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HPSCHD-L  June 2008

HPSCHD-L June 2008

Subject:

Re: two worlds of tuning

From:

Brad Lehman <[log in to unmask]>

Reply-To:

Harpsichords and Related Topics <[log in to unmask]>

Date:

Tue, 24 Jun 2008 13:27:39 -0400

Content-Type:

text/plain

Parts/Attachments:

Parts/Attachments

text/plain (92 lines)

Patrick Frye asked:
 > So---I began at the beginning and found myself here:
 > http://www.midicode.com/tunings/index.shtml
 > Does this seem a good site to research this aspect of the subject, as 
 > there is much to read and discover, I don't care to be misled.

Well...for starters, the analytical table of meantone makes the 
musically impossible choices of Db, Eb, F#, G#, and A# for the five 
accidentals.  (What, we're supposed to pick the notes so we have five 
wolves instead of one?  Db-G#, G#-Eb, F#-Db, A#-F, and Eb-A#?  And this 
selection of accidentals reduces our set of usable major triads from 
eight to only four!)

There is no section about modified meantone schemes (_temperament 
ordinaire_ strategies).

The section about "well temperament" is so superficial as to say almost 
nothing.  Furthermore, that section relies on almost nothing for its 
background, except a (good) medievalist article.

Better to start with the _New Grove_ "Temperaments" article than that 
web essay at midicode.com.

=====

Here's my own little summary of the different keyboard-temperament 
strategies, from this article (originally for _BBC Music Magazine_, 2006):
http://www-personal.umich.edu/~bpl/larips/art.html


Reference: Other keyboard tuning strategies, historically

Keyboard tuning methods fall historically into several competing styles, 
as to the size(s) of 5ths used to generate the notes. All of these 
styles have been brought back into use in the 20th and 21st centuries, 
for reasons that are variously historical, aesthetic, or practical.

     * Pythagorean (before the 17th century): Eleven pure 5ths or 4ths 
are easily tuned in succession. The resulting major 3rds are very wide. 
The cycle of 5ths does not meet itself exactly; a dissonance results 
between the twelfth note and the first note.

     * Just intonation (15th century forward): All the notes are related 
to one another variously by pure 5ths or pure major 3rds. The available 
intervals and chords are outstandingly resonant within the single home 
key, but strong dissonances arise in other keys (thereby restricting 
modulations). Melodies are bumpy, as the steps within the scale are of 
vastly different sizes.

     * Equal temperament (16th century forward, but not a universal 
"standard" until the 20th): The same amount of tempering is given to 
each of the eleven 5ths, so the beginning and end meet with a twelfth 
5th of the same size. All twelve notes are equally spaced, and all 
scales have the same character. The tempering of the 5th is so slight 
that it is difficult to control with precise equality, in practice.

     * Quasi-equal (18th century forward): These variously subtle 
methods make a neutral effect similar to equal temperament's, without a 
strongly recognizable character to any key. Some of these are easier to 
tune than equal temperament, as they have several pure 5ths spaced 
symmetrically among their tempered 5ths.

     * Meantone or "regular" systems (16th-19th centuries): Each tone 
(whole step) is placed at an exact mean (geometric average) position 
within the major 3rd, whatever its size. For example, C-D and D-E are 
equally spaced within C-E. All of these meantone systems are generated 
by tuning eleven identical (regular) 5ths of some selected size, until 
there are twelve different notes. There is a leftover gap, or "wolf" 
diminished 6th, from the twelfth back to the first note; this rift is 
usually placed at G#-Eb, D#-Bb, or C#-Ab. Notes such as Db, A#, and E# 
usually do not exist in this scheme; and they sound rough if they occur 
in the music. The most common meantone system had a strongly tempered 
5th so that the usable major 3rds worked out to be pure; but from the 
early 17th century forward this was gradually relaxed toward slightly 
sharper major 3rds and gentler 5ths, as practical compromises.

     * Modified meantone, or "circulating" or "irregular" or "ordinary" 
(17th-19th centuries): The series of 5ths is regular in the midsection 
(on the natural notes of the C major scale, ...C-G-D-A-E...), but 
increasingly wide toward the outsides. This sharpens the sharps and/or 
flattens the flats gradually, so they can serve passably as one another, 
and it reduces or eliminates the "wolf" intervals. Some of these schemes 
also have a flattened F or a raised B, as transition into the flats or 
sharps.

     * Split keys (15th century forward): Extra key-levers are added 
within the octave, e.g., having two separate keys to play G# and Ab. 
Such keyboards are usually tuned in a meantone style, taking advantage 
of excellent major 3rds.


Brad Lehman

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