Just to add to Ena's excellent post, people may like to listen to Tuesday's 'Essay' through listen again.  Marcus de Sautoy talked through the maths of why the harmonic series is so hard for modulations....

Tommo

A bear of very little brain writes:
Not all of us fully understand it even now


How many notes would there have to be, then, to permit all the intervals to remain 'pure'?

I thought the problem was that, even if you had that number of notes (whatever it is), then thirds and fifths wouldn't be among them. And that is the root of the problem. And that's why the problem emerges on stringed instruments (with as many notes as you can shake a stick at (ho, ho)). It's more a more fundamental fact about pitch relationships within a series of octaves rather than just not having enough notes available, as a matter of practical engineering fact, on a keyboard instrument. Not so?

[Sorry: This needn't delay the arrival of the meat course.]

But wouldn't you then need to split some of the white notes as well? Otherwise you won't have a B# to give you a nice major third with your G#...

and then wouldn't you...

er...

http://en.wikipedia.org/wiki/Archicembalo

Staying in tune when singing a cappella is certainly a challenge, which varies in degree according to the music. I confess that I can't understand why F major should be any more difficult than any other key.

I once heard a rather complicated disquisition from Rogers Covey-Crump on this matter. It seems that when the music modulates, the right singers have to adjust the crucial intervals so that Pythagorean tuning is maintained but the pitch does not drop. In Palestrina's music, the modulations are relatively straighforward and pitch drops don't generally occur. But in some other music of that period, special care must be taken at critical moments. Late Gesualdo is probably the ultimate test.

I rather think that the idea that there is some magical difference between C sharp major and D flat major was a fantasy of Mussorgsky's  - augenmusik, if you like.

This would be an interesting experiment: teach a choir a piece in F. Then one day send all members with absolute pitch out of the room, give the choir an E and pretend it's an F. Then let them sing the piece and see if the tuning is better.

F-major poses problems for violinists, too.  Much passagework in the key lies most comfortably in 2nd position, which is notoriously unstable.  Also, if a violinist has learned to tune in perfect fifths, the open E-string will be much too high.  This paradoxically forces many players to place the F too low, as many ("modern") violin teachers still adhere to the notion that leading tones ought always to be high and manage to indoctrinate their students with this rubbish.

(For anyone wondering why it's rubbish, consider that the leading tone is actually the third in the dominant chord.  Any tuning system that is trying to aim for pure-ish thirds will place that note low.  This failure to think/hear harmonically rather than linearly is a common violinistic failing, I'm afraid.)

(I'm not saying that high leading tones can't ever happen.  They can be squeezed high to very good emotional effect.  But to promulgate that as the default, within conventional harmonic language, is just wrong.)

So, to get back to F-major, violinists usually are wrong from the first note, placing their Fs too low.  A fifth (either pure or tempered) up to the C, and that's correspondingly is too low.  The too-bright Es will prevent the overall pitch from dropping, but it usually ends up sounding fairly dodgy.

What a minefield.   

Favourite key on the piano , sorry CD 

A

BACH TEMPERAMENT 2

Bradley Lehman’s interpretation of Bach’s scheme is explained inter alia HERE, and it is also worthwhile and rewarding to watch and hear him temper a harpsichord according to his tuning rationale (which can be done HERE). His method demonstrates clearly how consonant intervals are then carefully tempered, and this procedure would be followed by any such tuning scheme.

The only question remaining is this: does Lehman’s scheme successfully capture the sound world of Bach’s 48 Preludes and Fugues in the way the composer would have expected? While his interpretation is certainly a workable and interesting one, I do not believe it does fulfil Bach’s wishes. This is because I do not think he follows the exact method that Bach has indicated. Let us see why not.

The tuning procedure Bach specified appears on the title page as follows:



Lehman notes the appearance of the letter ‘c’ (next to the penultimate squiggle). Since one might expect the note C to occur at the beginning rather than the end of a tuning cycle, Lehman therefore assumes the diagram has to be read in reverse:



and that (despite being written backwards) the C is now placed as the second character. In order to set up a viable tuning sequence of 5ths and 4ths, Lehman therefore (in placing C as the second note) comes up with the following tuning order:



However, even before having reversed the diagram, Lehman edited out some vital parts of the graphic, thereby inadvertently confiscating (with the best will in the world) crucial evidence that bears upon its understanding. The complete graphic should have appeared as follows:



The reinsertion of the title Das Wohltemperirte Clavier shows not only that it applies as much to the tuning scheme under which it is placed as to the collection of pieces it goes on to describe, but also provides further direct indications about the tuning method to be adopted (which is different from Lehman’s).

