22TET

>>"Special Way?" You are a Yes fan, you must be a Genesis fan too :)

Actually, I haven't really heard much Genesis. Since the 70's were before
my time, I kinda miss the flow of stuff. I got the YES and ELP down, tho...

>>Wow, I strongly disagree. Virtually every music theorist who does not care
>>about tuning issues understands harmony in terms of the diatonic scale and
>>tonal functions. "Functional harmony" -- ring a bell? Unfortunately,
>>virtually every alternative-tuning theorist does not so understand harmony.
>>This is unfortunate. It seems that "music theory" is divided into two
>>schools, and Carl is perhaps only familiar with the alternative-tuning
>>school. There is much of value in the other school, Carl.

I guess what I meant is, the idea of generalizing the rules for use outside
of the diatonic scale is new. I had one semester of High School and three
semesters of Conservatory music theory, and one semester of Conservatory
composition. The composition class was all about wacky stuff, since the
prof was into wacky stuff. In the theory classes, we did learn the rules
of functional harmony, and maybe there is an analog for most criteria in
your "generalized" functional harmony...

>>}The root of 2 part is understandable, considering that we need strong low
>>}identies for our 7's to work. This seems contradictory to the rule that
>>}the higher identities are more sensitive to mistuning, since there are more
>>}low-numbered fractions near them.
>>
>>I don't see any contradiction.

There are apparently two ideas at work here...

A) That higher limit intervals are more sensistive to mis-tuning "since
they are more apt to be confused with other intervals".

B) What I call Tonality, and what Partch calls "Observation One". An
effect created by the harmonic series, such that the number of pitches a
given pitch will harmonize with is inversely proportional to the odd limit
of the given pitch.

Idea "A" seems to suggest that we temper the octave the most, the fifth the
next most, and so on up. But we can't do this without wrecking tonality,
since the low identities provide that, as explained in idea "B".

Both A and B are really one idea, in that they're both caused by the way
superparticular fractions get closer to eachother, so there's no
"contradiction", true. But when tempering, it presents a trade-off
situation, one which can't be well-addressed by just making the de-tuning
inversely proportional to limit.

>>}Paul's paper addresses this by making the standard deviation in
log->>}frequency detuning inversely proportional to the limit of >>}the
interval.
>>
>>That is not correct. I do offer this as an alternative model, but the first
>>model, in which the standard deviation is constant for all intervals, is the
>>one which yield the candidate tunings:

I don't have the charts! But it says that the candidate tunings are the
same, except that 22 & 26 are no longer better than 12 at the 5-limit.
Nowhere do you say your candidates must be better than 12 at the 5-limit.

>>}So the list of scales comes down to 22, 26, 27, and 31 tone equal
>>temperament.
>>
>>Can you suggest a way to make the paper less confusing on this point?

On the point of what scales are in the candidate list?

>>}The example of the diminished 5th is given, but why it should be
>>}considered a type of 5th, or why the P5 should not be considered a
>>}7th is not made clear.
>>
>>Are you serious? Count scale steps. Anyone in a traditional theory class
>>could answer this blinfolded; perhaps I presumed too much of the traditional
>>theory background when writing this paper.

1) Conventional theory is full of holes. The tritone is spelled as an
augmented forth in certain contexts, and as a diminished fifth in others,
but it's the same pitch. All the enharmonics should be thrown out in 12.
But that's just my opinion. Why you consider it a type of fifth in your
example was, in any case, not clear to me.

2) The smallest step in 12 is the semitone. Counting semitones, the
tritone and P5 do not share the same number of steps. Thus, the tritone is
not a characteristic dissonance of the 5th in semitones. So what kind of
step did you mean? I would expect a definition of "step" that would be
good for all the temperaments the paper was looking at.

Funny enough, I actually understand what you were trying to do with these
characteristic dissonances in your criteria, but the lack of a robust
definition from the beginning really hurt me.

