Re: just tuning (corrected version)

[Pat Missin <pat@xxxxxxxxxxxxx>]

Vern Smith wrote:
>
>This brings up a question about tunings.  I think that I know the answer,
>but maybe Pat Missin can give the definite answer.

I don't know about THE answer, but I'll try to give an answer...

>Is it not true that just tuning produces slightly different pitches in every
>key?  If true, this means that a harp can have just tuning only in the
>first-position key stamped on the cover?  Doesn't this mean that when you
>play it in any other position/key that you don't have just tuning any more?

Not really. There is a widespread belief that Just Intonation has to
be based in a given key. This is not exactly the case, although there
are many text books that talk about such things as "the Just
Intonation Scale in the key of G", or something like that. This is
misleading, as there is no single JI scale. JI can be used to generate
an infinite variety of scales. Nor do they have to be in a specific
key.

Just Intonation is more concerned with getting the optimal harmony for
a set of pitches to be played simultaneously. The harmonica makes this
a comparatively easy process, as there are only certain combinations
of notes that can be played at any given time. For example, on a C
diatonic, blowing produces the notes C, E and G. It is easy to tune
these notes to produce a perfectly pure chord. Now, it doesn't matter
whether you are using this C chord as the tonic chord in first
position, the subdominant chord in 2nd, or the dominant chord in 12th,
the subtonic in 3rd position, etc - in each case it is a perfectly
tuned C chord. 

If you draw on the lower end of the same harp, you get the notes G, B
and D. Again, it is easy to tune these notes as a pure G major chord
and again, it doesn't matter whether you are playing this as the tonic
chord in 2nd position, the dominant chord in 1st position, the
secondary dominant in 12th position, or whatever. In each case it is a
perfectly tuned G major chord.

Now, one of the problems with Just Intonation is that in order to have
every note perfectly in tune with every other note, you would have to
have an infinite range of notes. Instruments such as the trombone, a
choir of human voices, fretless stringed instruments, etc, have
precisely this capability, but many instruments have only a limited
range of notes. One immediate problem is that you cannot make up a
simple seven note diatonic scale in JI where each of the notes is in
perfect tune with each of the others. For example, if you took our C
chord above and tuned it purely, then built a G chord starting on the
fifth of the C chord, then rearranged them to form a scale, you would
have the notes C D E G B C. Now, the problems would start if you
wanted to add an A to this scale. If you were to tune the A to be a
perfect fifth above the D, it would be out of tune with the C. If you
were to tune it to be a pure minor third below the C, it would now be
out of tune with the D. Start adding more notes and you start adding
more of these problems. This is how tempering becomes a vitally
important factor in the history of fretted strings, harps (stringed
variety) keyboard instruments and the music associated with them.

However, the harmonica doesn't have these problems, for the simple
reason that there are only certain combinations of notes that can be
played simultaneously. For example, on our C harp, you can play the A
(6 draw) at the same time as the D (4 draw), but you cannot play it at
the same time as a C (4 or 7 blow). Therefore you can tune the D and
the A together, and not have to worry too much about whether it is
perfectly in tune with the C. As the difference between the first A
(the one that is a perfect 5th above the D) and the second A (the one
that is a pure minor 3rd below the C, or a pure major 6th above the C
- - it's the same note either way) is only about 20 cents, either one
will function quite well melodically. 

In fact, for the most part, the most common JI intervals do not vary
from their 12TET (12TET = 12 Tone Equal Temperament, the most common
tuning system used in modern world) equivalent by any painfully huge
amounts. Therefore it is mostly a simple case of getting the harp into
tune with itself to produce the sweetest chords, then tweaking the
whole tuning so that the average deviation from "standard concert
pitch" is minimal. This is what good harmonica technicians do on a
daily basis. Any small discrepancies that occur can be dealt with by
the player's technique and most good players do this almost
unconsciously. This is why I chuckle to myself when I read
recommendations that single note players would sound better using an
equal tempered harp. Actually, for the most part, if you are playing
single notes, the precise intonation of your harp barely matters at
all. Rather than 12TET sounding better for single note players, it is
more a case of 12TET sounding worse for chordal players.

However...

... there is one aspect of the standard diatonic tuning that does pose
a significant problem. When I said that "the most common JI intervals
do not vary from their 12TET equivalent by any painfully huge amounts"
I am referring to what is technically called 5-limit Just Intonation.
This is the theoretical basis of most Western art music (although in
practice things rarely achieve this ideal) and it is possible to tune
certain harmonica layouts perfectly using this system - natural minor
and Melody Maker, for example. However, the standard major diatonic
tuning (and also the harmonic minor tuning) pose certain problems, in
that they are impossible to tune purely by using 5-limit intervals.
The main problem is that pesky 5 draw (and its octave partner 9 draw).