Looking at this restored graphic, further evidence is clear. First the ‘C’ (near the end) is actually a decoration applied to the C of the word “Clavier”; second, another vital decoration is applied the D of “Das” – in this case an ascender pointing to the 4th squiggle, together with the letter “E” which is placed as an internal decoration to the D of “Das”; third, the very first squiggle looks very clearly (to me anyway) like a capital G. If these observations are valid (as I believe they are), the obvious tuning pattern that emerges is as follows:



This would mean that instead of beginning the tuning on F (Lehman), it should commence upon G as the axial point. This seems to be confirmed by another decoration that appears at the foot of the title page:



which seems to indicate a G- (or treble-) clef around which the lines and spaces of the stave (together with their pitches) describe a circular pattern drawn with some care and artistry.

I interpret the final squiggle in the tuning graphic to be an ornamental abbreviation for “ETC” (implying “and so on”). It could also signify the symbol “8ve” indicating that from this point the remaining notes are simply tuned in octaves with those that have already been set up.

Now it must be clear that the correspondences between ‘key’ and ‘affect’ will be quite different in a system begun on F as compared with one begun on G. Whatever ‘affect’ Bach had in mind for the sound of (e.g.) C major will in the former case now happen to Bb major. If, say, Bach intended the key of Bb minor to sound ‘sad’ or ‘plaintive’ (as I believe he did in Book 1), this will now match not with Bb minor, but instead with G# minor. So the point of origin and arrival is crucial in the tuning cycle.

Looking back to Bach’s diagram…



…it can be seen that (as recognised by Lehman) most of the large circles have internal ‘kinks’ (specifically the first 3 and the last 5) that indicate the notes to be tempered slightly flat. Since there are 8 of them, each must involve an adjustment of 1/8th of the comma to be lost (i.e. each pitch indicated must be lowered by 3 Cents in order for the 24-Cent mismatch mentioned in my last main posting to be addressed, since 8 X 3 = 24). The three remaining loops (without the squiggles) are to be left as ‘pure’ intervals. Now this represents the greatest difference between Lehman’s scheme and the one I am hereby proposing. Let’s look at the two:

LEHMAN


ME


In Lehman’s scheme (which works backwards through the graphic) the following intervals (+ their inversions) remain pure: Bb/F, E/B, B/F# and F#/C#.

In my scheme, however, the ‘pure’ ones are different: C/G, E/B, B/F# and F#/C#. Here the ‘pure’ intervals are arrived at earlier in the cycle since I read the graphic the right way round instead of back to front. It also means that C major (rather than F) has the ‘pure’ fifth.

Additionally, my scheme (I believe) explains why the last 5 tempered pitches appear with double loops inside the circles: they have to apply to pitches arrived at by opposite deductions, represented as C#/Db, G#/Ab, D#/Eb, A#/Bb, E#/F and B#/C (now brought together as common and interchangeable pitches).

But the internal properties of the semitones will also differ, not only between different keys, but also between the two systems just outlined. In my system, the following gives a complete breakdown of the relative values (in Cents) of all intervals. (Equally-tempered equivalents would need rounding up or down to the nearest 100.)



This is presented like some distance charts found in road altases – and allows any two pitches to be measured exactly by following the two appropriate co-ordinates. But in order to gain a quick view of the ways in which each chromatic scale differs from the others, I have compiled the following chart with the starting key-note indicated, and the relative differences of each interval-step for each key can easily be seen and compared:



The relative pitch-contents of the major keys can also be usefully compared:



Some of the leading notes (notably those in A, B, C, D, E and G) are much sharper than others (as in ‘black-note’ keys, and F). Similar differences of nuance can be observed in the pitches of major thirds – some of which are sharper even than their ET equivalents.

A similar comparison of minor-key layouts also provides some interest, especially with regard to the pitches of the minor thirds. C# Minor is quite unusual in two ways: the minor third C#-E is extremely narrow (as much as 6 Cents lower even than its ET equivalent), and the leading note also lies as much as 5 Cents lower also than the ET counterpart. This perhaps gives a rather stern or melancholic feel (perhaps as one might expect from the slow Prelude found in Book 1):



I do not think my own analysis (inasmuch as it can be thought ‘right’) in any way diminishes Lehman’s views. They differ, and it is up to each individual to decide which is the more convincing. This can only really be accomplished by actually listening to performances of both tunings and coming to a conclusion. Unfortunately that is the one thing I am unable to provide for this forum.