>>By the way, all the numbering and lettering in my paper is screwed up,
thanks
>>to Microsoft.

I noticed that. But why is it Microsoft's fault? Did you used some
automatic numbering scheme? I hate that trend in software nowadays! It's
not already so easy that you can't do it yourself?

>>I think 11TET would work much better for dissonant serialism than
>>12TET, since 11TET is the most effective tuning for random >>dissonance,
and 11TET is a subset of 22TET.

You just had to get that in :~)

>>}Far out avant garde music is certainly lacking of Special Way, if >>}not
other things.
>>
>>And I do love far-out avant garde music like Henry Cow, when I'm in the
mood for it. I love Phish when they go far out...

I actually had written that there's a time and a place for far out stuff,
but I deleted it for style considerations. Do you mean Henry Cowell? I
got a great Cd with Set of Five, Four Combinations for Three Instruments,
Hymn and Fuguing Tune #9, and Trio in Nine Short Movements. Great stuff.
And Junta!

>>I think 22TET can take you a lot farther out than 12TET.

Some of the far out music being done at Conservatories nowadays (like what
the prof of my composition class and his graduate students were doing)
really doesn't tune at all...

>>You did not address the rest of my paper, such as my demonstration that
22TET
>>is virtually the best tuning for the decatonic scales. It may seem circular
>>since I found the scales in 22TET, but one might have hoped for a better way
>>of tuning them, which (perhaps unfortunately) does not exist.

When I looked at the clock, I knew it was time to send.

>>Modulatory effects only possible in JI -- can you give a specific example?

I'm not the best person to ask, since the only JI I've done is on the
Cosmolyra (everything in root position) and on an conventional organ tuned
justly (only one key). It just stands to reason that there are all kinds
of effects, probably most of them un-discovered, based on the "anomalies"
of JI.

The Ben Johnston thing does come to mind, this quote of his appearing in
one of your ancient posts...

"I asked myself, suppose in writing some of the really very intense slow
movements that he did, Beethoven had not had the tempered scale to work
with, but had instead just intonation, what might that music have been?
Then I tried to write that piece, as an exercise for myself. It's actually
part of a piece--I liked it well enough that I put it into a piece. But at
first of course it's just to find out. The richness is one thing, the
flexibility is another. The typical Beethoven progression, for example will
cause you to drop by a microtonal amount every time the progression repeats
itself. What I did was to work it out that the pitch would drop, gradually,
until the whole thing drops almost a half step, and then I force it to come
back up by working out progressions that would make it do that. That is, of
course, quite different from what Beethoven was dealing with."

I've not heard the results of his efforts.

Probably the greatest expert on modulation in JI who ever lived was Partch,
and I'm sure I've heard effects not possible in temperament in Rotate the
Body and Delusion.

Bill Alves has said...

>Functionally, it may make sense to Paul to interpret the 22TET intervals as
>he has done above. However, I definitely hear the 10 step interval as being
>much more like an 11/8 (551 cents) than a 27/20. The 11/8 can have a
>beautiful blue-note quality, which I myself have exploited on occassion.
>The 19 step interval is almost dead-on a 20/11 (1035 cents), whose
>inversion, by the way, is a 5/4 down from the 11/8.

Here I publish an analysis of 22TET from a larger work of mine, still in
progress. It lists low-numbered rational approximations of 22TET
intervals, all within 7 cents deviation or better. The number of steps is
listed on the left, with the cents between that step and the tonic, rounded
to the nearest whole cent, appearing to the right....

