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Thanks for backing me up on that Jim.

I'd love to read it. Unfortunately the PTG does not make their past journals available to members.

However in the tenor range I think m3 and M3's are rare, as they sound muddy anyways. The most common intervals there are the P5 and octave, at least in my musical world.

Kees

Excerpted for your reading pleasure: http://goo.gl/SG9ue

Thanks Jim. His remark about 6:3 octaves and m3's is consistent with figure 3 in my previous post, and also with Jeff's intuition. I still don't understand why this is so, even though I can compute it.

Kees

I am back. I found myself making this Forum more important than it is and decided to let it go for a little while. Like turning off the TV when it interferes with more important things.

Doel, as always, thanks for crunching the numbers. But since you have used only two values for the iH above and below the break, rather than two fairly logarithmic curves, the numbers do not represent a real piano. iH doubling every 12 semitones above the break and doubling every 10 semitones below the break I think would be more realistic.

Regardless, with 4:2 octaves the P4s do not jump and with the 6:3 octave the m3s do not jump. And if you crunch more numbers I expect that with 8:4 octaves the m6s will not jump. But the upper interval in the tests (P5, M6 and M3, respectively) do jump. However, if the E2-E3 octave was the temperament octave and then expanded upward, I expect that the upper interval of the tests would not jump, but the lower interval would jump.

However, if the iH curves were more like a regular piano, I expect that the beatrate of the “complimentary” interval in the test would jump much less. My thoughts now extend to the P5-P8 test for a P12…

My point is that there is a relationship between the octave type that is chosen and the intervals that will have a progressive beatrate. This can be understood when considering what test intervals are available for the various octave types.

Thanks for the link!

I will read it when I have the time and may make a post about it.

Can you explain that? You already did but noone understood I think. The apparent refutation is that the partials involved in the test and in the interval are different, so anything could happen. Are you making use of a Young type model for the partials?

Kees

I’ll try to explain with an example. I see it as a geometric law sort of thing.

Let’s start with a perfect F3-F4 temperament based on a 6:3 octave. I am talking about where all RBIs and SBIs are truly progressive. This should always be mathematically possible unless there is a jump in iH. Now let’s expand the temperament downwards with the same 6:3 octaves but across a jump in iH. E3-G3 will beat the same as G3-E4, right? And since G3-E4 is progressive between G#3-F4 and F#3-D#4 then E3-G3 will be progressive between F3-G#3 and D#3-F#3, right? Or to clarify it, especially if I made a typo, when the upper M6’s are progressive and 6:3 octaves are tuned, then the lower m3s will also be progressive.

But here is the rub. There is no rule saying that when the m3 E3-G3 is progressive, that the M6 E3-C#4 will be also. So C#3-E3 is not necessarily progressive either. And this shows as a later, lower jump of beatrates in your graphs. But, again, I expect this jump in beatrates to be less if more realistic iH curves are used. The iH slope has much to do with beatrate progressions.

Now consider what would happen if a perfect temperament is set between E2-E3 and then expanded across the break upwards. The upper M6s will be progressive because the lower m3s are progressive, that is until you get to the F3-D4 M6.

And this brings up a very difficult question. What if the E2-E3 and the F3-F4 temperaments are tuned independently? Could we expect the resulting octaves to be 6:3? I do not think so. Since they are tuned independently, there is no way of knowing what pitches would be required for 6:3 octaves. Each temperament could only be tuned assuming that the iH curve continues in a similar manner.

I have more to say, but will hold off for now. I keep thinking about 12ths…

Inharmonicity in a real piano does not double every 12 semitones above the break nor does it double every 10 semitones below the break.

The illustrations shown by DoelKees would, perhaps, have been more useful had they been extended a few more notes on either side of the actual break but they are fairly representative of the problems encountered at the transition between the bass and the tenor sections of the piano.

There is no single technique used by pianomakers to transition between the bass section and the tenor section. Depending on the length of the piano the so-called “break” is usually placed between notes E-20/F-21 and notes E-32/F-33. (Some few pianos may place the break either below or above these limits but they are rare.) As well, there is no universally accepted way to design the transition. There may or may not be wrapped strings placed on the tenor bridge.

Whether wrapped strings are used in the tenor section of the scale or not, nearly all tenor bridges are foreshortened to some extent starting from around C-40 on down to the end of the bridge. Nothing in this part of the scale follows a consistent logarithmic progression.

What happens to inharmonicity in the bass section is anybody’s guess. It depends primarily on the diameters of the core wires used and on the amount of loading used. There are no consistent rules that have been followed over the years for these things. Some makers used very small core wires with heavy loading; others used relatively large core wires with light loading. Some makers, regardless the relative tensions of the tenor scaling, used high-tensions in the bass section, others used quite low tensions. There has been little agreement over the past century and a half of piano making.

What all this means in the real world is simply this: when discussion just what techniques might work best in tuning the transition between the bass and tenor sections of the piano one first has to decide—which piano?

ddf

Nice post Del, clear and straight!

