stuff (Paul E./Carl L.)

[Erlich:]
>> Actually, David, you have found the "problem of the
>> syntonic comma" which explains why 5-limit just intonation
>> never gained much importance in Western music.
>
[Lumma:]
> No! It explains why the diatonic scale was not tuned in
> 5-limit just intonation on most western keyboards and guitars.

I'm glad you spoke up against this.

> 5-limit JI is a big place, present in some form in many
> meantone tunings. Using the term synonymously with some
> 7-tone scale by Ptolemy is a big loss.

I agree with this, Paul.
I think if we deny the importance of 5-limit JI in
Euro-centric music for a least a few centuries, in many
different aspects of tuning and music-theory, we are
being less than totally accurate. Carl's statement
is a much more accurate way of putting it.

[Lumma:]
> The shortest-route metrics are equivalent to change-making
> problems. What is the least number of coins you can use
> to make 33 cents change? Forget a quarter, nickel, and
> three pennies, Erlich mints a 33-cent piece on the spot!

Isn't this the same thing I'm doing in my posting on
direct lattice connections? [TD 132, re Etta James MIDI-file]
The regular (Monzo-)lattice rungs show the 'quarter, nickel',
etc. 'change', and the direct rung is the 33-cent piece.

> Multiple locations may still be a pain visually, but probably
> not so much as wormholes...

But if my thinking about wormholes is correct,
they're unavoidable.

-monzo
http://www.ixpres.com/interval/monzo/homepage.html

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Carl Lumma wrote,

>> 5-limit JI is a big place, present in some form in many
>> meantone tunings. Using the term synonymously with some
>> 7-tone scale by Ptolemy is a big loss.

Joe Monzo wrote,

>I agree with this, Paul.
>I think if we deny the importance of 5-limit JI in
>Euro-centric music for a least a few centuries, in many
>different aspects of tuning and music-theory, we are
>being less than totally accurate. Carl's statement
>is a much more accurate way of putting it.

I think if we deny the importance of temperament as a practical solution for
incorporating 5-limit harmony with diatonic music, we are making a big
mistake.

>But if my thinking about wormholes is correct,
>they're unavoidable.

Wormholes don't occur if you have an axis for each odd number within the
limit, like in Erv Wilson's diagrams.

[me:]
>> I think if we deny the importance of 5-limit JI in
>> Euro-centric music for a least a few centuries, in many
>> different aspects of tuning and music-theory, we are
>> being less than totally accurate.

[Erlich:]
> I think if we deny the importance of temperament as a
> practical solution for incorporating 5-limit harmony with
> diatonic music, we are making a big mistake.

I absolutely agree.

The two statements are not mutually exclusive,
and in fact both 5-limit JI *and* temperaments
(of many types) must be considered in discussing
historical aspects of Euro-centric tuning theory.

[me:]
>> But if my thinking about wormholes is correct,
>> they're unavoidable.

[Erlich:]
> Wormholes don't occur if you have an axis for each
> odd number within the limit, like in Erv Wilson's diagrams.

OK, we're using 'wormhole' to mean two different things.

I thought that if you draw a lattice which incorporates
'bridging', i.e., shows the 'bridge' or 'unsion vector'
connections, then those are the 'wormholes'.

This would be true for a Wilsonian odd-lattice as well
as for one of my prime-lattices.

To use an overworked example, 225/224 would appear on
the odd-lattice exactly the same as on the prime-lattice.
(except for different angles, vector-lengths, etc.)
A composite odd like 9 doesn't even enter the picture.

So speaking of mutually exlcusive, this kind of
discrepancy is why I was thinking, a few TDs ago,
that our uses of the word 'wormhole' might be
mutually exclusive.

But to me (and you agreed), the word is a very apt
characterization of what happens on the lattice when
a 'bridge' is encountered. You sort of enter the
note at one ratio/lattice-point, and emerge somewhere
else in the lattice at a different point.

Or, what is really a better description, the
dimension in which one point exists becomes congruent
to the dimension in which the other exists.
(is congruent the correct word? perhaps 'a substitute'
is more accurate)

-monzo
http://www.ixpres.com/interval/monzo/homepage.html

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>I think if we deny the importance of temperament as a practical solution for
>incorporating 5-limit harmony with diatonic music, we are making a big
>mistake.

It comes down to this: What do you believe is lost by approximating the
diatonic scale on the 5-limit lattice? Well, what happens?

a) The tonic drifts. My take? Not an issue for music that modulates,
maybe an issue for music that keeps returning to tonic and ends there.

b) You get comma changes. My take? They add something, and they do take
something away. I enjoy what they add more than I miss what they take
away, but the effect is small overall.

c) You need 53 notes. My take? Guitars, no. Keyboards, pushing it. The
rest, my ear tells me, already use them.

