Manuel's files

Manuel wrote,

>But indeed, don't use the mills ref anymore. The file is still there,
>but the file permissions disallow me to remove it. The recent version is
>http://www.xs4all.nl/~huygensf/doc/bib.html

If the problem is with the Mills server, perhaps you should copy the
consistency table to another site?

Paul wrote:
>If the problem is with the Mills server, perhaps you should copy the
>consistency table to another site?

Okay, it's copied to
http://www.xs4all.nl/~huygensf/doc/consist_limits.txt and
http://www.xs4all.nl/~huygensf/doc/cons_limit_bounds.txt

Manuel

--- In tuning@egroups.com, <manuel.op.de.coul@e...> wrote:

http://www.egroups.com/message/tuning/17655

>
> Paul wrote:
> >If the problem is with the Mills server, perhaps you should copy
the
> >consistency table to another site?
>
> Okay, it's copied to
> http://www.xs4all.nl/~huygensf/doc/consist_limits.txt and
> http://www.xs4all.nl/~huygensf/doc/cons_limit_bounds.txt
>
> Manuel

Well, this is fascinating, but what exactly is it? (And I actually
tried to read it first this time before asking!)

Anybody up for giving me a clue??

__________ _____ _____ _
Joseph Pehrson

Manuel wrote,

>> Okay, it's copied to
>> http://www.xs4all.nl/~huygensf/doc/consist_limits.txt and
>> http://www.xs4all.nl/~huygensf/doc/cons_limit_bounds.txt
>>
>> Manuel

Joseph wrote,

>Well, this is fascinating, but what exactly is it? (And I actually
>tried to read it first this time before asking!)

>Anybody up for giving me a clue??

OK Joseph, let's look at the first table first. Let's say you are interested
in the consistency limits of 25-tone equal temperament. You scroll down the
table, keeping your eye on the third column ("Number of tones per octave")
until you get to the range you want -- these two lines:

5 48.09233 24.95201 6 9 8
6 47.80317 25.10294 4 9 6

See the "6" in the fourth column of the first line, and the "6" in the first
column of the second line? These two numbers will always be the same no
matter where you are in the table. They tell you the integer consistency
limit of the ET in question. Since it's 6 in this case, you know that if, in
25-tET, you wish to approximate any JI chord all of whose intervals are
ratios of numbers 6 or below, each interval can be represented by that
interval's closest equivalent in 25-tET without inconsistency. Since 25-tET
has perfect octaves, you know that it therefore has an odd consistency limit
of 5 -- but this table works for non-octave equal temperaments too, such as
the BP scale (which is 8.202-tET).

With me so far?

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/17661

Since 25-tET has perfect octaves, you know that it therefore has an
odd consistency limit of 5 -- but this table works for non-octave
equal temperaments too, such as the BP scale (which is 8.202-tET).
>
> With me so far?

Well, I hope so... let's see:

Let's take the Bohlen-Pierce scale for a moment since I'm "into" it
of late. If we are to follow the table we get:

6 149.05014 8.05098 7 4 4
7 145.45455 8.25000 3 4 4

So I see a "7" to the right and to the left. I remember the BP was
constructed from odd ratios, 3,5,7...

Is that what it's showing??

However,

See Bohlen, H. 13-Tonstufen in der Duodezime, Acustica 39: 76-86
(1978)
0: 1/1 0.000 unison, perfect prime
1: 27/25 133.238 large limma, BP small semitone
2: 25/21 301.847 BP second, quasi-tempered minor third
3: 9/7 435.084 septimal major third, BP third
4: 7/5 582.512 septimal or Huygens' tritone, BP fourth
5: 75/49 736.931 BP fifth
6: 5/3 884.359 major sixth, BP sixth
7: 9/5 1017.596 just minor seventh, BP seventh
8: 49/25 1165.024 BP eighth
9: 15/7 1319.443 septimal minor ninth, BP ninth
10: 7/3 1466.871 minimal tenth, BP tenth
11: 63/25 1600.108 quasi-equal major tenth, BP eleventh
12: 25/9 1768.717 classic augmented eleventh, BP twelfth
13: 3/1 1901.955 perfect 12th

There are a lot of higher primes here... I see a "27" right off the
bat... How does it account for that??... or doesn't that have
anything
to do with it??? I think I need a bit more "filling in..."

