about ET. What if i said that ET can only be tuned by machine which is
> just as reasonable, well throw out all those guitars (which i have to
> say as an aside, is one the worse instrument to realize a tuning,
though
> i can be done) .

The phrase "tuning by ear" is not the whole of my proposed definition
of justness (far from it). I merely used it as a snappy phrase to
distinguish it from the main alternative "tuning by ratios", which is
why I placed both in scare-quotes.

I am aware that it needs many conditions and caveats to make it a
useful defintion. I have mentioned several on this list recently.

At the moment I'd just be happy if I could convince people that the
"tuning by ratios" definition has a serious problem.

I personally love that special sound that you only get from just
intervals and chords, I don't expect a piece to consist of nothing
but, but if I pay for a CD or attend concert that claims to present
music in just intonation, I expect to hear at least _some_ examples of
this special sound. I do not hear this special sound at all in
12-equal or anything closely approximating it, except for its bare
octaves, fifths and fourths, but they are too boring for me to pay
specially to hear.

Now if you tell me that something that is audibly completely
indistinguishable from 12-equal can meaningfully be advertised as JI
then what can I do? I can't afford the time to audition every such CD
or concert before I pay my money.

>
> What about all the pieces by La Monte Young holding chords of high
> harmonics by machine. It is questionable that these could be tuned by
> ear, at least in a persons or instruments lifetime.
> There are all these JI scales that you are generating by omitting
> commas ( a most untunable method in its very conception) are these
not JI?
>

Yes, 12-equal can be tuned by ear, by counting beats. Obviously that's
not included as JI in my definition. Yes a JI microtemperament may
need to be tuned by machine, but the point is that its intervals still
sound much like they sound when tuned by ear.

Gene raise the same point. My definition of a just interval is one
that _can_ be tuned by ear so as to sound nearly beatless, or at least
to have "that special sound", which may be difficult to capture in
words but can easily be shown to anyone. If I set up one fixed note
and sweep another one, I can say to anyone listening with me, you hear
the special quality of this interval, and this one, and this one. If
they have normal human hearing they wil say yes. And then I can say,
now find me some others that have that same kind of quality, and they
will do it. And then we will probably start to find some (near ratios
of medium compelxity) where we are not quite sure, etc.

> What if we limited Ets to the number that we can actually realize? Well
> all of Gene's work would be meaningless.

Yes. Let's do that. ;-) What should the limit be? I remind you that I
have indeed pooh-poohed talk of very large ETs in the past.

I note that Johnny Reinhard once said on this list that he probably
can't distinguish pitches 1/3 of a cent apart. So that would mean we
shouldn't talk of ETs larger than 3600 since they are effectively
continua.

I note that George Secor and I are taking the distinguishability of
1/3 of a cent seriously enough to ensure that the Sagittal notation is
capable of that sort of resolution.

-- Dave Keenan

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

> > But what if instead of 5-limit JI I _intend_ the piano (which a moment
> > ago everyone agreed was tuned to an exceedingly accurate 12-equal) to
> > be tuned to certain complex rational ratios which differ from
> > theoretical 12-equal only in the hundredth decimal place of cents?
> > Does it then become a JI piano?
>
> It becomes a Kirnberger tuning, probably, with fifths tuned to
> 16384/10935.

That only differs in the 3rd decimal place of cents. And isn't this
tuning of Kirnberger's called equal temperament, not JI?

> How in the world do you propose to do this, though? An
> electronic piano?

I specifically said it was acoustic. I don't expect it to be tuned
accurately to the hundredth decimal place of cents, or even the third
place. Only as accurately as any other JI piano.

In other words I consider this to be a reductio ad absurdum of the
idea that 504:635 can be considered a "just" interval. Because if it
isn't a reductio ad absurdum, then I think I'm going to go into the
mail order business of reselling 12-equal instruments as JI. When the
customers complain I'll just send them a list of big ratios.

> > But if we were to insist that henceforth only such instruments could
> > be called JI we'd be flying in the face of an awful lot of historical
> > usage of the term. And there's also the point that when one combines
> > notes which are completely phase-locked like this, it can sound more
> > like one is just creating different timbres rather than intervals or
> > chords. It appears that it is the slow beat or drift in phase of one
> > note against the other that allows them to retain some degree of
> > independent existence when combined.
>
> Maybe the absence of any perceptible slow beat or drift *is* JI.

Not the complete absence, only its unobtrusiveness, otherwise people
would be asking for their money back on _any_ acoustic instrument that
was sold as JI, including the ones that most people agree _really_are_ JI.

-- Dave Keenan

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> I note that George Secor and I are taking the distinguishability of
> 1/3 of a cent seriously enough to ensure that the Sagittal notation is
> capable of that sort of resolution.

There you go. Go up to 2460, and then stop, calling the result JI.

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> > It becomes a Kirnberger tuning, probably, with fifths tuned to
> > 16384/10935.

> That only differs in the 3rd decimal place of cents. And isn't this
> tuning of Kirnberger's called equal temperament, not JI?

Right, so it is accurate to within 1/100 of a cent, which is what I
thought you meant. 10^(-100) cents is physically meaningless. As for
what to call it, that's the question.

> Not the complete absence, only its unobtrusiveness, otherwise people
> would be asking for their money back on _any_ acoustic instrument that
> was sold as JI, including the ones that most people agree
_really_are_ JI.

I dunno; I kind of like a Jonnie Cochran here--if it don't phase-lock,
you must acquit. But that opens up a big can of worms, as you note above.

From: "Dave Keenan" <d.keenan@bigpond.net.au> Subject: Re: Definitions of JI

Hi Kraig,

Why would I want to do that? because he claims he is doing JI. But it is very accurate, but does not sound like any other JI there is Try this scale. Does it sound like JI to you? Or will you say it is JI
without even listening, simply because it is expressed as ratios?

I do not believe that JI has a sound any more than any other family of scales does.
If someone uses this scale and calls it JI , it merely means it broadens are view and concept of what JI is.
Yes i do not have to listen at all Mystery scale
7
!
202/181
153/124
175/128
199/130
261/154
211/112
194/93

No true octave and 1/1 is implied as usual for Scala .scl files.

>> the question is ambiguity.
> >

I don't understand what you mean by this. Please explain.
For me JI produces definable intervals that lack the ambiguity that ETs do, where a single ET interval will functions as two different ratios. it use for me is to clearly define a structure.
Basically my problems and concern are the exact opposite of what most of those investigating ETs are looking for.
Where you can tune out comma, i want to have as many as possible. While you might want to tune out tones that are 4.5 Cents away in the eikosany, i am concern with keeping these differances as they are structurally and musically meaningful.
>> there is nothing in using ratios that requires the use or desire or >> no beats, at least for the last hundred years
> >

And if a JI scale has more than about 6 notes then it can't help but
contain many non-just intervals. You could still call these "JI scale
intervals", but that doesn't make them Just intervals.

I have found what you call non just intervals , some of my favotite.
Many times i prefer a 40/27 to any of the alternatives, in fact it defines where i am in my scale, just like a tritone does in a major scale.
Get rid of the tritopne and you do not know where you are. And a just interval doesn't have to be beatless, it only has to be
close enough to a simple enough ratio that at least with _some_
harmonic timbre, in _some_ chordal context, you can hear that the
beating of _some_ harmonics is slow enough so as to give rise to that
distinctive Just sound.

I don't know when a significant number of people started saying that
the justness of an interval wasn't any quality you can hear, but
instead a purely mathematical property. Maybe it was a hundred years
ago, but maybe it only started around when the first digital
synthesizers became available. In any case, there was an awful lot of
time before a hundred years ago, when the justness of an interval
meant something you could actually _hear_. I see no reason to give
that up.

>> And one again why should we not apply the same standard to ET, that >> is is only tunable by Machine.
> >

Because
(a) ET usn't the opposite of JI, non-JI is. Some JI scales are subsets
of large ETs. Some ETs are JI.
yes at the infinty of the continuum but that tells us nothing about what they are. then the terms have no meaning at all and we might as well close up the tuning list.
i do not agree in any way.

I personally do not see 19 ET as a JI scale, first of all because you are defining it as 19ET.
there is a question of gestalt in all this. The mind defines things in the simlppliest way.
i could define JI using every word in the dictionary to show it's interelationship with many things, but such a definition is meaningless.
At a certain point a chain of 19 just thirds will not be the same as 19ET.
That you say it is the same is only dbecause we lack enough tuning information. add one more third and see what happens. That you can makes it JI that you can't makes it 19ET.
This shows that the two exist in two completely different structural worlds, one is open , the other is closed beyond a certain limit.
I say again, "tunable by ear" is not my sole criterion for justness.
It was just a snappy phrase to contrast against "tuned to ratios".

Nothing is gained by dropping the tuned by ratios , yet everything is forever closed to progress further if you move away from it. One property that i can say about JI scales that has not been mentioned. It produces scales of unequal sized intervals, if it involves more than one ratio.With all the composers mentioned, this is true. therefore it might be more inclusive. Another property of JI scale can be added on to using the same material already used to construct it to infinity. that at certain points of a JI structure, it will resemble ET structures at the perceptable level.
There is a latter problem with the idea of perception though.
While one may not notice a perceptable differance immediately, With myself, the problems with 31 ET took quite a while of using it for me to run into a problem.
Personally it is a mistake i don't wish to duplicate.
Hopefully this might explain why i will resist. ________________________________________________________________________
________________________________________________________________________

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

________________________________________________________________________
________________________________________________________________________

Message: 16 Date: Tue, 07 Jun 2005 02:09:11 -0000
From: "Gene Ward Smith" <gwsmith@svpal.org>
Subject: Re: Definitions of JI

I've described just chords as having a granite-slab quality, of
steadiness and a kind of purity. Also interesting is the nearly-just
sound, but since 768-et--certainly nearly just, you might say very
nearly just--won't do for Kraig, clearly this isn't what some people
are looking for. Personally I think there is something to be said for
the view that slightly detuned chords, as you get from microtempering
which is not nanotempering, are musically more interesting for most
purposes than JI.

I will say that i like chords that beat more than dead
still sonorities that just sit there.

using Ets because of the properties of they give in this regard are a good way to deal with it.
Just as using higher ratios or in my case recurrent sequences to solve the problem are another,
The two approaches produce different qualites to these beats.
my main problem with 768 is that i just couldn't tell what it was i was hearing, phenomenon of the supposed ratios or an artifact of th e subdivision.
At the time, i wanted to atleast hear what the ratios sounded like.
If you go to an art store to buy oil paint, most people will take off the lid to see what the color looks really like, even if it seems to match what is one the label. OK most people might trust the white and black.
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

The wolfs have a real function and often imply what other intervals are in the scale.
I love wolves. like your dominants and other dissonances in ET , they keep the music going.
In fact the history of music might be the pursuit of people finding sound combinations to keep the music propelled forward.
Indonesian music is a great example of scales that refuse to resolve. that way they can keep going, even for days!

________________________________________________________________________
________________________________________________________________________

Message: 24 Date: Tue, 07 Jun 2005 04:11:28 -0000
From: "Igliashon Jones" <igliashon@sbcglobal.net>
Subject: Re: Definitions of JI

I've yet to see a JI system without a wolf. And rather than trying to set some sort of "threshold" as to how many wolf intervals are allowable in a Just Intonation (which is impractical and will likely never be agreed upon), why not just get rid of the troublesome misnomer and stick with something that can consistently describe these tuning systems? -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

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

> For me JI produces definable intervals that lack the ambiguity that
ETs do, where a single ET interval will functions as two different ratios.

Interestingly enough, this is a purely mathematical definition:
considered as mappings, an et has "puns"--there is something in the
kernel--and p-limit JI does not. However, in practice JI has puns
also. If you use Pythagorean tuning, entirely by thirds, you end up
with an Fb which is a good 5/4; this sort of thing is all over the
place in JI. Normally, the suggestion is that the difference be
averaged out, obtaining a temperament, but with accurate enough
temperaments you don't need to. I've given lots of nanotempered scales
in the form of just intonation scales, which serve to specify them
precisely. If you look at them in Scala, you will find chords with
errors of 0.721 cents, 0.396 cents, and the like. These errors can be
distributed, but they certainly don't have to be and the intervals
involved do indeed function as two different ratios.

> it use for me is to clearly define a structure.

Exactly. It defines a structure, which is to say, the definition is
mathematical. We are always talking about temperaments of various
ranks, meaning the various numbers of generators, and that does the
same thing.

However, the difference is that no one goes on about "pure meantone"
or "pure miracle" as if it was cosmically important that these are
*not* to be tuned to an equal temperament. People can quite happily
keep in mind the structural features of such a temperament while
tuning it in a way which in some theoretical sense totally demolishes
that structure. The structure, after all, is a theoretical
abstraction, where notes have been replaced by numbers, but a number
*is not* a note. You can't hear a number. There is a difference, in
other words, between the mathematical model and the music. The map is
not the territory.

> I have found what you call non just intervals , some of my favotite.
> Many times i prefer a 40/27 to any of the alternatives, in fact it
defines where i am in my scale, just like a tritone does in a major scale.

Well, that tells me you might object to meantone, which merges 40/27
with 3/2. But in a schismatic tuning, there is little difference; and
in one schismatic tuning, 1/12 schisma schismatic, there is no
difference at all in the sense that 81/80 has the same size. If all
you need is the structural presense of this tone, you can have it, and
still be without the structural presence of other tones you may not
really need.

> This shows that the two exist in two completely different
structural worlds, one is open , the other is closed beyond a certain
limit.

That is, once again, purely mathematical. Dave is not trying to define
something in mathematical terms. The abstract concepts are important
and I'm all for them, but there is another question one can ask, which
is what is it that we can hear?

> Nothing is gained by dropping the tuned by ratios , yet everything
is forever closed to progress further if you move away from it.

This seems amazingly dogmatic. How does this answer Dave's point that
you can, after all, tune equal temperament by ratios?

> One property that i can say about JI scales that has not been
mentioned. It produces scales of unequal sized intervals, if it
involves more than one ratio.With all the composers mentioned, this is
true. therefore it might be more inclusive.

This comes down to number theory, and once again is a mathematical
property following from a mathematical definition. However, it is also
true of, for instance, 1/4 comma meantone.

> Another property of JI scale can be added on to using the same
material already used to construct it to infinity.

Ditto the above.

> While one may not notice a perceptable differance immediately, With
myself, the problems with 31 ET took quite a while of using it for me
to run into a problem.

But this really goes to Dave's point--he is not claiming 31 is JI. It
clearly is tempering in any sense of the word.

> Personally it is a mistake i don't wish to duplicate.
> Hopefully this might explain why i will resist.

Not to me.

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

> I will say that i like chords that beat more than dead
> still sonorities that just sit there.

Hey! Now that's interesting. How much detuning is about right, do you
think?

> using Ets because of the properties of they give in this regard are a
> good way to deal with it.

Not just ets, but I think there is a lot to be said in the 7-limit for
99, 130 and 140 equal on those grounds, or for a temperament like
hemiwuerschmidt, come to that.

> The two approaches produce different qualites to these beats.
> my main problem with 768 is that i just couldn't tell what it was i was
> hearing, phenomenon of the supposed ratios or an artifact of th e
> subdivision.

768 is hardly ideal for the purpose, but acoustic instruments are not
a good choice either. Csound will nail it for you. If your sound card
is up to the task of really accurate tuning, so will midi.

Why, the beauty of the acronym of JI would then be transformed into RI, which is hardly aesthetical or compelling for me.

Cordially,
Ozan
----- Original Message -----
From: Igliashon Jones
To: tuning@yahoogroups.com
Sent: 07 Haziran 2005 Salı 7:11
Subject: [tuning] Re: Definitions of JI

> Yes I noticed it says simply JI = rational.

I agree that the two should not "equal" each other. Music can
sound "Just"; it cannot sound "Rational". As far as I can see, the
term "Just" is an adjective to describe the sound of an interval or
chord. The trouble began when it was realized that these Just-
sounding chords were representative of simple integer frequency
ratios: people started reasoning that any chord or interval that
represents simple integer frequency ratios should be called "Just".
Let me put this reasoning into standard form to illustrate its
invalidity:

Premise A: chord X has a certain audible quality which we call
Justness.
Premise B: chord X represents simple integer frequency ratios.
Conclusion: all chords that represent simple integer frequency ratios
will have the quality of Justness.

The conclusion does not at all follow from the premises.

I personally think that the term "Just Intonation" should be replaced
with "Rational Intonation", seeing as how in even the simplest "JI"
systems not all the intervals are uniformly Just in their sound
quality. I've yet to see a JI system without a wolf. And rather
than trying to set some sort of "threshold" as to how many wolf
intervals are allowable in a Just Intonation (which is impractical
and will likely never be agreed upon), why not just get rid of the
troublesome misnomer and stick with something that can consistently
describe these tuning systems?

I challenge anyone to give me a practical reason why the term "Just
Intonation" should not simply be replaced with "Rational
Intonation". What would be lost? What would be compromised? What
effect would it have on the music?

-Igs

________________________________________________________________________ ________________________________________________________________________ Message: 7 Date: Tue, 07 Jun 2005 08:16:00 -0000 From: "Gene Ward Smith" <gwsmith@svpal.org> Subject: Re: Definitions of JI --- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:

>> For me JI produces definable intervals that lack the ambiguity that
> >
ETs do, where a single ET interval will functions as two different ratios.

Interestingly enough, this is a purely mathematical definition:
considered as mappings, an et has "puns"--there is something in the
kernel--and p-limit JI does not. However, in practice JI has puns
also. If you use Pythagorean tuning, entirely by thirds, you end up
with an Fb which is a good 5/4; this sort of thing is all over the
place in JI. Normally, the suggestion is that the difference be
averaged out, obtaining a temperament, but with accurate enough
temperaments you don't need to. I've given lots of nanotempered scales
in the form of just intonation scales, which serve to specify them
precisely. If you look at them in Scala, you will find chords with
errors of 0.721 cents, 0.396 cents, and the like. These errors can be
distributed, but they certainly don't have to be and the intervals
involved do indeed function as two different ratios. *yes i am quite aware of these. and I am quite aware that these are the type of problems you wish to focus on.
Personally i have others. If i want say scales with unequal amount of steps, for instance , why would one use Ets.
>> it use for me is to clearly define a structure.
> >

Exactly. It defines a structure, which is to say, the definition is
mathematical. We are always talking about temperaments of various
ranks, meaning the various numbers of generators, and that does the
same thing. The type of structures generated in JI are different than those generated by ETs, except if we take it to infinity, but that is unusable.
The question is that the two approaches produce different results, although as you point out one can be used to appoximate the other. In reality it fosters different types of music , which is artistically useful

However, the difference is that no one goes on about "pure meantone"
or "pure miracle" as if it was cosmically important that these are
*not* to be tuned to an equal temperament.