In JI, with the exception of octaves (and to a less extent fourths and
fifths) two consecutive intervals of exactly the same size simply do
not sound good - they produce harsh dissonance. So, if you were to
tune 3 and 4 draw (B and D on our C harp) and 4 and 5 draw (D and F)
so that there were both pure minor thirds, they would sound fine if
you were to play either B and D, or D and F. However, play any
combination that includes both B and F and it will sound like an angry
hornet's nest. The way to solve this is to tune the D-F "minor third"
interval so that is slightly smaller than the B-D minor third. This
gives you a beautifully smooth and rich G7 chord, but the F note at
the top of that G7 chord is much flatter than the F that is the fourth
note of the C major scale. Whereas the typical 5-limit JI intervals
are usually less than 20 cents away from their 12TET equivalents, this
"flat" 5 draw (technically it is a 7-limit minor third, also called a
subminor third) is more than 30 cents from the tempered version, which
is a lot more noticeable. This means that if you tune your harp like
this (which is how all diatonics were traditionally tuned), it will
sound great for playing cross harp blues, but when you play first
position stuff (especially if you use single notes rather than a
chordal approach) it might not sound so good. It probably will also
sound a little odd when you play third position, as you are
technically playing a subminor minor mode rather than the typical
dorian minor and it will probably sound pretty horrible if you play in
12 position, as your root note will be substantially "flat". There are
various ways to deal with this, however.

If anyone is interested, the harmonic minor tuning presents even more
of a problem. Not only do you have the minor third between 3 and 4
draw, you also have minor thirds between 4 and 5 draw AND 5 and 6
draw. The solution to this involves using higher level JI intervals
drawn from the 17-limit, but I'll save that for a later (probably much
later!) post...

>Just tuning is the reason barbershop and string quartets sound so good,
>because the instruments used (voice and strings) can be tuned by the ears of
>the players who automatically use just regardless of the key.

This is often described as "adaptive JI". Rather than having a fixed
palette of pitches, singers and players of certain instruments can
continuously tweak each note for the optimum intonation at any point
in the tune. Again, to some extent, a good harmonica will be able to
do micro-bends to keep their instrument in good relative tune with the
accompaniment.

>Tempered/equal (twelfth-root-of-two) tuning is a compromise that doesn't
>sound quite as good in any key as just but does sound the same in all keys?
>Thus equal tuning is the obvious choice for any instrument that uses fixed
>pitches and is played in all keys.

It is a choice, but it is not the only one, nor is it always the best
one, although it is the commonest. There are various other
possibilities, depending on the context.

>It would seem possible to have an electronic keyboard that could adjust the
>pitches to give just tuning in any key.  You would only have to push a
>button to tell it which key to tune to.

There are such things, although you then run into the problems that I
mention above, so in anything but the most harmonically simple piece
of music, you would have to keep pushing that button all the way
though the tune.

Hope this isn't too technical and of course, if anyone has any further
questions, I'd be happy to try to answer them.

However, as I am here anyway...

I did have a couple of offlist questions regarding the term "limit" in
JI theory. For the most part, it's not something you need to worry
about too much, but here is a quick cursory coverage.

All intervals in JI are determined by ratios, with a preference for
simpler ones. Basically, the simpler the relationship between two
notes, the easier it is for our brain to process them, so the sweeter
they sound to us.

If you take two notes of identical pitch, they are said to be in the
ratio of 1:1, ie a single helping of one note is the same size as a
single helping of the other note. The highest prime number (in fact,
the only number) in this ratio is 1, therefore this could be said to
have a prime limit of 1. A 1-limit tuning system would not be very
exciting, as you would only have one note!

You probably all know that if you take a note and double its
frequency, you will raise its pitch by an octave. Therefore, two notes
an octave apart are said to be in the ratio 1:2. The highest prime
number here is 2. If you were to have a tuning system with a prime
limit of 2, then you could have only one note, but you could have it
in any octave you wanted. Obviously not a very practical tuning system
for most music.