Del:

Young’s paper states that the iH on a Stienway D doubles about every 8 semitones: http://www.afn.org/~afn49304/youngnew.htm “The order of magnitude may be remembered by noting that the coefficient of inharmonicity is 0.1 cent/n² at C3 and that it doubles every 8 semitones.”

This is confirmed in one of the many graphs from the Pscale site: http://www.goptools.com/gallery.htm



I suggested the values of 12 and 10 semitones for the doubling of iH based on a fair collection of Verituner files that were graciously given to me by a fellow member here. Smaller pianos have an iH curve that is shallower and it is smaller pianos that have the type of scaling break that is being discussed.

My point is that one cannot generalize either about scaling or about tuning across the bass/tenor break. There are simply too many variables.

By my calculation the Model D scale has an inharmonicity coefficient of 0.058 at F-21 and 0.127 at F-33. This is reasonably close to your doubling. Of course, at F#-22 it is 0.062 and up an octave to F#-34 it is 0.169. Not so close to doubling. And this is for a nice, long concert grand where the bass/tenor transition is quite forgiving.

Looking at the Model B is a whole other story. F-21 has an inharmonicity coefficient of 0.126 and at F-32 it is 0.146 with a significant dip in between.

A third example might be the Model M which has its bass/tenor transition between notes A#-26 and B-27. (Both the Model B and the Model D have their transitions between notes E-20 and F-21.) The Model M scale uses two unisons of wrapped strings at the end of the tenor bridge. The inharmonicity coefficient of note B-27 (a wrapped string) is 0.085 and for note B-39 it is 0.267. Nowhere close to doubling in one octave. Indeed, between notes C-28 (a wrapped string) and C#-29 (a plain steel string) it goes from 0.087 to 0.224. And that is between two adjacent notes on the scale. Even if we want to ignore the two unisons of wrapped strings and consider the first octave of plain steel strings we’re not much better off. As mentioned earlier, the inharmonicity coefficient for C#-29 is 0.224 and for C#-41 it is 0.328. Again, with a significant dip in the middle.

As I said earlier, when discussing just how to tune beat rates at, or close to, the bass/tenor break you first have to decide just which piano you want to use as your model. A beat rate progression that works for the Model D will be quite different than the beat rate progression that will be suitable for a Model B or a Model M.

And we haven’t even gotten into the Model S or the Baldwin Acrosonic or the Whitney spinet yet, none of which are anything like the three already discussed.

ddf

I plotted the range where the intervals span the break. Nothing interesting happens for intervals that don't span the break. (At least under my assumptions.)

If requested I'll be happy to recompute for ih doubling per octave in both directions. I don't think the break is a big deal for tuning. The timbre of the sound across the break changes so drastically on the small uprights we are considering that this is the main thing that is heard, completely overwhelming any tuning issues.

Of course that is just my opinion.

Kees

All:

I agree that each piano, and not just each model, must be taken as it comes. (Actually I think the Whitney spinet is one of the better designs for a spinet. They have a straight bass bridge! How they were actually put together is another matter.) But there are some generalities that can be noticed, and when these generalities do not work then there is a basis for making adjustments.

For instance, the Kimball spinet has the typical lower iH in the highest two wound strings, which are on the tenor bridge, when compared to the lowest unwound strings. But the Kimball console has higher iH on those wound strings. In either case, depending on the octave type that is used, certain intervals will have a better beat progression than others. Doel’s excellent graphs show this. And by noting which intervals jump in progression in which direction better decisions can be made.

As far as timbre, Doel, I have to agree that this is what is really perceived and is part of the reason for my preference to tune with 4ths and 5ths. For that matter I think it is what makes tuning with 4ths and 5ths possible. These intervals have a color separate from mere beat speeds. The lowest 5th on unwound strings for a small piano can be really obnoxious. So to avoid having a “raison” in the rabbit stew, I will adjust other intervals to make those low 5ths sound no worse than necessary.

And yes please Doel, could you post some graphs with a logarithmic slope to the iH for comparison? And if you would like to see what happens when the lower notes have the greater iH, you could consider crunching those numbers too.

Here are some more computed beatrates. Break at E3/F3. IH below break 0.2, above 0.46, or reversed. IH doubles per octave.
We then tune F3-B4 standard ET, and extend down by one of 4 methods (called "octave" in the plots).
1) 4:2, 2) 6:3, 3) proportional beating P4/P5, 4) Beatless P12's.










Kees

omg Kees is doing more pretty pictures for us to SEE,... of course, everyone, or most everyone, understands your post. It's just that it is inconsequential in day to day life dealing with various scalings. SM

ok Kees, I'm going to have to DO some sort of painting using your graphs and colors, overlay, should be interesting. SM

"If you can't keep up with the conversation you better not try to join in at all."

Kees

Inconsequential is right. In the end, it's how the tuning sounds, not how it looks.

Yes I heard you the first time. What happens now when you tune by progressive P4/P5 across the break, as you seem to do Jeff? Not sure how to simulate that.

Kees