________ _____ __ __
JP

--- In tuning@egroups.com, "Joseph Pehrson" <joseph@c...> wrote:

http://www.egroups.com/message/tuning/17669

Wait a minute, though. The BP scale I cite in the previous message
is NOT AN ET!!! Does that have something to do with solving the
"higher prime" riddle??

__________ ______ ____
Joseph Pehrson

Joseph wrote,

>Well, I hope so... let's see:

>Let's take the Bohlen-Pierce scale for a moment since I'm "into" it
of late. If we are to follow the table we get:

>6 149.05014 8.05098 7 4 4
>7 145.45455 8.25000 3 4 4

>So I see a "7" to the right and to the left. I remember the BP was
>constructed from odd ratios, 3,5,7...

>Is that what it's showing??

It's actually showing that it's consistent with respect to the integers 1,
2, 3, 4, 5, 6, and 7 -- that is, in BP, you wish to approximate any JI chord
all of whose intervals are ratios of numbers 7 or below, each interval can
be represented by that
interval's closest equivalent in BP without inconsistency. Now BP wasn't
designed to handle the even numbers (2, 4, 6), and they aren't represented
very accurately, but what this tells you is that they are represented
consistently.

>There are a lot of higher primes here... I see a "27" right off the
>bat...

Well 27 isn't prime.

>How does it account for that??...

Consistency is defined in terms of a _limit_, in this case, an integer
limit, in this case, 7. What that means is that if you use numbers higher
than 7 (specifically, 8, for example), you _might_ get inconsistency.
However, you won't always, and since BP does so well with ratios involving
1, 3, 5, and 7, it's not too surprising that it does pretty well with ratios
involving _products_ of these numbers, such as 27, 63, and 75.

Joseph wrote,

>Wait a minute, though. The BP scale I cite in the previous message
>is NOT AN ET!!!

OK, but it's very well approximated by the ET version of BP, which is what
we were referring to.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/17692

>
> It's actually showing that it's consistent with respect to the
integers 1, 2, 3, 4, 5, 6, and 7 -- that is, in BP, you wish to
approximate any JI chord all of whose intervals are ratios of numbers
7 or below, each interval can be represented by that interval's
closest equivalent in BP without inconsistency.

I've really been trying to understand the concept of "consistency"
and have read over the Monzo dictionary entry several times. Is it
saying, then, that you could put intervals and chords together in
this scale with whatever ordering as long as they include ratios made
from the integers 1-7 and, in that case, they would come out with the
same level of "accuracy?" (Which, in the case of the BP scale if you
used ratios from 1-7 would be very good...)

And, therefore, intervals using the "even" ratio pitches that you
mentioned, could still be "inaccurate" and yet consistent at a
certain level in their inaccuracy??.... That's a slight bit humorous
in a way, if, indeed that is the correct concept...

________ ____ ____ _
Joseph Pehrson

Joseph Pehrson wrote,

>I've really been trying to understand the concept of "consistency"
>and have read over the Monzo dictionary entry several times. Is it
>saying, then, that you could put intervals and chords together in
>this scale with whatever ordering as long as they include ratios made
>from the integers 1-7 and, in that case, they would come out with the
>same level of "accuracy?" (Which, in the case of the BP scale if you
>used ratios from 1-7 would be very good...)

No, the accuracy would only be very good if you stuck to the odd numbers.
Whay I'm saying is that no ratio formed from the integers 1-7 would ever be
forced to be represented by anything but its best approximation in the BP
scale.

An example of inconsistency is 24-tET in the 7-limit. In 24-tET, the best
approximation of 4:5 is 400 cents, and the best approximation of 5:7 is 600
cents. If you stack these intervals on top of each other, you get 4:7
approximated by 1000 cents, but 24-tET has a better approximation of 4:7 at
950 cents. So you can't _consistently_ represent a 4:5:7 chord in 24-tET.

>And, therefore, intervals using the "even" ratio pitches that you
>mentioned, could still be "inaccurate" and yet consistent at a
>certain level in their inaccuracy??.... That's a slight bit humorous
>in a way, if, indeed that is the correct concept...

Yes, Dan Stearns has made this same criticism of the consistency concept,
and I've always replied that the importance of consistency is simply to
prevent you from ascribing more accuracy to an ET than is really there . . .
like the above example with 24-tET . . . it's not very useful for very low
ETs, where the accuracy will tend to be very poor, nor is it very useful for
very high ETs, where even the second- or third- best approximation of a
given interval will be sufficiently accurate . . .

