The Prime 5ths
The Prime 5hs are the lowermost and uppermost 5ths contained in the Prime Octave: A3/E4 & D4/A4
The prime 5ths are used to determine the best prime octave width for the particular piano being mapped and eventually tuned. When the widths of the prime 5ths are added together, when that total is -3.0, the prime octave width is about right.
Once a template has been found positioning both A3 and A4 that gives us a combined prime 5th width of -3.0 c., the 5ths can then be ‘balanced’ to achieve more correct beat rates in the prime octave. Balancing the prime 5ths, involves tweaking the shape of the prime octave ‘curve’ by slightly raising or lowering the mid point of the prime octave, while leaving A3 and A4 unmoved.
(Click on the chart for a larger view)
This chart has A3 @ 2.5 c. and A4 @ 10 c. The top most blue line is a straight line connecting A3 and A4. Note the locations of D#4 on the red line (5.8), the green line (5.3), and the black line (4.8).
This is what can be done using mapping, templates, and the LC spreadsheet. If the lower 5th is not as narrow as the upper 5th, we can balance the prime 5ths by lowering the D# in LC and adjust the prime octave curve to make the lower 5th more narrow and the upper 5th less narrow.
Balancing the Prime 5ths results in more correctly beating prime 5ths and 4ths. It also helps the 3rds progress more smoothly.
Raising or lowering the middle of the prime octave tuning curve, will make the prime octave curve either slightly deeper or slightly more shallow. Once the prime 5ths are about the same -1.5 c. width, the best tuning curve for the prime octave will have been found.
Since using D#4 is not practical to be used as a mapping note, D4 and E4 are used instead. D#4 can easily be placed between the settings of those D4 and E4.
For example, if the lower 5th is -1.0 (not narrow enough), and the upper 5th is -2.0 (too narrow), lowering both D4 and E4 by .5 will make the lower 5th more narrow and the upper 5th less narrow – both by .5 c., which will make them each -1.5 c. narrow and balanced.
Lowering the middle of the curve puts a little more ‘dip’ in the prime octave tuning curve. But that is what is needed for this particular piano.
(It’s easy to check this with whatever software you are using. Use the 6th/10th aural check to see if the prime 5ths are beating (mostly) the same, or better yet, you can measure them with a SAT).
Once both prime 5ths are tuned to this -1.5 c. width, not only will the 5ths sound good, but so will the octave, the resultant prime 4ths (A3/D4 and E4/A4), and the M3rds will also progress very smoothly.
(Note: Even though both of these prime 5ths are tuned to the same width, the upper one will still beat very-very, ever so slightly, faster than the lower one. That’s inHarmonicity. The same thing happens even when tuning the F3/A3 M3rd, the A3/C#4 M3rd, and the C#4/F4 M3rd each to an equal width of 13.0 c.. Even though those M3rds are tuned to the same ‘exact’ width, their beat rates gradually get faster. Again, that’s inHarmonicity.
Al Sanderson’s Two Octave Temperament, when tuned using Direct Interval tuning, starts by tuning the 5 lower M3rds all to the same width. The accelerating beat rates is due to iH, not to constantly wider (in terms of cents) intervals.)
The prime 4ths are really good aural check indicatorss for all of this, since they beat a little faster than the 5ths, which makes them easier to hear than the 5ths. The upper prime 4th, should be the fastest beating of all, but when everything is well done, that upper 4ths beating shouldn’t be objectionable at all.