No one used the word Pure in this conversation, and these are temperments anyways

People can quite happily
keep in mind the structural features of such a temperament while
tuning it in a way which in some theoretical sense totally demolishes
that structure. The structure, after all, is a theoretical
abstraction, where notes have been replaced by numbers, but a number
*is not* a note. You can't hear a number. There is a difference, in
other words, between the mathematical model and the music. The map is
not the territory.

You can hear the shape of a structure which is one thing i noticed about the eikosany , especuially after i had it tuned up in Just.
I could hear a single chord, and since that chord only exist in on place in the structure, i knew exactly where i was. Now the very last thing one wants in such a situation is to increase the puns in such a structure as once you start using them in this context, you destroy what you are attempting to do >> I have found what you call non just intervals , some of my favotite.
>> Many times i prefer a 40/27 to any of the alternatives, in fact it
> >
defines where i am in my scale, just like a tritone does in a major scale.

Well, that tells me you might object to meantone, which merges 40/27
with 3/2. As i mentioned i don't like meantone because i can't stand the second or at least the mean tonne second

But in a schismatic tuning, there is little difference; and
in one schismatic tuning, 1/12 schisma schismatic, there is no
difference at all in the sense that 81/80 has the same size. If all
you need is the structural presense of this tone, you can have it, and
still be without the structural presence of other tones you may not
really need. I am aware that you have many examples of ETS that have the 811/80.
It seems expedient that if this is what i want, i might as well just use it in this form. >> This shows that the two exist in two completely different
> >
structural worlds, one is open , the other is closed beyond a certain
limit.

That is, once again, purely mathematical. Dave is not trying to define
something in mathematical terms. The abstract concepts are important
and I'm all for them, but there is another question one can ask, which
is what is it that we can hear?

As Dan Wolf points out, we do not know. and also on what a time scale we are talking about.
There arequestion of memory feedback, where what one hears is contintioned by what one has heard before. >> Nothing is gained by dropping the tuned by ratios , yet everything
> >
is forever closed to progress further if you move away from it. This seems amazingly dogmatic. How does this answer Dave's point that
you can, after all, tune equal temperament by ratios?

oh but has any one done it? there are as i mentioned JI scales that one cannot tune by ear.
to tune 19ET by ear, one must suppress the over all tuning to do so, one can only listen to the last in the chain to hear it as the others will through you off. >> One property that i can say about JI scales that has not been
> >
mentioned. It produces scales of unequal sized intervals, if it
involves more than one ratio.With all the composers mentioned, this is
true. therefore it might be more inclusive. This comes down to number theory, and once again is a mathematical
property following from a mathematical definition. However, it is also
true of, for instance, 1/4 comma meantone.

>> Another property of JI scale can be added on to using the same
> >
material already used to construct it to infinity. Ditto the above.
how is that if you have a closed chain?

>> While one may not notice a perceptable differance immediately, With
> >
myself, the problems with 31 ET took quite a while of using it for me
to run into a problem.

But this really goes to Dave's point--he is not claiming 31 is JI. It
clearly is tempering in any sense of the word. Soon or later, what another temperment does will show it colors. >
>_------------ >
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

Message: 8 Date: Tue, 07 Jun 2005 08:37:25 -0000
From: "Gene Ward Smith" <gwsmith@svpal.org>
Subject: Re: Definitions of JI

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

>> I will say that i like chords that beat more than dead
>> still sonorities that just sit there.
> >

Hey! Now that's interesting. How much detuning is about right, do you
think?

But things can be done with detuning that can be muscially useful. Having the beat rate generate a pitch in the scale for one. >> using Ets because of the properties of they give in this regard are a >> good way to deal with it.
> >

Not just ets, but I think there is a lot to be said in the 7-limit for
99, 130 and 140 equal on those grounds, or for a temperament like
hemiwuerschmidt, come to that.

I wil take you word on these
>> The two approaches produce different qualites to these beats.
>> my main problem with 768 is that i just couldn't tell what it was i was >> hearing, phenomenon of the supposed ratios or an artifact of th e >> subdivision.
> >

768 is hardly ideal for the purpose, but acoustic instruments are not
a good choice either. Here i confess to being a pagan and desire the physicality of an instrument, which also i find easier to listen to over loger periods of time.
I have had to go back after all my instruments are tuned and adjust them all in the same day to get them to match

Csound will nail it for you. If your sound card
is up to the task of really accurate tuning, so will midi.

I have a copy of Logic 6 which i am trying to learn at the moment. >---------------------------- >
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

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

> *yes i am quite aware of these. and I am quite aware that these are
the type of problems you wish to focus on.
> Personally i have others. If i want say scales with unequal amount
of steps, for instance , why would one use Ets.

By far the most famous scale with unequal steps is not JI at all, but
is the diatonic major scale. People use ets for that because it is
convenient.

>
>
> >> it use for me is to clearly define a structure.
> >
> >
>
> Exactly. It defines a structure, which is to say, the definition is
> mathematical. We are always talking about temperaments of various
> ranks, meaning the various numbers of generators, and that does the
> same thing.
>
> The type of structures generated in JI are different than those
generated by ETs, except if we take it to infinity, but that is unusable.

In fact they are two examples of the exact same kind of thing, which
mathematically are called "free abelian groups". This means that
there are a certain number of generators, and everything else is a sum
of those generators. For 5-limit JI, the generators would be
1200 cents (2), 1901.955 cents (3) and 2786.314 cents. For 12-et, you
get only a single generator of 100 cents. The first is called rank 3,
because it has three independent generators, and the second rank 1. In
between is rank 2; for example 1200 cents and 1896.578 cents, which is
1/4 comma meantone.

The latter has, in fact, exactly the same structure as Pythagorean
tuning, but simply tunes the 3 differently. This sort of thing is
quite common. Marvel temperament, the 225/224 planar temperament, has
exactly the same structure as 5-limit JI, but has its tuning adjusted
so that 225/32 serves for a 7. Hence a certain quadrilateral in the
lattice of 5-limit note-classes suddenly becomes an important
additional structural feature, but not in a way which makes any of the
5-limit structure change in any way. It is all still there. If now you
make an additional comma out of 1029/1024, the 5-limit plane wraps
itself into a cylinder, so you do have a different structure, but a
related one. This is miracle.

> The question is that the two approaches produce different results,
although as you point out one can be used to appoximate the other. In
reality it fosters different types of music , which is artistically useful

It's true that adding commas, nanotempering commas, can foster one
clear difference--the comma pump. JI harmony tends to always relate
itself to the 1/1; commas lead to comma pumps, which allow for a kind
of harmonic restlessness which you'd probably have to allow
micromodulations by very small intervals to get from JI. I've tested
it, and to my ears there isn't much to choose between a sequence of
chords which modulates by 2401/2400 in JI, and one which returns to
its starting point in breed temperament or ennealimmal. Someone with a
very refined absolute pitch sense might react differently, I suppose,
and it would be interesting to test that.

> >> Another property of JI scale can be added on to using the same
> material already used to construct it to infinity.
>
> Ditto the above.
> how is that if you have a closed chain?

Not every temperament is an equal (rank 1) temperament. Any
temperament with rank above one will have this property; that would
mean "logical" meantone, or meantone tuned to 1/4 comma, for instance.

> But this really goes to Dave's point--he is not claiming 31 is JI. It
> clearly is tempering in any sense of the word.
>
> Soon or later, what another temperment does will show it colors.

I don't think if you used 2460 it would ever show its colors. I don't
think for your work it would ever make any difference. Even if I am
wrong, there is going to be a point where it becomes true. The midi
tuning standard divides the octave into 196608 equal parts. Do you
really expect this to "show its colors"? How would that be possible?

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Why, the beauty of the acronym of JI would then be transformed into
RI, which is hardly aesthetical or compelling for me.
>

I sincerely hope you're being facetious here, Ozan. If you're not,
well, I'm afraid that this is the least compelling argument I can
possibly fathom. To prefer the term "Just" over "Rational" simply
because you think "JI" *looks nicer* than "RI"? What? That's your
opinion and you're certainly entitled, but it's no basis for an
argument.

-igs

> Cordially,
> Ozan
> ----- Original Message -----
> From: Igliashon Jones
> To: tuning@yahoogroups.com
> Sent: 07 Haziran 2005 Salý 7:11
> Subject: [tuning] Re: Definitions of JI
>
>
>
> > Yes I noticed it says simply JI = rational.
>
> I agree that the two should not "equal" each other. Music can
> sound "Just"; it cannot sound "Rational". As far as I can see,
the
> term "Just" is an adjective to describe the sound of an interval
or
> chord. The trouble began when it was realized that these Just-
> sounding chords were representative of simple integer frequency
> ratios: people started reasoning that any chord or interval that
> represents simple integer frequency ratios should be
called "Just".
> Let me put this reasoning into standard form to illustrate its
> invalidity:
>
> Premise A: chord X has a certain audible quality which we call
> Justness.
> Premise B: chord X represents simple integer frequency ratios.
> Conclusion: all chords that represent simple integer frequency
ratios
> will have the quality of Justness.
>
> The conclusion does not at all follow from the premises.
>
> I personally think that the term "Just Intonation" should be
replaced
> with "Rational Intonation", seeing as how in even the
simplest "JI"
> systems not all the intervals are uniformly Just in their sound
> quality. I've yet to see a JI system without a wolf. And rather
> than trying to set some sort of "threshold" as to how many wolf
> intervals are allowable in a Just Intonation (which is
impractical
> and will likely never be agreed upon), why not just get rid of
the
> troublesome misnomer and stick with something that can
consistently
> describe these tuning systems?
>
> I challenge anyone to give me a practical reason why the
term "Just
> Intonation" should not simply be replaced with "Rational
> Intonation". What would be lost? What would be compromised?
What
> effect would it have on the music?
>
> -Igs

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
>
> The wolfs have a real function and often imply what other intervals are
> in the scale.
> I love wolves. like your dominants and other dissonances in ET , they
> keep the music going.
> In fact the history of music might be the pursuit of people finding
> sound combinations to keep the music propelled forward.
> Indonesian music is a great example of scales that refuse to resolve.
> that way they can keep going, even for days!

I totally agree, and I expect Igs does too.

> From: "Igliashon Jones" <igliashon@s...>
> I've yet to see a JI system without a wolf.

Agreed. Probably the closest you can get is this pentatonic
1/1 9/8 5/4 3/2 7/4 2/1
which is simply this harmonic series segment
6:7:8:9:10 with the 8 as tonic.

> And rather
> than trying to set some sort of "threshold" as to how many wolf
> intervals are allowable in a Just Intonation (which is impractical
> and will likely never be agreed upon),

It can be a fuzzy threshold and still be useful. I note that my "more
than linearly connected by just intervals" still allows the number of
"wolves" to increase as almost the square of the number of notes in
the scale.

> why not just get rid of the
> troublesome misnomer and stick with something that can consistently
> describe these tuning systems?

I totally agree that it would be far better if what Kraig and others
are calling JI were to be called RI (R for rational). But then I would
still want to define what it means to be a JI scale, based on audible
properties. And they certainly would not be wolf-less.

I should however point out that the term "RI" still has a problem - in
the "I" part, not the "R". And that is that "intonation" is definitely
an audible thing. There really is no such thing as Rational
Intonation. It seems to be more the rational _structure_ than the
intonation that is important in this case, but unfortunately "RS" has
other connotations. :-) So I guess I could live with "RI" for this.

But let's face it, we're probably never going to convince everyone to
stop using "JI" for this inaudible structural property.

All I ask is that we recognise that different people use these two
very different definitions of JI and when someone talks about "JI" and
you can't figure out which one they mean, you just have to ask. And if
anyone thinks his or hers is the only possible definition, calmly
point out the other one.

Monz, how about making this difference clear in your encyclopedia.

-- Dave Keenan

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> I will say that i like chords that beat more than dead
> still sonorities that just sit there.

Me too. I still call them Just if each beat takes more than about a
second or maybe a half second. i.e. if you can actually _count_ the
beats. Of course this translates to needing more accurate tuning the
higher the register, and also more accurate tuning the more complex
the ratio (because it is higher harmonics that are beating).

>
> using Ets because of the properties of they give in this regard are a
> good way to deal with it.
> Just as using higher ratios or in my case recurrent sequences to solve
> the problem are another,
>
> The two approaches produce different qualites to these beats.
> my main problem with 768 is that i just couldn't tell what it was i was
> hearing, phenomenon of the supposed ratios or an artifact of th e
> subdivision.

Good point.

> At the time, i wanted to atleast hear what the ratios sounded like.
> If you go to an art store to buy oil paint, most people will take off
> the lid to see what the color looks really like, even if it seems to
> match what is one the label. OK most people might trust the white and
> black.

Good analogy.

-- Dave Keenan

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> I do not believe that JI has a sound any more than any other family
of scales does.
> If someone uses this scale and calls it JI , it merely means it
broadens are view and concept of what JI is.
> Yes i do not have to listen at all
>
> Mystery scale
> 7
> !
> 202/181
> 153/124
> 175/128
> 199/130
> 261/154
> 211/112
> 194/93
>
> No true octave and 1/1 is implied as usual for Scala .scl files.

OK. Well that's the most un-Just (by the Dionysian definition) 7-note
scale I can come up with.

That doesn't mean I don't think someone can't do something interesting
with it.

Your (Apollonian?) definition of Just has no meaning to me other than
a way of writing down scales. Any scale can be written down either in
cents or ratios and I'm very comfortable with converting between them.
As Gene said, this is about the map, not the territory.

I remember some complaints when Joseph Pehrson called his music in
Blackjack (tuned as a subset of 72-equal) JI. I have no idea whether
that incuded you. He changed to calling it near-JI. Would you (or
anyone else) be happy for him to call it JI if we describe it in terms
of ratios of 3-digit numbers (even though it won't sound any
different)? If not, why not?

In fact I agreed he should call it near-JI, but that had nothing to do
with ratios and everything to do with what I _heard_.

-- Dave Keenan

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
There really is no such thing as Rational
> Intonation. It seems to be more the rational _structure_ than the
> intonation that is important in this case, but unfortunately "RS" has
> other connotations. :-) So I guess I could live with "RI" for this.
>
> But let's face it, we're probably never going to convince everyone to
> stop using "JI" for this inaudible structural property.

The inaudible strucutral property is more important to theory than
anything audible, and it seems it is that structure which interests
Kraig also. Given all that, it seems to me we need a term for it, and
JI is what people have been using.

i quite appreciated some of your comments on this thread, and am still processing some of these. While it is one thing to see ones tunings it is another to get the overall approach that is the background from which it comes.
I appreciate the way you approach it, even if it is different than my own , but then again that forces to stumble upon different things. Message: 25 Date: Thu, 09 Jun 2005 04:26:38 -0000
From: "Gene Ward Smith" <gwsmith@svpal.org>
Subject: Re: Definitions of JI

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
There really is no such thing as Rational

>> Intonation. It seems to be more the rational _structure_ than the
>> intonation that is important in this case, but unfortunately "RS" has
>> other connotations. :-) So I guess I could live with "RI" for this.
>> >> But let's face it, we're probably never going to convince everyone to
>> stop using "JI" for this inaudible structural property.
> >

The inaudible strucutral property is more important to theory than
anything audible, and it seems it is that structure which interests
Kraig also. Given all that, it seems to me we need a term for it, and
JI is what people have been using.

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

This is exactly the kind of argument that took place between me and Paul Erlich concerning the application of such pitches as 31/25, 36/29, 21/17 that I claimed were sought after in certaing Maqam scales.

Cordially,
Ozan

----- Original Message -----
From: Dave Keenan
To: tuning@yahoogroups.com
Sent: 23 Temmuz 2005 Cumartesi 6:27
Subject: [tuning] Re: Definitions of JI

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
> So, the "mathematical JI" provides us with a fairly simple criterion
> for justness of intonation, namely that the fundamental tones are
> in _exact_ small integer ratios, where the limits to "small" might be
> fixed by some conventional agreement.

Well no, Kraig Grady and Daniel Wolf and others made it quite clear
that their definition of JI will brook no limits on the size of the
numbers in the ratios (or if they do, they are at least into 3
digits). So "mathematical JI" is simply having notes tuned to exact
ratios. But then they have the problem that very few real-world
instruments can guarantee _exact_ ratios, so then it comes down to
something like the _intention_ to tune to exact ratios.

They made it quite clear that you could have something that was
audibly (or even measurably) indistinguishable from 12-equal, and they
would still call it JI, provided it was intended to be tuned to exact
ratios (no matter how large the numbers in the ratios).

> Compared to this, "perceptible JI" is a bit more complex and subtle,
> isn't it?

That depends on whether you are considering the actual psycho-physical
musical event, or the mathematical modelling of it.

Mathematical JI is of course extremely simple to model mathematically.
But as a psycho-physical event it is so complex and subtle as to be
completely non-existent. :-) i.e. As Kraig said, there's no way to
tell by listening whether or not something is (mathematical) JI.

Apparently one must simply take the composer's word for it.

Perceptible JI is complex and subtle to model mathematically (so is
colour). But as a psycho-physical event it is quite easy to
demonstrate "that special sound" to someone on any instrument capable
of playing two notes simultaneously and continuously varying the pitch
of one of them. This is little different from the way a child learns
how dark a colour has to be before it can be called black. (Although
to me, synaesthetically, (perceptible) JI harmonies are more akin to
_saturated_ colours, a term children are rarely taught).

> > Let me quote the meat of your definition again:
> > ... perceptibly just
> > intervals occur near ratios of small whole numbers. You ...
> > refuse to be nailed down on exactly how small the numbers in the ratio
> > have to be and how close to the ratio you have to be. Since these are
> > highly context dependent.

That's not a definition.

My definition is currently injuctive. It is of the form "Go and do
things A, B, C; L, M, N and X, Y, Z and listen, then know that
Justness is the most salient audible quality that is common to A, B
and C and is absent from X, Y and Z and is almost or just barely
present in L, M and N."

How would you define Redishness?