Paul Erlich:
> An example of inconsistency is 24-tET in the 7-limit. In
> 24-tET, the best
> approximation of 4:5 is 400 cents, and the best approximation
> of 5:7 is 600
> cents. If you stack these intervals on top of each other, you get 4:7
> approximated by 1000 cents, but 24-tET has a better
> approximation of 4:7 at
> 950 cents. So you can't _consistently_ represent a 4:5:7
> chord in 24-tET.

Something strikes me as odd about this example, but I can't quite put my
finger on it. I think I'm thinking it's kinda like comparing apples to
oranges. I think it also depends upon where you want your consistency.
Here's another example:
Construct a 7-limit scale with, lets say, twelve tones. I don't know what
the appropriate comma would be for that, but lets just say we have that
scale.
Now, from the tonic, we'd get a consonant 4:5:7 sound, however, if we move
up one tone in the scale, we won't be able to produce a just 4:5:7 using the
other notes of the scale. From that point of view, we can't consistently
represent a 4:5:7 in the 7-limit scale.
With the 24-tET, sure we won't be able to hit a just 4:5:7 anywhere, but at
least our error will be the same no matter what tone we use as the root of
the chord.

Another thing about that example: If you stack 400 cents on top of 600 cents
in 24-tET, you get 1000 cents. If you stack 4:5 and 5:7 in a 7-limit scale,
you get 4:7. Just because 400 cents is the closest approximation of 4:5 and
600 is the closest approximation of 5:7 doesn't mean that it makes sense to
compare their conjunctions across the two tuning methods. Faulting 24-tET
for not being able to produce a just 4:5:7 makes about as much sense as
faulting a 7-limit scale for not being able to produce an interval of 1000
cents.

Todd

Todd Wilcox wrote,

>Something strikes me as odd about this example, but I can't quite put my
>finger on it. I think I'm thinking it's kinda like comparing apples to
>oranges. I think it also depends upon where you want your consistency.
>Here's another example:
>Construct a 7-limit scale with, lets say, twelve tones. I don't know what
>the appropriate comma would be for that, but lets just say we have that
>scale.
>Now, from the tonic, we'd get a consonant 4:5:7 sound, however, if we move
>up one tone in the scale, we won't be able to produce a just 4:5:7 using
the
>other notes of the scale. From that point of view, we can't consistently
>represent a 4:5:7 in the 7-limit scale.
>With the 24-tET, sure we won't be able to hit a just 4:5:7 anywhere, but at
>least our error will be the same no matter what tone we use as the root of
>the chord.

Todd, you're missing the point. Consistency is only defined for ETs. An
example of an ET close to 24-tET which is consistent in the 7-limit is
22-tET. In 22-tET, 5:4 is approximated by 384 cents, 7:5 is approximated by
600 cents, and 4:7 is approximated by 384+600=984 cents. These are all the
best approximations of these intervals that 22-tET can provide, and they are
all very good, to boot.

>Another thing about that example: If you stack 400 cents on top of 600
cents
>in 24-tET, you get 1000 cents. If you stack 4:5 and 5:7 in a 7-limit scale,
>you get 4:7. Just because 400 cents is the closest approximation of 4:5 and
>600 is the closest approximation of 5:7 doesn't mean that it makes sense to
>compare their conjunctions across the two tuning methods.

It does, because 4:5:7 is a clear and distinct musical phenomenon, and you
may wish to know how well you can reproduce it in a given tuning system.

>Faulting 24-tET
>for not being able to produce a just 4:5:7 makes about as much sense as
>faulting a 7-limit scale for not being able to produce an interval of 1000
>cents.

That's not quite right. There are several psychoacoustical phenomena in
which frequency ratios play a strong role and with respect to which various
intervals will be heard "as" a frequency ratio with more or less accuracy.
For example, the 12-tET major triad is heard "as" an out-of-tune 4:5:6, and
the closer you tune it to 4:5:6, the clearer it will sound (though you may
find this clarity boring).

