Newbie questions - "modulation",

Hi,

I've been lurking here for a while, and used the posted url's to help scoure
the web, but I have some questions I haven't found anything specific on.
I'm programming alternate tuning capability into my sequencer. I apologize
to anyone offended by the low level of my questions on this list. Please
provide references when you tell me where to go :)

1. For illustrative purposes, I'll use a small prime limit JI, starting
in C. If I play a C major chord, the E is played about 14 cents lower than
12TET. If I move to a E major (using non-diatonic note for the 3rd (I don't
know if non-diatonic is the right word in this case)), holding the E, does
one usually (i) keep the E at that position and adjust each of the other
notes accordingly so their intervals are identical to the original C major
chord, or (ii) shift the E down slightly and keep the intervals the same so
that there is less "drift" over time, or (iii) use the intervals for the E
major that are implied by the original . And if I move to a C major again,
would I use the C in the original pitch, or the pitch based on E as being
the last "tonic" (I use the word loosely) (ie: is it like modulating keys
when one changes chords, or does one stay within the original C tuning?

And what if two notes are held from a previous chord. If they form a
different interval in the two chords, which is almost always the case, one
or both have to shift. Any recommendations?

Another example: if I modulated keys in fifths around the full circle, would
I accumulate the 2 cent discrepancy all the way around so that I would be 24
cents off from my previous C when I get back to it, or would I somehow shift
pitch slightly at convenient places to try and stop the drift? If shifting
is the answer, any recommendations for good places or ways to do it?

My prejudice: for maximum harmony, I was intending on playing each new
chord as if the root of that chord is the 1 of the JI scale. So chord
changes change the tuning of all the keys - like one would expect with
modulation. The only problem is this means the tuning actually becomes
quite fluid, with intervals always "correct", but no consistent pitches for
notes. Or another way to look at it is I'ld always be playing an
approximation of a small prime limit JI subset of a large note scale (like
53 circle of primes or something).

2. I've seen an old article in Electronic Musician talk about playing
melodies differently than harmonies (ie: using different tunings).
Unfortunately, the article was short, imprecise, and all that, and I haven't
seen mention of this anywhere else. I presume he means unacommpanied
melodies, since any accompanied melodies have harmony? Any info on what is
often done here? I understand the physical basis for using small prime
limit JI for harmonic situations. Are there physical basis for melodic
intervals as opposed to harmonic?

Perhaps such melodic intervals are the appropriate interval to be used for
chord changes and modulation if the root or 1 position is not held from a
previous chord?

All recommendations, relevant references, etc. are greatly welcomed.

Jim Savage

Jim Savage wrote,

>1. For illustrative purposes, I'll use a small prime limit JI, starting
>in C. If I play a C major chord, the E is played about 14 cents lower than
>12TET. If I move to a E major (using non-diatonic note for the 3rd (I don't
>know if non-diatonic is the right word in this case)), holding the E, does
>one usually (i) keep the E at that position and adjust each of the other
>notes accordingly so their intervals are identical to the original C major
>chord, or (ii) shift the E down slightly and keep the intervals the same so
>that there is less "drift" over time, or (iii) use the intervals for the E
>major that are implied by the original.

I am unclear as to what exactly these alternatives mean. Can you please
give some ratios, etc. to clarify?

>And if I move to a C major again,
>would I use the C in the original pitch, or the pitch based on E as being
>the last "tonic" (I use the word loosely) (ie: is it like modulating keys
>when one changes chords, or does one stay within the original C tuning?

That is controversial -- we've had some debate on this topic in the last few
months. My view is that works of classical music operate on a sort of medium-term
pitch memory (of which perfect pitch would be the long-term analogy) and so when
the music returns to the home key, the musical effect is lost unless the final
pitch is quite close to the original pitch. There are works of Mozart which would
drift downward by almost half an octave if strict JI were adhered to in all
comma-dependent progressions (including, but not limited to, modulations) and
that drift would certainly destroy the feeling of "return," as well as a host of
other pitch-memory dependent features on shorter scales.