>>>Strictly speaking, only 1/3 or 1/4 of the notes in a meantone tuning can
>>>coincide with 5-limit JI.
>>
>>By huge margin the two most popular meantones in history :)
>
>What is that supposed to mean?

That I knew which meantones had pure 5-limit intervals in them.

>>Well, you said "could it hurt me", so I was just saying "no, but I can't
>>help it".
>
>I'm lost.

You asked if it would hurt me to combine a lattice visualiztion of a
harmonic complexity metric (ie your algorithm) with a lattice-route metric
(ie Paul Hahn's algorithm). I was trying to say, "yes".

Don't sweat -- just the lack-of-tone-in-speach e-mail bug again.

>That would be identical to Tenney's algorithm, discussed recently (only
>prime axes needed, length of one step along axis p = log(p)). I'm not sure
>what the reference is . . . Daniel? And once again, octave-equivalence
>shouldn't be allowed in the context of this algorithm.

Going back to what you said about the triangular lattice...

>I do feel strongly that this improves upon Tenney's and Barlow's
>formulations, especially in the typical case where factors of two are
>ignored

Would you change this to "only when the factors of two are ignored"? Or
would you now say it doesn't improve at all?

C.

Carl Lumma wrote,

>What do you believe is lost by approximating the
>diatonic scale on the 5-limit lattice?

a) Tetrachordality. The Pythagorean or meantone diatonic scale has two
identical tetrachords in every octave span, making melodic motion
exceptionally comprehensible and flexible at the same time. Now differences
of a comma won't disturb this too much, unless the tetrachords are combined
harmonically, but I (and Zeke Hoskin on his harp page) can certainly hear
the differences and find melodies smoother without the differences. Notice
how the basic scale in India, in 5-limit JI, is tuned with two identical
tetrachords between the tonic and the octave above. The 27/16 is melodically
necessary despite the fact that a 5-limit lattice would find 5/3 more
closely related to the rest of the scale.

b) Chords by fourths/fifths. Due to passing tones, suspensions, and styles
with thick harmony, chords with four or more notes on the chain of fifths
occur. Observing 5-limit ratios for the thirds would mean one of the fifths
would be off by a comma, which is often unacceptable even in passing.

c) The constancy of diatonic pitches and interval-sizes in a tuning where
the comma vanishes has an aesthetic effect distinct from that where the
pitches and interval-sizes are variable by a comma. Perhaps either is
equally valid, but the fact is that most music in the West so far was
written with the former aesthetic effect intended is important for those of
us who place ourselves within that tradition.

>a) The tonic drifts. My take? Not an issue for music that modulates,
>maybe an issue for music that keeps returning to tonic and ends there.

Even music that modulates usually returns to the tonic.

>c) You need 53 notes. My take? Guitars, no.

You haven't heard Garibaldi's ensemble? They do it!

>The
>rest, my ear tells me, already use [53 notes].

By ear tells me that equal-tempered thirds have "brainwashed" most musicians
most of the time. Most published attempts to measure pitch tendencies of
musicians have found no evidence that the deviations from equal temperament
are more often in the direction of 5-limit JI that not, though I certainly
believe that they should be.

>>That would be identical to Tenney's algorithm, discussed recently (only
>>prime axes needed, length of one step along axis p = log(p)). I'm not sure
>>what the reference is . . . Daniel? And once again, octave-equivalence
>>shouldn't be allowed in the context of this algorithm.

>Going back to what you said about the triangular lattice...

>>I do feel strongly that this improves upon Tenney's and Barlow's
>>formulations, especially in the typical case where factors of two are
>>ignored

>Would you change this to "only when the factors of two are ignored"? Or
>would you now say it doesn't improve at all?

With regard to the Tenney, I would say that if factors of two are ignored,
the issue of triangular vs. rectangular comes down to whether you use a
numerator limit or a numerator times denominator limit, and since I don't
particularly have a problem with the latter, and since it leads to fairly
predictably-spaced Farey-type series, I could stomach a rectangular lattice
in this case. I still have other problems with Barlow's formulations.

>a) Tetrachordality. The Pythagorean or meantone diatonic scale has two
>identical tetrachords in every octave span, making melodic motion
>exceptionally comprehensible and flexible at the same time. Now differences
>of a comma won't disturb this too much, unless the tetrachords are combined
>harmonically, but I (and Zeke Hoskin on his harp page) can certainly hear
>the differences and find melodies smoother without the differences. Notice
>how the basic scale in India, in 5-limit JI, is tuned with two identical
>tetrachords between the tonic and the octave above. The 27/16 is melodically
>necessary despite the fact that a 5-limit lattice would find 5/3 more
>closely related to the rest of the scale.