> So, perceptibly just intonation (PJI) might be achieved using any one
> of an enormous number of tempering schemes; can avoid the wolf
> fifths, such as D:f and F:d in Cmajor (typically 680 cents instead of
> 702);

Any tempering scheme that is gentle enough to leave a tuning
perceptibly just is unlikely to be able to eliminate all wolves. It
may reduce their number however. But microtemperaments can also be
used to eliminate the need for very small steps in a JI scale or to
allow it to modulate a little more widely, or as you saw in my
microtempered guitar article, to straighten out frets, while doing
minimal damage to the existing JI harmonies.

> can supply thirds and sixths that are indistinguishable from
> mathematically just (MJ) intervals; and can produce regular scales
> with fewer different step sizes.

Yes. Although you simply can't make a diatonic scale with all thirds
and sixths perceptibly just. A meantone somewhere near 2/7-comma is
the best you can do, with a little over 3 cents error in all thirds
and sixths.

> Whereas MJ intonation (MJI) uses no tempering whatsoever; cannot
> avoid the wolves; can supply exact thirds and sixths; and produces the
> least regular scales possible.

Oh no. That's quite wrong. You can do absolutely anything you like in
mathematical JI, since the only requirement is that the result is
expressed using (frequency or wavelength) ratios, never cents or other
logarithmic units. You can even do _tempering_ in mathematical JI, but
you'd better not call it that. :-)

> (Ah! Woodsman! Come temper your axe, to save poor little Red Riding
> Hood, bewildered in a forest of mathematically just intervals, and
> pursued by wolves!)

Hee hee.

-- Dave Keenan

And once again wish to point out that these same 'real -world' problems exist also for ET or MOS scales based on generators, or any method one can devise for generating scales.
Only a few machine can give us any of these things exactly, yet we call them by the method in which they are constructed.

> But then they have the problem that very few real-world
>instruments can guarantee _exact_ ratios, so then it comes down to
>something like the _intention_ to tune to exact ratios.
>
>They made it quite clear that you could have something that was
>audibly (or even measurably) indistinguishable from 12-equal, and they
>would still call it JI, provided it was intended to be tuned to exact
>ratios (no matter how large the numbers in the ratios).
>
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

Dave,

You wrote:

> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" wrote:
> > So, the "mathematical JI" provides us with a fairly simple criterion
> > for justness of intonation, namely that the fundamental tones are
> > in _exact_ small integer ratios, where the limits to "small" might be
> > fixed by some conventional agreement.
>
> Well no, Kraig Grady and Daniel Wolf and others made it quite clear
> that their definition of JI will brook no limits on the size of the
> numbers in the ratios (or if they do, they are at least into 3
> digits). So "mathematical JI" is simply having notes tuned to exact
> ratios.

So, I have apparently defined "mathematical small-integer JI" ...
Hate to proliferate terms unnecessarily, but that's the kind of JI
that most appeals to me - nothing beyond those ratios generated in
a 5-limit regular heptatonic scale with 5/4 thirds or 3/2 fifths.
Well, maybe some 7-limit flavour of a seventh ...

> But then they have the problem that very few real-world
> instruments can guarantee _exact_ ratios, so then it comes down to
> something like the _intention_ to tune to exact ratios.
>
> They made it quite clear that you could have something that was
> audibly (or even measurably) indistinguishable from 12-equal, and they
> would still call it JI, provided it was intended to be tuned to exact
> ratios (no matter how large the numbers in the ratios).

I followed that conversation, and suspect that they were
set up, and sandbagged ... ! :-)

God may know the intentions of your heart, but of what
possible use is that to hearers of your music?

>
> > Compared to this, "perceptible JI" is a bit more complex and subtle,
> > isn't it?
>
> That depends on whether you are considering the actual psycho-physical
> musical event, or the mathematical modelling of it.
>
> Mathematical JI is of course extremely simple to model mathematically.

As is "mathematical small-integer JI".

> But as a psycho-physical event it is so complex and subtle as to be
> completely non-existent. :-) i.e. As Kraig said, there's no way to
> tell by listening whether or not something is (mathematical) JI.
>
> Apparently one must simply take the composer's word for it.

This is not true of "mathematical small-integer JI", is it?

> Perceptible JI is complex and subtle to model mathematically (so is
> colour). But as a psycho-physical event it is quite easy to
> demonstrate "that special sound" to someone on any instrument capable
> of playing two notes simultaneously and continuously varying the pitch
> of one of them. This is little different from the way a child learns
> how dark a colour has to be before it can be called black. (Although
> to me, synaesthetically, (perceptible) JI harmonies are more akin to
> _saturated_ colours, a term children are rarely taught).

"Rich" was the term used in my schools, both for saturated
colours and saturated harmonies.

> > > Let me quote the meat of your definition again:
> > > ... perceptibly just
> > > intervals occur near ratios of small whole numbers. You ...
> > > refuse to be nailed down on exactly how small the numbers in the ratio
> > > have to be and how close to the ratio you have to be. Since these are
> > > highly context dependent.
>
> That's not a definition.
No, it's not ...

> My definition is currently injuctive.
Injunctive? As in, giving an injunction or instruction?

> It is of the form "Go and do
> things A, B, C; L, M, N and X, Y, Z and listen, then know that
> Justness is the most salient audible quality that is common to A, B
> and C and is absent from X, Y and Z and is almost or just barely
> present in L, M and N."

Even so, fraught with potential pitfalls. That you have even
allowed for things L, M and N means you have a grey area
where different people will perceive different things; in short,
where you cannot confidently assert that justness is perceptible.
The best you could hope for is some kind of statistical norm.
This, too, has difficulties. The brains of musicians have been
demonstrated to grow in those areas responsible for pitch
discrimination. Yes, we really do "grow a brain"!

> How would you define Redishness?

"An object exhibits reddishness if it emits or reflects any
light of wavelengths in that region of the EM spectrum where
the Red colour receptors are most responsive."

I've worked with colour for a long time, as a visual artist.
I do take your point - perception is subtle, conditioned on
all manner of interfering circumstances, on culture and on
training. But it wasn't I who was trying to argue for the
existence of "perceptible JI"; so it's not up to me to either
define it or demonstrate it.

> > So, perceptibly just intonation (PJI) might be achieved using any one
> > of an enormous number of tempering schemes; can avoid the wolf
> > fifths, such as D:f and F:d in Cmajor (typically 680 cents instead of
> > 702);
Correction: D:a and A:d

> Any tempering scheme that is gentle enough to leave a tuning
> perceptibly just is unlikely to be able to eliminate all wolves.

Aha! As I suspected! The wolf sneaks in the back door ...

> It
> may reduce their number however. But microtemperaments can also be
> used to eliminate the need for very small steps in a JI scale or to
> allow it to modulate a little more widely, or as you saw in my
> microtempered guitar article, to straighten out frets, while doing
> minimal damage to the existing JI harmonies.

Yes; an excellently _musical_ application of tempering!

> > can supply thirds and sixths that are indistinguishable from
> > mathematically just (MJ) intervals; and can produce regular scales
> > with fewer different step sizes.
>
> Yes. Although you simply can't make a diatonic scale with all thirds
> and sixths perceptibly just. A meantone somewhere near 2/7-comma is
> the best you can do, with a little over 3 cents error in all thirds
> and sixths.
>
> > Whereas MJ intonation (MJI) uses no tempering whatsoever; cannot
> > avoid the wolves; can supply exact thirds and sixths; and produces the
> > least regular scales possible.
>
> Oh no. That's quite wrong. You can do absolutely anything you like in
> mathematical JI, since the only requirement is that the result is
> expressed using (frequency or wavelength) ratios, never cents or other
> logarithmic units.

I almost believe you're right ... In an earlier message, I produced a
scale with an imperfect fifth of 299/200 as generator, replacing 3/2
in a cycle of fifths stopping after seven notes: as regular as can be,
with all whole-tones equal in size, and both semitones equal in size.
Now that's MJI, but clearly not PJI, since every fifth is almost 6
cents flat. And it's not MSIJI, so I'd be unlikely to use it.

> You can even do _tempering_ in mathematical JI, but
> you'd better not call it that. :-)

You can??? Suppose you want to temper out a comma, say
2401/2400. Do you simply adjust the ratio of the next nearest
interval by that ratio? Of course, in this flavour of MJI, it
doesn't matter how large the integers get, or how irregular
the scale, does it? But it wouldn't do me at all.

>
> > (Ah! Woodsman! Come temper your axe, to save poor little Red Riding
> > Hood, bewildered in a forest of mathematically just intervals, and
> > pursued by wolves!)
>
> Hee hee.
>
> -- Dave Keenan

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--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> > Any tempering scheme that is gentle enough to leave a tuning
> > perceptibly just is unlikely to be able to eliminate all wolves.
>
> Aha! As I suspected! The wolf sneaks in the back door ...

Not necessarily. Any large enough equal divsion will be sensibly just
and have no wolves. 171 would probably be fine for your purposes, and
if it wasn't, 441 or 612 would be.

> I almost believe you're right ... In an earlier message, I produced a
> scale with an imperfect fifth of 299/200 as generator, replacing 3/2
> in a cycle of fifths stopping after seven notes: as regular as can be,
> with all whole-tones equal in size, and both semitones equal in size.
> Now that's MJI, but clearly not PJI, since every fifth is almost 6
> cents flat. And it's not MSIJI, so I'd be unlikely to use it.

It's a perfectly fine meantone, between 31 and 50 and closer to 50;
it's a poptimal generator for 5-limit meantone, in fact. Hence, you
get a nice diatonic scale from your proceedure, which I personally
would not consider JI, strictly speaking.

> > You can even do _tempering_ in mathematical JI, but
> > you'd better not call it that. :-)
>
> You can??? Suppose you want to temper out a comma, say
> 2401/2400. Do you simply adjust the ratio of the next nearest
> interval by that ratio?

2401/2400 is so small you can pretty much do anything or nothing. That
is, you can simply leave it as is, or retune, for instance by tuning
to 612 equal.

Of course, in this flavour of MJI, it
> doesn't matter how large the integers get, or how irregular
> the scale, does it? But it wouldn't do me at all.

Why not?

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
> Dave,
>
> You wrote:
>
> > --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" wrote:
> > > So, the "mathematical JI" provides us with a fairly simple criterion
> > > for justness of intonation, namely that the fundamental tones are
> > > in _exact_ small integer ratios, where the limits to "small"
might be
> > > fixed by some conventional agreement.
> >
> > Well no, Kraig Grady and Daniel Wolf and others made it quite clear
> > that their definition of JI will brook no limits on the size of the
> > numbers in the ratios (or if they do, they are at least into 3
> > digits). So "mathematical JI" is simply having notes tuned to exact
> > ratios.
>
> So, I have apparently defined "mathematical small-integer JI" ...
> Hate to proliferate terms unnecessarily, but that's the kind of JI
> that most appeals to me - nothing beyond those ratios generated in
> a 5-limit regular heptatonic scale with 5/4 thirds or 3/2 fifths.
> Well, maybe some 7-limit flavour of a seventh ...

Fine. That's clearly perceptible JI as well. But presumably it appeals
to you primarily because of the way it _sounds_? And presumably you
have _some_ tolerance for mistuning? What instruments do you like it on?

What I'm getting at is that maybe what you actually like is
"perceptible small integer JI" with a very small tolerance.

> > But then they have the problem that very few real-world
> > instruments can guarantee _exact_ ratios, so then it comes down to
> > something like the _intention_ to tune to exact ratios.
> >
> > They made it quite clear that you could have something that was
> > audibly (or even measurably) indistinguishable from 12-equal, and they
> > would still call it JI, provided it was intended to be tuned to exact
> > ratios (no matter how large the numbers in the ratios).
>
> I followed that conversation, and suspect that they were
> set up, and sandbagged ... ! :-)

I'm not sure what "sandbagged" means in this context, but I sure
_thought_ I was setting them up. However, when I came to the point
where Kraig (or any other mathematical-JI-er following the
conversation -- presumably including Daniel Wolf -- should have said,
er now wait a minute, I wouldn't call something indistinguishable from
12-ET, "JI" -- or worse, that scale I made with the least
perceptibly-just intervals I could come up with -- they didn't bat an
eyelid.

Kraig specifically said that he didn't need to listen to it. It was
expressed in ratios so it was JI.

> God may know the intentions of your heart, but of what
> possible use is that to hearers of your music?

Well, my point exactly.

> > > Compared to this, "perceptible JI" is a bit more complex and subtle,
> > > isn't it?
> >
> > That depends on whether you are considering the actual psycho-physical
> > musical event, or the mathematical modelling of it.
> >
> > Mathematical JI is of course extremely simple to model mathematically.
>
> As is "mathematical small-integer JI".

Yes. And indeed this is the obvious point of departure in the
mathematical modelling of perceptible JI. The first thing to add to
this is _tolerance_. Most of us on this list can agree that departing
from the intended ratio by 1/3 of a cent does not change the
perceptible quality of a harmony, for most musical purposes.

Of course one can always sustain an electronically-generated chord
long enough that you can hear beats that take even several minutes to
cycle. But this isn't typical of music, and most instruments aren't
that accurate. We can make exceptions for things like La Monte Young's
Dream House. Its perceptible justness apparently _does_ require
extremely accurate tuning because it uses ratios of such large numbers.

> > But as a psycho-physical event it is so complex and subtle as to be
> > completely non-existent. :-) i.e. As Kraig said, there's no way to
> > tell by listening whether or not something is (mathematical) JI.
> >
> > Apparently one must simply take the composer's word for it.
>
> This is not true of "mathematical small-integer JI", is it?

No. Because it is also a subset of perceptible JI.

> "Rich" was the term used in my schools, both for saturated
> colours and saturated harmonies.

I like that.

> > My definition is currently injuctive.
> Injunctive? As in, giving an injunction or instruction?

Yes.

If you'd never experience a sponge cake (or a sponge anything) and I
wanted to define "sponge-cake" for you, the simplest thing might be
for me to give you a recipe.

I'm currently reading Oliver Sacks' "The Island of the Colour Blind".
He describes a friend who drove with his 3-year old son into the
countryside for the first time and his son said, "Hey dad, look at the
orange grass". His dad said. "No son, orange is the colour of an
orange." His son said "Yes, its the colour of an orange". That was the
first he realised his son was red-green colour-blind.

We have to be just as prepared that some people may be able to
perceive justness that we can't or vice versa, but this is something
that can be tested objectively.

> Even so, fraught with potential pitfalls. That you have even
> allowed for things L, M and N means you have a grey area
> where different people will perceive different things; in short,
> where you cannot confidently assert that justness is perceptible.

Sure. But I can live with that. We do it with colour. And there's an
awful lot that we will agree on.

> The best you could hope for is some kind of statistical norm.
> This, too, has difficulties. The brains of musicians have been
> demonstrated to grow in those areas responsible for pitch
> discrimination. Yes, we really do "grow a brain"!

That doesn't sound unlikely to me at all. However there is a big
difference between someone claiming they can tune 13:17 by ear and
claiming they can tune 131:171.

>
>
> > How would you define Redishness?
>
> "An object exhibits reddishness if it emits or reflects any
> light of wavelengths in that region of the EM spectrum where
> the Red colour receptors are most responsive."

A lot of people consider spectral violet to be reddish.

> I've worked with colour for a long time, as a visual artist.
> I do take your point - perception is subtle, conditioned on
> all manner of interfering circumstances, on culture and on
> training. But it wasn't I who was trying to argue for the
> existence of "perceptible JI"; so it's not up to me to either
> define it or demonstrate it.

Right. But my point was, that we live with this ambiguity in the case
of colour, we don't try to force it into an oversimplified
mathematical model.

> > Any tempering scheme that is gentle enough to leave a tuning
> > perceptibly just is unlikely to be able to eliminate all wolves.
>
> Aha! As I suspected! The wolf sneaks in the back door ...

What's wrong with a few wolves anyway?

Gene mentioned nano-temperaments that are also equal temperaments,
such as 171-ET as an example with no wolves. This is true, and at the
7-limit, I personally consider even 72-ET to be a microtemperament
(perceptibly-JI). but I assumed you were tanking of scales with far
fewer notes.

If a 5-limit western diatonic is all you need, and maximum perceptible
justness is your aim, then you either live with the D:A wolf or you
use an optimal meantone (possibly with tempered octaves). There are no
other options. You have to hide a 21.5 cent comma, and this comma can
only be distributed in certain ways such that you're bound to have an
error of at least a quarter of this comma in some consonant interval. See
http://dkeenan.com/music/DistributingCommas.htm

> > > Whereas MJ intonation (MJI) uses no tempering whatsoever; cannot
> > > avoid the wolves; can supply exact thirds and sixths; and
produces the
> > > least regular scales possible.
> >
> > Oh no. That's quite wrong. You can do absolutely anything you like in
> > mathematical JI, since the only requirement is that the result is
> > expressed using (frequency or wavelength) ratios, never cents or other
> > logarithmic units.
>
> I almost believe you're right ... In an earlier message, I produced a
> scale with an imperfect fifth of 299/200 as generator, replacing 3/2
> in a cycle of fifths stopping after seven notes: as regular as can be,
> with all whole-tones equal in size, and both semitones equal in size.
> Now that's MJI, but clearly not PJI, since every fifth is almost 6
> cents flat. And it's not MSIJI, so I'd be unlikely to use it.

Correct. As Gene said, it's an excellent meantone and this is a prime
example of tempering by ratios. Have you listened to it.

A problem some folks have with meantone is that the major ninths are
way too far from 4:9 for any semblance of justness. This can be
improved slightly by widening the octaves by up to a sixth of a comma,
but the fourths and major sixths suffer.

> > You can even do _tempering_ in mathematical JI, but
> > you'd better not call it that. :-)
>
> You can???

Sure. You just did it above with your 200:299 fifths.

> Suppose you want to temper out a comma, say
> 2401/2400. Do you simply adjust the ratio of the next nearest
> interval by that ratio?

See my paper above, but instead of dividing your comma into
logarithmically equal parts by first converting to cents, calculate
rational pseudo-roots (I don't know what they are really called).

Lets say you want the Syntonic/Didymus comma (81/80) in four
nearly-equal parts, i.e. find pseudo fourth roots for it. Multiply
both sides by 4 to get 324/320 then you can write it as

324 323 322 321
--- * --- * --- * ---
323 322 321 300

> Of course, in this flavour of MJI, it
> doesn't matter how large the integers get, or how irregular
> the scale, does it?

That's right.

