Showing Enharmonic Notes
Users familiar with tuning
systems of the Renaissance and Baroque will know that what modern
terminology calls "enharmonic notes", for example Eb and D#, will be
tuned differently. If you have music that modulates far enough to
involve more than 12 separate pitch classes, you may wish to display
the location of notes that lie outside your selected scale (which is
limited to 12 notes).
This is implemented by creating a second temperament
that relates to the main one you want to use, but that defines, for
example, D# where the main one defines Eb. Two such complementary pairs
of temperaments
have been created with this use in mind for you to try. For Baroque
ensemble work there is a 1/6 Comma
(Eb - G#) temperament,
which defines the naturals, two flats, and three sharps. [An
interesting discussion of ensemble temperament, with some aural
exercises as well, is at http://music.cwru.edu/duffin/BaroqueTemp/XMT.intro.html
]. You would
select that as your main temperament, under Options | Temperament. Then under View | Secondary Temperament Lines
you find the same pop-up list of temperaments; now select 1/6 Comma extended. [If
any Secondary temperament is selected, View | Secondary Temperament Lines
will be checked. Select None to
revert to the normal, 12 note mode of display]. This temperament has
been defined with Db instead of C#, D# instead of Eb, Fb instead of E,
E# instead of F, Gb instead of F#, Ab instead of G#, and A# instead of
Bb. [We could also have changed the B to Cb, and the C to B# if we
needed those notes, too, for example. All told, you could have 24
different pitch classes, but that would be quite unusual for any
Baroque music].
You will then see the
placement of these extra notes
in the upper graph, delineated by thin black
lines. There you get a good image of the placement of all the
notes in the
extended scale. For better resolution you might wish to select View | Upper Graph Range Display... |
Single Octave, as has been done for these illustrations. The
right-hand graph above is taken from the other
included pair of temperaments: Mean
Tone (1/4 comma) (Eb - G#) - Mean
Tone (1/4 comma) extended. This is
strict 1/4 comma, which produces exactly pure major thirds. The
distribution of notes follows the same pairing as the 1/6 comma pair
just explained. [Fb was not included.] Strict meantone makes
a
very dramatic demonstration, as the upper graph easily shows the wide
spacing between "enharmonic" notes. It is totally appropriate to
Renaissance and early Baroque music. The major thirds are pure. Quite a
few harpsichords survive that have (or show evidence of having once
had) extra notes for some of these enharmonic equivalents; other
instruments (organs as well) are mentioned in treatises or other
documents. And the most
radical is the surviving 31-note/ octave harpsichord. Thirty-one fifths
of 1/4 comma meantone add up very closely to an octave (well, many
octaves).
The Cs have
been exactly aligned side-by-side on this pair
of graphs, so that you can get an accurate sense
of how the two temperaments compare in their note placements.
To elucidate, a list of all note lines in the upper
graph will be helpful. Ascending from C, next comes C#, then Db, D, D#,
Eb, E, Fb, E#, F, F#, Gb, G, G#, Ab, A, A#, Bb, B, and C.
Now, how will we see the secondary lines when
looking at the magnified trace in the
lower graph? Tuning Meister will add in a colored dotted line
corresponding to the
placement of the enharmonic equivalent of the note being tracked. For
instance, if you play a D#, where an Eb is defined in the main
temperament and D# is distinct and defined in the secondary temperament
[as in the 1/6 comma example you should be trying now] an olive-colored
dashed line will draw at about -20 cents in the lower graph, as D# is
20 cents lower than Eb in 1/6 comma meantone, and that is where your
note lies, on the added colored line, not on the center line, which is
where the Eb of the primary temperament resides. [The extra dashed line
will draw whether you intend
D# or Eb, it can't tell which you may play, only that you are playing
that semitone of the 12. You will know the difference by your
intention].
[These images were a quick fix, using a recording
that has no bearing on
either of these temperaments, but with them you can see the lines added
into the lower graph that mark the enharmonic possibilities. In the 1/6
comma image, the E shows also the position of Fb (about 20 cents higher
than E), and the Bb and F show A# and E#, both about 21 cents lower
than the main notes. In the 1/4 comma image, again Bb shows A# but here
it is about 41 cents lower than the Bb; and the F# indicates also the
Gb by the dashed dark red lines about 41 cents above the center of
F#.][plan here is to get someone to record a few scales in flat
and sharp keys, attempting to match 1/6 and 1/4 comma
scales, for a more useful illustration]
Now in this
1/6 temperament all the enharmonic equivalents happen to be the same 20
cents distant, some above, some below, but you might have a temperament
where those distances vary from note to note. The color of the added
dashed line of course relates to the particular semitone being traced,
just
like that same color in the trace defines the semitone. [If you are
perfectly in tune, right on the dashed line, that line might be a bit
difficult to see but there is very little chance of that!!!] In the 1/4
comma temperament, enharmonic equivalents are some 41 cents apart.
(just to add to the confusion, let me add in the
fact that the comma of 1/4 and the comma of 1/6 comma are not the
same comma! the former is 1/4 of the space between a Pythagorean third
and a pure 5/4 third; the other is the space between 12 fifths and an
octave. Equal temperament can also be thought of as 1/12 comma
meantone; so the fifths in 1/6 comma are tempered twice as much as in
equal temperament.) ( you can spend a lot of time studying about all of
this, but I don't recommend it - you will most likely go crazy.)
Back to the Index
