Robert Conrad, Registered Piano Tuner/Technician, Tucson, AZ

# Mapping the Prime Octave (A3/A4) with a SAT “980” & Templates

This post assumes a SAT IV “980” with Conrad Templates installed.

1: Measure the A4 numbers:

Tune A4 to A440 (SAT: A4 @ 0.0)

Measure the locations of A4’s 2nd, 4th, & 8th Partials.

Tune A3 as a pure A3/A4 4:2
(Tune A3 to A4’s 2nd Partial.)

Measure the A3/A4 4:2 as a 2:1
Set the SAT to listen to A4. Play A3, Stop the lights.   That’s the width of the A3/A4 2:1)

2.  Select a template:

Go to the page in the SAT that contains a template with
a.  the 4th P. number of A4 for A4, and
b.  the 2nd P. number of A4 for A3.

The templates are organized by A4 numbers from smallest to largest – top to bottom – Pg. 141:  A4 = 4.2,  Pg. 980:  A4 = 18.0.

Scroll thru the template headers to the templates with the correct A4 numbers and then further refine the template selection by selecting a template with an A3 number that matches (or is as close to) the 2nd P. number of A4 as measured above.

(The SAT’s Page numbers are shown in the leftmost column.)

3.  Using the selected template, tune A3, D4, E4, (and A4)

4.  Using Direct Interval Tuning, measure the widths of both Prime 5ths.

A.Measure the width of the A3/E4 5th.   (Lower prime 5th)
Set SAT to E5, play A3 and stop the lights using the cents buttons.  Zero out the display by storing that number as an offset – Press Blue Shift/RST.   Now, play E4, and use cents buttons to stop the lights.  That number is the width of the A3/E4 5th.   It will be a negative number because 5ths are narrow.  Make note of or just remember that number.

B. Measure the width of the D4/A4 5th.  (Upper prime 5th)
Set SAT to A5, play D4, and stop the lights using the cents buttons.  Zero out the display by storing that number as an offset – Press Blue Shift/RST.   Now, play A4, and use cents buttons to stop the lights.  That number is the width of the D4/A4 5th.   It too will be a negative number because 5ths are narrow.  Make note of or just remember that number.

C. Add the widths of the two prime 5ths together.

The combined width of the two prime 5ths should be -3.0 c.

If they equal -3.0 c. the template’s prime octave width is about right for the piano being tuned.
If the combined widths of the 5ths is NOT  -3.0 c., a different template will need to be selected.

If the combined width is -4.0, the A3/A4 octave is too narrow by about 1.0 c.
A template with a lower A3 (but with the same A4) will expand the octave and make the combined widths of the 5ths less narrow.

If the combined width is -2.0, the A3/A4 octave is not narrow enough by about 1.0 c.
A template with a higher A3 (but with the same A4) will contract the octave and make the combined widths of the 5ths more narrow.

5. If the combined widths of the prime 5ths is too wide or too narrow, select a different template with a higher or lower A3 number.

If the combined width is -3.5, select a template with a .5 lower A3.   If the combined width is -2.5, select a template with a .5 higher A3.

Again, the target width for the combined widths of the prime 5ths is -3.0 c.  Continue to work with the templates and the tuning and re-tuning of A3,D4 and E4 until the combined width of the prime 5ths is -3.0 c.

This routine does require good tuning and measuring techniques.   But with practice, consistent repeatable results can be found.

{If there is not a template with a low enough A3 number, use DOB and the template with the lowest A3 to lower A3 down to the correct setting.   Templates are not used for tuning. They are only used for mapping – finding the settings for the target notes.  Once the target notes are mapped, the templates have served their purpose.   The mapped settings are used by the LC Spreadsheet to generate the tuning.}

6. Using the 2nd selected template, re-tune A3, D4, and E4 (and check A4).

A. Measure the A3/E4 5th and the D4/A4 5th as before.

B. Add the widths of the two prime 5ths together.

If the combined width equals -3.0 c.,  the template contains the correct width for the prime octave.

If the combined width of the prime 5ths still doesn’t equal -3.0, select a different template with a different A3 number, and try again.

(Continuing repeating until the combined width of the prime 5ths equals -3.0 c.)

C. Once the combined widths of the prime 5ths is -3.0, the width of each prime 5ths should be close to -1.5 c.
Chances are they will be different widths: one slightly too narrow and the other not quite narrow enough.