Here's one way to get 12-equal in MJI:

The Pythagorean comma is 3^12 / 2^19 = 531441/524288
The difference between top and bottom is 7153. Unfortunately this has
no common factors with 12 so we multiply both sides by 12 and
increment by 7153 each time to get 12 nearly equal parts by which to
adjust the 12 fifths making the cycle. I'll list only the largest and
smallest. They differ by less than 0.03 of a cent.

6377292 6298609
------- ... -------
6370139 6291456

This is not an approximation of 12-equal, this _is_ 12-equal, for all
_musical_ purposes. And yet Kraig and others would call it JI.

Personally, I think that doing so makes a mockery of the whole idea of
"just intonation".

-- Dave Keenan

Gene,

You wrote:
--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
> > > Any tempering scheme that is gentle enough to leave a tuning
> > > perceptibly just is unlikely to be able to eliminate all wolves.
> >
> > Aha! As I suspected! The wolf sneaks in the back door ...
>
> Not necessarily. Any large enough equal divsion will be sensibly just
> and have no wolves. 171 would probably be fine for your purposes, and
> if it wasn't, 441 or 612 would be.

Well, those would certainly be "gentle" tempering schemes.
I rather thought we'd need to go to higher integer EDOs
than those to eliminate wolves. I'm not as fond of large-integer
JI as of small-integer JI, which has stronger overtones.

> > I almost believe you're right ... In an earlier message, I produced a
> > scale with an imperfect fifth of 299/200 as generator, replacing 3/2
> > in a cycle of fifths stopping after seven notes: as regular as can be,
> > with all whole-tones equal in size, and both semitones equal in size.
> > Now that's MJI, but clearly not PJI, since every fifth is almost 6
> > cents flat. And it's not MSIJI, so I'd be unlikely to use it.
>
> It's a perfectly fine meantone, between 31 and 50 and closer to 50;
> it's a poptimal generator for 5-limit meantone, in fact. Hence, you
> get a nice diatonic scale from your proceedure, which I personally
> would not consider JI, strictly speaking.

Sorry, I don't understand what you mean by "between 31 and 50"
here.

Why wouldn't you consider it JI; which "strict" definition of JI do
you have in mind?

> > > You can even do _tempering_ in mathematical JI, but
> > > you'd better not call it that. :-)
> >
> > You can??? Suppose you want to temper out a comma, say
> > 2401/2400. Do you simply adjust the ratio of the next nearest
> > interval by that ratio?
>
> 2401/2400 is so small you can pretty much do anything or nothing. That
> is, you can simply leave it as is, or retune, for instance by tuning
> to 612 equal.

Doing nothing is hardly tempering ... Retuning to 612-EDO
certainly is, and I presume you mean that 2401/2400 is
0 steps of 612-EDO?

> Of course, in this flavour of MJI, it
> > doesn't matter how large the integers get, or how irregular
> > the scale, does it? But it wouldn't do me at all.
>
> Why not?

Because I prefer MSIJI, with its greater resonances.
When playing an unfretted string instrument, tuning
to low-integer ratios produces quite a bit of sympathetic
vibration on the unplucked strings when holding notes -
and I like that sound. "Just" personal preference.

Regards,
Yahya

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Hi Kraig,

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> And once again wish to point out that these same 'real -world' problems
> exist also for ET or MOS scales based on generators, or any method one
> can devise for generating scales.

Yes. But no one I know of claims that 12-ET ceases to be 12-ET if
constructed by ratios. In fact I understand that's how it was first
constructed (or at least how it was constructed at one time), and I
outlined one possible such construction at the end of this post:
/tuning/topicId_58865.html#59471

Also top-octave-divider organs produce 12-ET in precise ratios, and no
one I know disputes that they are in 12-ET (assuming their divisors
are large enough - typically between 256 and 512).

But you appear to be claiming that if one deliberately tempers a set
of ratios to produce an irrational scale, then it can no longer be
called JI, even if the tempering causes no change greater than
0.00..as many zeros as you like..001 of a cent in any interval. While
an accidental mistuning (in the real world) of a much larger amount
(How large?) does not disqualify it.

Did I get that right?

> Only a few machine can give us any of these things exactly, yet we
call
> them by the method in which they are constructed.

Well that's exactly where we disagree.

I don't care so much how they are constructed. I care more about what
they _sound_ like. This is apparently a consumer's, rather than a
producer's point of view.

I don't care too much if your medicine is produced by chemical
synthesis or extracted from Tuvaluan bat droppings. I mainly care if
it cures me without any nasty side-effects.

I also don't understand why _you_ would be worried by a perceptible
definition of JI. As I understand it, all your mathematical-JI music
is also perceptibly-JI.

-- Dave Keenan

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> Here's one way to get 12-equal in MJI:
>
> The Pythagorean comma is 3^12 / 2^19 = 531441/524288
> The difference between top and bottom is 7153. Unfortunately
> this has no common factors with 12 so we multiply both sides
> by 12 and increment by 7153 each time to get 12 nearly equal
> parts by which to adjust the 12 fifths making the cycle.
> I'll list only the largest and smallest. They differ by
> less than 0.03 of a cent.
>
> 6377292 6298609
> ------- ... -------
> 6370139 6291456
>
> This is not an approximation of 12-equal, this _is_ 12-equal,
> for all _musical_ purposes. And yet Kraig and others would
> call it JI.
>
> Personally, I think that doing so makes a mockery of the
> whole idea of "just intonation".

I agree completely ... which is precisely why i came up
with the idea of using "RI" / "rational intonation" back
in 2000, for tunings like this.

I know you can appreciate my argument, Dave, because back
then, you were the one who dug up the Oxford English
Dictionary definition of "just intonation".

The definition of "just" implies some type of "correctness".
Granted, many authors and theorists *do* see corrections
as a virtue of 12-edo (i could name Barbour, for one),
but in general, "just intonation" is taken to mean a
tuning in which the individual notes of intervals and chords
are perceived as blending together smoothly -- a property
that i think we should all be calling "concordance".
(yes, i already know that there are dissenters out there ...)

As those of us here know by now, this doesn't necessarily
mean small-integer ratios -- but i buy the line of reasoning
that says that small-integer ratios are a sort of
perceptual archtype by which a listener (consciously or
unconsciously) measures the "justness" of an interval,
chord, or scale ... i suppose Paul Erlich's "relative
harmonic entropy" is one of the best models.

I personally have my own theory about accordance, in
which both the size of the prime-factors and the size
of their exponents are taken into account. I've never
tried to carefully develop a mathematical model for
this, but i believe that this is the essential problem
being debated in the "lattice metric" thread right now
-- if each prime-factor is a certain angle in the
lattice space, and each exponent is a certain distance
away from the origin, then the straight line which
connects any two notes is a geometrical model of the
Monzo-accordance measure.

The primary quesion i can see is: how does one properly
set up the relationship between the prime-factors
and the exponents, so that varying size in the two
domains correlates in some audibly perceptually
meaningful way.

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

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> Hi Kraig,
>
> --- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
>
> > And once again wish to point out that these same
> > 'real -world' problems exist also for ET or MOS scales
> > based on generators, or any method one can devise
> > for generating scales.
>
> Yes. But no one I know of claims that 12-ET ceases to be
> 12-ET if constructed by ratios. In fact I understand that's
> how it was first constructed (or at least how it was
> constructed at one time), and I outlined one possible such
> construction at the end of this post:
> /tuning/topicId_58865.html#59471
>
> Also top-octave-divider organs produce 12-ET in precise
> ratios, and no one I know disputes that they are in 12-ET
> (assuming their divisors are large enough - typically
> between 256 and 512).

I've speculated that the Sumerians, with their sophisticated
sexagesimal (base-60) math, could have derived a rational
form of 12-edo back around 3000 BC.

http://tonalsoft.com/monzo/sumerian/sumerian-tuning.htm

http://tonalsoft.com/monzo/sumerian/simplified-sumerian-tuning.htm

In the latter page, i speculate that even a simple approximation
in the sexagesimal calculations, would yield perceptually
"satisfactory" approximation to 12-edo:

------- begin quote from Tonalsoft Encyclopedia webpage -----

This can even be simplified further, by using simpler approximations
in the second sexagesimal place: 24 represents 2/5, 30 represents 1/2,
36 represents 3/5, 40 represents 2/3, 48 represents 4/5.

--------- string-length ----------
notesexagesimal fraction decimal ~cents ~cents error

F 60 1/1 0 0 0
F#/Gb 56,36 283/300 0.943 100.9925372 + 0.993
G 53,30 107/120 0.892 198.508331 - 1.492
G3/Ab 50,30 101/120 0.842 298.4149354 - 1.585
A 47,36 119/150 0.793 400.8011126 + 0.801
A#/Bb 45 3/4 0.75 498.0449991 - 1.955
B 42,24 53/75 0.707 601.0778831 + 1.078
C 40 2/3 0.667 701.9550009 + 1.955
C#/Db 37,48 63/100 0.63 799.8915195 - 0.108
D 35,40 107/180 0.594 900.4633319 + 0.463
D#/Eb 33,40 101/180 0.561 1000.369936 + 0.370
E 31,48 53/100 0.53 1099.122882 - 0.877
F 30 1/2 0.5 1200 0

------- end quote ------------------------------------------

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

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> Here's one way to get 12-equal in MJI:

A slick method due to Kirnberger is to flatten a fifth by a schisma,
that is, 32805/32768. This gives a 5-limit version of 12-equal, which
is a 5-limit Fokker block and so by some points of view
paradigmatically JI.

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> Sorry, I don't understand what you mean by "between 31 and 50"
> here.

299/200 is 696.17 cents, which is shaper than the 5--et fifth of
exactly 696 cents, but flatter than the 31-et fifth of 696.77 cents.

> Why wouldn't you consider it JI; which "strict" definition of JI do
> you have in mind?

In practice, using this as a generator results in a tempered scale, so
that's what I'd consider it to be, unless I really wanted to go up to
the 23-limit and regard it not as a fifth, but as something you get to
by way of a pair of 23-limit consonances like (13/10)(23/20). But I'm
not much of a believer in the 23 limit.

> > 2401/2400 is so small you can pretty much do anything or nothing. That
> > is, you can simply leave it as is, or retune, for instance by tuning
> > to 612 equal.
>
> Doing nothing is hardly tempering ...

I think in the case of 2401/2400, where you get chords off by less
than a cent, it really is a sort of tempering. It acts as if it was,
at any rate.

Retuning to 612-EDO
> certainly is, and I presume you mean that 2401/2400 is
> 0 steps of 612-EDO?

Right, so this is a way of tempering out 2401/2400.

>
> > Of course, in this flavour of MJI, it
> > > doesn't matter how large the integers get, or how irregular
> > > the scale, does it? But it wouldn't do me at all.
> >
> > Why not?
>
> Because I prefer MSIJI, with its greater resonances.

What the heck is MSIJI, and why does it have more resources?

> When playing an unfretted string instrument, tuning
> to low-integer ratios produces quite a bit of sympathetic
> vibration on the unplucked strings when holding notes -
> and I like that sound. "Just" personal preference.

And you think nanotempering won't do that? I think it would do this
equally well in 612-equal tuning.

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> I don't care so much how they are constructed. I care more about what
> they _sound_ like. This is apparently a consumer's, rather than a
> producer's point of view.

That may well be. Kraig cares about how the notes are structured; he
wants 5-limit to have three generators, 7-limit four, and so forth. Of
course you can take music with only one generator, such as the
theoretical output of midi with pitch bending, and use it to produce
music structured with four generators.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > Here's one way to get 12-equal in MJI:
>
> A slick method due to Kirnberger is to flatten a fifth by a schisma,
> that is, 32805/32768. This gives a 5-limit version of 12-equal, which
> is a 5-limit Fokker block and so by some points of view
> paradigmatically JI.

I hasten to point out to the viewers that that's 5-_prime_-limit. It
of course has no 5-odd-limit just intervals (except maybe the
3-odd-limit ones).

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > I don't care so much how they are constructed. I care more about what
> > they _sound_ like. This is apparently a consumer's, rather than a
> > producer's point of view.
>
> That may well be. Kraig cares about how the notes are structured; he
> wants 5-limit to have three generators, 7-limit four, and so forth. Of
> course you can take music with only one generator, such as the
> theoretical output of midi with pitch bending, and use it to produce
> music structured with four generators.

Right, but in the case of a nanotemperament he could easily treat it
that way, ignoring the fact that some of the generators are derivable
from each other. It isn't exactly something that forces itself on your
attention.

-- Dave Keenan

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> > You can even do _tempering_ in mathematical JI, but
> > you'd better not call it that. :-)
>
> You can???

Yes; I tried to point this out to you off-list. Plus, your recent
example which is very close to Woolhouse-optimal meantone was an
example of mathematical JI functioning as temperament.

> Suppose you want to temper out a comma, say
> 2401/2400. Do you simply adjust the ratio of the next nearest
> interval by that ratio?

You'd take a chain of simple-integer intervals that together form
2401/2400, and adjust each of them by an interval *considerably
smaller than* 2401/2400 (such that all these small intervals together
make up a 2400/2401) so that when you construct the full chain of
(formerly simple-integer, still perceptibly just) intervals, the net
result will be not 2401/2400 but 1/1. And it's quite possible to
divide 2401/2400 into a series of smaller intervals that are all
rational; there's no need for them to be irrational if you don't want
them to be (though you'd never hear the difference).

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> I personally have my own theory about accordance, in
> which both the size of the prime-factors and the size
> of their exponents are taken into account. I've never
> tried to carefully develop a mathematical model for
> this, but i believe that this is the essential problem
> being debated in the "lattice metric" thread right now
> -- if each prime-factor is a certain angle in the
> lattice space, and each exponent is a certain distance
> away from the origin, then the straight line which
> connects any two notes is a geometrical model of the
> Monzo-accordance measure.

Well, as you know if you've read my writings, I believe that rather
than straight-line distance, the *taxicab* distance works much better
(in conjunction with the Tenney lattice in particular), and one of
the wonderful things about the taxicab distance is that the angle
between the axes becomes irrelevant.

> The primary quesion i can see is: how does one properly
> set up the relationship between the prime-factors
> and the exponents, so that varying size in the two
> domains correlates in some audibly perceptually
> meaningful way.

I think the Tenney lattice, when a taxicab metric is used on it,
automatically takes care of both of these considerations in a very
nice way.

If you need to impose octave-equivalence, the van Prooijen lattice,
along with a hexagonal-norm (or higher-dimensional analogue) metric,
is the best way to adapt the Tenney picture.

Of course, no lattice/metric combination will tell you much about
accordance once you get beyond the simplest ratios. 3000:2001 is much
more concordant than any of the lattice/metric combinations would
predict, since it's less than 1 cent off from 3:2. But as long as
your are constructing your music via a connective web of concordant,
simple-integer-ratio intervals, the lattice/metric approach will
indeed give you a sense of how far your music wanders, harmonically
speaking.

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> And it's quite possible to
> divide 2401/2400 into a series of smaller intervals that are all
> rational; there's no need for them to be irrational if you don't want
> them to be (though you'd never hear the difference).

You can also express everything in the 7-limit in terms of 2 and two
other rational numbers: 2, 1079/881 and 2516/1761, for example. Doing
this gives rational numbers which are less than a sixth of a cent off,
for the 9-odd limit, and much closer for the 5-limit, and such that
2401/2400 goes away.

Dave Keenan wrote:

> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
> >
> > Dave,
> >
> > You wrote:
> >
> > > --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" wrote:
> > > > So, the "mathematical JI" provides us with a fairly simple criterion
> > > > for justness of intonation, namely that the fundamental tones are
> > > > in _exact_ small integer ratios, where the limits to "small"
> might be
> > > > fixed by some conventional agreement.
> > >
> > > Well no, Kraig Grady and Daniel Wolf and others made it quite clear
> > > that their definition of JI will brook no limits on the size of the
> > > numbers in the ratios (or if they do, they are at least into 3
> > > digits). So "mathematical JI" is simply having notes tuned to exact
> > > ratios.
> >
> > So, I have apparently defined "mathematical small-integer JI" ...
> > Hate to proliferate terms unnecessarily, but that's the kind of JI
> > that most appeals to me - nothing beyond those ratios generated in
> > a 5-limit regular heptatonic scale with 5/4 thirds or 3/2 fifths.
> > Well, maybe some 7-limit flavour of a seventh ...
>
> Fine. That's clearly perceptible JI as well. But presumably it appeals
> to you primarily because of the way it _sounds_?

Yes. So you may be right in calling it PJI as well.

> And presumably you
> have _some_ tolerance for mistuning?

Very little in slow, sparse music; very great in fast, dense music.
Most importantly - it depends on the context. Untuned percussion
doesn't bother me a bit. So - yes, I do have SOME tolerance for
mistuning.

> What instruments do you like it on?

Voice, pipes such as simple flutes or recorders,
plucked string instruments.

> What I'm getting at is that maybe what you actually like is
> "perceptible small integer JI" with a very small tolerance.

I think you've made your case. :-) Tho I do enjoy other
styles of music besides. I doubt I could enjoy Chopin's piano
music played on anything BUT a 12-EDO piano (haen't tried
it yet, tho).

> > > But then they have the problem that very few real-world
> > > instruments can guarantee _exact_ ratios, so then it comes down to
> > > something like the _intention_ to tune to exact ratios.
> > >
> > > They made it quite clear that you could have something that was
> > > audibly (or even measurably) indistinguishable from 12-equal, and they
> > > would still call it JI, provided it was intended to be tuned to exact
> > > ratios (no matter how large the numbers in the ratios).
> >
> > I followed that conversation, and suspect that they were
> > set up, and sandbagged ... ! :-)
>
> I'm not sure what "sandbagged" means in this context, but I sure
> _thought_ I was setting them up. However, when I came to the point
> where Kraig (or any other mathematical-JI-er following the
> conversation -- presumably including Daniel Wolf -- should have said,
> er now wait a minute, I wouldn't call something indistinguishable from
> 12-ET, "JI" -- or worse, that scale I made with the least
> perceptibly-just intervals I could come up with -- they didn't bat an
> eyelid.
>
> Kraig specifically said that he didn't need to listen to it. It was
> expressed in ratios so it was JI.

Yes, I thought that strange.

> > God may know the intentions of your heart, but of what
> > possible use is that to hearers of your music?
>
> Well, my point exactly.

We seem to agree on this.

