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In the very lowest Bass, especially on smaller pianos, you may begin to hear a faint resonance when tuning a single octave. This is actually the result of a large difference in inharmonicity between partials beyond the 8th partial of the note being tuned and the note an octave above it. Oddly enough, even though it is actually a rapid beat, it does not sound objectionable but rather pleasing to the ear. That is why I called it a resonance rather than an octave with a rapid beat in it, even though technically, that is what it is. Believe me, there is no such thing as a “pure” octave on a piano!

You may hear this starting on or about F1. Now is the only time I would ever listen to any RBIs when tuning the octaves. Typically, I would find the note an octave above the note being tuned and play the M10 interval and listen to whatever rapid beat there is. Now, when I tune that low octave, I cause that octave to sound in tune but also cause that resonance to mimic the rapid beating of the M10 that I just played. You can also go to the temperament octave and play a M3.

For example, you are tuning F1. When you make a reasonable sounding octave with F2 above it, you hear that resonance. Play F3-A3 and then "copy and paste" that rapid beat to the resonance of F1-F2 octave you are tuning. If you do this, you will get a different resonance for each of the low Bass octaves you are tuning because the RBIs above them are each a different speed. If you were tuning ET, you would want that resonance to be evenly digressive but you are not tuning ET! So, a slightly different character in each of the low Bass octaves is correct, expected, intended and deliberate.

This will cause the low Bass to enhance the sound of the entire rest of the piano. When you finish tuning the low Bass and have all of your unisons tuned, play the long, C Major arpeggio from the bottom to the top while pressing the pedal and you will hear and witness the splendid, "pipe organ" effect. After striking all the keys, listen to the sound that emanates from the piano. It will sound to your amazement just like a pipe organ. This is because you have used the canceling effect of equal beating to suppress the “noise” of ET and inharmonicity. It never fails to impress any customer that the piano is now supremely in tune with itself. I have never heard anyone who denied that it really does sound like a pipe organ.

six pure 5ths and six fifths tempered twice as much as they are in ET. That means that the tempered fifth are each 4 cents narrow. If I compare an octave with a pure fifth and make both sound "pure", the octave won't be a very wide one; probably a 4:2 type, at least in the midrange. However, if I make an octave beat equally with a 4 cent narrow 5th, it will definitely have a beat in it, probably a 6:3 type at a minimum. That much stretch in an octave is still tolerable for modern piano tuning.

Bill and David,

Thanks again. Bill. After reading the article and trying some experiments, I want to ask you both--what tests would you suggest for extending the temperament of Werckmeister III? My thought is just that a focus on balancing octaves and M12's will throw things off, since the 5ths vary a bit. Or am I getting lost?

Bill, how far up and down from the temperament octave (assumed to be F3-F4) do you apply this equal beating 5th and 8th scheme? It is an interesting scheme, applicable to all "late baroque" 1/6 or less ' tunings and I want to do a theoretical analysis of how a temperament "morphs" across the piano compass with this method.

Kees

Tuning from E3 to C3 (or lowest Tenor note)
After completing the F3 to F4 temperament octave, begin tuning the rest of the low tenor starting with E3. First, tune a reasonable sounding octave from E4, then compare E3 with the P4 and P5 above it and adjust E3 so that the octave still sounds reasonable but the P4 and P5 beat exactly the same or as nearly to that as possible. You can cause the P5 to be slightly less tempered sounding than the P4 but not at the expense of creating an obvious beat in the octave or the P4. The important thing is to have all three, octave, P4 and P5 sound reasonable.

You do not need to check any Rapidly Beating Intervals (RBI). The M3s, M6s and minor thirds (m3) will all sound uneven if played chromatically. No RBI test that would be necessary in ET would ever be valid when tuning the EBVT. Just as a 17th or 18th Century tuner who would not have known those tests, you do not need to use them at all, just skip that entirely. Whatever happens to the RBIs does not matter.

Continue likewise for D#3 and D3. At C#3, the F#3-C#4 P5 is beatless in the temperament octave, so when you tune C#3 to C#4 as a reasonable sounding octave, you should find that the C#3-F#3 P4 also sounds beatless. You may flatten C#3 just enough to slightly improve the C#3-G#3 P5 but not at the expense of creating an obvious beat in the octave nor the P4. The P4 need not remain perfectly beatless but it should also not have an obviously tempered sound.