1- 55
2- 109
3- 164 11/10 (1)
4- 218 17/15 (1)
5- 273 7/6 (6)
6- 327 23/19 (4)
7- 382 5/4 (4)
8- 436 9/7 (1)
9- 491
10- 545 11/8 (6)
11- 600 31/22 (6)
12- 655
13- 709 3/2 (7)
14- 764
15- 818
16- 873
17- 927
18- 982
19- 1036
20- 1091 15/8 (3)
21- 1145 31/16 (0)
22- 1200

..Even more to the right appear the rational approximations, with
intervals listed only once (as otonality or utonality), so as to reveal the
structure of the scale, if there is any. The cent deviations are in () to
the most right, and are of course the same for the inversions. I used my
own judgement for balancing cent deviation, prime limit, and odd limit when
picking the rational approximations.

Bill Alves assertion that the 11/10 is a 5/4 down from the 11/8 is
confirmed by, and consistently represented in the above scheme.

Please Note: How the intervals are heard depends on context!!! I'm just
throwing in my analysis here because sometimes a particular context seems
to be assumed when giving rational approximations for the 22TET intervals...

Carl


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Graham Breed wrote,

}I would be very surprised if any temperament significantly
}different to 22 equal supported decatonic scales. Vanishing
}septimal commas is a difficult trick to pull off.

You're right. If the decatonic approximation to the 3/2 is much less
than 707 cents, the decatonic approximation to the 7/4 becomes too
out-of-tune. If the decatonic 3/2 is larger than 711 cents, the
decatonic 5/3 becomes too out-of-tune. Compare this 4-cent range with a
range of about 8 cents (692-700) for the diatonic approximation to the
3/2.

}The octave is the best interval to define a harmonic scale around.
}Once you've done that, the octave should be very accurate.
}Although you should think about the octaves, it would be annoying
}if "the same" chords in different registers were out of tune with
}each other.

This "sameness" and "in-tune-ness" are surprisingly flexible, especially
when it comes to multiple octaves. Playing the same melody on opposite
ends of a piano simultaneously, you can get away with a semitone
transposition without much disturbing the "sameness" or "in-tune-ness"
(Yasser gives a melody to use for this, and I didn't believe it until I
tried it!)
>
>}I tend to think that low primes should be tuned best. The more
>}complex harmony you use, though, the more accuracy is required
>}for _all_ intervals.
>
>I agree with this in a certain sense. However, perhaps returning a bit to
>some of our earlier discussions, there is a sense in which harmony can become
>more complex without imposing any additional requirements on tuning. For
>example, consider a major 7th chord, 8:10:12:15. One might be tempted to
>choose a tuning for this chord, or evalute the quality of approximation of
>this chord in various tunings, by considering all its intervals, including
>15:8. That would not be correct. 15:8 is a _resultant_ interval in this
>chord, arising from the "stacking" or "linking" of consonant intervals. That
>these consonant intervals are well-represented is enough to ensure that the
>chord will have a pleasing quality. The 15:8 itself need not be tuned close
>to its just value; this interval is dissonant and simply lends a roughness to
>the chord that is independent of its exact tuning; this roughness is
>tolerated because there are so many (5) consonant intervals in the chord.
>
>For example, in 12TET the major seventh is much, much closer to a 17:9 than
>to a 15:8. 17:9 is not much more complex a ratio than 15:8, so that it seems
>certain that the "17:9-ness" of the 12-equal major seventh is much greater
>than its "15:8-ness". And yet the major seventh chord "works" just fine in
>12-equal. That is because the 5 consonant intervals in the chord are
>well-represented in 12-equal.

For another example, take 26TET. Here the major seventh is nowhere near
15:8 -- in fact it nails 24:13, halfway between 13:7 and 11:6. Both
these intervals would be much more important in describing how the 26TET
major 7th is heard than 15:8. And yet, since the consonances are still
OK, the major seventh chord sounds smooth, perhaps smoother than in
12TET. This is mainly because the roughness between the 2nd harmonic of
the root and the fundamental of the major 7th is lessened as the major
7th gets smaller.

This is one of my arguments against using anything beyond level-1
odd-limit considerations in evaluating temperaments. Also this hints at
why I don't think it can be relevant whether a tuning nails 25:18.


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