SAT - Learning aural tuning Part IV

[Jim]

Dear Associates and others interested:

In this 4th part of my series, we will see how to fit in the last two notes
of the F to F part of the Baldassin-Sanderson-Tremperment.

G#3 can be tuned down as a tempered 4th from C#4, or as a tempered 5th down
from D#4.  Since all 4ths are widened and all 5th are narrowed, it is
fairly easy to locate G#3 this way, however, it must also work as a M3rd
below C4 and fit between the G3 3rd and the A3 3rd.  It should also fit in
as the top interval of the sequence of 3 M6ths namely: F#3-D#4, G3-E4, and
G#3-F4.

D4 can be tuned as a 4th above A3, or as a 5th below A4, or as a 5th above
G3, or as M6th above F3, or as a M3rd above A#3.  Seldom have I found these
last two notes to not fit in where they should.  However, if you are tuning
a piano which has a rough scale break anywhere above the F3, there could be
problems. Just know that it is not your fault.

Here are some basic tests for the various intervals which have been used so
far:

4ths	The test for a tempered 4th involves the use of a note which may or
	may not have been tuned previously.  It is located 4 half steps or
 or     a M3rd below the bottom note of the 4th.  This trial note will
 5      also make a M6th with the upper note of the interval.  In order to
half    insure that the 4th is wide or expanded, the M6th should be about
steps   1 bps faster than the M3rd.  Remember that when tuning the bottom
	note of a 4th, you must tune flat of the zero beat location.  When
	you are tuning the upper note of a 4th, you must raise it above the
	zero beat location.

	When tuning a 4th, listen to the beat two octaves above the bottom
	note.

5ths	The test for a tempered 5th involves the use of a note in common to
	both of the notes of the interval.  A M6th below the bottom note
 or     (9 half steps) should always beat slightly faster than the M10th
 7      which this same note will form with the upper note of the 5th.
half    If the two test intervals were equal, the 5th would be without
steps   beats.  If you are trying to have perfect - zero beat 5ths, you can
	also use a test note a m3rd (3 half steps) above the bottom note of
	the 5th.  This same note will make a M3rd (4 half steps) below the
	upper note of the 5th.  This can be used when tuning the zero beat
	5ths as trial 5ths used to locate the B3.  Just remember that if
	you are tuning the bottom note of a 5th, you must tune it sharp of
	zero beat in order for the 5th to be tempered on the narrow side.
	If you are tuning the upper note of a 5th, it must be tuned flat of
	zero beat.  For tempered 5ths, always use the 6th-10th test as
	described above.  The m3rd-M3rd test of a 5th is good for pure 5ths
	but is in reality a test for the second set of coincident partials
	of the 5th interval and gives you a 5th that beats more than twice
	the normal speed of the the lowest coincident partials.  If you slow
	down that beat, the 5th will be too pure.

	When tuning a tempered 5th, listen to the beat at one octave above
	the upper note.

M3rds	Just remember that all M3rds are widened.  If you are tuning the
	bottom note of a M3rd, it must be tuned on the flat side.  If you
	are tuning the upper note of a M3rd, it must be tuned on the sharp
	side of zero beat.

	When tuning a tempered M3rd, listen to the beat two octaves above
	the upper note.

m3rds	Remember that all minor 3rds (3 half steps) are tuned on the narrow
	side.  If you are tuning the bottom note of a m3rd, it must be on
	the sharp side in order to make the interval narrow.  Likewise, if
	you are tuning the upper note of a m3rd, it must be on the flat
	side of zero beat.

	When tuning a tempered m3rd, listen to the beat two octaves and a
	major 3rd above the upper note, or two octaves and a 5th above the
	lower note.

M6ths	All M6ths are wide.  If you are tuning the upper note of a M6th, you
	must tune it sharp of zero beat.  If you are tuning the bottom note
	of a M6th, it must be tuned flat of zero beat.

	When tuning a M6th, listen to the beat one octave and a 5th above
	the upper note or two octaves and a M3rd above the bottom note.

3rd-10th Test	This is usually used to test the width of an octave.  If
		the M3rd below the lower note of an octave beats the same
		speed as that same note compared to the top note of the
		octave (thereby forming a 10th) the octave will be pure at
		the location of the 4th partial of the bottom note of the
		octave and the 2nd partial of the top note.  If the 10th
		beats faster than the 3rd, the octave is said to have a
		little additional stretch.  Most of the better aural tuners
		prefer from 1/3 to 1/2 bps difference.  This gives a little
		headstart on the necessary octave stretching which is needed
		as you tune up the scale later.  If this is not done, it is
		more difficult to balance out the single, double, and triple
		octaves.

		When all of the contiguous M3rds are completed in the BST
		temperament, you can play 3 parallel 10ths to check for
		evenness: A2-C#4, C#3-F4, and F3-A4.  This is an additional
		check on all of the 3rds progressing evenly.

M6th-M3rd test	Some call this the outside-inside test, where the M3rd is
		inside the the M6th interval.  An example is that the F3-D4
		M6th beats at about the same speed as the M3rd one step
		higher or G3-B3.  This test will point out any discrepancies
		in the exactness of the tempering.  This test can modulate
		up by half steps and still should sound equal.

The next series will show how to complete the rest of the BST two octave
temperament and will give some valuable contiguous intervals tests.

Jim Coleman, Sr.