6.  Balancing the Prime 5ths:

A. The prime 5ths are balanced by raising or lowering the mid point (D#4) in the prime octave.

B.  Balancing the Prime 5ths will establish the location of D#4.

The correct location for the mid point of the prime octave (D#4) is found by working with the 2 prime 5ths (A3/E4 & D4/A4).   Once the locations of D4 and E4 are known, it’s easy to find D#4.   This correct location of D#4 will be incorporated into the tuning using the LC spreadsheet.

Once the octave is the right width, D#4 can be adjusted up or down in the LC spreadsheet, and will adjust both 5ths to the correct balanced width at the same time.

C. Each prime 5th should be -1.5 c. (narrow).
If the lower 5th (A3/E4) is -2.0 narrow, it’s too narrow by .5 c..
If the upper 5th (D4/A4) is -1.0 c. narrow,it’s not narrow enough by .5 c..

For example: if the lower 5th is -2.0 and the upper 5th is -1.0 c., raising the midpoint of the prime octave by .5 c. will make the lower 5th less narrow, and the upper 5th more narrow.

Raising both D4 and E4 by .5 c. adjusts both 5ths at the same time in the right direction – making one more narrow and the other less narrow, by the same amount!

To confirm and hear to how this sounds, simply re-tune D4 and E4 to the new settings.

To do this, just look at the template’s setting for D4 and E4 and do a little math.  Use the cents buttons to add .5 c. to the setting for D4 (2.8) , and then tune D4 to that 3.3 c. setting.    3.3 c. (2.8 + .5)

Do the same with the template’s setting for E4.   E4 is 3.9 c. on the template, use the cents buttons to make it 4.4 c. (3.9 + .5) and then re-tune E4 to that 4.4 c. setting.

Listening to the octave, 4ths, and 5ths, should confirm these new settings for the target notes of the prime octave.  If each prime 5ths is -1.5 c., the octave, 4ths, and 5hs should sound pretty good.

Make sure to make note of the new setting for both D4 and E4.   They will be needed by the LC Spreadsheet so they can be incorporated into the tuning.

D. Use these completed mapped settings to re-tune A3, D4 , E4, and A4.

Listen to the octave, the prime 5ths, and the resultant 4ths, and use whatever aural checks necessary to confirm the prime octaves mapping.   Everything should be sounding pretty good.   If it doesn’t recheck all the measurements and the tuning of each note.   But that’s easy at this point, since the settings for A3, D4, E4, and A4 are all known.

## A3 and the prime octave is now mapped.   The hardest part of the mapping process for the entire piano is complete.

With the LC spreadsheet, the tweak for the mid point adjustment of the A3/A4 octave will be made on D#4.

Adjusting the position of D#4 up or down reshapes the curve of A3/A4 octave to accurately fit the tuning curve to the piano’s scaling in the prime octave.

But since there are really no workable or usable tuning checks or measurements for tuning the A4/D#4 tri-tone, we use the prime 5ths to locate the mid point of the prime octave for each piano.

The shape of the ‘curve’ in the prime octave differs from piano to piano.   Some pianos need more of a ‘dip’ and some pianos need less of a dip to sound their best.

The best curve for the prime octave will result in 2 well balanced Prime 5ths.

Some pianos call for a fairly rapid fall from A4 down to D#4, and then level out.  Others are just the opposite, having a fairly quick rise from A3 up to E4 and then leveling out from E3 to A4.

This is 100% piano dependent, and it’s why we go to all the trouble of measuring and balancing the prime 5ths.

Here’s an easy aural check for you to try with your current tuning software or tuning system.   Just tune A3, D4, E4, and A4 to whatever program you’re using at the moment.   Then listen to the prime 4ths and 5ths.

If the prime 4ths and 5ths beat the same or almost exactly the same, the program tuning you’re using is a good match for the prime octave on that piano.  The more pianos you check this, the more you will hear it.   But more likely than not, uneven beating will often be heard.

### (Click image to enlarge)

Above is a screenshot of the midrange view of the piano in the LC Spreadsheet.

The note names at the very top, the A’s are in Red, and the other notes are in Black.

The numbers below those note names are the settings for the notes.

In the graph, the bars represent the ‘differences’ in cents between each note.
(The numbers below each bar are the cents difference).

The RED bars are the A’s.

The big black numbers in the boxes I’ve outlined in red are the D, D#, and E settings.

The red outlined box on the left is for the sub-prime octave showing the settings for D3 =- 0.9, D#3 = -0.6, and E3 = -0.3.
The red outlined box on the right is for the prime octave showing the settings for D4 = 3.3, D#4 = 3.8, and E4 =4.4.

Underneath each of those 3 note settings are two big blue arrows, one pointed up and the other pointed down.   The blue arrows are used to adjust D#4 up or down.