Tuning C3 is similar. The F3-C4 P5 is beatless in the temperament octave. Therefore when you tune C3 from C4 as a reasonable sounding octave, you should find that that the C3-F3 P4 is also beatless. Similar to tuning C#3, you can slightly flatten C3 to slightly improve the C3-G3 P5 but again, not at the expense of creating an obvious beat in the octave nor the P4. The P4 does not need to remain perfectly pure but it should also not have an obviously tempered sound.

I consider A0 to B2 to be the Bass but depending on how large or small piano you are tuning, tune whichever notes remain in the low tenor below the F3-F4 temperament octave similarly. Seven and nine foot concert grands will often go down to F2, some even lower. Whatever remains in the low tenor, tune that as I described above, then before tuning the strings on the Bass bridge, move to the first note above the temperament octave, F#4. I suggest leaving the Bass notes for last, not first.

Tuning the "Killer" octave

The region just above the temperament octave is often called the "Killer" octave because of its difficulty in making compromises both in tuning and voicing. I think of the "Killer" octave as being F4 to F5 but some may think of it as an octave and a half or as much as two full octaves.

Begin with F#4 similarly to the way you tuned down from the temperament octave. When you tune F#4 from F#3 first as a reasonable sounding octave, the C#4-F#4 P4 should sound beatless but the B3-F#4 P5 will sound tempered. You may sharpen F#4 slightly so that the B3-F#4 P5 is slightly improved but not at the expense of creating an obviously wide octave or an obviously tempered P4. The P4 need not remain perfectly beatless but it should also not be obviously tempered.

When you tune G4 from G3 as a reasonable sounding octave, both the P4 and P5 below it will sound tempered. Now, you will definitely want to improve the P5 as much as possible by sharpening G4 slightly. Check to see that the octave still sounds reasonable but it can be allowed to have a slight beat in it. I do not specify an exact size of octave but if you do this properly, the size of the octave would be in the 6:3 range. There will be more about that later. Naturally, the speed of the P4 will be active and you don't want it to be excessive but it is not nearly as important as the sound of the P5. So, favor slightly the sound of the P5 over the P4 while trying to make the octave and the P5 beat equally (probably less than one beat per second for each). The P5 will still be tempered but what you want is a tempered sound that is barely perceptible. If you then play the C3-G4 octave-fifth, that will sound quite good, close to beatless, perhaps apparently beatless to the ear.

Tuning G#4 is easy. A reasonable sounding octave with G#3 will create a beatless C#3-G#4 P5. Leave it just like that; beatless octave, beatless P5th. The P4 doesn't matter but it will sound just fine.

If you tuned A4 initially and tuned A3 from it as a 6:3 octave, it will already be in place. However, if you started from A3, it is alright. If you tuned A3 from an A440 pitch source, the A4 will either end up right on pitch or mere tenths of a cent from it. When you tune A4, do that similarly to the way you tune G4. The result will be a slightly improved P5 below it, a slightly faster P4 below it and the octave will inevitably be a wider 6:3 type if you care to test it but you need not do that. If you followed the instructions literally, A4 will already be where it needs to be.

Tuning A#4 is also easy. Tuning a reasonable sounding octave from A#3 will produce a P4 and P5 below it which also sound reasonable. You may very slightly sharpen A#4 so that the D#3-A#4 P5th sounds virtually beatless but the F4-A#4 P4 is only slightly tempered.

Tuning B4 is similar to tuning G4 and A4. Favor the P5 below it slightly over the P4. The E3-B4 octave and P5th will sound virtually beatless.

When you tune C5 from C4 as a reasonable sounding octave, the F4-C5 P5 will be beatless. Leave it that way. The G4-C5 P4 will be tempered but not excessively. The C3-C5 double octave will also sound virtually or perfectly beatless. Play a C Major arpeggio from C3 to C5 , hold it with the pedal and you will hear a beautiful chord that largely or entirely suppresses all of the RBIs within it. It will sound as though you have tuned that chord in 1/4 or 1/5 comma Meantone but you have not. It only sounds like you have. The equal beating RBIs cancel themselves from your perception.

Tuning C#5 is easy. When you tune a reasonable sounding octave from C#4, the F#4-C#4 P5 will sound virtually beatless. Since you may have sharpened F#4 slightly, simply tune the F#4-C#5 P5th as beatless and check to see that the C#4-C#5 octave still sounds reasonable. The G#4-C#4 P4 may beat but certainly not excessively. This is the range where any perception of beating of P4’s begins to fade.