If you were to take two numbers in the ratio of 2:3, the second number
would be a perfect fifth higher than the first. So, if you were to
take a note and raise its pitch by 3/2 (ie multiply it by one and a
half) you would raise it by a fifth. For example, if you were to start
with 440Hz and multiply by 3/2, you would get 660Hz. In our standard
system, 440Hz is an A, so 660Hz would be a perfect 5th above it. By
taking any note as a starting point and multiplying it by 3/2, then
multiplying the result by 3/2, over and over, you can generate a
series of perfect fifths. Unlike in our typical tempered tuning
system, there is no ~circle~ of perfect fifths, it is an infinite
spiral. Starting with C you would have C, G, D, A, E, B, F#, C#, G#,
D#, A#, E#, B#, F##, C##, etc., etc. The B# in this system would be a
little sharper than C, the C## would be a little sharper than D, etc.
In fact, this is how most Western music was tuned for several
centuries until someone decided to make B# the same as C and limit the
scale to 12 notes. Tuning this way by perfect fifths is often called
Pythagorean tuning, although Pythagoras was by no means the first or
the only person to discover it. As the scale is generated by the ratio
2:3 and the highest prime number in this ratio is 3, then this is
termed a 3-limit system. It makes a fine scale for melodic purposes,
but the "major thirds" produced by the 3-limit are a little harsh when
used harmonically. For this reason, early Western art music treated
the major third as a dissonant interval.

OK. So we've had 1:2 and 2:3, so the next obvious pair of numbers
would be 3:4. If you take any frequency and raise it by 4/3 (or
multiply by 1 and a third), you raise the pitch of the note by a
perfect fourth. Actually, the highest prime in the ratio 3:4 is also
3, so this is still a 3-limit system, because a perfect fourth is
simply a perfect fifth going off towards infinity in the opposite
direction: C, F, Bb, Eb, Ab, Db, Gb, Cb, Fb, Bbb, etc. 

So, after 1:2, 2:3, and 3:4, we would logically arrive at 4:5 and if
you were to raise a frequency by 5/4 (ie multiply it by one and a
quarter) you would get a note a major third higher. So if we take our
440Hz and multiply it by 5/4, we get 550Hz. This would be a justly
intonated C#. If we were to throw in our E of 660Hz, we would have a
chord of 440Hz, 550Hz and 660Hz. This would be a pure major chord in
the ratios 4:5:6. The highest prime number in this ratio is 5, so this
is part of a 5-limit system. As I said, this is the theoretical basis
of much Western music. Both 3-limit and 5 limit JI give acceptable
diatonic and chromatic scales, but 5-limit gives nice smooth harmonies
in thirds and sixths.

The next ratio after 1:2, 2:3, 3:4, and 4:5 would be 5:6. Actually, we
already encountered that in our chord of 440Hz 550Hz and 660Hz. The C3
and E in this A major chord are in the ratio of 5:6, so multiplying
any frequency by 6/5 raises it by a minor third. This is still part of
the 5-limit.

So our next basic ratio would be 6:7. This is where the "problems"
come in. If we take our major chord in the ratio of 4:5:6 and add a
note to make it 4:5:6:7, we get a pure seventh chord, the same as you
would find in holes 2, 3, 4 and 5 draw on a traditionally just
intonated harp. So, added to our A of 440Hz, our C# of 550Hz and our E
of 660Hz, we would have a slightly flat G of 770Hz. The highest prime
in the ratio 6:7 is 7, so we are now dealing with a 7-limit system.

Now pause for second and remember that to raise a note by a perfect
fourth you multiply it by 4/3. So, lets take our A, raise it by a
fourth to get a D, the raise it by another fourth to get a G. 

440Hz x 4/3 = 586.666Hz
586.666Hz x 4/3 = 782.222Hz

Hm. So the 7-limit G (the upper note of our pure A7 chord) is actually
more than 12Hz flatter than the 3-limit G (two perfect fourths above
the A). In fact, although both the 3-limit and 5-limit systems give us
stuff that sounds like our familiar diatonic scales, the 7-limit
introduces notes that are often about 1/3 of a semitone away from our
familiar scales. This can make for nice sweet barbershop harmony, or
for piquant blues notes, but you really don't want to be playing a
7-limit interval when you ears (or your audiences ears) want to hear a
5-limit or 3-limit note.

For the most part, any relevance to the harmonica comes to an end with
the 7-limit, however I'll throw in some stuff about the higher prime
limits just for the hell of it. 

The next prime numbers after 7 are 11 and 13. The 11-limit and
13-limit intervals involved even greater deviation from familiar
territory, often involving intervals of around a quartertone (half a
tempered semitone), giving such exotic things as neutral thirds and
sevenths (ie neither major nor minor, but somewhere in between). Not
particularly useful for harmonic purposes, but you do often encounter
such melodic intervals in blues and some Middle Eastern music.

The next couple of primes are 17 and 19. These start to take us back
to more familiar sounds, as 17-limit and 19-limit intervals often come
quite close to approximating our common 12TET notes. Beyond the
19-limit, things tend not to have much of an identifiable quality of
their own.

Ok - that was considerably longer than I was planning, but hopefully
it is useful to somebody.

Any questions?

 -- Pat.