Aural Pure 5ths Equal Temperament Part II

[Jim]

Hi to all:

This is the second installment. Thanks for your comments. I'll
try to include any suggestions or corrections in the article
before it comes out in the Journal in August.

               PURE 5ths Temperament, Aurally, Part II

Tune A4 to A-440 Tuning Fork
   Test F2-A4 with F2 and the Tuning fork for equality of beat.
   Test D3-A4 with D3 and the Tuning fork for equality of beat.
   Test G1-A4 with G1 and the Tuning fork for equality of beat.
Tune D4 pure to A4
   Test F3-D4 6th equals F3-A4 10th.
Tune G3 pure to D4
   Test G3-A# min 3rd equals A#3-D4 Maj 3rd.
Tune A3 to D4 at 1.5 beats wide
   Check that A3 - A4 octave is almost 1.5 beats wide
   Check that F3-D4 6th is 1.5 beats faster that F3-A3 3rd.
Tune F3 from A3 at 7 beats wide temporarily.
Tune F4 from F3 at almost 1.5 beats wide
   Check that C#3-F4 10th is 1.5 beats faster than C#3-F3 3rd.
Tune C#4 from F4 wide so that the A-C#4 3rd is half way between
   the F-A 3rd and the C#4-F4 3rd. If the C#4-F 3rd is not
   halfway between the A3-C#4 3rd and the F4-A4 3rd, then the
   two Fs must both be adjusted, but they must maintain their
   almost 1.5 beat octave spread.
Tune A#3 from F4 as a pure 5th
   Test C#3-A# 6th equals C#3-F4 10th
   Check that F3 4th is only slightly faster than F3-F4 octave.
   Check that A#-D4 3rd fits with A-C#4 3rd.
Tune F#3 from C#4 as pure 5th
   Test A2-F# 6th equals A2-C#4 10th
   Check that F# 3rd fits with F3 3rd.  If it doesn't then the
   speed of the 4ths and octaves need to be changed or else the
   C#4 was not balanced properly between A3 and F4.
Tune C4 as pure 5th from F3
   Test G#2-F3 6th equals G#2-C4 10th.
   Check that C4-F4 4th is not faster than F3-F4 octave.
Tune E4 as pure 5th from A3
   Test C3-A3 6th equals C3-E4 10th.
   Check that C-E4 3rd fits with C#-F4 3rd
   Check that E4-A4 4th is not much over 1.5 beats per second.
   Check that G-E 6th is slightly slower than C4-E4 3rd.
Tune B3 between F#3 and E4 as balanced 4ths, each being 1.5 beats
   Check that G-B 3rd fits between F-A and A-C#
   Check that G-B is slower than G-E4 6th by about 1.5 beats.
Tune D#4 to A#3 as a 4th that is as wide as its tuned neighbors
   Test F#3-D#4 6th to be 1.5 beats faster than F#3-A#3.
   Check that B-D# 3rd fits with its neighbor 3rds A# and C4.
   Check that F#3 6th fits with its neighbors F3 and G3 6ths.
Tune G#3 as a pure 5th from D#4
   Test B2-G#3 6th equals B2-D#4 10th.
   Check that the G# 4th fits with its neighbors
   Check that G# 3rd fits with its neighbors.
   Check evenness of 4 min 3rds: F3, G#, B3, D4, F

This completes the F-F temperament, but should be filled out in
both directions, tuning F#4 as a pure 5th to B3 and checking
to see that the octaves are stretched as much as the near 4ths.
Continue with G4, G# in similar fashion.

Tune down from B3 to E3 as a pure 5th maintaining equal
6th 10ths as above and balancing with descending 3rds and 10ths.
Another test for pure 5ths which works is the complementary
minor and Major 3rds like E3-G3 equals G3-B3. This works better
when tuning pure 5ths. Continue downward in similar fashion to
the bass break.

As one continues above A4, eventually balance the 5ths and octave
5ths checking all along to keep the 10ths moving up evenly. The
octaves will be as wide as you can tolerate them and eventually
utilizing double-octave-5ths balancing them with the single
octave 5ths. By the time you arrive at C6, you should have
pure triple octaves. 17ths will be faster than 10ths and 3rds
giving you wide double-octaves and wider single top octaves.
In the top octave, the double-octave-5ths are easier to gauge
than the fast speed of the 17ths to maintain evenness.

You may be amazed at how evenly your 10th and 17th turn out because
the pure octave-5ths really zero in with greater accuracy than most
of us achieve with tuning by the 17ths up high in the scale.

Next time I'll show a very simple method for achieving this amount
of stretch using the Sanderson Accutuner.

Jim Coleman, Sr.