Tuning D5 is always the most difficult compromise to make. The G3-D4 P5 in the temperament octave is tempered more than twice as much as in ET ( a little more than 4 cents narrow) in all three versions of the EBVT. You have already sharpened G4 slightly to improve the C4-G4 P5. So, now to create a G4-D5 P5 that does not offend the ear, you must compromise the D4-D5 octave. When tuning electronically, I have found that it most often means adding one full cent over what the default stretch would provide. That is not an excessive amount.

It will cause the D4-D5 octave to have about one beat per second in it. Played as an isolated octave, an ET tuner may find that objectionable but remember, you are not tuning ET! You are not taking the tuning exam! What is most important is that the G4-D5 P5 not sound too narrow and have an obvious beat in it. As a practical matter, you could consider that the octave is one beat per second wide and the P5 is one beat per second narrow. What happens to the P4 below D5 does not matter. This is high enough in the scale that the sound of the P4 as previously mentioned, is beginning to fade from perception. The D3-D5 double octave may also be slightly wide. Isolated octaves and double octaves rarely, if ever occur in actual music in the midrange. They may occur higher lower but not here. Therefore, what is in between the octave and double octave is more important.

The audible width of the octave and double octave are not nearly as important as creating a P5 that does not offend the ear. Virtually anyone but especially a string player would not want to hear an obvious beat in the G4-D5 P5 but they wouldn't care at all that the octave and double octave are slightly wide. You can also consider that the width of the D4-D5 octave is about the same as would be created when tuning the “ET with pure fifths” idea. So, some ET tuners would, in fact have about the same width in this octave at this point, so it is not extreme.

By now you must have seen that each of these octaves tuned so far is of a different size (width). That is, in fact, correct and it is in fact, intended. No ETD program, at least as far as I know, can do this. If you want to use an ETD calculated program, you can do so but what you will want to do is examine each of these combinations of octaves, P4’s and P5’s. You will want to sharpen some notes slightly and flatten others slightly. When you find that a note needs to be sharpened or flattened, either do that by ear, find the pitch with the ETD and enter the new value in the program or estimate how much flatter or sharper the note should be with the ETD, tune the note to that pitch and check it aurally. If it now seems correct, enter that value in the program or make another estimate until you are satisfied, then enter it in the program. The amount of change to each note would be in the range of 0.5 cents to 1.5 cents. I cannot imagine needing to change any note any more than 2.0 cents at the very most.

Continue with D#5 which will be easy. Tune a reasonable sounding octave and check with the P5 below it. If the P5 sounds too tempered, sharpen D#5 slightly so that it is improved but the octave still sounds reasonable. At this point in the scale, the P4 below the note being tuned is completely irrelevant. Only check to make sure the double octave is not overly wide. The octave-fifth should sound quite good.

Tune E5 in the same manner as you have been. Pay special attention to the A4-E5 P5 and make sure it does not beat objectionably.

Now, before proceeding to F5, play all of the octaves you have tuned beginning at F#4 and play all of the fifths from F#4 to E5. They will not have an entirely consistent sound; that is not the goal. You only want to be especially sure that none of the fifths in this area beat objectionably. Bear in mind that a fifth may be improved not only by sharpening the top note of the interval but also by lowering the bottom note. Sometimes the perfect solution is to do both, each by a very small amount. If you were tuning ET, you would be looking for complete consistency but you are not tuning ET! You are tuning the EBVT or EBVT III with Tempered Octaves.

Notice that I have said nothing about the RBIs. You do not need to check any of them. You also do not need to use any octave tests. If you did use octave tests, you would find that each octave varied in size, one to the next. That is correct, that is expected and that is intended.

The "Killer" octave was difficult but the entire rest of the piano is so easy to do as to be "mindless".

Tuning "Mindless" Octaves

Beginning at F5 and using the sostenuto pedal, play and hold the F3-F5 double octave. (If the piano has no sostenuto pedal or it doesn't work, use the damper pedal the same way you would use the sostenuto pedal. Play the notes first and then press the pedal. If you are tuning a vertical piano and using muting strips, the strip mute will hold open the treble dampers, so you do not need to use a pedal at all.)

Temporarily tune the F3-F5 double octave beatless. Now play and hold the A#3-F5 octave-fifth. You should hear a beat in the octave-fifth. Sharpen F5 slightly until both the double octave and the octave-fifth have virtually the same quality. Neither interval will have much of any audible beat. They will both sound apparently or very nearly beatless.