Todd -- I have to clear something up here. The point of consistency is that
many theorists evaluted ETs in terms of how well their intervals approximate
JI intervals. Let's leave aside for the moment the debate over whether this
is a useful evaluation -- let's assume that it is. Now, typically these
theorists would look as 24-tET and say, "Well, 4:5 is 14 cents off, 5:7 is
17 cents off, and 4:7 is 19 cents off, so 24-tET can approximate JI ratios
involving 4, 5, and 7 to an accuracy of better than 20 cents". The flaw in
this reasoning is that the typical music for which JI is proposed often uses
more than two voices simultaneously, so chords like 4:5:7 will occur. But in
24-tET, you can't play a 4:5:7 within an accuracy of 20 cents, since the
14-cent-off 4:5 plus the 17-cent-off 5:7 doesn't add up to the 19-cent-off
4:7, but rather to an interval 31 cents off 4:7.

The point of consistency is to ensure that this situation _won't_ occur for
a particular ET and a particular (integer or odd) harmonic limit.

Paul Erlich again:
> >Faulting 24-tET
> >for not being able to produce a just 4:5:7 makes about as
> much sense as
> >faulting a 7-limit scale for not being able to produce an
> interval of 1000
> >cents.
>
> That's not quite right. There are several psychoacoustical
> phenomena in
> which frequency ratios play a strong role and with respect to
> which various
> intervals will be heard "as" a frequency ratio with more or
> less accuracy.
> For example, the 12-tET major triad is heard "as" an
> out-of-tune 4:5:6, and
> the closer you tune it to 4:5:6, the clearer it will sound
> (though you may
> find this clarity boring).

I'm not at all unfamiliar with the psychoacoustical phonemenon that you
mention, although I might leave off the psycho- prefix. Certainly the
difference between an ET triad and a JI triad is acoustically significant.
The real question I'm asking is which should be considered more "correct."
That is to say, the acoustic relationship between the notes of a triad don't
necesarily lead to a specific psychological relationship.
Lets use yet another example:
Two twins are born and raised in isolation on separate islands. Both twins
are heavily exposed to recorded music, but are not allowed to play or tune
any instruments as they grow up. Twin A only listens to music recorded on
12-tET instuments. Twin B is only exposed to a JI tuning, which may be
adaptive for the purposes of this argument, if you prefer.
On their 23rd birthdays, each twin is exposed to the other's tuning system.
If I understand you correctly, you would wager that the twin raised in the
ET system might hear the JI recordings and think "ahh.. that's much easier
to listen to," albeit maybe subconciously, while the twin raised JI will
find the ET recordings at least mildly distasteful.
*I* would say that both twins would find each other's system equally
distasteful.

I base that on the fact that whereas I know that certain frequency ratios do
have important and consistent psychoacoustical effects (the most important
and famous being the octave), preferring just intervals is not inborn. You
assume that a 4:5:6 is "correct" and that the ear would hear an equivalent
ET chord as an "out of tune 4:5:6." I don't agree. I'd say a major chord has
a certian sound in an ET system, and a certain sound in a JI system, and
that if you're raised on ET major chord, you probably won't like the JI
major chord, and vice-versa and so on for other harmonic constructions (like
minor chords). To go back to the example, the JI child might head an ET
chord as an "out of tune 4:5:6" whereas the ET child would hear an "out of
tune 0,400,700" (cents of course).

Later post from Paul:
> more than two voices simultaneously, so chords like 4:5:7 will occur. But
in
> 24-tET, you can't play a 4:5:7 within an accuracy of 20 cents, since the
> 14-cent-off 4:5 plus the 17-cent-off 5:7 doesn't add up to the 19-cent-off
> 4:7, but rather to an interval 31 cents off 4:7.

Once again, this thinking presupposes the correctness of the just ratios.
Note the use of the word "accuracy." I may be flying in the face of
psychoacoustical convention, but I just don't agree that the human brain
prefers just intervals at the biological level.

Todd

Todd wrote,

>I'm not at all unfamiliar with the psychoacoustical phonemenon that you
>mention, although I might leave off the psycho- prefix.

Not at all. Acoustics by itself tells us nothing about how different sounds
and chords will affect us. It is the interplay of acoustics with the human
ear, brain, and mind which is of interest. I would recommend you go to the
library and read _Introduction to the Physics and Psychophysics of Music_ by
Juan Roederer, and/or search the archives of this list for "roughness",
"virtual pitch", and "combination tones".

