A good tuning enhances whatever else is done for a piano.
I started out with a Braid White F temperament, struggling around the circle of fifths with tests and practiced beat-speeds - a process of blood, sweat, and tears... I took the challenge very seriously, though, and ended up going to NBSS in Boston MA, suffering further under Bill Garlick's very literal approach. The more pianos I tuned with this torturous system, the more entrenched I became in the quest for its mastery and the more married to making it work.
Then, I came across what I thought was a radical, perhaps heretical, insight: that fourths in the temperament area defied the speeding up logic of ascending thirds and sixths and wanted to all beat at the same speed (with different speeds for differently scaled pianos). Shortly thereafter, I attended a Boston Chapter PTG technical featuring Dr. Al Sanderson and he noted, as an aside in his presentation, the same observation. My attention was peaked. He also took a classroom survey and to my astonishment, more than half those present confessed to not knowing what pitches they were listening to when they tuned. I was now opened to considering a different approach.
Dr. Sanderson suggested a temperament very much like the one below - this was my interpretation, anyway, of what he said, as I (painfully) re-taught myself to tune - one of the best things I ever did for myself as a piano technician.
The tuning-unisons-as-you-go part of what follows was a European approach that I learned from Bob Glazebrook of Steinway London. Thinking how great it would be to ply my new trade in England, my wife being English, I went to New York for a job interview – but without my tools! No problem, Bob said, I could use his. And these consisted of the smallest tuning hammer I had ever seen (handle maybe 5" long) and two mutes. Apologies Bill, but I learned more in those 45 minutes, than a year of your tests and exercises. (Bill later provided me with a not dissimilar kick in the pants for concert preparation, but that's a different story.)
Prepare overall string tension to be at pitch (no sun on strings or other pitch-shifting influences, if possible).
The matrix grid - two 8ves of contiguous major thirds, indexed to concert A4:
1 A4 (440-5) - I use a Sanderson Accu-fork to accurately set the pitch
2 A3 to A4 (I do all of the grid and mini-temperament by ear, with no tests or beat-counting)
3 A2 to A3 (new 8ve and double 8ve sound good - will naturally be refined later)
4, 5 C#3 to A2 and F3 to C#3 (major thirds beat approximately 4:5 lower to upper at the 5:4 coincident partials) – just a little faster one to the other (don’t try to count it out) – with F3 to A3, there are now three contiguous (share a note between them) major thirds that need to increase in speed within the lower 8ve A2-A3
6, 7 C#4 to A3 and F4 to C#4 (if these thirds, good 8ves above their already-tuned counterparts below, along with F4-A4, all increase in speed as you ascend and the double 8ve sounds good, you're close as dammit, plus the major tenths they form will also speed up 4:5 each, lower to higher)
4:5 contiguous major third beat-speed relationship note:
The marvelous near-math subdivisions of vibrating wire has the whole string length producing the fundamental pitch, two half string lengths producing the second partial at (approximately) half the length but twice the speed of the whole, three third string lengths producing three times the frequency (a twelfth above the fundamental), four fourth string lengths at four times (double octave), five fifth string lengths at five times (two octaves and a major third), etc until the string is too stiff to continue. And stiffness at both bearing points and nodes, of course, makes all these divisions and frequencies respectively shorter and faster, i.e., produces what we call inharmonicity.
But the point of this excursion into details (and BTW, all these speeds and subdivisions are happening simultaneously) is to note partial 4 and partial 5, which beat at 4X and 5X the fundamental, respectively and produce a major third between them. And because of inharmonicity, application of this interval in equal temperament has it beating at clearly discernable speeds in the temperament area. And ascending major thirds beat faster as their frequencies get higher – a valuable asset to the tuner.