> > > > Compared to this, "perceptible JI" is a bit more complex and subtle,
> > > > isn't it?
> > >
> > > That depends on whether you are considering the actual psycho-physical
> > > musical event, or the mathematical modelling of it.
> > >
> > > Mathematical JI is of course extremely simple to model mathematically.
> >
> > As is "mathematical small-integer JI".
>
> Yes. And indeed this is the obvious point of departure in the
> mathematical modelling of perceptible JI. The first thing to add to
> this is _tolerance_. Most of us on this list can agree that departing
> from the intended ratio by 1/3 of a cent does not change the
> perceptible quality of a harmony, for most musical purposes.

Right.

> Of course one can always sustain an electronically-generated chord
> long enough that you can hear beats that take even several minutes to
> cycle. But this isn't typical of music, and most instruments aren't
> that accurate. We can make exceptions for things like La Monte Young's
> Dream House. Its perceptible justness apparently _does_ require
> extremely accurate tuning because it uses ratios of such large numbers.

Agreed, it would materially alter the sound of the piece to be even
fractionally mistuned. Very little music uses interacting pure sine
waves, either; the closer it gets to that (as with soprano voices, simple
flutes or thin plucked strings), the less tolerance most of us have for
mistuning.

> > > But as a psycho-physical event it is so complex and subtle as to be
> > > completely non-existent. :-) i.e. As Kraig said, there's no way to
> > > tell by listening whether or not something is (mathematical) JI.
> > >
> > > Apparently one must simply take the composer's word for it.
> >
> > This is not true of "mathematical small-integer JI", is it?
>
> No. Because it is also a subset of perceptible JI.

I thnk the appropriate model may be that PJI and
MJI _intersect_ in small-integer just intonation (SIJI).
That is, that SIJI, and only SIJI, is both percebtibly and
mathematically just.

> > "Rich" was the term used in my schools, both for saturated
> > colours and saturated harmonies.
>
> I like that.
>
> > > My definition is currently injuctive.
> > Injunctive? As in, giving an injunction or instruction?
>
> Yes.
>
> If you'd never experience a sponge cake (or a sponge anything) and I
> wanted to define "sponge-cake" for you, the simplest thing might be
> for me to give you a recipe.

And this approach has now-venerable antecedents in mathematics,
too. Call me old-fashioned if you will, but I prefer analytic ideas
when possible.

> I'm currently reading Oliver Sacks' "The Island of the Colour Blind".
> He describes a friend who drove with his 3-year old son into the
> countryside for the first time and his son said, "Hey dad, look at the
> orange grass". His dad said. "No son, orange is the colour of an
> orange." His son said "Yes, its the colour of an orange". That was the
> first he realised his son was red-green colour-blind.
>
> We have to be just as prepared that some people may be able to
> perceive justness that we can't or vice versa, but this is something
> that can be tested objectively.

Indeed!

> > Even so, fraught with potential pitfalls. That you have even
> > allowed for things L, M and N means you have a grey area
> > where different people will perceive different things; in short,
> > where you cannot confidently assert that justness is perceptible.
>
> Sure. But I can live with that. We do it with colour. And there's an
> awful lot that we will agree on.
>
> > The best you could hope for is some kind of statistical norm.
> > This, too, has difficulties. The brains of musicians have been
> > demonstrated to grow in those areas responsible for pitch
> > discrimination. Yes, we really do "grow a brain"!
>
> That doesn't sound unlikely to me at all. However there is a big
> difference between someone claiming they can tune 13:17 by ear and
> claiming they can tune 131:171.

Again, such claims can be tested objectively.

> > > How would you define Redishness?
> >
> > "An object exhibits reddishness if it emits or reflects any
> > light of wavelengths in that region of the EM spectrum where
> > the Red colour receptors are most responsive."
>
> A lot of people consider spectral violet to be reddish.

I was wondering if you'd mention that! :-)

Most normally-sighted people do. And that's because
the Red colour receptors are also responsive there,
but to a smaller degree than in the "red" wavelengths
at the other end of the visible spectrum. If you delete
the word "most" from my attempted definition, that
should suffice. I think that improves it.

> > I've worked with colour for a long time, as a visual artist.
> > I do take your point - perception is subtle, conditioned on
> > all manner of interfering circumstances, on culture and on
> > training. But it wasn't I who was trying to argue for the
> > existence of "perceptible JI"; so it's not up to me to either
> > define it or demonstrate it.
>
> Right. But my point was, that we live with this ambiguity in the case
> of colour, we don't try to force it into an oversimplified
> mathematical model.

Sure we do! And to good purpose, too. Most list members
are reading these messages on a computer monitor that
reconstitutes many colours in the eye using just three "primary"
colour guns - Red, Green and Blue. No way you can see a
pure yellow light or a pure violet light emanate from your
screen. Yet the simulation is, for most practical purposes,
good enough.

Not only are simplified models (mathematical or otherwise)
useful, they also test and expand our understanding, by the
very ways they fail in particular cases. That is, they are a
tool of the scientific method, with its cycle of observation,
hypothesising and testing.

We all use simplified models all the time. They enable us to
cross the street safely, on our walk to the corner store to
pick up a newspaper which we confidently expect to find
there, in the same place as yesterday's.

Whether our models are _over_-simplified is entirely a
question of our purposes when we apply them. If they
fail to help us achieve those purposes, they are inadequate
and need revision. Their inadequacy is sometimes, but not
always, due to being overly simple. Sometimes we have to
refine and complicate our models; at others, we need to
change or challenge our basic assumptions.

> > > Any tempering scheme that is gentle enough to leave a tuning
> > > perceptibly just is unlikely to be able to eliminate all wolves.
> >
> > Aha! As I suspected! The wolf sneaks in the back door ...
>
> What's wrong with a few wolves anyway?

Try holding one in two- or three-part vocal harmony for
two seconds! Fingernails on a blackboard can't compete ...

I remember when (too long ago, I won't say) we sang
John Joubert's "Let Us Now Praise Famous Men".
Extended passages of whole-tone (6-EDO) singing
give me the heebie-jeebies. The last word I'd apply
to this kind of music is "harmony"! :-( And wolves are
even more discordant.

> Gene mentioned nano-temperaments that are also equal temperaments,
> such as 171-ET as an example with no wolves. This is true, and at the
> 7-limit, I personally consider even 72-ET to be a microtemperament
> (perceptibly-JI). but I assumed you were tanking of scales with far
> fewer notes.
>
> If a 5-limit western diatonic is all you need, and maximum perceptible
> justness is your aim, then you either live with the D:A wolf or you
> use an optimal meantone (possibly with tempered octaves). There are no
> other options. You have to hide a 21.5 cent comma, and this comma can
> only be distributed in certain ways such that you're bound to have an
> error of at least a quarter of this comma in some consonant interval. See
> http://dkeenan.com/music/DistributingCommas.htm

Thanks; been there!

> > > > Whereas MJ intonation (MJI) uses no tempering whatsoever; cannot
> > > > avoid the wolves; can supply exact thirds and sixths; and
> produces the
> > > > least regular scales possible.
> > >
> > > Oh no. That's quite wrong. You can do absolutely anything you like in
> > > mathematical JI, since the only requirement is that the result is
> > > expressed using (frequency or wavelength) ratios, never cents or other
> > > logarithmic units.
> >
> > I almost believe you're right ... In an earlier message, I produced a
> > scale with an imperfect fifth of 299/200 as generator, replacing 3/2
> > in a cycle of fifths stopping after seven notes: as regular as can be,
> > with all whole-tones equal in size, and both semitones equal in size.
> > Now that's MJI, but clearly not PJI, since every fifth is almost 6
> > cents flat. And it's not MSIJI, so I'd be unlikely to use it.
>
> Correct. As Gene said, it's an excellent meantone and this is a prime
> example of tempering by ratios. Have you listened to it.

No; why should I want to?

> A problem some folks have with meantone is that the major ninths are
> way too far from 4:9 for any semblance of justness. This can be
> improved slightly by widening the octaves by up to a sixth of a comma,
> but the fourths and major sixths suffer.
>
> > > You can even do _tempering_ in mathematical JI, but
> > > you'd better not call it that. :-)
> >
> > You can???
>
> Sure. You just did it above with your 200:299 fifths.

OK ...

> > Suppose you want to temper out a comma, say
> > 2401/2400. Do you simply adjust the ratio of the next nearest
> > interval by that ratio?
>
> See my paper above, but instead of dividing your comma into
> logarithmically equal parts by first converting to cents, calculate
> rational pseudo-roots (I don't know what they are really called).
>
> Lets say you want the Syntonic/Didymus comma (81/80) in four
> nearly-equal parts, i.e. find pseudo fourth roots for it. Multiply
> both sides by 4 to get 324/320 then you can write it as
>
> 324 323 322 321
> --- * --- * --- * ---
> 323 322 321 300
>
That's 320, not 300, of course.

In this case they are sequential epimoric or superparticular
ratios. In the case you give below, the sequence is formed
similarly, with a fixed surplus of numerator over denominator,
but this is no longer equal to 1. We might call them "surplus
roots" if they have no other name.

> > Of course, in this flavour of MJI, it
> > doesn't matter how large the integers get, or how irregular
> > the scale, does it?
>
> That's right.
>
> Here's one way to get 12-equal in MJI:
>
> The Pythagorean comma is 3^12 / 2^19 = 531441/524288
> The difference between top and bottom is 7153. Unfortunately this has
> no common factors with 12 so we multiply both sides by 12 and
> increment by 7153 each time to get 12 nearly equal parts by which to
> adjust the 12 fifths making the cycle. I'll list only the largest and
> smallest. They differ by less than 0.03 of a cent.
>
> 6377292 6298609
> ------- ... -------
> 6370139 6291456
>
> This is not an approximation of 12-equal, this _is_ 12-equal, for all
> _musical_ purposes.

Very clever. But surely, for all _musical_ purposes, it's
indistinguishable from 12-EDO?

> And yet Kraig and others would call it JI.
>
> Personally, I think that doing so makes a mockery of the whole idea of
> "just intonation".
>
> -- Dave Keenan

It certainly does seem unjust ...

Regards,
Yahya

--
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--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
> Dave Keenan wrote:

> > What instruments do you like it on?
>
> Voice, pipes such as simple flutes or recorders,
> plucked string instruments.

OK. So you are _never_ actually hearing Small Integer Mathematical JI,
otherwise known as small integer _phase-locked_ JI.

> > What I'm getting at is that maybe what you actually like is
> > "perceptible small integer JI" with a very small tolerance.
>
> I think you've made your case. :-) Tho I do enjoy other
> styles of music besides. I doubt I could enjoy Chopin's piano
> music played on anything BUT a 12-EDO piano (haen't tried
> it yet, tho).

Sure.

> > > As is "mathematical small-integer JI".
> >
> > Yes. And indeed this is the obvious point of departure in the
> > mathematical modelling of perceptible JI. The first thing to add to
> > this is _tolerance_. Most of us on this list can agree that departing
> > from the intended ratio by 1/3 of a cent does not change the
> > perceptible quality of a harmony, for most musical purposes.
>
> Right.

> > Of course one can always sustain an electronically-generated chord
> > long enough that you can hear beats that take even several minutes to
> > cycle. But this isn't typical of music, and most instruments aren't
> > that accurate. We can make exceptions for things like La Monte Young's
> > Dream House. Its perceptible justness apparently _does_ require
> > extremely accurate tuning because it uses ratios of such large
numbers.
>
> Agreed, it would materially alter the sound of the piece to be even
> fractionally mistuned. Very little music uses interacting pure sine
> waves, either; the closer it gets to that (as with soprano voices,
simple
> flutes or thin plucked strings), the less tolerance most of us have for
> mistuning.

I'm not sure about the sine-wave thing. It depends how loud they are.

I find I can tolerate less mistuning with quiet sawtooths than with
quiet sines. With loud enough sines, harmonics are generated by
nonlinearities in the ear (not to mention the amps and speakers).

> > > > Apparently one must simply take the composer's word for it.
> > >
> > > This is not true of "mathematical small-integer JI", is it?
> >
> > No. Because it is also a subset of perceptible JI.
>
> I thnk the appropriate model may be that PJI and
> MJI _intersect_ in small-integer just intonation (SIJI).
> That is, that SIJI, and only SIJI, is both percebtibly and
> mathematically just.

Yes.

Of course there is no sharp cutoff for justness in smallness of
integers. The best simply-expressed cutoffs for dyads seem to be in
terms of the "product complexity" i.e. the product of both sides of
the ratio when expressed in lowest terms, however there seems to be a
special case needed for ratios where the smaller number is a power of
2 (i.e. low note octave-equivalent to the virtual fundamental). Like
maybe throw out a factor of two in that case.

> > If you'd never experience a sponge cake (or a sponge anything) and I
> > wanted to define "sponge-cake" for you, the simplest thing might be
> > for me to give you a recipe.
>
> And this approach has now-venerable antecedents in mathematics,
> too. Call me old-fashioned if you will, but I prefer analytic ideas
> when possible.

Me too. But the important phrase is "when possible".

> > > > How would you define Redishness?
> > >
> > > "An object exhibits reddishness if it emits or reflects any
> > > light of wavelengths in that region of the EM spectrum where
> > > the Red colour receptors are most responsive."
> >
> > A lot of people consider spectral violet to be reddish.
>
> I was wondering if you'd mention that! :-)
>
> Most normally-sighted people do. And that's because
> the Red colour receptors are also responsive there,
> but to a smaller degree than in the "red" wavelengths
> at the other end of the visible spectrum. If you delete
> the word "most" from my attempted definition, that
> should suffice. I think that improves it.

Unfortunately no. It never stops responding. Its a probabilistic
thing. But yes, you can refine this and we've got it pretty well
worked out now for "normally sighted" people under standard
conditions. But there's been a hell of a lot of work done and money
spent to get that far. We just haven't got that far yet with justness
(maybe because there's no money in it). ;-)

> > Right. But my point was, that we live with this ambiguity in the case
> > of colour, we don't try to force it into an oversimplified
> > mathematical model.
>
> Sure we do! And to good purpose, too. Most list members
> are reading these messages on a computer monitor that
> reconstitutes many colours in the eye using just three "primary"
> colour guns - Red, Green and Blue. No way you can see a
> pure yellow light or a pure violet light emanate from your
> screen. Yet the simulation is, for most practical purposes,
> good enough.

In that case, it isn't _over_simplified.

Oversimplified is saying "Justly intoned means tuned to ratios of
small integers" when that covers hardly anything we actually hear that
we call Just (in fact nothing prior to electronics). It only covers
phase-locked JI.

All one has to do is add to that "or anything within some small
tolerance (typically 0.3 to 3 cents, depending on the listener) of
such a ratio of small integers" and you've suddenly got a practical
analytic definition that covers most of what anyone has ever heard and
called a Just interval.

You may have heard me say previously that I think we have a damn fine
start on an analytical definition of Just (for dyads at least) with
the second derivative of Paul Erlich's Harmonic Entropy function.

The beauty of this is that it has a single parameter which you can
vary to correspond to the listener's discrimination. This
simultaneously adjusts both the tolerance and the complexity cutoffs
and different intervals get appropriately different tolerances. More
tolerance on simpler ratios.

See
/tuning/files/Erlich/keenan.jpg
for the curve with one specific value of the discrimination parameter.
You can take everything above zero to be perceptibly just for a
listener with this discrimination.

Note that an interval can be dissonant but still just, e.g. near 8:13.

> Not only are simplified models (mathematical or otherwise)
> useful,
...

You're preaching to the converted here. Science is my religion ...
with a little Theravada Buddhism sprinkled on top :-).

> Whether our models are _over_-simplified is entirely a
> question of our purposes when we apply them. If they
> fail to help us achieve those purposes, they are inadequate
> and need revision. Their inadequacy is sometimes, but not
> always, due to being overly simple. Sometimes we have to
> refine and complicate our models; at others, we need to
> change or challenge our basic assumptions.

Indeed. And if your purpose is to predict when intervals in a scale
you've constructed will _sound_ just, then "if they're ratios they're
just" is utterly useless, and "if they're small integer ratios they're
just" is useless for anything you're going to implement with voice or
acoustic instruments (and most electronic ones).

> > What's wrong with a few wolves anyway?
>
> Try holding one in two- or three-part vocal harmony for
> two seconds! Fingernails on a blackboard can't compete ...

Right. I meant wolves in the tuning (that you can choose to avoid) not
wolves in the music, although that's sometimes fun.

Meantone gets rid of the D:A wolf, but still has the G#:Eb wolf, and
somehow folks managed to work around it, for a while, ... like about
400 years. :-)

> > Correct. As Gene said, it's an excellent meantone and this is a
> > prime example of tempering by ratios. Have you listened to it.
>
> No; why should I want to?

No need. It's just that I'd have been surprised if you'd _listened_ to
it and not realised it was a temperament, despite being rational.

> > 324 323 322 321
> > --- * --- * --- * ---
> > 323 322 321 300
> >
> That's 320, not 300, of course.

Yes. Thanks.

> > This is not an approximation of 12-equal, this _is_ 12-equal, for all
> > _musical_ purposes.
>
> Very clever. But surely, for all _musical_ purposes, it's
> indistinguishable from 12-EDO?

Yes. And I thought that's what I said. :-)

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> You're preaching to the converted here. Science is my religion ...
> with a little Theravada Buddhism sprinkled on top :-).

You're shouting into the wind. Art is my religion ... with a side-dish
of Quakerism thrown in, and that accompaniment is what keeps me from
going into any more detail.

Best,
Jon

Hi Paul,

Glad to see you back with gusto on this list!

Some comments ...

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > I personally have my own theory about accordance, in
> > which both the size of the prime-factors and the size
> > of their exponents are taken into account. I've never
> > tried to carefully develop a mathematical model for
> > this, but i believe that this is the essential problem
> > being debated in the "lattice metric" thread right now
> > -- if each prime-factor is a certain angle in the
> > lattice space, and each exponent is a certain distance
> > away from the origin, then the straight line which
> > connects any two notes is a geometrical model of the
> > Monzo-accordance measure.
>
> Well, as you know if you've read my writings, I believe
> that rather than straight-line distance, the *taxicab*
> distance works much better

Well, i like the taxicab measure too, since as i stated,
i'm not so sure how to weight the relative sizes of
prime-factor vs. exponent. The taxicab metric removes
the problem of how to relate them to each other, by
simply letting them be separate.

> (in conjunction with the Tenney lattice in particular),

Remind me again ... what's the dope on the Tenney lattice?
... i really should make an Encyclopedia page for that.

Are you able to write up something that can be added
to the Encyclopedia "lattice" page, comparing different
types of lattice geometries?

> and one of the wonderful things about the taxicab distance
> is that the angle between the axes becomes irrelevant.