>If I understand you correctly, you would wager that the twin raised in the
>ET system might hear the JI recordings and think "ahh.. that's much easier
>to listen to,"

No I would never wager that. If the music was written _specifically_ for
12-tET, then it should be played in 12-tET. However, a lot of Western music
_wasn't_ originally written for 12-tET! (slightly off-topic . . .)

>Once again, this thinking presupposes the correctness of the just ratios.
>Note the use of the word "accuracy." I may be flying in the face of
>psychoacoustical convention, but I just don't agree that the human brain
>prefers just intervals at the biological level.

It's not a matter of convention but of many different experiments. Let me
give you some examples. In the psychoacoustical literature you will come
across the well-known phenomemon of "roughness". This is what happens when
two pitches are close enough together so that they excite overlapping
portions of the cochlea (the organ in the inner ear which sorts incoming
sounds along a pitch continuum). Roughness is easy to demonstrate in
musically trained or untrained subjects. Roughness is often accompanied by
beating, which is a simple wavering phenomenon that happens when two pitches
are very close together (see the Feynman Lectures on Physics) -- the
resulting sound is a pitch halfway between the two actual pitches, whose
amplitude varies from zero to maximum at a frequency equal to the difference
between the frequencies of the two actual pitches.

Now let's compare what happens when you hear a just major third vs. an ET
major third. In the ET major third, the 5th partial of the lower note will
be 14 cents away from the 4th partial of the higher note. This is very
close, and will hence be accompanied by some roughness and beating. On the
other hand, in the JI major third, the two frequencies will coincide
_exactly_, and though there may be some constructive or destructive
interference, there will be absolutely no roughness or beating. Hence the
interval sounds "smoother", more "clearly defined" -- whether or not this is
aesthetically preferable depends of course on the background of the listener
and how (if at all) the sensation related to any musical experience the
listener may have. In fact, it's been shown that of musically untrained
listeners, there are two groups: some listeners consistently prefer major
thirds that are about 15 cents out-of-tune, while other listeners
consistenly prefer just ones.

The relation of musical tuning theory to beating is most fully developed in
the book _On the Sensations of Tone_ by Helmholtz, which you should read.
However, other psychoacoustical phenomena also lead one to treat just
intervals and chords as "special", though the precise conclusions will
differ according to which phenomena are involved . . . I'll let you step in
with questions now.

Paul:
> Todd wrote,
> >I'm not at all unfamiliar with the psychoacoustical
> phonemenon that you
> >mention, although I might leave off the psycho- prefix.

> It's not a matter of convention but of many different
> experiments. Let me
<snip acoustical concepts>
> the difference
> between the frequencies of the two actual pitches.

Hmmm... I see you don't believe me. You know, I think the Feynman Lectures
is a bit of a highbrow reference just to look up beating. Lets not resort to
name-dropping now. Yes I know he was a drummer and once played the frigidera
or whatever it's called in Brazil. Now that I've said that, lets see if
you'll post a huge biography on Feyman to show that you don't believe I know
him and his work as well.

Acoustics notwithstanding, I frankly suspect any experimentation that claims
to reveal a lot of useful knowledge about how people interpret things. To
me, acoustics is a science that makes a lot of sense. Psychoacoustics is
not. I also hate MP3s, so there! :)

> Now let's compare what happens when you hear a just major
> third vs. an ET
<snip>
> beating. Hence the [JI]
> interval sounds "smoother", more "clearly defined"

Actually, I'd wager that with some timbres a JI harmony might make the chord
LESS defined, since the upper intervals might just fade into the harmonics
of the root. Again, let me stress that I'd only expect to see this in a few
special timbres, but theoretically it certainly sounds possible.

I think a preference for non-perfect consonance is actually pretty well
established in the musical listening community without needing
experimentation. Consider: piano UNISONS are deliberately mistuned slightly
to enrichen the timbre; twelve string guitars and mandolins sound different
partly because it's almost impossible to keep the unisons perfect, again
producing a more interesting timbre; Robert Smith of The Cure deliberately
detuned his guitar to create a signature sound that proved very popular;
many pop and rock acts use chorus effects that create imperfect unisons
electronically; there's no way a whole symphony orchestra is within 3 cents
of the same note when playing, but the fact that they're not actually sounds
better; vibrato.

> whether or not this is
> aesthetically preferable depends of course on the background
> of the listener
> and how (if at all) the sensation related to any musical
> experience the
> listener may have. In fact, it's been shown that of musically
> untrained
> listeners, there are two groups: some listeners consistently
> prefer major
> thirds that are about 15 cents out-of-tune, while other listeners
> consistenly prefer just ones.

