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Text File | 1990-12-28 | 9KB | 111 lines

                          ┌┬────────────────────┬┐
                          ││ FM-Tune: Tempering ││
                          └┴────────────────────┴┘

  Function key usage                                         Mouse

   F1     - Display this help information.                  [Help]
   F3     - Exit program.                                   [Exit]
   F4     - Save tempering information to disk.             [Save]
   F5     - Retune notes using current tempering data.      [Retune]
   F6     - Return to note frequencies screen.              [Freq]
   F9     - Decrease value of current field.                <left button>
   F10    - Increase value of current field.                <right button>
   Home   - Move to first screen cell.
   Alt-F3 - Push to DOS.                                    [Push]

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 │ Background │
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 Temperament refers to how the spacing between steps in a scale is adjusted
 to achieve optimal relationships in certain larger intervals (ex: thirds,
 fifths, etc.), generally to improve the accuracy of selected chords.  For
 this discussion, only 12-step scales are considered.

 Most common is the equal or even tempered scale, to which most fixed-scale
 instruments are tuned.  Each step is based on a multiplier of 1.05946 (the
 12th root of 2).  If A is 440 Hz, then A# will be 440 x 1.05946, or
 466 Hz.  Continuing to multiply each successive frequency by the same factor
 will result in a frequency of 880 Hz after the 12th multiplication, or
 exactly twice the original -- one octave higher.

 A pleasing consonance is achieved when an interval such as a fifth, for
 example, A to E, approaches the theoretical "pure fifth", defined as a
 frequency ratio of 3:2 (E=660, A=440).  A pure major third would have a
 ratio of 5:4, a perfect fourth, 4:3, and a major sixth, 5:3.  These are
 among the most consonant intervals, having ratios based on small whole
 numbers.  Others may have less consonance, as indicated by greater beating,
 or low frequency pulsations, when the interval is played.

 The equally tempered scale approaches pure fifths and fourths, at the
 expense of inaccurate thirds.  If you've never noticed, listen to a third in
 equal temperament, and then the same third in the same key in "just"
 temperament, which is based on pure thirds, fourths and fifths.  The
 disadvantage of other temperaments is that transposition is generally not
 possible except by octaves.  This is due to unequal step sizes.  In fact,
 notes which are normally considered the same (enharmonic), such as C# and
 Db, may actually have different frequencies.  In general, flats will be
 higher, and sharps lower than their even tempered equivalents.  So selecting
 the right one for tuning a fixed scale instrument depends on the intended
 musical use.

 ┌───────────┐
 │ Operation │
 └───────────┘

 A set of predefined temperaments is listed on the left side of the screen.
 They include equal temperament, and a number of common alternatives.  Space
 for additional user-defined temperaments is provided.  To select one, move
 to it with the arrow keys and press return twice (or double click if using a
 mouse).  An arrow head () will indicate the current selection.

 Step data for the selected temperament appears on the right.  The "Cents"
 column indicates the relative position in the scale of each step in cents
 (100th of an equally tempered semitone).  For equal temperament, these
 increase in exact increments of 100 cents.  The "Delta" column shows the
 difference from the equally tempered step in cents.

 [Note: Just as semitones are logarithmic increments (each larger than the
 last), so are cents.  An increase of 1 cent is the same as multiplying the
 frequency by the 100th root of 1.05946 (or 1.0005778).]

 Once a temperament is selected and any changes to the steps are made, the
 instrument tuning can be updated by pressing the F5 (Retune) function.  This
 may take several seconds.  Be sure to check the "Base" note at the top of
 the screen prior to retuning.  This is the note on which the scale will be
 based, and is critical in non-equal tempered scales which are generated for
 playing in a specific key.  The default is the last note selected on the
 Note Frequency screen.  All other note frequencies will be computed relative
 to the base note frequency (which may be set to any value prior to
 retuning).

 For example, to generate a "just" tempered tuning for the key of C:

   - Move to the "Just" temperament cell and select it.
   - Make sure some C value is indicated as the "Base" note.
   - Press F5 (Retune).
   - Return to the Note Frequency screen (F6) to view the generated values.

 As another example, to generate an equally tempered tuning using a reference
 frequency of A3 = 442 Hz (instead of 440 Hz):

   - On the frequencies screen, select A3 and type in 442.
   - Switch to the tempering screen.
   - Select "Equal" temperament.
   - Press F5 (Retune).

 ┌────────┐
 │ Saving │
 └────────┘

 There is a single data file (temper.dat) which contains information on each
 of the 18 available temperaments.  If you make any changes and press F4
 (Save), they will be stored in this file for future use.  The temperament
 data is loaded automatically each time the program is started.  Be careful
 not to confuse the save option here with the Note Frequencies save option;
 that saves the whole frequency table.  This save function only replaces the
 temperaments information file.  The file temper.sav is a copy of the
 original temper.dat file should it be necessary to restore it.

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