That's most kind of you Mr Grew. You can read the work HERE

I hope you enjoy it!

Looks like LERVE to me!

My time of the month comes around in about... oh, a month's time, Ros.  Just wear tight-fitting mittens till then, ok?

BACH TEMPERAMENT 1

In this posting, I shall consider WHY keyboards need to be tempered. It will then be clear (in my next posting) how and why BACH tempered his own instrument in the way he did. The conclusion I shall then posit  will turn out to be rather different from the views of Bradley Lehman, and reasons will be given for this.

Why temper at all?
The reason why keyboards provided with only 12 notes to the octave need to be tempered is simple: there are not enough notes provided to permit all the intervals to remain 'pure'! This can be shown by simple maths:

The octave covers an interval comprised of 1200 cents;
The pure fifth contains 702 cents;
The pure fourth has 498 cents;
The pure major third has 386 cents;
The pure minor third covers 316 cents;
The Pythagorean tone has 204 cents.

If the octave were to contain 6 tones (which we require on a keyboard instrument) we find that 6 X 204 = 1224. This is greater than the octave (1200) by 24 cents (= the Pythagorean comma). It would therefore be necessary to compress the size of the tones, each by one-sixth of a comma.

Equal Temperament does this by reducing the size of each tone by one-sixth of a comma. More than that, each of its semitones is reduced in size by one-twelfth of a comma. In that way, Equal Temperament should properly be considered as ‘Regular 1/12th-comma Mean Tone Temperament’. But this gives one nasty drawback: with the exception of the octave, every single interval is impure. Some (especially the thirds) are particularly harsh and sour when compared with the euphony previously attained (which explains why early 20th-c music theoreticians like Kitson began to prohibit the doubling of major thirds in harmonic writing – something Bach did all the time with no ill effects – because this emphasised harshness was then unacceptable to their ears).

Tempering keyboards, then, has always been concerned with compressing six tones to fit into the octave. But problems arise in dealing with the other pure intervals, and the manner of accommodating those within an octave.

The major third is the most problematic (as proved by its difficulty even in Equal Temperament). The problem is that the ‘pure’ major third only has 386 cents. If a major-third sequence is played using only ‘pure’ intervals (i.e. C-E, E-G#, G#-B#) the last note (B#) lies horribly flat from the expected C. (This is because 386 x 3 = 1158; falling short of 1200 by 42 cents, which is nearly a quarter-tone.) These major thirds therefore have to be stretched (in Equal Temperament each must be opened by no less than 14 cents, which is more than half a Pythagorean comma in itself). The only way that tempering can address these mismatches is by taking the interval of the perfect fifth and very slightly narrowing it. This must be done in such a way that while it remains ‘acceptable’ to the ear, the slightly out-of-phase combination of the two pitches produces a slow beat (roughly 3 beats per second). That will equate approximately with a reduction of between 1 and 3 cents. When carried through an octave, these adjustments should result in a pure octave that contains 3rds, 4ths, 5ths and 6ths that are acceptable while not being perfect.

In comparison with Equal Temperament (‘Regular 1/12th-comma Mean Tone’), those other temperaments that will be more suitable to the music of Bach will have a tendency towards smaller major thirds, larger minor thirds, smaller major 6ths and larger minor 6ths. Such temperaments were created by people like Werkmeister, but usually imposed a limitation upon modulation and key usage.

Bach, however, seems to have provided his own Well Temperament for the express purpose of performing his Well Tempered Clavier – which is not surprising since it was the very first collection that employed (twice over indeed) a sequence of pieces covering all 24 major and minor keys for which the instrument had to be serviceable without retuning for each new piece.

In the next posting I shall examine the technicalities of Bach’s tuning.

(to be continued…)

Not to me it ain't Bryn... what's your problem? I found that very interesting Ena , I shall read it again tomorrow when I am more awake !!

A

Ena, I am highly interested and entertained by your postings; I will have to read them more closely.

(I know a few mathematical theorems relating to 12-note tuning that I will drag up hopefully at the weekend; also, I'm delighted to know of this recording, as I have a musicologist friend who has spent some time wondering about the existence of a non-equal tempered WTC recording).

Specifically so - and also a vast amount of acoustic data that spring directly from every twitch of Bach's quill!