Please elaborate on why, other than just simplifying things,
making the axis angles irrelevant is "wonderful".

My feeling is that axis angle is yet another piece
of the geometry, which can be used to code data about
some pyschoacoustic phenomenon ... whatever it might be.
As you know, my "Monzo lattice" attempts to do this,
encoding the pitch-height data for each prime into the
angle of its axis on the lattice.

> > The primary quesion i can see is: how does one properly
> > set up the relationship between the prime-factors
> > and the exponents, so that varying size in the two
> > domains correlates in some audibly perceptually
> > meaningful way.
>
> I think the Tenney lattice, when a taxicab metric
> is used on it, automatically takes care of both
> of these considerations in a very nice way.
>
> If you need to impose octave-equivalence, the
> van Prooijen lattice, along with a hexagonal-norm
> (or higher-dimensional analogue) metric,
> is the best way to adapt the Tenney picture.

If you are able to contribute the bit i asked for
about lattice geometries, would you elaborate on
these two paragraphs? Thanks.

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

What with all the comma worship and religious banter, am I allowed to propound Islamic fundamentalism on top of it?

I'm not a scholastic and I believe in the necessity of rationality. Therefore, I certainly think that the tuning list does not require to uphold any dogmas during its investigation of musical theory and mathematics.

`Science... commits suicide when it adopts a creed.` (T. H. Huxley)

Cordially,
Ozan

----- Original Message -----
From: Jon Szanto
To: tuning@yahoogroups.com
Sent: 26 Temmuz 2005 Salı 7:19
Subject: [tuning] Re: Definitions of JI

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> You're preaching to the converted here. Science is my religion ...
> with a little Theravada Buddhism sprinkled on top :-).

You're shouting into the wind. Art is my religion ... with a side-dish
of Quakerism thrown in, and that accompaniment is what keeps me from
going into any more detail.

Best,
Jon

Dave,

You wrote:

> > > What instruments do you like it on?
> >
> > Voice, pipes such as simple flutes or recorders,
> > plucked string instruments.
>
> OK. So you are _never_ actually hearing Small Integer Mathematical JI,
> otherwise known as small integer _phase-locked_ JI.

You can prove that assertion, of course? :-)
Including justifying your insertion of the epithet
"phase-locked".

> > Agreed, it would materially alter the sound of the piece to be even
> > fractionally mistuned. Very little music uses interacting pure sine
> > waves, either; the closer it gets to that (as with soprano voices,
> > simple
> > flutes or thin plucked strings), the less tolerance most of us have for
> > mistuning.
>
> I'm not sure about the sine-wave thing. It depends how loud they are.
>
> I find I can tolerate less mistuning with quiet sawtooths than with
> quiet sines. With loud enough sines, harmonics are generated by
> nonlinearities in the ear (not to mention the amps and speakers).

Sure it does. Loudness distorts anything. I don't claim to enjoy
JI in heavy-metal music! Not even "power chords" with their
open fifths.

> > I thnk the appropriate model may be that PJI and
> > MJI _intersect_ in small-integer just intonation (SIJI).
> > That is, that SIJI, and only SIJI, is both percebtibly and
> > mathematically just.
>
> Yes.
>
> Of course there is no sharp cutoff for justness in smallness of
> integers. The best simply-expressed cutoffs for dyads seem to be in
> terms of the "product complexity" i.e. the product of both sides of
> the ratio when expressed in lowest terms,

There's something like that in the "harmonic distance" � la Tenney,
isn't there? HD = log (numerator * denominator)/log 2. That's
one thing I picked up from reading Paul Erlich's paper "A Middle
Path ..."

> ... however there seems to be a
> special case needed for ratios where the smaller number is a power of
> 2 (i.e. low note octave-equivalent to the virtual fundamental). Like
> maybe throw out a factor of two in that case.

Why only " ... where the _smaller_ number ..."? Wouldn't, say,
(2^n + 1)/ 2^n have nearly the same "justness" as 2^n /( 2^n - 1)?
(Both have HD ~ = 2n.)

> > > If you'd never experience a sponge cake (or a sponge anything) and I
> > > wanted to define "sponge-cake" for you, the simplest thing might be
> > > for me to give you a recipe.
> >
> > And this approach has now-venerable antecedents in mathematics,
> > too. Call me old-fashioned if you will, but I prefer analytic ideas
> > when possible.
>
> Me too. But the important phrase is "when possible".

Agreed.

> > > > > How would you define Redishness?
> > > > "An object exhibits reddishness if it emits or reflects any
> > > > light of wavelengths in that region of the EM spectrum where
> > > > the Red colour receptors are most responsive."
> > >
> > > A lot of people consider spectral violet to be reddish.
> > I was wondering if you'd mention that! :-)
> >
> > Most normally-sighted people do. And that's because
> > the Red colour receptors are also responsive there,
> > but to a smaller degree than in the "red" wavelengths
> > at the other end of the visible spectrum. If you delete
> > the word "most" from my attempted definition, that
> > should suffice. I think that improves it.
>
> Unfortunately no. It never stops responding. Its a probabilistic
> thing. But yes, you can refine this and we've got it pretty well
> worked out now for "normally sighted" people under standard
> conditions. But there's been a hell of a lot of work done and money
> spent to get that far. We just haven't got that far yet with justness
> (maybe because there's no money in it). ;-)

There's no justness! ... er , justice ...

> > > Right. But my point was, that we live with this ambiguity in the case
> > > of colour, we don't try to force it into an oversimplified
> > > mathematical model.
> >
> > Sure we do! And to good purpose, too. Most list members
> > are reading these messages on a computer monitor that
> > reconstitutes many colours in the eye using just three "primary"
> > colour guns - Red, Green and Blue. No way you can see a
> > pure yellow light or a pure violet light emanate from your
> > screen. Yet the simulation is, for most practical purposes,
> > good enough.
>
> In that case, it isn't _over_simplified.
>
> Oversimplified is saying "Justly intoned means tuned to ratios of
> small integers"

Did I say that? I musta been _much_ younger then ...

> ... when that covers hardly anything we actually hear that
> we call Just (in fact nothing prior to electronics). It only covers
> phase-locked JI.

How do two singers - or a barbershop quartet, for that
matter - produce "perceptibly just" harmonies? Are you
saying they must lock phase? How can they possibly do so?
8-0

> All one has to do is add to that "or anything within some small
> tolerance (typically 0.3 to 3 cents, depending on the listener) of
> such a ratio of small integers" and you've suddenly got a practical
> analytic definition that covers most of what anyone has ever heard and
> called a Just interval.

OK ... could be useful!

> You may have heard me say previously that I think we have a damn fine
> start on an analytical definition of Just (for dyads at least) with
> the second derivative of Paul Erlich's Harmonic Entropy function.

No, I'm fairly sure I'd have remembered something that
so smacks of analytical mechanics ...

> The beauty of this is that it has a single parameter which you can
> vary to correspond to the listener's discrimination. This
> simultaneously adjusts both the tolerance and the complexity cutoffs
> and different intervals get appropriately different tolerances. More
> tolerance on simpler ratios.
>
> See
>
/tuning/files/Erlich/keenan.jp
g
> for the curve with one specific value of the discrimination parameter.
> You can take everything above zero to be perceptibly just for a
> listener with this discrimination.

OK, I will. Thanks!

> Note that an interval can be dissonant but still just, e.g. near 8:13.

Dissonant ... No, I won't start down that road again right now!

> > Not only are simplified models (mathematical or otherwise)
> > useful,
> ...
>
> You're preaching to the converted here. Science is my religion ...
> with a little Theravada Buddhism sprinkled on top :-).

Sorry!!! 8-( Didn't mean to preach. When I read it back,
I realised it sounded a bit ... didactic. And pompous! No
offence taken, I hope?

> > Whether our models are _over_-simplified is entirely a
> > question of our purposes when we apply them. If they
> > fail to help us achieve those purposes, they are inadequate
> > and need revision. Their inadequacy is sometimes, but not
> > always, due to being overly simple. Sometimes we have to
> > refine and complicate our models; at others, we need to
> > change or challenge our basic assumptions.
>
> Indeed. And if your purpose is to predict when intervals in a scale
> you've constructed will _sound_ just, then "if they're ratios they're
> just" is utterly useless,

I can see that. Who can tell when you're singing
a 131:171? Really!

> ... and "if they're small integer ratios they're
> just" is useless for anything you're going to implement with voice

I don't see that. At all!

> or
> acoustic instruments (and most electronic ones).
>
> > > What's wrong with a few wolves anyway?
> >
> > Try holding one in two- or three-part vocal harmony for
> > two seconds! Fingernails on a blackboard can't compete ...
>
> Right. I meant wolves in the tuning (that you can choose to avoid) not
> wolves in the music, although that's sometimes fun.

Yes, if you're a conservationist and not a woodcutter.

> Meantone gets rid of the D:A wolf, but still has the G#:Eb wolf, and
> somehow folks managed to work around it, for a while, ... like about
> 400 years. :-)

It's called "an avoidance tactic" ...

> > > Correct. As Gene said, it's an excellent meantone and this is a
> > > prime example of tempering by ratios. Have you listened to it.
> >
> > No; why should I want to?
>
> No need. It's just that I'd have been surprised if you'd _listened_ to
> it and not realised it was a temperament, despite being rational.

Exactly. If I listen to sustained flat fifths, I know
they're flat. Within a limit of a few cents.

> > > 324 323 322 321
> > > --- * --- * --- * ---
> > > 323 322 321 300
> > >
> > That's 320, not 300, of course.
>
> Yes. Thanks.
>
> > > This is not an approximation of 12-equal, this _is_ 12-equal, for all
> > > _musical_ purposes.
> >
> > Very clever. But surely, for all _musical_ purposes, it's
> > indistinguishable from 12-EDO?
>
> Yes. And I thought that's what I said. :-)

And I thought the point worth repeating. It's just
a rational tempering rather than an irrational one.
That still doesn't make it PJI or SIJI, but I guess
we could call it MJI.

Regards,
Yahya
--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.338 / Virus Database: 267.9.5/58 - Release Date: 25/7/05

Jon Szanto wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> > You're preaching to the converted here. Science is my religion ...
> > with a little Theravada Buddhism sprinkled on top :-).
>
> You're shouting into the wind. Art is my religion ... with a side-dish
> of Quakerism thrown in, and that accompaniment is what keeps me from
> going into any more detail.
>
> Best,
> Jon

:-) You're both tossing words into the bitstream.
Islam is my religion ...
with a foundation of blind optimism.

Peace!
Yahya

--
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Checked by AVG Anti-Virus.
Version: 7.0.338 / Virus Database: 267.9.5/58 - Release Date: 25/7/05

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> Remind me again ... what's the dope on the Tenney lattice?
> ... i really should make an Encyclopedia page for that.

Unfortunately the dope on the Tenney lattice is a little confusing,
with Paul and I defining it differently. Paul defines it as a
rectangular lattice in ordinary space, with the lenths of the lines
between adjacent lattice points proportional to log(p), where p is the
prime corresponding to the axis in question. Then, instead of using
the as-the-crow-flies distance between two lattice points, you use the
taxicab distance. This is a distance measure between lattice points
only, in other words.

I define the Tenney lattice as the lattice of integer coordinates in
Tenney space, where Tenney space is a particular normed vector space
which is not Euclidean. This has advantages over Paul's definition, in
that distance in the space between lattice points is Tenney distance,
but the distance can also be used for other purposes, such as defining
TOP tuning. My definition appears here:

http://66.98.148.43/~xenharmo/top.htm

I'm giving Paul's definition by memory; I was going to make some
comments on his paper, and had it on the floor next to my computer,
but I think my cleaning lady may have thrown it out. If Paul thinks he
can spare another copy...

> My feeling is that axis angle is yet another piece
> of the geometry, which can be used to code data about
> some pyschoacoustic phenomenon ... whatever it might be.

I'm the main person who uses axis angle around here; most recently in
order to project higher limit lattices down on to lower limit ones.
Alas, in Paul's defintion of the Tenney lattice, the lattice angle is
irrelevant, whereas in my definition, it doesn't exist. You need
Euclidean space to have angles.

> As you know, my "Monzo lattice" attempts to do this,
> encoding the pitch-height data for each prime into the
> angle of its axis on the lattice.

Which never made sense to me. Why do that?

> If you are able to contribute the bit i asked for
> about lattice geometries, would you elaborate on
> these two paragraphs? Thanks.

The Kees height is found by taking the odd part of a positive rational
q, reduced to lowest terms, and then the maximum of the numerator and
denominator. Take the log of that to some base (eg, 3) and you have a
distance measure on the octave equivalence classes. It might be
interesting to do what I did in the case of the Tenney lattice, and
put it into a corresponding space.

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> Agreed, it would materially alter the sound of the piece to be even
> fractionally mistuned. Very little music uses interacting pure sine
> waves, either; the closer it gets to that (as with soprano voices,
simple
> flutes or thin plucked strings), the less tolerance most of us have
for
> mistuning.

All the psychoacoustic evidence points to the opposite conclusion. Not
to mention my own experience, and that of others who've posted here. If
you haven't read Sethares' book yet, you could search the web for Plomp
& Levelt . . . their classic experiments found that listeners discern
*no* minima of discordance other than 1:1 for pairs of pure sine waves.
Best of all, try some experiments yourself!

BTW thin strings have more high upper partials, not less.

-Paul

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> > (in conjunction with the Tenney lattice in particular),
>
>
> Remind me again ... what's the dope on the Tenney lattice?

*Please* read the 'Middle Path' paper I snail-mailed you. Pretty
please.

> ... i really should make an Encyclopedia page for that.
>
> Are you able to write up something that can be added
> to the Encyclopedia "lattice" page, comparing different
> types of lattice geometries?

Sure, especially if you'll be providing nice graphics for all of
them!! Are you saying you are actually accepting corrections/changes
to the Encyclopedia now? You know I've been waiting a long time for
this moment . . .

> > and one of the wonderful things about the taxicab distance
> > is that the angle between the axes becomes irrelevant.
>
> Please elaborate on why, other than just simplifying things,
> making the axis angles irrelevant is "wonderful".

Simplifying things is wonderful enough. From a mathematical point of
view it's great; if you don't need angles to be defined, you can get
away with a simpler axiomatic system to geometrize your algebra; and
of course the fewer arbitrary assumptions the better. But it's also
nice because you can change the angles to help view different things -
- for example to bring scales with a certain number of notes per
octave into focus, or to help see your way around a very high-
dimensional lattice -- while leaving the taxicab distances unchanged.

> My feeling is that axis angle is yet another piece
> of the geometry, which can be used to code data about
> some pyschoacoustic phenomenon ... whatever it might be.
> As you know, my "Monzo lattice" attempts to do this,
> encoding the pitch-height data for each prime into the
> angle of its axis on the lattice.

That "pitch-height data" being the fraction of an octave left over
after subtracting enough octaves from the prime.

Anyway, you said at one point you guys were working with the pitch
axis itself (in the lattice). The paper I mailed you introduces a
related concept, the "pitch contours," which function like a pitch
axis but don't require the concept of angle to be defined.

> > I think the Tenney lattice, when a taxicab metric
> > is used on it, automatically takes care of both
> > of these considerations in a very nice way.
> >
> > If you need to impose octave-equivalence, the
> > van Prooijen lattice, along with a hexagonal-norm
> > (or higher-dimensional analogue) metric,
> > is the best way to adapt the Tenney picture.
>
>
>
> If you are able to contribute the bit i asked for
> about lattice geometries, would you elaborate on
> these two paragraphs? Thanks.

Of course I will, with accompanying graphics by you of course. But I
*beg* you, please read my paper -- I think being away from your
computer and on a comfy couch or something might be the best way to
get this stuff, including Tenney Harmonic Distance, to sink in.

-Paul

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> There's something like that in the "harmonic distance" à la Tenney,
> isn't there? HD = log (numerator * denominator)/log 2. That's
> one thing I picked up from reading Paul Erlich's paper "A Middle
> Path ..."

That's right! Monz and Aaron take note. There's another way of
writing this distance measure, which clarifies its nature as a
taxicab metric. If we factor the ratio as

2^a * 3^b * 5^c * 7^d * 11^e * . . .

then the Tenney Harmonic Distance is equal to

|a|*log(2) + |b|*log(3) + |c|*log(5) + |d|*log(7) + |e|*log(11)
+ . . .

Thus it's the taxicab distance on the Tenney lattice, where the
length of a rung for prime p (*including* 2) is log(p).

We can take these to be logs base 2. No matter which type of log
you're using, this can be accomplished by dividing these logs by the
log of 2, as long as you use the same base for both logs. Of course
the log base 2 of 2 is 1.

A line or two math should then show you that Yahya's expression, and
mine above, are equivalent.

But then again, this isn't the tuning-math list . . . :)

-Paul

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > Remind me again ... what's the dope on the Tenney lattice?
> > ... i really should make an Encyclopedia page for that.
>
> Unfortunately the dope on the Tenney lattice is a little confusing,
> with Paul and I defining it differently. Paul defines it as a
> rectangular lattice in ordinary space,

It's not exactly ordinary space because angles are not defined in it.
You could think of it as a latticework with freely rotating
connections at the joints, like one of those doohickeys you use to
grab things from far away. All the joints are constrained to move
together. So imagine that the angles, while always remaining
identical to one another, change randomly from moment to moment. This
is one way of using a real-world intuition to get at a more abstract
mathematical concept. The 'real' properties of the Tenney lattice are
the ones that stay constant over all these angle-changes; hence,
abstractly, they don't depend on the notion of angle being defined at
all. (Parallelism, though, does have to be defined.)

> I'm giving Paul's definition by memory; I was going to make some
> comments on his paper, and had it on the floor next to my computer,
> but I think my cleaning lady may have thrown it out. If Paul thinks
he
> can spare another copy...

Sure . . . e-mail me your address again, while I wait for Jacques
Dudon to do the same . . .

> > My feeling is that axis angle is yet another piece
> > of the geometry, which can be used to code data about
> > some pyschoacoustic phenomenon ... whatever it might be.
>
> I'm the main person who uses axis angle around here; most recently
in
> order to project higher limit lattices down on to lower limit ones.
> Alas, in Paul's defintion of the Tenney lattice, the lattice angle
is
> irrelevant, whereas in my definition, it doesn't exist. You need
> Euclidean space to have angles.

