Mapping A3 and the width of the prime octave.
Like setting the temperament, finding the best position for A3 is the basis for the tuning.
A good way to determine the width of the prime octave is by using the combined widths of the prime 5ths (A3/E4 & D4/A4). The prime octave width in terms of both 4:2 and 2:1 will vary from piano. However the widths of the prime 5ths, are quite consistent and can therefore be used to find the width of the prime octave.
The prime octave needs to be wide enough for the 5ths to sound good, but not so wide as to create excessive beating in the prime 4ths, or in the prime octave itself.
The tuning will sound better when the prime 5ths are ‘balanced’. Balanced simply means the prime 5ths are the same width.
My suggested width for the prime 5ths is -1.5 c. (1.5 c. narrow).
Being able to map A3 using the prime 5ths requires a template with a consistent curve between A3 and A4. As long as there are no partial changes within that A3/A4 octave, almost any ‘template’ can be used.
An FAC tuning or other tuning curve can be used to tune A3, D4, E4, and A4. Once those notes are tuned, the prime 5ths can be measured.
The number we’re looking for is -3.0. When the combined widths of the two prime 5ths equals -3.0, the placement of A3 is about right. If the width of the lower prime 5th is -1.3 and the width of the upper prime 5th is -1.7, since -1.3 + -1.7 = -3.0, the width of the prime octave is good.
Ideally, each of the prime 5ths, are -1.5 c. narrow, but for now all we need to know is their combined width.
If the combined widths of the two prime 5ths equals -2.0, the octave isn’t narrow enough. It’s too wide – A3 is too low by about 1.0 c.
If the combined widths of the two prime 5ths equals -4.0, the octave is too narrow – A3 is too high by about 1.0 c.
Now that we know the prime octave is too wide or too narrow, and by about how much, it is easy to select a different template with a higher or lower A3 setting.
When using my templates, sometimes there isn’t a template with a low enough A3 setting. When that happens DOB (Double Octave Beat) can be used to widen the octave. This works really well, and maintains the integrity of the curve of the prime octave.
Each .1 DOB lowers A3 by .2 cents. So, if the template with the lowest A3 setting has A3 @ 1.0, but A3 needs to be .4, a DOB setting of .3 will lower A3 by .6 cents and place A3 @ .4 c. instead of 1.0.
That’s really all there is to mapping A3 using templates. Using the prime 5ths and a template in this way will result in a nice sounding prime octave for that piano.
The A3/A4 2:1 should always be wide.
And much of the time the prime 4:2 will also be wide.
But there are times when the A3/A4 4:2 will end up being pure, and on occasion, it needs to be narrow!
Here are some guidelines for what to do based on the width of the A3/A4 2:1 when tuned as a pure 4:2:
1. If, after tuning the prime octave as a pure 4:2, the A3/A4 2:1 width is between 0.0 c. and .8 c. wide, A3 will almost always need to be lowered, creating a wide A3/A4 4:2 and a wider 2:1.
2. If, after tuning the prime octave as a pure 4:2, the A3/A4 2:1 width is between 0.8 & 2.5 c. wide, the pure 4:2 may be a good width for the prime octave and a good place to start mapping A3.
Most of the time, on most pianos, the pure 4:2 is a good place to start mapping A3.
Measuring the prime 5ths (A3/E4, D4/A4) will let us know if any adjustment needs to be made to the width of the A3/A4 octave:
As stated above, my suggested width for each of the prime 5ths, is -1.5. If the combined widths of the two prime 5ths, is -3.0 (-1.5 + -1.5 = -3.0), A3 is probably well located.
The combined width of the prime 5ths, is directly related to the width of the prime octave.
If the combined width of the 5ths, is -3.5, the octave is .5 too narrow. Simply lowering A3 by .5 and then re-tuning and re-measuring the 5ths, will give the octave a good width that will accommodate 2 good sounding 5hs.
But of course there are exceptions. Inharmonicity has it’s way of making everything in piano tuning open for exceptions.
The 4th P of A3 and the 2nd P of A4 are used for tuning the A3/A4 4:2. If either of those partials are unusual and out of the ordinary, due to scaling, string length, or tension, or whatever, challenging anomalies can occur.
That’s why tuning a pure A3/A4 4:2 and then measuring it as a 2:1 is so helpful and revealing.
If A4’s iH causes the 2nd P of A4 to be unusually low or unusually high, the width of the A3/A4 2:1 will be effected.
Or, if A3’s iH causes the 4nd P of A3 to be unusually low or unusually high, the width of the A3/A4 2:1 will be effected.
And the width of the prime octave is critical for fitting in 2 good sounding 5ths as well as having a good sounding A3/A4 octave – in terms of both the 4:2 and the 2:1.
For example, I tuned a Yamaha U1E about 25 years old on which the 2nd P of A4 was only .7 c.. The first time I measured it I got .5 c.! That’s quite low for the 2nd P location of A4 when A4 is tuned to A440.