To me, this sounds like you're agreeing with my assertion that could be
stated: "'Correct' intervals are in the ear of the listener."

> However, other psychoacoustical phenomena also lead one to treat just
> intervals and chords as "special", though the precise conclusions will

I would never argue that just intervals are not special when compared to
other intervals. Clearly, only precise sets of frequencies are just, any
others are not. Personally I kinda think of the set of just intervals above
a certain frequency as a countable subset of the uncountable set of possible
frequencies of sound, and that's certainly noteworthy. The real question is
whether they are special to the human ear in general.

Todd

Paul H. Erlich wrote,

<< it's not very useful for very low ETs, where the accuracy will tend
to be very poor, nor is it very useful for very high ETs, where even
the second- or third- best approximation of a given interval will be
sufficiently accurate . . . >>

Agreed. But I think it is also somewhat misleading for those
temperaments where quite miniscule manipulations can change a very low
consistency into a surprisingly high one; say 13.975 for instance,
where 14-tET would theoretically shimmy from a three odd limit on up
to a seven.

I think consistency is an interesting and useful measure (and I'd sure
be darned proud if I'd dreamed it up), but I also think that its very
terms -- "consistent" and inconsistent" (much like "just" and
tempered") -- set up a distorted sense hierarchical worth!

--Dan Stearns

Todd wrote,

>Hmmm... I see you don't believe me. You know, I think the Feynman Lectures
>is a bit of a highbrow reference just to look up beating. Lets not resort
to
>name-dropping now.

That's not what I'm doing at all. It's just the simplest, clearest
explanation of beating that I can remember seeing.

>Yes I know he was a drummer and once played the frigidera
>or whatever it's called in Brazil. Now that I've said that, lets see if
>you'll post a huge biography on Feyman to show that you don't believe I
know
>him and his work as well.

Jeez oh man.

>Acoustics notwithstanding, I frankly suspect any experimentation that
claims
>to reveal a lot of useful knowledge about how people interpret things. To
>me, acoustics is a science that makes a lot of sense. Psychoacoustics is
>not.

Alright -- I suggest you at least look at Roederer's book before making such
a claim. Psychacoustics deals with many quantifiable and verifiable
phenomena.

>> beating. Hence the [JI]
>> interval sounds "smoother", more "clearly defined"

>Actually, I'd wager that with some timbres a JI harmony might make the
chord
>LESS defined, since the upper intervals might just fade into the harmonics
>of the root. Again, let me stress that I'd only expect to see this in a few
>special timbres, but theoretically it certainly sounds possible.

It depends what you mean by that (be clearer if you can), but it is true
that on an old Hammond organ, many of the partials are tuned in 12-tET

>I think a preference for non-perfect consonance is actually pretty well
>established in the musical listening community without needing
>experimentation. Consider: piano UNISONS are deliberately mistuned slightly
>to enrichen the timbre; twelve string guitars and mandolins sound different
>partly because it's almost impossible to keep the unisons perfect, again
>producing a more interesting timbre;

This agrees with what I said about Dave Keenan and I recommending at least a
1-cent detuning from JI to prevent phase-lock.

>Robert Smith of The Cure deliberately
>detuned his guitar to create a signature sound that proved very popular;
>many pop and rock acts use chorus effects that create imperfect unisons
>electronically; there's no way a whole symphony orchestra is within 3 cents
>of the same note when playing, but the fact that they're not actually
sounds
>better; vibrato.

Yes, all good points -- and yes, the 15-cent-off thirds of 12-tET have a
certain _quality_ which musicians miss if it's gone -- the thirds sound
dead, stale -- but once they get used to purer thirds, the ET thirds sound
out-of-tune . . . it's largely a matter of experience . . . remember, 12-tET
has only been "our" tuning system for 100-150 years . . .

>I would never argue that just intervals are not special when compared to
>other intervals. Clearly, only precise sets of frequencies are just, any
>others are not. Personally I kinda think of the set of just intervals above
>a certain frequency as a countable subset of the uncountable set of
possible
>frequencies of sound, and that's certainly noteworthy. The real question is
>whether they are special to the human ear in general.

