A good tuning enhances whatever else you can do for a piano. A really good tuning can launch the player to a higher place…
I started out learning the Braid White F temperament, struggling around the circle of fifths with tests and practiced beat-speeds, hoping to arrive in the right place. This process, as anyone familiar with the method knows, involves blood, sweat, and tears. The more pianos I tuned with this system, the more determined I was to make it work or die trying.
But really, I was totally ready when Dr Al Sanderson offered an alternative one evening at our local PTG chapter meeting. The version I offer below was learned from Al in that hour-long presentation. The reprogramming necessary to change systems admittedly brought me some extra pain, but ultimately it was so worth it!
Set concert pitch. Tune down one octave and then a second octave. Fill in the double octave with contiguous (sharing a note) major thirds – there are six, the top three being an octave above the bottom three. If all major thirds speed up evenly as they ascend and all octaves and the double octave sound good, the resulting matrix provides a structure that will help to best compromise all your intervals.
Once this matrix is in place, tuning fourths and fifths from the central pair of contiguous major thirds creates a mini temperament of impeccable integrity. A fifth up from the lowest note and a fourth down from the highest note produces a major third one halfstep down from the upper major third. A fifth down from the highest note and a fourth up from the lowest note creates a major third one halfstep up from the lower major third. And finally, a fourth down from the newly tuned second-to-highest note and a fourth up from the newly tuned second-to-lowest note produce a central major third.
If from bottom to top, these five major thirds increase smoothly in beat speed, you have exactly what you need. And if they do not, the needed corrections are only a couple of steps away.
The fifths in this solution are virtually pure and the fourths noisy. And neither fifths nor fourths speed up. Pure fifths extrapolate to beatless twelfths as the tuning expands. Use pure fifths, first with major thirds and major sixths in the temperament octave, then with octaves for all remaining notes. By applying musical judgement to fifth and octave in each case, making both sound their best, not favoring one over the other, the notes tune themselves.
Testing for smoothly rising or falling beat speeds in chromatic major thirds, major sixths, and major tenths and for clean, good-sounding octaves, fifths, double octaves, and twelfths can validate or troubleshoot, but constant testing will not be needed.
Next, tune unisons as you go. First, adjust overall pitch as needed ahead of time. If flat or sharp, aim one third as far in the other direction (a Sanderson Accu-fork makes this calculation easy) and stretch a little extra the further you get from the middle. This will land you where your tuning doesn’t destabilize your tuning.
If tuning unison strings shifts a note’s pitch, which is better – to correct the error right away, while surrounding notes are valid, or to erode the tuning later, when time is running out?
More. Open unisons force better intervals, which force better unisons. And both improve stability.
One of our great challenges as tuners is self-honesty. Is that note still in its best place? Is its unison the best it can be? Can I face checking surrounding relationships or detuning strings to hear if they could be better? Unisons-as-you-go takes care of such uncertainties and develops skills for faster and better outcomes.
One last note. What happens if you get interrupted by the artist needing to rehearse (or some other unexpected circumstance) that means you must stop where you are? If you still have the piano strip-muted, you will leave behind an unmusical mess. With unisons tuned, at least what you've done is musical and the rest is what it was – no blame.
The intervals for both tuning and making music are embedded in the harmonic series. In the case of stretched wire, the overall speaking length vibrates as a whole (the fundamental), in two halves (an octave up), in three thirds (up a fifth), in four quarters (up a fourth), in fifths (a major third), in sixths (a minor third), in sevenths (a less than major second/more than minor second), and so on. Higher partials become less and less useful.
In mitigation, nature offers volume reduction: the second partial is half the volume of the fundamental, the third a third, etc. The reality may be slightly different and inconsistent, but your ear will hear what it hears, invoking a hierarchy of which pitches sound more and most important. Generally, interval importance goes from simple to less simple. So, consonance erodes down a slope from unison, through octave, fifth, fourth, major third, and minor third to greater dissonances beyond, with major sixth and minor sixth fitting in (in that order) and with compounds more or less matching their simpler forms (i.e., a major tenth matching a major third). Music needs and enjoys dissonance, but only in relation to consonance. Where the seventh partial takes us is out of our known musical universe. Backing down the series, we adapt intervals by stretch or contraction to musically fit.
