Tuning Fifths in Just Intonation

April 26, 2012 at 09:07 AM · Two extremes:

- "Meantone", lovely smooth pure major thirds, highly compromised fifths;

- "Pythagorean", pure fifths, gritty major thirds, (and maybe, stretched octaves...)

What to do?

- Tune each string to the harpsichord/organ/piano/lute etc.

- "Constuct" the intonation in fifths, in the key of the music, and "soften" the thirds when our sensitised "baroque" ears demand it!

Fifths are "spatial", thirds sensual..

In double-stops, I have sometimes had to raise the lower note of a major third, rather than lower the upper note, e.g. C with open E.

A comma in first position is good 1/8th inch wide, wider than my usual romantic vibrato!

A surprising number of musicians don't understand what on earth we are talking about!

Take frequent breaks for a nice cup of tea!

Adrian

April 26, 2012 at 11:11 PM · Maybe I wasn't entirely clear in my original post on what I was looking for. I am conversant with just intonation principles and have progressed beyond "equal temperament ears", having played baroque violin for several years now. What I'm asking is, objectively speaking, which notes of the scale need to be differently pitched when functioning as a fifth instead of a third.

I am not concerned here with keyboard temperaments; these, as I understand, of necessity have variable fifths (some wider than perfect/pure, some narrower than perfect/pure). I'm concerned solely with just or pure intonation and perfect or pure fifths.

To put the question as precisely as possible, should, e.g., F be played differently as the fifth of B flat than it should be as the third of D? When I play F as the third of D I play it quite high so as to make a wide and pure minor third. Should this pitch then be exactly the same to tune a perfect/pure fifth with B flat? This is the crux of my question.

April 27, 2012 at 04:40 AM · Michael, I think this (just intonation) is the most difficult subject in music. There are examples on line (I will post later) that demonstrate that you can either play linear motio justly or you can play harmonies justly but you cannot do both. You cannot even tune 4 successive thirds without some compromise. My guide on this subject is figuratively a full professor Nobel Laureate to my Pre-K but I have come to believe that this is the Mount Everest of artistry. Good luck! You will need it.

April 27, 2012 at 07:02 AM · All right, let's take A in the key of C:

If the root of the chord is D, it's a pure perfect fifth;

If the root is F, we lower the A by a comma to have a pure major third;

Most important: if the A is sustained during changing harmonies, we should adjust its pitch as we play it;

If in double-stops we must use open-string A's, we may want to raise the F by a comma..

But to answer your question (!), in a B-flat chord, F should be a pure fifth above; in a D minor chord, the F must adapt, as we go along, to the D & A. All this depends on where the B-flat came from: probably a pure 3rd below open D? Or A pure 5th below F a pure 4th above C? And so on and so on..

Every note of every scale, is subject to this dichotomy; but as I suggested above, we should usually "build" with fifths and "enjoy" the thirds.

In one chamber-music workshop, (a Haydn quartet in C major), the instructor asked us to tune the E & A strings down one comma: the D-A 5th was horrible but C and G chords were lovely!

Semitones are often wider than people think: a B leading-note to C is very often a pure third above G; a melodic B may be much higher.

One other thing: in a context favoring pure thirds, G-sharp is usually lower than A-flat.

April 28, 2012 at 08:50 PM · Is this the case in all tunings? If you were tuning to A 412, 415,435, 444, 455 is the principal still the same? If the instrument resonates differently, would things still turn out?

Can you this while in scordatura? Say you are in A major and you bring your g and d up one tone, you should tune those strings in 5ths, and you tune your e string a perfect octave from the d string, dosn't that do enough?

Was this only done in the baroque? It seems to give the music an entirely different sort of voice; and give more artistic freedom! Has anyone ever done this with adagio from bach sonata no.1? How did it work out!

April 29, 2012 at 05:48 AM · Katrina, I confess I haven't used much scordatura, except for a brief flirtation with the viola d'amore, which is usually tuned to a chord of D minor - or D major! How to tune the F/F# string in a way that allows modulations? I had to examine the pieces carefully beforehand.

But the same constant adjustments to fingered notes apply to any tuning - and any tuning fork. Playing with gambas (adjustable frets) or lutes (fixed frets) complicates life even more..

The composer's style comes into play as well: Vivaldi loves to use the ringing of open strings, either bowed or "sympathethic"; Bach's complex and ever-changing motifs don't give so much time for listening to resonance.

Minor keys are also a problem; their tuning often derives from the relative major, but not always. And the final chord is often an open octave or fifth, or even a "Picardy" major chord.

In his 5th solo 'cello suite in C minor, Bach lowers the A-string to G: a more silvery resonance, and perhaps a remeniscence of the (preceding?) version for lute.

Try playing Vivaldi's (in)famous A minor concerto in G minor, without retuning: it looses its "bloom", and becomes almost plaintive. Now tune the violin down a tone but play normally: the lower tension lets the strings "shimmer", and the open strings and their harmonics ring out.

At least "baroque" performance practice makes us really listen!

April 30, 2012 at 05:19 PM · I am going to continue to do intonation experiments with my group to examine this issue. I am going to sound a note, say F, and keep it constant while the cellist plays first third (D) and then a fifth (B flat) and see whether my F (which I will not have changed) sounds good in each context. I will do this with various notes.

One thing I've seen in discussions about violin intonation is that people make a distintion between "melodic" and "harmonic" intonation. With my baroque training, I am not inclined to agree with this distinction. My training has taught me that every melodic note fits into a harmonic framework, and to my ears just intervals sound perfectly fine in a linear context as harmonically. The example of "melodic" intonation that is often used is that of the sharpened leading tone, but this is a modern notion originating with Casals. I don't see it having any application to earlier music.

April 30, 2012 at 06:48 PM · Michael, I quite agree about the dominance of the harmony, but there can be magic moments when the melody escapes to have its own logic,e.g. quick passing notes, échappés and cambiati etc. Or in Purcell's Fantasias where a rising B-natural can clash with a falling B-flat..

My favorite demonstration with students:

On the A-string, play B double-stopped with open D; then with open E: it has to rise by a (syntonic) comma.

Now tune C and C-sharp with E: very close together.

Try a B-flat against D: rather high.

Last, play the resulting chromatic scale, choosing one of the B's, and having an inspired guess to "invent" the D-sharp.

All quite wierd to my "tempered" ears, but a kind of expanded conciousness..

Pleasant dreams..

Adrian