I'm trying to say that in my definition, angles aren't even defined;
hence you're not really dealing with Euclidean space. However, that
shouldn't stop us from drawing pictures to help people understand
this stuff and how it works. I hope you won't complain that, because
pictures are drawn on a real-world piece of paper, they are therefore
suggesting Euclidean space! The big black textbook _Gravitation_ is
all about non-Euclidean space but it has plenty of pictures in it.

> The Kees height is found by taking the odd part of a positive
rational
> q, reduced to lowest terms, and then the maximum of the numerator
and
> denominator. Take the log of that to some base (eg, 3) and you have
a
> distance measure on the octave equivalence classes.

Kees van Prooijen calls it expressibility. He defines it in the 5-
limit case here:

http://www.kees.cc/tuning/perbl.html

and discusses the 7-limit and {3,5,7} cases as well.

> It might be
> interesting to do what I did in the case of the Tenney lattice, and
> put it into a corresponding space.

This is exactly what I was asking the tuning-math list about, to no
avail, about two years ago. But I see it now. The way to visualize it
is as a regular hexagonal norm over the second-to-last lattice on
this page:

http://www.kees.cc/tuning/lat_perbl.html

The paradox Kees mentions goes away because the metric is neither a
Euclidean nor a taxicab one, but one in which concentric regular
hexagons define contours of constant 'distance' from a given center.
One could visualize this as a taxicab metric, though, by
superimposing a very fine regular triangular 'grid' of streets over
the lattice, and insisting that one travel along these streets rather
than along the lattice rungs themselves. But I think it's easy enough
to consider concentric regular hexagons as defining a 'distance' of a
special kind.

Isn't this a topic for the tuning-math list?

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> > Unfortunately the dope on the Tenney lattice is a little confusing,
> > with Paul and I defining it differently. Paul defines it as a
> > rectangular lattice in ordinary space,
>
> It's not exactly ordinary space because angles are not defined in it.

In your paper you did not say the space was not Euclidean, and this is
*always* assumed unless you specify otherwise. If you say the lattice
is of points {n2*log(2) + n3*log(3) + ... + np*log(p)}, for integers
ni, then unless you say it is *not* a lattice in Rn, of course that is
what it means to say it is a lattice.

If you don't mean for the space to be Rn, you need to say so, at least
in a footnote. However, if you go that route, isn't it more sensible
to do what I do, and define a normed vector space in which the Tenney
lattice is embedded, and for which the metric is Tenney distance?
Again, that could be left to a footnote, but the Tenney space would
allow you to define TOP tuning in general, which I think would be a
good thing for your paper anyway.

> Isn't this a topic for the tuning-math list?

No doubt. I'm not seeing where you get hexagons.

Gene,

You wrote:

> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
> > Sorry, I don't understand what you mean by "between 31 and 50"
> > here.
>
> 299/200 is 696.17 cents, which is shaper than the 5--et fifth of
> exactly 696 cents, but flatter than the 31-et fifth of 696.77 cents.

Got it.

> > Why wouldn't you consider it JI; which "strict" definition of JI do
> > you have in mind?
>
> In practice, using this as a generator results in a tempered scale, so
> that's what I'd consider it to be, unless I really wanted to go up to
> the 23-limit and regard it not as a fifth, but as something you get to
> by way of a pair of 23-limit consonances like (13/10)(23/20). But I'm
> not much of a believer in the 23 limit.

OK.

> > > 2401/2400 is so small you can pretty much do anything or nothing. That
> > > is, you can simply leave it as is, or retune, for instance by tuning
> > > to 612 equal.
> >
> > Doing nothing is hardly tempering ...
>
> I think in the case of 2401/2400, where you get chords off by less
> than a cent, it really is a sort of tempering. It acts as if it was,
> at any rate.

In that you can choose as a substitute for any JI ratio,
a whole number of steps of 2401/2400.

> Retuning to 612-EDO
> > certainly is, and I presume you mean that 2401/2400 is
> > 0 steps of 612-EDO?
>
> Right, so this is a way of tempering out 2401/2400.

Right.

> > > Of course, in this flavour of MJI, it
> > > > doesn't matter how large the integers get, or how irregular
> > > > the scale, does it? But it wouldn't do me at all.
> > >
> > > Why not?
> >
> > Because I prefer MSIJI, with its greater resonances.
>
> What the heck is MSIJI, ...

Mathematical Small-Integer JI.

> ... and why does it have more resources?

I wrote "resonances", not "resources"!

> > When playing an unfretted string instrument, tuning
> > to low-integer ratios produces quite a bit of sympathetic
> > vibration on the unplucked strings when holding notes -
> > and I like that sound. "Just" personal preference.
>
> And you think nanotempering won't do that? I think it
> would do this equally well in 612-equal tuning.

I don't know for sure. I'd have to experiment to find
out. 12-EDO certainly has less sympathetic vibration
than small-integer JI. But 612-EDO is a very fine
division of the octave - 1200/612 =
1.960784313725490196078431372549 cents, so
we're looking at errors of less than 2c from JI. The
sympathetic resonances should be strong; I don't
claim to be able to play any stringed instrument to
that degree of accuracy anyway!

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.338 / Virus Database: 267.9.5/58 - Release Date: 25/7/05

"wallyesterpaulrus" wrote:
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
> > Agreed, it would materially alter the sound of the piece to be even
> > fractionally mistuned. Very little music uses interacting pure sine
> > waves, either; the closer it gets to that (as with soprano voices,
> > simple
> > flutes or thin plucked strings), the less tolerance most of us have
> > for
> > mistuning.
>
> All the psychoacoustic evidence points to the opposite conclusion. Not
> to mention my own experience, and that of others who've posted here. If
> you haven't read Sethares' book yet, you could search the web for Plomp
> & Levelt . . . their classic experiments found that listeners discern
> *no* minima of discordance other than 1:1 for pairs of pure sine waves.
> Best of all, try some experiments yourself!

Paul,
I've tried many such experiments over the years, using
my Moog analog synth. Clearly, I'm not one of the listeners
in the "classic experiments" you cite. Tell me, were their
listeners musicians?

If "all the evidence" says I'm wrong, then must I doubt
my own ears??

If the evidence of my ears supports my conclusions - and
it does - then obviously I have over-generalised my own
experience. On rereading the parts you quoted, I must
say that I have no evidence _whatsoever_ that "most of
us" have less tolerance for mistuning in purer sounds.

A complete failure of induction!

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.338 / Virus Database: 267.9.5/58 - Release Date: 25/7/05

--- In tuning@yahoogroups.com, "Jon Szanto" <jszanto@c...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> > You're preaching to the converted here. Science is my religion ...
> > with a little Theravada Buddhism sprinkled on top :-).
>
> You're shouting into the wind. Art is my religion ... with a side-dish
> of Quakerism thrown in, and that accompaniment is what keeps me from
> going into any more detail.
>
> Best,
> Jon

My brother Jon has given me a timely reminder that I may have let my
love of rationality overcome my compassion. Indeed I may have been a
little cruel. For which I wish to apologise to Kraig Grady and Daniel
Wolf and anyone else I may have offended.

I can understand that to some the mathematics of JI itself may have a
spiritual dimension. And so to _deliberately_ deviate from the strict
ratios, even by micro or nano-tempering, or to use anything other than
integer arithmetic in the design of a JI scale, may be felt as a
violation. While the inevitable similar-sized deviations due to
real-world instruments may not.

Who was the famous mathematician who said "Only the natural numbers
[positive integers and zero] are the work of God, all else is the work
of man"? (Gene will know)

However, I can't help feeling that there must be _some_ way to define
JI so as to respect this, while not admitting scales that are audibly
indistinguishable from 12-ET into its hallowed halls.

And now I really must take a break from the list for a while, to get
some other things done. I've got a heap of photovoltaic modules (solar
panels) arriving tomorrow, among other things.

Many thanks to all who engaged in these enjoyable conversations, and
all who were willing to listen.

-- Dave Keenan

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> Who was the famous mathematician who said "Only the natural numbers
> [positive integers and zero] are the work of God, all else is the work
> of man"? (Gene will know)

Leopold Kronecker. In spite of saying that, he had his own way of
constructing abstract algebraic extension fields, for instance. In
terms of math relevant to music theory, we have the Kronecker Basis
Theorem, so I guess we can claim a tuning theory connection.

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> > And you think nanotempering won't do that? I think it
> > would do this equally well in 612-equal tuning.
>
> I don't know for sure. I'd have to experiment to find
> out. 12-EDO certainly has less sympathetic vibration
> than small-integer JI. But 612-EDO is a very fine
> division of the octave - 1200/612 =
> 1.960784313725490196078431372549 cents, so
> we're looking at errors of less than 2c from JI.

It's a lot closer than that. The fifth is sharp by 0.0058 cents, the
major third flat by 0.039 cents, the 7/4 is 0.20 cents flat and the
11/8 is 0.34 cents flat. These are awfully small, especially in the 3
or 5 limit.

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

> If you say the lattice
> is of points {n2*log(2) + n3*log(3) + ... + np*log(p)}, for integers
> ni, then unless you say it is *not* a lattice in Rn, of course that is
> what it means to say it is a lattice.

I can't parse this sentence.

> If you don't mean for the space to be Rn, you need to say so, at least
> in a footnote. However, if you go that route, isn't it more sensible
> to do what I do, and define a normed vector space in which the Tenney
> lattice is embedded, and for which the metric is Tenney distance?

I don't see any crucial difference right now.

> Again, that could be left to a footnote, but the Tenney space would
> allow you to define TOP tuning in general, which I think would be a
> good thing for your paper anyway.

I can define it just fine without going into your definition of Tenney
space, so I don't know what you mean.

> > Isn't this a topic for the tuning-math list?
>
> No doubt. I'm not seeing where you get hexagons.

If the pictures on Kees' page don't make it clear, I'll have to draw
some other pictures for you.

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
> "wallyesterpaulrus" wrote:
> > --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...>
wrote:
> >
> > > Agreed, it would materially alter the sound of the piece to be
even
> > > fractionally mistuned. Very little music uses interacting pure
sine
> > > waves, either; the closer it gets to that (as with soprano
voices,
> > > simple
> > > flutes or thin plucked strings), the less tolerance most of us
have
> > > for
> > > mistuning.
> >
> > All the psychoacoustic evidence points to the opposite
conclusion. Not
> > to mention my own experience, and that of others who've posted
here. If
> > you haven't read Sethares' book yet, you could search the web for
Plomp
> > & Levelt . . . their classic experiments found that listeners
discern
> > *no* minima of discordance other than 1:1 for pairs of pure sine
waves.
> > Best of all, try some experiments yourself!
>
> Paul,
> I've tried many such experiments over the years, using
> my Moog analog synth. Clearly, I'm not one of the listeners
> in the "classic experiments" you cite. Tell me, were their
> listeners musicians?

I believe they were all nonmusicians, but Bill Sethares and others
should speak up with the actual experimental details.

> If "all the evidence" says I'm wrong, then must I doubt
> my own ears??

No, but there may be something wrong with your experimental setup.
Pure sine waves are not easy to produce, for several reasons. Still I
find it odd that your tolerance or accuracy wouldn't be finer with
other waveforms. Almost everyone I've discussed this stuff with, for
example Daniel Wolf, finds it a lot harder to pin down the tuning
when sine waves are used as opposed to any other (periodic) waveform.

The accepted explanation is: sine waves have no harmonics; tuning
simple-integer ratios is done by (and is always the result when)
eliminating beating between coincident harmonics; hence there's
little chance of accurately tuning simple-integer ratios (except 1:1)
when using sine waves. This jibes very well with my experience but I
can't categorically deny that you're experiencing something else.
Nonetheless, this 'experiment' or 'observation' is a fundamental
assumption behind Sethares' entire book, since he assumes that simply
replacing harmonic partials with inharmonic partials in the timbres
used will completely replace any 'attraction to' or 'consonance of'
simple-integer ratios with an 'attraction to' or 'consonance of' the
relevant inharmonic ratios. I've tried to point out some limitations
to this way of thinking; perhaps your experience will point to others.

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
>
> Gene,
>
> You wrote:

> > I think in the case of 2401/2400, where you get chords off by less
> > than a cent, it really is a sort of tempering. It acts as if it was,
> > at any rate.
>
> In that you can choose as a substitute for any JI ratio,
> a whole number of steps of 2401/2400.

That's not what Gene meant. He meant that if you have a simple-integer
JI chord in mind, but in your realization, one of the notes is off by a
2401:2400 (perhaps because you've restricted yourself to a finite JI
set, for your instrument), then it's much like having a tempered chord
yet without any commas having been split and distributed (the way
tempering normally works). In other words, you could temper out the
2401:2400, but even if you don't, the 'wolf' intervals will only be off
by 2401:2400 or 0.7 cents, which is so small that you could consider
the wolf 'tamed' even though you didn't do any explicit tempering.

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> I can define it just fine without going into your definition of Tenney
> space, so I don't know what you mean.

Didn't you define it only for one comma?

> > > Isn't this a topic for the tuning-math list?
> >
> > No doubt. I'm not seeing where you get hexagons.
>
> If the pictures on Kees' page don't make it clear, I'll have to draw
> some other pictures for you.

I screwed up on the hexagons, there are indeed hexagons.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > I can define it just fine without going into your definition of
Tenney
> > space, so I don't know what you mean.
>
> Didn't you define it only for one comma?

The particular method explained in the latest version of my 'Middle
Path' paper for *finding* a TOP (Tenney-OPtimal) tuning is only
defined for one comma, but TOP tuning itself is defined the same way
no matter how many commas are tempered out -- it's the tuning that
minimizes the maximum damage (where damage = ratios's mistuning
divided by ratio's Harmonic Distance) over all ratios in the infinite
lattice. Early on, I discussed (on tuning-math) how to calculate the
TOP tuning for ETs, even though more than one comma is tempered out
in those cases:

/tuning-math/message/8512

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

>In other words, you could temper out the
> 2401:2400, but even if you don't, the 'wolf' intervals will only be off
> by 2401:2400 or 0.7 cents, which is so small that you could consider
> the wolf 'tamed' even though you didn't do any explicit tempering.

Among interesting 7-limit commas, there seems to be a distinct gap
between the upper end of the really small range, at 65625/65536, or
2.35 cents, and the lower end of the small range, at 6144/6125, or
5.36 cents; the best thing in there being 589824/588245. I think for
65625/65536 or less we can get away pretty well with not tempering the
comma--certainly that works excellently when we get down to 2401/2400,
but I think in practice it's worked well for 32805/32768.

I wonder if this could be worked up into something definite enough to
help on the comma boundry question. The gap fills in from a pure
11-limit point of view until we get to 5632/5625, however, and there
are high-complexity 5-limit commas we might stick in there from a
5-limit point of view, such as the 19-comma, parakleisma, or semisuper
comma.

Gene Ward Smith wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
> > > And you think nanotempering won't do that? I think it
> > > would do this equally well in 612-equal tuning.
> >
> > I don't know for sure. I'd have to experiment to find
> > out. 12-EDO certainly has less sympathetic vibration
> > than small-integer JI. But 612-EDO is a very fine
> > division of the octave - 1200/612 =
> > 1.960784313725490196078431372549 cents, so
> > we're looking at errors of less than 2c from JI.
>
> It's a lot closer than that. The fifth is sharp by 0.0058 cents, the
> major third flat by 0.039 cents, the 7/4 is 0.20 cents flat and the
> 11/8 is 0.34 cents flat. These are awfully small, especially in the 3
> or 5 limit.
>

Sure. I was only setting an upper limit on the possible error.
The fifth and major third would have to be good enough for
most practical purposes, excepting possibly La Monte Young's
kind of slow evolution of consonances over a long time.

Regards,
Yahya

________________________________________________________________________
________________________________________________________________________

Message: 4
Date: Wed, 27 Jul 2005 16:19:39 -0000
From: "centeroftownkh" <centeroftownkh@yahoo.com>
Subject: salinas 19tone

Hi Everyone -

I'm a little confused about the enharmonic 19-tone scale by Salinas. Barbour
says it's an
ET scale, but when I open it up in Maxmicrotuner all the intervals are
Just - 1/1, 24/25,
16/15, 9/8, 75/64, 6/5, 5/4, 125/96, 4/3, 25/18, 64/45, 3/2, 25/16, 8/5,
5/3, 125/72,
16/9, 15/8, 125/64, 2/1. Pardon my simple-mindedness.

Thank you.

________________________________________________________________________
________________________________________________________________________

Message: 5
Date: Wed, 27 Jul 2005 19:34:02 -0000
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
Subject: Re: Lattice metrics and geometries (was: Definitions of JI)

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

> If you say the lattice
> is of points {n2*log(2) + n3*log(3) + ... + np*log(p)}, for integers
> ni, then unless you say it is *not* a lattice in Rn, of course that is
> what it means to say it is a lattice.

I can't parse this sentence.

> If you don't mean for the space to be Rn, you need to say so, at least
> in a footnote. However, if you go that route, isn't it more sensible
> to do what I do, and define a normed vector space in which the Tenney
> lattice is embedded, and for which the metric is Tenney distance?

I don't see any crucial difference right now.

> Again, that could be left to a footnote, but the Tenney space would
> allow you to define TOP tuning in general, which I think would be a
> good thing for your paper anyway.

I can define it just fine without going into your definition of Tenney
space, so I don't know what you mean.

> > Isn't this a topic for the tuning-math list?
>
> No doubt. I'm not seeing where you get hexagons.

If the pictures on Kees' page don't make it clear, I'll have to draw
some other pictures for you.

________________________________________________________________________
________________________________________________________________________

Message: 6
Date: Wed, 27 Jul 2005 19:38:24 -0000
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
Subject: Re: Microtonal meeting in France

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > Perhaps "concordant" with or without some qualification?
>
> Hmmm...
>
> 4-8 cents harmonious
> 2-4 cents concordant
> 1-2 cents millitempered
> 1/2-1 cents microtempered
> 1/4-1/2 cents nanotempered
> < 1/4 cents sensibly JI

Clearly some ratios are more concordant than others so this doesn't
sound like a reasonable suggestion as it stands. And millitempered? Has
anyone heard of a millicomputer, or millifiche, or milliwaves? :)

________________________________________________________________________
________________________________________________________________________

Message: 7
Date: Wed, 27 Jul 2005 19:40:58 -0000
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
Subject: Re: another "special" comma and a puzzle

--- In tuning@yahoogroups.com, Jeremy Targett <jeremy.targett@g...>
wrote:

> It seems to me that
> given random sets of numbers below a similar bound, you'd be unlikely
> to find such close fits. Of course the prime numbers aren't "random"
> so I have no way of measuring how unlikely these coincidences are.