You can continue this very same procedure to C8. Some double octave and octave and 5th combinations will beat very slightly more than others because the octave sizes you have created below them are all of different sizes but none will beat very obviously.

If you desire more stretch in the high treble, you can choose to start favoring the octave-fifth at or about F6. That will mean a wider double octave but in the high treble, that is often found to be acceptable. You can delay such favoring until C6 or higher is you choose. The limitation you may find is in how wide of a single octave that will create in that area of the piano. If the single octave sounds objectionably wide to you, you may wish to keep tuning equal beating double octaves and octave-fifths until such a point in the scale that some slight beating in a single octave no longer sounds objectionable.

You can also do this with an ETD with virtually the same results as you get aurally. This is what I highly recommend. At F5, the ETD program will probably read F5 on its second partial. Change the partial selection to the first partial (F5 read on F5). Play F3 and stop the pattern. Now play A#3 and the pattern will roll or otherwise displays somewhat sharp. Adjust the pattern to the point where it displays equally flat when you play F3 as it does sharp when you play A#3. Whatever value that is, enter it into the program and tune F5 to that.

You may continue that same procedure all the way to C8. However, if you desire more stretch in the high treble, you may begin favoring the octave-fifth beginning on or about F6. You can simply play the note which is an octave-fifth below the note to be tuned, stop the pattern, enter it and tune the note to whatever value is found. For an even sharper high treble, you can use the note which is a double octave-fifth below the note to be tuned. Play that note, stop the pattern, enter that value and tune to it.
Typically, at F6, I tune the octave-fifth beatless. In programming the ETD, I simply play A#4, stop the pattern, enter that value and tune to it. I do that until B6. At C7, I use the note three octaves below and tune to that (triple octave).. At F7 to C8, I use the double octave-fifth, sometimes the triple octave, sometimes the triple octave-fifth.

In making any of these choices, you are tuning the piano to or making a compromise based upon the actual inharmonicity which the piano has. If you choose to tune beatless octave-fifths or pure double octave-fifths beginning on or about F6, you are still using the piano's actual inharmonicity to determine a pitch for the note to be tuned, not a calculation. However, anytime you favor one interval over another, the interval you have disfavored becomes compromised. In the high treble, that essentially means that you will have audible beats in the single octaves. It is up to you to determine how far you can or should go with that depending on the circumstances.

A maximally stretched high treble will sound very bright and is often appealing to many or most people. To others, however, those kind of single octaves "scream" with dissonance. In any case, the amount of beating in a single octave is never very extreme, even with maximum stretch. Remember that sustain in the high treble is short (that is why there are no dampers). The actual musical context played in the high treble is not the kind of harmony played in the midrange. Therefore, a brilliant high treble that projects well is often a good choice in many circumstances. You simply need to use your professional judgment or respond to the suggestions or complaints of the person or people for whom you tune. The complaint may well be that the high treble sounds "flat" and if so, you will know what to do about that, regardless of your own opinion.

Tuning the Bass

Beginning with B2 or whichever is the highest note on the Bass bridge, do a "mirror image" of what you did in the treble. Tune first a reasonable sounding octave, then compare the P4 and P5 above it. You may favor the P5 slightly over the P4 but not at the expense of creating an overly wide octave.

At or about F2, you no longer need to consider the P4 above the note to be tuned. Simply concentrate on Octaves and fifths. You can also use the sostenuto pedal and compare double octaves and octave-fifths. Single octaves, fourths and fifthss (in the high Bass) should all sound reasonable but a fifth may be favored over a fourth. In the low Bass, you will want to be sure that single octaves, double octaves, fifths and octave-fifths all sound reasonable. Each should sound balanced with each other so that none has any more noticeable beat than any other. All should sound nearly beatless but none perfectly so.

In the very lowest Bass, especially on smaller pianos, you may begin to hear a faint resonance when tuning a single octave. This is actually the result of a large difference in inharmonicity between partials beyond the 8th partial of the note being tuned and the note an octave above it. Oddly enough, even though it is actually a rapid beat, it does not sound objectionable but rather pleasing to the ear. That is why I called it a resonance rather than an octave with a rapid beat in it, even though technically, that is what it is. Believe me, there is no such thing as a “pure” octave on a piano!