Paul, you're assuming a fixed scale.

Also, I have never heard anything convincing on that Indian musicians
actually play what they (or we) claim they do. In fact, I once watched a
real-time pitch-tracking algorithm plot a sitar against the scale it was
supposed to be playing. Not convincing at all.

>b) Chords by fourths/fifths. Due to passing tones, suspensions, and styles
>with thick harmony, chords with four or more notes on the chain of fifths
>occur. Observing 5-limit ratios for the thirds would mean one of the fifths
>would be off by a comma, which is often unacceptable even in passing.

You're *really* assuming a fixed scale. The 3-limit is a subset of the
5-limit.

>>a) The tonic drifts. My take? Not an issue for music that modulates,
>>maybe an issue for music that keeps returning to tonic and ends there.
>
>Even music that modulates usually returns to the tonic.

Usually? Anyway, I stick with that it doesn't matter if there's been
enough modulation betwixt.

>>c) You need 53 notes. My take? Guitars, no.
>
>You haven't heard Garibaldi's ensemble? They do it!

Ensemble maybe. I was thinking individual instruments.

>>The rest, my ear tells me, already use [53 notes].
>
>By ear tells me that equal-tempered thirds have "brainwashed" most musicians
>most of the time.

I don't think that musicians are _at all_ drawn to the 12tET 5/4 in
harmonic context, without a guitar or keyboard present, and then _usually_
not.

>Most published attempts to measure pitch tendencies of musicians have
found >no evidence that the deviations from equal temperament are more
often in the >direction of 5-limit JI that not, though I certainly believe
that they >should be.

More often, probably not. More often among good musicians, you bet!

>>a) Tetrachordality. The Pythagorean or meantone diatonic scale has two
>>identical tetrachords in every octave span, making melodic motion
>>exceptionally comprehensible and flexible at the same time. Now
differences
>>of a comma won't disturb this too much, unless the tetrachords are
combined
>>harmonically, but I (and Zeke Hoskin on his harp page) can certainly hear
>>the differences and find melodies smoother without the differences. Notice
>>how the basic scale in India, in 5-limit JI, is tuned with two identical
>>tetrachords between the tonic and the octave above. The 27/16 is
melodically
>>necessary despite the fact that a 5-limit lattice would find 5/3 more
>>closely related to the rest of the scale.

>Paul, you're assuming a fixed scale.

Without a fixed scale you get the comma differences I mentioned.

>Also, I have never heard anything convincing on that Indian musicians
>actually play what they (or we) claim they do. In fact, I once watched a
>real-time pitch-tracking algorithm plot a sitar against the scale it was
>supposed to be playing. Not convincing at all.

The sitar and most other Indian instruments and vocalists use a very
expressive form of intonation where pitch-tracking would be of little value.
The placing of frets, however, was found to be in line wih the theoretical
ratios, at least when the work of certain master musicians were measured.
Moreover, the santoor is a fixed-pitch instrument and the recordings I've
heard are clearly in 5-limit JI. Perhaps Daniel Wolf can comment?

>>b) Chords by fourths/fifths. Due to passing tones, suspensions, and styles
>>with thick harmony, chords with four or more notes on the chain of fifths
>>occur. Observing 5-limit ratios for the thirds would mean one of the
fifths
>>would be off by a comma, which is often unacceptable even in passing.

>You're *really* assuming a fixed scale. The 3-limit is a subset of the
>5-limit.

Ah, but in this case, when the passing tone steps or the suspension
resolves, one of the _other_ notes would have to move by a comma even though
it does not have a new attack. That kind of adjustment is really unlikely to
occur in practice.

>>>c) You need 53 notes. My take? Guitars, no.
>
>>You haven't heard Garibaldi's ensemble? They do it!

>Ensemble maybe. I was thinking individual instruments.

Do you know of what you speak? Each individual guitarist (of the duo) plays
a 53-tone guitar and articulates cleanly.

>I don't think that musicians are _at all_ drawn to the 12tET 5/4 in
>harmonic context, without a guitar or keyboard present, and then _usually_
>not.

I don't know about you, but I am a musician, a serious musician, and most
singers I've worked with tend to exaggerate the differences between
semitones and whole tones, leading to Pythagorean-like intonation. String
players have the same tendencies, even many "world-class" ensembles. I don't
like it, but it's true.