Re: Newbie questions - "modulation",

>From: Jim Savage <waldpond@oanet.com>
>
>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 .

If I understand what you write, a cappella singers would do number i.

>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?

A cappella singers would form a C chord that incorporates the E from the
previous key of E. If you are simply alternating between the keys of C and
E there would be no drift.

Now there may be times when a piece modulates to the wrong key, so to
speak, and that would be the correct thing to happen. For example, the
Chopin Prelude no. 9 in E major returns to the original key of E=1 in bar 9
after several modulations. But then the progression goes as follows: E B E
A- F Bb G- D G B E. At this point the key is E=80/81. Draw the lattice
diagram for major thirds and perfect fifths and you'll see that that is the
case. If a group of singers were to sing the final four bars of Chopin's
prelude in tune they would drift down by the comma of Didymus, because they
are driven there by the progression. So actually I shouldn't call it drift.
It is just movement to a new key that by convention has the same name as
the original key. I would like to know whether there is any sort of uneasy
feeling at the end when we don't finish in the original key. I have
arranged that prelude for my group but we haven't sung it. To carry out my
experiment I'll have to get a fretless guitar or sing the guitar part, or
get the equipment to do a sequence.

>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?

It depends upon the particular example.

>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?

A cappella singers, if there were really in tune, would do the former.

>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.

YES! Flexible tuning. Hindemith alluded to it and rejected it in Book 1 of
The Craft of Musical Composition. But good a cappella singers do it all the
time, as do good string quartets. See Gerald Eskelin's book Lies My Music
Teacher Told Me. Eskelin's website is http://home.earthlink.net/~stg3music/
and his email address is stg3music@earthlink.net.

>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).

If you play in tune, there is no approximation to anything.

>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?

I believe that when singing solo unaccompanied we sing pitches according to
the harmony that we imagine.

John Link
ALMOST ACAPPELLA

>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?

These are aesthetic questions, and the freedom of choice seems to be what
we've requested. I recommend playing, and listening. The recordings
listed at http://lumma.org/topten.html offer a good taste of what's been done.

>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?

You could spread it out uniformly by rooting chords in 12tET. In some
progressions, there are convenient places to insert the shift, rather than
spreading it out by choosing roots from a temperament. These are usually
spots where the root motion is large (say, on the circle of fifths). In
the progression A E C F, for example, a jump would probably be least
noticed between E and C. But in the case of the uniform circle-of-fifths
progression, there's hardly a better solution than equal distribution. Of
course, you can always leave the drift -- you might like it.

>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).

The common-tone v. harmony problem is one of the oldest in music theory.
Temperament is a very interesting solution -- the range of temperaments
has, just in the last decade, become well-understood.

One solution from a JI perspective is offered by David Doty, in a
recently-released landmark recording: http://www.syntonic-rec.com/ucp.html

I wrote about a solution to the problem from a JI perspective in a series of
posts around the beginning of October. I've summarized these posts at:
http://lumma.org/adaptive.txt

You'll also want to check out John deLaubenfels' adaptive tuning page:
http://www.idcomm.com/personal/jadl

>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?

I would say yes. The melodic basis is proving much harder to discover than
the harmonic. I believe there are several independent components to the
melodic basis, some but not all of which are mutually exclusive. And some,
but not all of which are exclusive with the harmonic basis.

One interesting thing I suggest you try: melody and harmony coming from
different tunings.

-Carl

>From: Jim Savage <waldpond@oanet.com>
>
>>From John Link:
>
>>>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?
>>
>>It depends upon the particular example.
>
>
>Thanks a lot for the answers. It's cleared up most of what I want to do.
>Only one thing left for now :)
>
>Is there an enumeration of the cases and what is usually done for each
>somwhere, or is it largely by instinct since there's a huge number of cases?
>
>Jim Savage

A cappella singers do it by ear all the time, probably without knowing
exactly what they're doing. I'm working on a book about my theories.

A very important example is the movement from IImin7 to V7 to Imaj7. For
concreteness, let's consider the key of C, so we're talking about D-7 going
to G7 going to Cmaj7. Here's a lattice diagram of the tones related by 5/4
(up and to the left) or 3/2 (up and to the right):

B D

E G

A C

D F

I won't waste my time trying to draw the usual lines. Since we're in C we
have C=1/1. Here is my conjecture of what good a cappella singers do:

C 1/1 B 15/16 B 15/16
A 5/8 G 3/2 G 3/2
F 4/3 F 21/16 E 5/4
D 10/9 D 9/8 C 1/1

Going from D-7 to G7 note that BOTH common notes adjust. Also note that
while the two different D's are both in the lattice diagram, the second F
is not, because relative to G it is 7/4, and I drew the lattice only for
5/4 and 3/2.

John Link
ALMOST ACAPPELLA

>I won't waste my time trying to draw the usual lines. Since we're in C we
>have C=1/1. Here is my conjecture of what good a cappella singers do:

That's also what I'd expect they'd do! Assuming you meant A=5/3, I've
taken the liberty of doing up a version showing relative ratios on the
left of each chord, and common tones with ----, showing cents difference
in the adjusted notes. . .

9/5 C 1/1 5 B 15/8 ------ 15 B 15/8
3 A 5/3 1 G 3/2 -------- 3 G 3/2
6/5 F 4/3 -(-27)-- 7 F- 21/16 5 E 5/4
1 D 10/9 -+22--- 3 D+ 9/8 1 C 1/1

>Going from D-7 to G7 note that BOTH common notes adjust. Also note that
>while the two different D's are both in the lattice diagram, the second F
>is not, because relative to G it is 7/4, and I drew the lattice only for
>5/4 and 3/2.

Here's how I'd draw the lattice -- I've named the adjusted F and D notes
F- and D+ . . .

D---------A E---------B
\ / \ / \ /|\
\ / \ / \ / | \
\ / \ / \ / F- \
\ / \ / \ /,' `.\
F---------C---------G--------D+

-Carl

In a recent message, I wrote:

>>I won't waste my time trying to draw the usual lines. Since we're in C we
>>have C=1/1. Here is my conjecture of what good a cappella singers do:

And Carl Lumma responded:

>That's also what I'd expect they'd do! Assuming you meant A=5/3,

Of course. In my haste to respond I mad a mistake.

> I've
>taken the liberty of doing up a version showing relative ratios on the
>left of each chord, and common tones with ----, showing cents difference
>in the adjusted notes. . .
>
> 9/5 C 1/1 5 B 15/8 ------ 15 B 15/8
> 3 A 5/3 1 G 3/2 -------- 3 G 3/2
> 6/5 F 4/3 -(-27)-- 7 F- 21/16 5 E 5/4
> 1 D 10/9 -+22--- 3 D+ 9/8 1 C 1/1
>
>>Going from D-7 to G7 note that BOTH common notes adjust. Also note that
>>while the two different D's are both in the lattice diagram, the second F
>>is not, because relative to G it is 7/4, and I drew the lattice only for
>>5/4 and 3/2.
>
>Here's how I'd draw the lattice -- I've named the adjusted F and D notes
>F- and D+ . . .
>
> D---------A E---------B
> \ / \ / \ /|\
> \ / \ / \ / | \
> \ / \ / \ / F- \
> \ / \ / \ /,' `.\
> F---------C---------G--------D+
>

My preference is to orient the diagram with 5/4 up and to the left and 3/2
up and to the right, but the orientation obviously carries no information.
With pencil and paper I would draw F- as Carl has (but still label it "F"),
but draw a line between G and F but not from F to either B or D+ (which I
would label "D").

John Link
ALMOST ACAPPELLA