The second point of this note is the value of trial and error. The exact math of the complexities produced by these characteristics lives in a land of crazy numbers that still only serve as approximations. But – by just tweaking the two middle notes in their 8ve, i.e. C#3 and F3 between A2 and A3 and C#4 and F4 between A3 and A4 and linking them all together, you virtually cannot get all six of the thirds produced to ascend through both 8ves without being extremely close to the exact answer, numbers or no! This is a powerful manipulation of math and physics made possible not by numbers, but by the strategy of trial and error. In fact, whatever our source of pitches to tune, the physically getting strings into place and stable depends on us being willing to go out on the trial-and-error limb with our tuning lever, muscle strength, ears, brain, and muscle memory to achieve this 235 string place (arguably impossible to achieve) every time in well under two hours!!! So…
The mini-temperament - F3 to C#4 (fifths and fourths in easiest-to-hear pitch range fill in missing notes):
8, 9 C4 to F3 and G#3 to C#4 produce G#3(Ab3)-C4 (check ascending thirds) and
10, 11 F#3 to C#4 and A#3(Bb3) to F3 produce F#3-A#3 (check ascending thirds)
12 B3 to F#3 and G3 to C4 produce G3-B3 (completing 5 chromatic major thirds, which should speed up evenly as they ascend - actual speeds will be determined by the piano’s string scale)
The characteristics so far:
8ves sound stable - generally pure at the 12:6 coincidentals
Double 8ve the same at the 24:6 coincidentals
Contiguous major thirds relate 4:5 in beat-speeds lower to higher at the 5:4 coincidentals
Contiguous major tenths do the same at the 10:4 coincidentals
Fifths are virtually pure throughout this range (and the rest of the piano…)
Chromatic fourths equal each other in beat speed through this range
Tune all unisons and verify the mini-temperament (you're never more than 2 steps away from a mistake!):
13, 14, 15 Add chromatic notes up to F4 (major sixths should increase in speed as they ascend) - tune unisons
16, 17, 18 Add chromatic notes down to C#3 (don't test with minor sixths) - tune unisons
19, 20, 21 Add chromatic notes down to A2 (grid matrix is now complete) - tune unisons
Note that all intervals are self-correcting, particularly when heard with open tuned unisons:
Proceed down to A0 chromatically (major tenths, fifths, and 8ves suffice) - tune unisons as you go
Proceed up to first treble break (fifths, major sixths, 8ves, and major tenths) - tune unisons
Complete the treble (fifths, 8ves, double 8ves) - tune unisons as you go
Play and tweak unisons, trouble-shoot anything that music dislikes...
The strength of this non-specific-beat-speed system is that a piano's individual specifics produce its beat-speeds for given intervals, i.e., they may be different from those of same model and scale pianos. If you establish and fill in a piano's naturally-produced grid with intervals that just aurally fit in and maintain the characteristics listed above, you'll naturally produce the best equal-tempered solution for that piano. And musically, the resultant integrity turns out to be really useful, and beautiful.
Also, this system has no trouble adapting to short or long scales, or to crossing breaks, including moving from plain wire to wound or vice versa. For two-piano tunings, the grids should be tweaked to match - a series of compromises that requires good sense. Particularly, the mini-temperament notes should match when played chromatically together. Then, tune each piano to its own best compromises. From C4 up, most scales are very similar and, generally, the lower ends of the pianos are less likely to be played in unison and are generally forgiven for being a little growly when they are.
Tuning unisons as-you-go fine tunes the tuning of your intervals. And your interval tuning, in turn, fine tunes the tuning of your unisons. If you are interrupted, say by the performer arriving early needing to practice, or by something very noisy taking place, or by a blackout, etc, the core of the piano at least will be in good tune and stable. Tuning unisons later, on the other hand, will twist or dislodge what you worked hard to get right, just when you should be finishing up.
Tuning unisons as-you-go also forces you to be flexible and willing to bring back in what has been changed, and forces you to recognize and admit what has been changed, and forces you to move on quickly and keep going. This is true with aural tuning or ETD tuning. Aural tuning, however, has the advantage of following a drift in pitch due to atmospheric changes. Also, you don't have to fiddle with knobs, buttons, foot switches, or moving extra things around. But whatever your preference, tune unisons as you go.
Postscript on the best time to tune:
I advise my customers to call me when the piano sounds its best. As this odd piece of advice sinks in, they realize I’m telling a joke, however earnestly and sincerely rendered. They are unlikely, of course, to ever call me for a tuning when the piano was sounding its best, so in the end we schedule an arbitrary time that is likely to have temperate weather.
But it is good advice: the closer to in-tune the piano is when you come to tune it, the less you have to move strings around, and the more stable you will leave it when you go (and the faster your best tuning will be accomplished).
So, tune the piano in the same conditions each time, preferably in temperate (comfortable) conditions, and sell the customer a Dampp-chaser to keep pulling it back into tune. Grands should get Undercovers, as well, and sometimes string covers, too – but do not undo an Undercover for the Dampp-chaser service until after the tuning.
An upshot of employing this "best-interest" down-sell – you will likely get hired to do other, also-needed work on the piano, then filling intemperate days of the year with well-paid work!