You may hear this starting on or about F1. Now is the only time I would ever listen to any RBIs when tuning the octaves. Typically, I would find the note an octave above the note being tuned and play the M10 interval and listen to whatever rapid beat there is. Now, when I tune that low octave, I cause that octave to sound in tune but also cause that resonance to mimic the rapid beating of the M10 that I just played. You can also go to the temperament octave and play a M3.

For example, you are tuning F1. When you make a reasonable sounding octave with F2 above it, you hear that resonance. Play F3-A3 and then "copy and paste" that rapid beat to the resonance of F1-F2 octave you are tuning. If you do this, you will get a different resonance for each of the low Bass octaves you are tuning because the RBIs above them are each a different speed. If you were tuning ET, you would want that resonance to be evenly digressive but you are not tuning ET! So, a slightly different character in each of the low Bass octaves is correct, expected, intended and deliberate.

This will cause the low Bass to enhance the sound of the entire rest of the piano. When you finish tuning the low Bass and have all of your unisons tuned, play the long, C Major arpeggio from the bottom to the top while pressing the pedal and you will hear and witness the splendid, "pipe organ" effect. After striking all the keys, listen to the sound that emanates from the piano. It will sound to your amazement just like a pipe organ. This is because you have used the canceling effect of equal beating to suppress the “noise” of ET and inharmonicity. It never fails to impress any customer that the piano is now supremely in tune with itself. I have never heard anyone who denied that it really does sound like a pipe organ.

You can also use an ETD to tune the Bass. You may use the calculated program to get the pitches close to what you want first. The partial selection will probably already be on the 6th partial but if not, change it to that. Play the note two octaves above the note to be tuned and stop the pattern. Now play the note which is an octave-5th above the note being tuned and the pattern will again display sharp. Adjust the pattern so that it rolls equally sharp and flat when the double octave and octave-fifth are compared. Enter that value and tune the note to whatever it is.

You may continue the same all the way to A0 but just as with the high treble, you may begin to favor the octave-fifth or the double octave-fifth at some point in the low bass, beginning on or about C2. Check to make sure that whichever decision you have made does not create overly wide single octaves. Remember the resonance often encountered in the low Bass. Only a nine foot concert grand may not produce that resonance and even one of those may do so in the very lowest few notes.

Hi!

Restringing has progressed on the Emerlich Betsy
http://www.youtube.com/watch?v=pk12p0eJ9uI

The pianist playing here tried the Bechstein earlier in the evening and, expert in Chopin Mazurkas, commented that the temperament I use does the job of expression that a pianist otherwise has to insert consciously into playing.

In my continuing examination of Chopin there are a number of areas of Chopin's work where perfect fifths and thirds which "sing" seem to be very natural fitting in with the emphasis of the rhythm.

Reaction to the temperament is very extraordinary - on the one hand musicians such as my visitor the other evening rave about it and others, I guess of the nature of whom Ross Duffin complains, who metaphorically run from the room screaming. It's still a mystery to me why Ross Duffin falls for the Lehman Bradley temperament not merely for reasons referred to earlier, a debate on which I don't want to restart, but for the reason that I'm not greatly aware of the Lehman temperament leading to many if any perfect intervals which one would assume to be a prerequisite for Duffin's return to finding better harmony . . . So the book is a good proposition, makes a great point with significant validity but its conclusion is a mystery to me - to the point of being a non-sequiture as far as I'm concerned.

To find harmony, in my mind having played the 1775 organ at St Maximin in France and been looking at a lot of the French Baroque repertoire using Meantone, a quest from harmony has to start from there, relaxing it into playability in all keys, so giving nicely purish intervals, particularly major thirds in Bb F C G D and letting the far-flung keys take the strain. If the validity of the musicoligist's experience the other night has any weight, then this sort of tuning appears to accord with the spirit of Chopin and one must therefore start to ask how universal it was in terms of geographical spread within Europe and up to what date.

Best wishes

David P

Interesting. I must be honest and say that I like your earlier, narrower pitching of the first octave above A440, however, on your two pianos. Your thought that the human ear is particularly sensitive in that area struck me as right, or at least as right for your well temperament on those pianos. Delaying the stretch somehow gives them a quality I find hard to describe. (Human? Humble? Plaintive? None of these terms seems exactly right.)

On the other hand, have you followed the thread on expanding the temperament with M12's? I wonder how your temperament, and other well temperaments, would sound following those guidelines.