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
  ]. 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].

Secondary Temperaments 1/4 and 1/6

    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.)

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