As piano tuners, we know about stiffness of wire. Steel resists bending, curving into and away from changes of direction. A string’s fundamental is shortened by stiffness at agraffe and bridge, sounding slightly sharp to what its measured length would suggest. And each subdivision of the whole vibrates shorter than its literal length should, causing sharper and sharper results the higher up the series you go.
But inharmonicity is our friend. Because the human ear discriminates less well at very low and very high pitches, it feels more comfortable hearing highs a little higher and lows a little lower. Inharmonicity naturally stretches our tuning to accommodate.
This leads me back to using pure fifths. Fifths theoretically need to be contracted to fit a temperament. But it turns out that by resisting that theory, by tuning fifths aurally pure and extra-stretching fourths, both the mathematical needs of tempering and the needs of our musical inclinations benefit.
One detail about the naturally derived intervals can be helpful. Two pitches sounding together, one vibrating 4 times faster than a given pitch and one 5 times faster than the same pitch (i.e., the pitch's fourth and fifth partials), produce a major third. Three contiguous major thirds that are unstretched add up to an unattractively contracted octave, so they must be expanded and will beat quite fast when adding up to a musical octave.
A second major third built on top of the upper note of the first will beat approximately 5 times for every 4 times the lower third beats. Each of the contiguous major thirds that fill the double octave of our tuning matrix will beat approximately 5:4 faster than the one below and 4:5 slower than the one above. And approximately is fine since trial and error will find the real relationships.
To enhance your estimating powers, tap a beat with one hand and four beats with the other. Keeping the first beat the same, accelerate the faster beat to five times the first. Go back and forth to feel the significance of change. This will help you make the lowest third slow enough and the highest third fast enough in that first octave. One way or the other, though, as soon as you enter the second octave and add the fourth major third, you will be forced to fish until you get the first octave’s thirds right. And very quickly, you will become skilled at fishing.
A final word about test notes. To place a pitch at a precise number of vibrations per second (not just in its best place relative to another pitch), a test note helps. For concert A, the F two octaves and a major third below provides a readily audible fifth partial that will beat against both the pitch source and the string being tuned (and its unison strings). Tweak that F to produce an easily recognizable beat speed with the source A, then tune the strings to match that beat speed. Other test notes, though helpful in training, will slow down your tuning.
Order of Matrix and Mini Temperament Steps
- Tune A4 to PitchSource: Match pitch.
- Tune A3 from A4: Octave down.
- Tune A2 from A3 and A4: Octave down / 2 octaves down.
- Tune C#3 from A2: M3 up.
- Tune F3 from C#3 and A3: M3 up / M3 down.
- Tune C#4 from C#3 and A3: Octave up / M3 up.
- Tune F4 from F3 and C#4 and A4: Octave up / M3 up / M3 down.
- Tune C4 from F3: P5 up.
- Tune G#3 from C#4: P4 down.
- Tune F#3 from C#4: P5 down.
- Tune A#3 from F3: P4 up.
- Tune B3 from F#3: P4 up.
- Tune G3 from C4: P4 down.
Not in the mini-temperament. E4 gets tuned moving up from the mini-temperament using “pure” fifths and checking with ascending major thirds and minor sixths to complete the temperament octave. The position of E4 will then be challenged or validated by tuning down from the mini-temperament by “pure” fifth, noisy fourth, descending major third and minor sixth. The E’s (like the rest of the notes) are self-adjusting as there is really only one place for each that agrees with the rest. Thanks for your question, Whit. One of the beauties of this approach is how few notes it takes to tap the best array for a given piano.
Oh, so I guess there’s no E4 in this temperament?
Thanks for posting the procedure. I’m quite new to music notation, I bought an old verticalpiano and I’m working my way to the tuning. Electronically (entropy piano tuner) has gone smoothly, but for aural I still need to learn a lot. A clean algorithm, with enough checks along the way, was really what I was looking for. My main fear is getting correctly the 6 major thirds. I hope it will be feasible. I prepared a graphical synthesis of the procedure, you can find the image file here, hopefully it may help somebody else too: https://drive.google.com/file/d/1UDYh176LZPSZ56xHvYAop_BH3xobS5hd/view?usp=drivesdk
Chris, thanks for the well stated, elegant procedure. Thou many roads lead to Rome, the one less traveled – more revealing.
Great article, please let me know if can I find videos of it? thank you.
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