So when tuning A3 as a pure 4:2 using the .7 c. setting for the 2nd P of A4, and then measuring the width of the pure 4:2 as a 2:1, the 2:1 was 1.8 c. wide. That 1.8 c. wide was not all that bad by itself, when it comes to a fairly decent sounding prime octave (pure 4:2. 1.8 c. wide 2:1). But when the prime 5ths were tuned using a template that provided a pure 4:2 relationship for the prime octave, the 5th were each about -2.0 c. (narrow).
In order for me to achieve 2 prime 5ths each -1.5 c., I would have had to widen the prime octave by about 1.0 c. That would have resulted in a pair of nice 5ths, but the prime 2:1 would have been around -2.8 c. wide.
I may have been able to live with a -2.8 c. wide 2:1 but instead, I decided to compromise with a pair of slightly more narrow 5ths. Instead of -1.5, I ended up with one 5th @ -1.8 and the other @ -1.7. That little bit of widening of the 5ths, (from -2., to -1.7 and -1.8) did make them sound a tiny bit better, and the slighly wider 2:1 octave (now 2.2 c.) was not objectionable either.
Another consideration here involves the tuning of A5. Since the 2nd P of A4 is so low (.7), lowering A3 down too much, might make for a too wide A3/A5 4:1 when the A4/A5 2:1 is tuned wide enough to accommodate 2 good sounding 5ths in the 5th octave (A4/E5, D5/A5).
Both the A4/A5 2:1 and the A3/A5 4:1 need to be wide, but if A3 is lowered too much when mapping A3 and the prime 5ths, when finding a good sounding location for A5 in terms of the A4/A5 2:1 and the 5th octave upper and lower 5ths, the A3/A5 4:1 may be too wide and start beating objectionably.
So the compromise of having 2 slightly more narrow prime 5ths kept all the other relationships in tact.
A similar approach can be used when the iH of A3 might be higher than usual, which happens more often than the above example where it’s the low iH on the 2nd P of A4.
High iH @ A3 is actually quite common on many of the short scales and American made consoles and spinets from the 60’s – 80’s. The scaling on A3 can often produce a high iH @ A3.
On those pianos, after tuning the pure 4:2, the 2:1 can be quite wide but this time it’s because of A3’s iH rather than A4’s.
Those pianos are exposed when measuring the width of the 2:1
What is really interesting on those pianos, is that most of the time, the 5ths just won’t let me contract the octave much at all. Contracting the octave of course also contracts the 5ths. If I contract the octave to clean up the beating in the 2:1, the 5ths can quickly get too narrow and start beating too much.
Here’s how that seems to work. Since we are working with the combined widths of the 5ths, and using them as our guide to finding the best width for the prime octave, if we contract the prime 5ths, by .1 c. each, (-1.6 c. each 5th) our target number for the combined total would be -2.8 c., which would give us a .2 c. more narrow prime octave.
Believe it or not, sometimes just that .2 cent reduction in the prime octave makes that 2:1 beating more tolerable.
Carrying this a step farther, we can also contract the prime 5ths by a total of .2 c. each. That results in 2 prime 5ths, about -1.7 c. each and shrink the 2:1 by .4 c. That .4 c. difference is easily audible in the prime octave’s 2:1 beating.
I believe in maintaining what I call the hierarchy of the beating of the intervals. I still think the octave should beat slower than the 5ths, and the 5ths should beat slower than the 4ths, and so on. Or going the other way, the 4ths should beat faster than he 5ths, and the 5ths should beat faster than the octave. That’s exactly what is being done here. The .4 c. difference in the 2:1 is a result of only .2 c. in each of the prime 5ths is a good compromise in that the 5ths, are only slightly moe narrow, but I’m able to shrink the octave by .4 c.
Balancing the prime 5ths involves moving the mid point of the prime octave up or down.
When the prime 5ths are ‘balanced’, they are the same width. I think that if the prime 5ths, are with .1 or .2 c. or each other, further ‘balancing’ isn’t really audibly necessary. But if one of the prime 5ths, is -2 and the other -1., an audible difference can be heard, and it will mostly be heard in their resultant 4ths.
During mapping of the prime octave, the first interval I check is the upper 4th (E4/A4). I just do a quick listen to that E4/A4 4th and it lets me know if I’m close or not.
Let me say here that accurately tuning and accurately measuring the prime octave and 5ths, is not easy. Of course some pianos are easier than others, but some can put us thru our accurate tuning and our accurate measuring paces.
The Accutuner is a very accurate device, and it hears it. So if things are moving around on us, the Accutuner will let us know.
There have been pianos, that I thought I’d never get good numbers and good sounding intervals in there. But keeping at it has always paid off. I often end up trying every string of each of the unison to before I find the best one. Eventually I’ve gotten numbers I could use, and I always felt it was worth the extra time spent. I think it makes me a better tuner.
Balancing the prime 5ths is possible using the LC spreadsheet.
When mapping the prime octave, the prime 5ths are measured and the settings for D4 and E4 that give us a pair of ‘balanced’ 5ths are written down and eventually entered into the spreadsheet. The prime octave’s mid point is D#4, but when creating the tuning using the LC spreadsheet, the settings for D4 and E4 are used to find the position of D#4.