Number theorists have ways of measuring these things. Jeremy, this is
truly a topic for the tuning-math list, which split off from this list
a few years ago because some people were sick of seeing such
mathematically-oriented posts here.

________________________________________________________________________
________________________________________________________________________

Message: 8
Date: Wed, 27 Jul 2005 19:53:19 -0000
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
Subject: Re: Definitions of JI

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
> "wallyesterpaulrus" wrote:
> > --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...>
wrote:
> >
> > > Agreed, it would materially alter the sound of the piece to be
even
> > > fractionally mistuned. Very little music uses interacting pure
sine
> > > waves, either; the closer it gets to that (as with soprano
voices,
> > > simple
> > > flutes or thin plucked strings), the less tolerance most of us
have
> > > for
> > > mistuning.
> >
> > All the psychoacoustic evidence points to the opposite
conclusion. Not
> > to mention my own experience, and that of others who've posted
here. If
> > you haven't read Sethares' book yet, you could search the web for
Plomp
> > & Levelt . . . their classic experiments found that listeners
discern
> > *no* minima of discordance other than 1:1 for pairs of pure sine
waves.
> > Best of all, try some experiments yourself!
>
> Paul,
> I've tried many such experiments over the years, using
> my Moog analog synth. Clearly, I'm not one of the listeners
> in the "classic experiments" you cite. Tell me, were their
> listeners musicians?

I believe they were all nonmusicians, but Bill Sethares and others
should speak up with the actual experimental details.

> If "all the evidence" says I'm wrong, then must I doubt
> my own ears??

No, but there may be something wrong with your experimental setup.
Pure sine waves are not easy to produce, for several reasons. Still I
find it odd that your tolerance or accuracy wouldn't be finer with
other waveforms. Almost everyone I've discussed this stuff with, for
example Daniel Wolf, finds it a lot harder to pin down the tuning
when sine waves are used as opposed to any other (periodic) waveform.

The accepted explanation is: sine waves have no harmonics; tuning
simple-integer ratios is done by (and is always the result when)
eliminating beating between coincident harmonics; hence there's
little chance of accurately tuning simple-integer ratios (except 1:1)
when using sine waves. This jibes very well with my experience but I
can't categorically deny that you're experiencing something else.
Nonetheless, this 'experiment' or 'observation' is a fundamental
assumption behind Sethares' entire book, since he assumes that simply
replacing harmonic partials with inharmonic partials in the timbres
used will completely replace any 'attraction to' or 'consonance of'
simple-integer ratios with an 'attraction to' or 'consonance of' the
relevant inharmonic ratios. I've tried to point out some limitations
to this way of thinking; perhaps your experience will point to others.

________________________________________________________________________
________________________________________________________________________

Message: 9
Date: Wed, 27 Jul 2005 19:59:31 -0000
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
Subject: Re: Definitions of JI

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
>
> Gene,
>
> You wrote:

> > I think in the case of 2401/2400, where you get chords off by less
> > than a cent, it really is a sort of tempering. It acts as if it was,
> > at any rate.
>
> In that you can choose as a substitute for any JI ratio,
> a whole number of steps of 2401/2400.

That's not what Gene meant. He meant that if you have a simple-integer
JI chord in mind, but in your realization, one of the notes is off by a
2401:2400 (perhaps because you've restricted yourself to a finite JI
set, for your instrument), then it's much like having a tempered chord
yet without any commas having been split and distributed (the way
tempering normally works). In other words, you could temper out the
2401:2400, but even if you don't, the 'wolf' intervals will only be off
by 2401:2400 or 0.7 cents, which is so small that you could consider
the wolf 'tamed' even though you didn't do any explicit tempering.

________________________________________________________________________
________________________________________________________________________

Message: 10
Date: Wed, 27 Jul 2005 20:07:25 -0000
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
Subject: Re: salinas 19tone

--- In tuning@yahoogroups.com, "centeroftownkh" <centeroftownkh@y...>
wrote:

> Hi Everyone -
>
> I'm a little confused about the enharmonic 19-tone scale by
Salinas. Barbour says it's an
> ET scale, but when I open it up in Maxmicrotuner all the intervals
are Just - 1/1, 24/25,
> 16/15, 9/8, 75/64, 6/5, 5/4, 125/96, 4/3, 25/18, 64/45, 3/2, 25/16,
8/5, 5/3, 125/72,
> 16/9, 15/8, 125/64, 2/1. Pardon my simple-mindedness.
>
> Thank you.

Salinas's 19-note scale was definitely not just. He did mention a 24-
note just scale but this was for theoretical, not practical, use. The
three tunings Salinas considered for practical use were all
meantones: 1/4-comma meantone, 2/7-comma meantone, and 1/3-comma
meantone. I believe Salinas used all three carried out to 19 notes,
with his 19-note keyboard. He found the 1/3-comma version 'languid'
but seems not to have noticed that it is nearly identical to 19-note
equal temperament, a system without wolves and with much more
modulational freedom than the other tunings.

________________________________________________________________________
________________________________________________________________________

Message: 11
Date: Wed, 27 Jul 2005 20:13:58 -0000
From: "Gene Ward Smith" <gwsmith@svpal.org>
Subject: Re: Lattice metrics and geometries (was: Definitions of JI)

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> I can define it just fine without going into your definition of Tenney
> space, so I don't know what you mean.

Didn't you define it only for one comma?

> > > Isn't this a topic for the tuning-math list?
> >
> > No doubt. I'm not seeing where you get hexagons.
>
> If the pictures on Kees' page don't make it clear, I'll have to draw
> some other pictures for you.

I screwed up on the hexagons, there are indeed hexagons.

________________________________________________________________________
________________________________________________________________________

Message: 12
Date: Wed, 27 Jul 2005 20:57:56 -0000
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
Subject: Re: Lattice metrics and geometries (was: Definitions of JI)

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > I can define it just fine without going into your definition of
Tenney
> > space, so I don't know what you mean.
>
> Didn't you define it only for one comma?

The particular method explained in the latest version of my 'Middle
Path' paper for *finding* a TOP (Tenney-OPtimal) tuning is only
defined for one comma, but TOP tuning itself is defined the same way
no matter how many commas are tempered out -- it's the tuning that
minimizes the maximum damage (where damage = ratios's mistuning
divided by ratio's Harmonic Distance) over all ratios in the infinite
lattice. Early on, I discussed (on tuning-math) how to calculate the
TOP tuning for ETs, even though more than one comma is tempered out
in those cases:

/tuning-math/message/8512

________________________________________________________________________
________________________________________________________________________

Message: 13
Date: Wed, 27 Jul 2005 21:59:35 -0000
From: "Gene Ward Smith" <gwsmith@svpal.org>
Subject: Re: Definitions of JI

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

>In other words, you could temper out the
> 2401:2400, but even if you don't, the 'wolf' intervals will only be off
> by 2401:2400 or 0.7 cents, which is so small that you could consider
> the wolf 'tamed' even though you didn't do any explicit tempering.

Among interesting 7-limit commas, there seems to be a distinct gap
between the upper end of the really small range, at 65625/65536, or
2.35 cents, and the lower end of the small range, at 6144/6125, or
5.36 cents; the best thing in there being 589824/588245. I think for
65625/65536 or less we can get away pretty well with not tempering the
comma--certainly that works excellently when we get down to 2401/2400,
but I think in practice it's worked well for 32805/32768.

I wonder if this could be worked up into something definite enough to
help on the comma boundry question. The gap fills in from a pure
11-limit point of view until we get to 5632/5625, however, and there
are high-complexity 5-limit commas we might stick in there from a
5-limit point of view, such as the 19-comma, parakleisma, or semisuper
comma.

________________________________________________________________________
________________________________________________________________________

Message: 14
Date: Thu, 28 Jul 2005 07:21:28 -0000
From: "monz" <monz@tonalsoft.com>
Subject: Re: salinas 19tone

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> --- In tuning@yahoogroups.com, "centeroftownkh" <centeroftownkh@y...
>
> wrote:
>
> > Hi Everyone -
> >
> > I'm a little confused about the enharmonic 19-tone scale
> > by Salinas. Barbour says it's an ET scale, but when I
> > open it up in Maxmicrotuner all the intervals are
> > Just - 1/1, 24/25, 16/15, 9/8, 75/64, 6/5, 5/4, 125/96,
> > 4/3, 25/18, 64/45, 3/2, 25/16, 8/5, 5/3, 125/72,
> > 16/9, 15/8, 125/64, 2/1. Pardon my simple-mindedness.
> >
> > Thank you.
>
> Salinas's 19-note scale was definitely not just. He did
> mention a 24-note just scale but this was for theoretical,
> not practical, use. The three tunings Salinas considered
> for practical use were all meantones: 1/4-comma meantone,
> 2/7-comma meantone, and 1/3-comma meantone. I believe
> Salinas used all three carried out to 19 notes, with
> his 19-note keyboard. He found the 1/3-comma version
> 'languid' but seems not to have noticed that it is nearly
> identical to 19-note equal temperament, a system without
> wolves and with much more modulational freedom than the
> other tunings.

I have a lot about Salinas's 19-tone 1/3-comma meantone here:

http://tonalsoft.com/enc/number/19edo.aspx#salinas

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

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"wallyesterpaulrus" wrote:
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
> >
> >
> > Gene,
> >
> > You wrote:
>
> > > I think in the case of 2401/2400, where you get chords off by less
> > > than a cent, it really is a sort of tempering. It acts as if it was,
> > > at any rate.
> >
> > In that you can choose as a substitute for any JI ratio,
> > a whole number of steps of 2401/2400.
>
> That's not what Gene meant. He meant that if you have a simple-integer
> JI chord in mind, but in your realization, one of the notes is off by a
> 2401:2400 (perhaps because you've restricted yourself to a finite JI
> set, for your instrument), then it's much like having a tempered chord
> yet without any commas having been split and distributed (the way
> tempering normally works). In other words, you could temper out the
> 2401:2400, but even if you don't, the 'wolf' intervals will only be off
> by 2401:2400 or 0.7 cents, which is so small that you could consider
> the wolf 'tamed' even though you didn't do any explicit tempering.

Paul,
Thanks for that explanation! It makes more sense than my
attempt at one.

Regards,
Yahya

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"wallyesterpaulrus" wrote:
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
> >
> > "wallyesterpaulrus" wrote:
> > > --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...>
> > > wrote:
> > >
> > > > Agreed, it would materially alter the sound of the piece to be
> > > > even fractionally mistuned. Very little music uses interacting
> > > > pure sine waves, either; the closer it gets to that (as with
> > > > soprano voices, simple flutes or thin plucked strings), the less
> > > > tolerance most of us have for mistuning.
> > >
> > > All the psychoacoustic evidence points to the opposite
> > > conclusion. Not to mention my own experience, and that of
> > > others who've posted here. If you haven't read Sethares' book
> > > yet, you could search the web for Plomp & Levelt . . . their classic
> > > experiments found that listeners discern *no* minima of
> > > discordance other than 1:1 for pairs of pure sine waves.
> > > Best of all, try some experiments yourself!
> >
> > Paul,
> > I've tried many such experiments over the years, using
> > my Moog analog synth. Clearly, I'm not one of the listeners
> > in the "classic experiments" you cite. Tell me, were their
> > listeners musicians?
>
> I believe they were all nonmusicians, but Bill Sethares and others
> should speak up with the actual experimental details.

Paul,
It's fair to state that, despite scientific training, I've never
approached music-making on the Moog scientifically! :-) I've
certainly tried to be a systematic observer, but haven't set
up rigorous experiments to test my theories. But now I see
there might be some value in doing so.

> > If "all the evidence" says I'm wrong, then must I doubt
> > my own ears??
>
> No, but there may be something wrong with your experimental
> setup. Pure sine waves are not easy to produce, for several
> reasons.

There may be. The chief error would lie in my musician's
attitude of "I'm here to play". But if I'm to make statements
of scientific, repeatable, fact, I would need to approach the
instrument as just a tool of the experiments I make. So I'll
try to make some time to devise and conduct REAL experiments.

It's quite easy to produce sine waves, I believe, using elementary
electronics; as to their purity, I have no easy way to gauge.
Perhaps I could measure the THD in the recorded waves, using
some audio software? Of course, what the electronics produces
then has to pass through amplification and loudspeakers, which
may introduce significant distortion.

> ... Still I find it odd that your tolerance or accuracy
> wouldn't be finer with other waveforms. Almost everyone I've
> discussed this stuff with, for example Daniel Wolf, finds it a lot
> harder to pin down the tuning when sine waves are used as opposed
> to any other (periodic) waveform.

That's what I intend to test.

> The accepted explanation is: sine waves have no harmonics; tuning
> simple-integer ratios is done by (and is always the result when)
> eliminating beating between coincident harmonics; hence there's
> little chance of accurately tuning simple-integer ratios (except
> 1:1) when using sine waves. This jibes very well with my experience
> but I can't categorically deny that you're experiencing something
> else.

Because most pitched musical instruments do produce overtones
of the fundamental frequency, I believe, we automatically compare
the sum tone of two pure sine sources with the expected overtones,
and thus can tune two pure sine sources using their combination tones.
According to this theory, we should be able to readily tune 1:2, 2:3,
3:4 and maybe 4:5 ratios of sine tones. The theory suggests we can
tune whatever intervals have combination tones near those
frequencies we would expect to find as sufficiently strong overtones.
I note that the same theory applies equally to tuning 1:1, since the
sum tone is then an octave, one of the commonest (tho not always
strongest) overtones in 'natural' instruments.

Including the sum and difference tones for 1:2, 2:3, 3:4 and 4:5
ratios gives timbres that include 1:1:2:3, 1:2:3:5, 1:3:4:7 and 1:4:5:9
ratios. If we take 110Hz as the fundamental, or 1, the corresponding
9 is only 990Hz, so all these tones and combinations lie well within a
normal hearing range, easy to test for.

Another possible mechanism for tuning performance is kinaesthetic:
it involves subvocalisation. In short, it posits that we may tune
intervals by (almost) singing their component notes. Muscle memory
helps us to find the perfect fifth, or just major third, above the first
tone. On this theory, we should get better at tuning the more we sing,
and possibly also the more recently we have sung. This theory requires
no awareness of the overtone structure of common timbres, but of
course people who sing a lot are aware of the timbre of their own
voices, so perhaps singers could use both mechanisms.

So now I want to design experiments to distinguish what effect, if
any, these possible mechanisms have on tuning performance.

> ... Nonetheless, this 'experiment' or 'observation' is a
> fundamental assumption behind Sethares' entire book, since he
> assumes that simply replacing harmonic partials with inharmonic
> partials in the timbres used will completely replace any 'attraction
> to' or 'consonance of' simple-integer ratios with an 'attraction to'
> or 'consonance of' the relevant inharmonic ratios. I've tried to
> point out some limitations to this way of thinking; perhaps your
> experience will point to others.

If I'm right, and we automatically interpret 'musical' sounds
according to our stored experiences of music, we'll find
consonances where we expect to - and dissonances, too.

Regards,
Yahya

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--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> It's quite easy to produce sine waves, I believe, using elementary
> electronics;

From what I've read and seen it's far easier to produce sawtooth,
triangle, and pulse waves than accurate sine waves.

> as to their purity, I have no easy way to gauge.
> Perhaps I could measure the THD in the recorded waves, using
> some audio software? Of course, what the electronics produces
> then has to pass through amplification and loudspeakers, which
> may introduce significant distortion.

Exactly. You'd have to take *all* the distortion into account. Even
combinational tones are a kind of distortion (occuring primarily in
the ear and less in the brain), and although I allow them below, I'm
pretty sure the Plomp & Levelt experiments were deliberately done at
low volume levels so that the combinational tones would be relatively
inaudible. (Most combinational tones have a loudness which increases
as the 2nd, 3rd, 4th, or 5th power of the loudness of the 'real'
tones.)

> > The accepted explanation is: sine waves have no harmonics; tuning
> > simple-integer ratios is done by (and is always the result when)
> > eliminating beating between coincident harmonics; hence there's
> > little chance of accurately tuning simple-integer ratios (except
> > 1:1) when using sine waves. This jibes very well with my
experience
> > but I can't categorically deny that you're experiencing something
> > else.
>
> Because most pitched musical instruments do produce overtones
> of the fundamental frequency, I believe, we automatically compare
> the sum tone of two pure sine sources with the expected overtones,
> and thus can tune two pure sine sources using their combination
tones.

Sum tones are very elusive compared to several of the different-order
difference tones that are produced. But even granting your sum tones,
there's no beating with "expected" overtones. Only real overtones can
beat. Without the beating, I don't see how you propose to acheive
great accuracy in the tuning. You may be able to tune with accuracy
on the order of the *melodic* just noticeable difference (since
you're relying on your memory of intervals or pitches just as you
would when listening to a melody) but probably not on the order of
the *harmonic* just noticeable difference, which relies on active
perception of the interaction between actually sounding tones.

I believe a far more likely explanation for any success in tuning
sine waves to small-integer ratios involves aligning the virtual
pitches with the combinational tones (difference tones). I'm pretty
sure I've discussed virtual pitches with you before, either here or
on the harmonic entropy list. Basically, the brain 'creates' a
fundamental, even if the fundamental is physically absent, when it
encounters a set of pure tones suggestive of a harmonic series.
Whenever you hear a male voice on the telephone, this virtual pitch
phenomenon is a large part of your perception -- the pitches you
think you're hearing are not even physically present. The virtual
pitch is *not* any kind of difference tone, but it will be identical
in pitch to one of the difference tones if the pure tones are in
exact low-harmonic proportions.

> > ... Nonetheless, this 'experiment' or 'observation' is a
> > fundamental assumption behind Sethares' entire book, since he
> > assumes that simply replacing harmonic partials with inharmonic
> > partials in the timbres used will completely replace
any 'attraction
> > to' or 'consonance of' simple-integer ratios with an 'attraction
to'
> > or 'consonance of' the relevant inharmonic ratios. I've tried to
> > point out some limitations to this way of thinking; perhaps your
> > experience will point to others.
>
> If I'm right, and we automatically interpret 'musical' sounds
> according to our stored experiences of music, we'll find
> consonances where we expect to - and dissonances, too.

Well, at this point I'd suggest obtaining Sethares's book (which
comes with a CD), because your reactions to the sound examples he
presents will probably startle you; or if not, they may show
limitations in Sethares's arguments (which it would be good for him
to know about).