One thing that makes them special is that you can tune them precisely, say
by eliminating the beating, by ear, when using normal harmonic timbres. You
can't tune non-JI intervals by ear with anything like the same degree of
accuracy.

Another thing that makes them special is that the harmonic series is, to a
large extent, the template for harmony. This goes to one of the other
psychoacoustical phenomena which you should read about: virtual pitch. We
have evolved (or, some argue, prenatally learned) to recognize harmonic
series in our environments, and a few harmonics can be sufficient to evoke
the pitch of a physically absent fundamental (no, this is not difference
tones). Much of tonal harmony makes use of this mechanism, primarily by
evoking the 2nd, 3rd, 4th, 6th, 8th, and sometimes 5th and even 7th, 9th,
and 10th harmonics over a "root", an implied fundamental. Although 12-tET is
close enough for this effect still to work, it is important to understand
that the harmonic series is actually the origin of this phenomemon,
particularly if you are trying to create new styles of music where the
phenomemon can be evoked under more specious circumstances, imply higher
harmonics, or differ in other ways from harmony as we know it. Although the
"smoothness" may be unpleasant at first, it's clear that the closer your
tuning is to a harmonic series, the more easily you'll be able to imply a
rooted harmony, often of a kind totally unheard in 12-tET, and that is the
attraction of JI, or tuning systems close to JI, to many of us on this list.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/17703

>
> An example of inconsistency is 24-tET in the 7-limit. In 24-tET,
the best approximation of 4:5 is 400 cents, and the best
approximation
of 5:7 is 600 cents. If you stack these intervals on top of each
other, you get 4:7 approximated by 1000 cents, but 24-tET has a
better
approximation of 4:7 at 950 cents. So you can't _consistently_
represent a 4:5:7 chord in 24-tET.
>
>
>I've always replied that the importance of consistency is simply
to prevent you from ascribing more accuracy to an ET than is really
there . . . like the above example with 24-tET . . .

Thanks, Paul.... Got it!

_______ _____ ___ _
Joseph Pehrson

Paul wrote:
> Todd wrote,
> >Yes I know he was a drummer and once played the frigidera
> >or whatever it's called in Brazil. Now that I've said that,
> lets see if
> >you'll post a huge biography on Feyman to show that you
> don't believe I
> know
> >him and his work as well.
>
> Jeez oh man.

Well, I'll give you points for NOT giving me a bio on Feynman when I say I
know a lot about him, however, you lose all those points and more by giving
yet ANOTHER primer on acoustics that I don't need. Lets just hope others on
the list found this edifying, especially after you writing "Jeez oh man."
And I'm thinking at this point it's time to agree to disagree, at least
until I make all those MP3s of John's MIDI files and accidentally convert
myself to a JI-lover.

Just in case it's not clear to anyone on the list: YES I HAVE A WORKING
KNOWLEDGE OF ACOUSTICS. In fact, I could probably freehand a so-so
Fletcher-Munson curve based on 1KHz, OK?

> Another thing that makes them special is that the harmonic
> series is, to a
> large extent, the template for harmony. This goes to one of the other

<snip another acoustics primer this time on the harmonic series and implied
fundamentals, and how that relates to JI harmony>

> attraction of JI, or tuning systems close to JI, to many of
> us on this list.

--- In tuning@egroups.com, "Todd Wilcox" <twilcox@p...> wrote:

>however, you lose all those points and more by giving
> yet ANOTHER primer on acoustics that I don't need.

This was all psychoacoustics, not acoustics; and, I assumed I needed
to explain this stuff because of the questions you were asking, about
what psychoacoustical phenomena set just intervals apart. The
statements you were making made it seem as if you were unfamiliar
with these things -- especially the statement that you don't consider
psychoacoustics a science. Well, apparantly I misunderstood where you
were coming from, but I always feel it is safer to over-explain when
others following the discussion may be edified.

Anyway, welcome to the list Todd, and I hope you stick around for a
while! It's not often that I get to _defend_ JI around here!

Paul wrote:
> Anyway, welcome to the list Todd, and I hope you stick around for a
> while! It's not often that I get to _defend_ JI around here!
>

Hehe... no joke! I feel like a spy in the enemy camp here, but we'll see how
I feel after I've heard more, and maybe I just need to hear some adaptive
tuning. Of course, I'm all about live performance, so adaptive tuning has a
problem there, but I could see a real-time algorithm working at some point.

Todd