>And what if two notes are held from a previous chord. If they form a
>different interval in the two chords, which is almost always the case, one
>or both have to shift. Any recommendations?

Again, please give some specific examples. In general, some shift is appropriate
if it results in greater consonance for the two chords, but the shift should not
be so abrupt as to be noticeable. John deLaubenfels has recently discussed
various strategies for subtly "sliding" pitches into tune.

>Another example: if I modulated keys in fifths around the full circle, would
>I accumulate the 2 cent discrepancy all the way around so that I would be 24
>cents off from my previous C when I get back to it, or would I somehow shift
>pitch slightly at convenient places to try and stop the drift? If shifting
>is the answer, any recommendations for good places or ways to do it?

Again, I feel that pitch memory is important in classical music, but in this case
the discrepancies are so small that almost any strategy would be acceptable. By
contrast, a doo-wop song from the 50's, with a I-vi-ii-V progression
repeated over and over, with no modulation, would drift down 22 cents each time
the progression went around, if you kept all common tones and tuned each chord in
JI. That could add up to a huge drift over the course of the song, which would be
totally unacceptable. See the archive from a few months back on the subject
"adaptive JI" for a discussion of strategies to prevent this drift.

>My prejudice: for maximum harmony, I was intending on playing each new
>chord as if the root of that chord is the 1 of the JI scale.

That doesn't work so well for certain chords. For example, a C6/9 chord
(C E G A D) would have the fifth between A and D as 40:27 if you tuned it
according to "the JI major scale" on C, and an Am7add11 chord (A C E G D) would
have the fifth between G and D as 40:27 if you tuned it according to "the JI
minor scale" on A.

>So chord
>changes change the tuning of all the keys - like one would expect with
>modulation. The only problem is this means the tuning actually becomes
>quite fluid, with intervals always "correct", but no consistent pitches for
>notes. Or another way to look at it is I'ld always be playing an
>approximation of a small prime limit JI subset of a large note scale (like
>53 circle of primes or something).

Circle of primes? You mean circle of fifths? Anyway, most common-practice music
can be played in meantone temperament, which would require only 21 notes per
octave if the music contains no double-sharps or double-flats, and all the
consonant intervals would be less than 6 cents out-of-tune. There would be no
drift due to progression or modulation, except if you tried to go all the way
around a circle of 12 fifths, such as from F# to Gb. That's not nearly as common
as the "syntonic" type of drift. So to perform most music in JI, you could use
meantone temperament as your basis, correct the 6-cent errors on the fly if
desired, and for the most part not have to worry about drift.

>2. I've seen an old article in Electronic Musician talk about playing
>melodies differently than harmonies (ie: using different tunings).
>Unfortunately, the article was short, imprecise, and all that, and I haven't
>seen mention of this anywhere else. I presume he means unacommpanied
>melodies, since any accompanied melodies have harmony? Any info on what is
>often done here? I understand the physical basis for using small prime
>limit JI for harmonic situations. Are there physical basis for melodic
>intervals as opposed to harmonic?

If you look at most musical cultures where melody is accompanied, you see a wide
variety of scales. These scales tend to have reasonably good (by harmonic
standards) octaves, fourths, and fifths, but the thirds and sixths are peculiar
to local tradition and show no correlation with any simple-integer ratio basis.
Simply put, unaccompanied melody should be modeled by 3-limit JI at most. Ratios
of 5 and higher have no influence over a purely melodic music. I would agree with
Johnny Reinhard that Pythagorean tuning is ideal for diatonic melody when
divorced from harmonic considerations. However, 5-limit harmonic considerations
are a part of Western musical culture since 1480, so any unaccompanied melody
written in the West during that period would be influenced by this harmonic
environment.

>Perhaps such melodic intervals are the appropriate interval to be used for
>chord changes and modulation if the root or 1 position is not held from a
>previous chord?

The only melodic intervals that really have any "pull" as to tuning are
the octave, fourth, and fifth, as well as whetever intervals are familiar due to
context and/or culture. Depending on the particular example of what you are
describing (care to provide an example?), melodic considerations may or may not
be able to determine the "correct" interval for such changes.