Mark:

Our experiences differ. I have found in all cases that when pure 12ths are tuned (except with wacko wound strings...) wide double octaves and narrow triple octaves are the result. The single octaves progress from 6:3 (or wider) in the bass to 4:2 in the middle and between 4:2 and 2:1 higher up. Where these changes occur depends on the scaling. Generally a studio upright will have 4:2 octaves in the temperment, a spinet between 2:1 and 4:2 in the temperment and a decent sized grand between 4:2 and 6:3 in the temperment. I have also analyzed these relationship mathematically and it confirms what I hear. OK, so you hear something different.

Hi Jeff,

You are right. In general, higher octaves are taught to be tuned less narrow, 2:1 for example in the treble, because the higher coincidental partial, 4:2 and 6:3, are so faint that their beating is not heard, hence the 2:1 sounds good by itself.

However, it is not just the quality of the octave we should concern ourselves with. These higher notes must be in tune with the lower intervals. I concentrate on those intervals so the undamped treble strings ring sympathetically with the lower intervals. So the actual size of the octaves does not concern me. The byproduct of my approach is not just less narrow octaves on top, but consistently less narrow by the same amount, each one.

My checks use M3, M6, M17, m3, m6. So when comparing to lower interval beats which were all set even, all the higher ones beat even as well. See? I don't try for even M17 for example, they just happen. Trying for even M17 is putting the cart before the horse. You can get even M17 and bad compound octaves for example.

As for pure 12ths creating wide double octaves and narrow triple octaves, that is not what I've found. The pure 12th creates wide double octaves and slightly wide triple octaves. Tempering the 12th creates less wide double octaves (beatless) and pure triple octaves. I am working on a proof for that and will post it if I figure it out.

You said you have already figured this out mathematically. Can you post your proof?

If you want the undamped treble strings to ring sympathetically with the lower interals, how about pure twelfths?



Since you show true interest, give me a week or three to work something up. What I have is a little out of date and not really clean. I'll start a new Topic when the time comes.

To give a preview, it has to do with the partial number, the semitone span of intervals, and the iH slope. Just like the theoretical pitches do not double each octave because of iH, neither do the beatrates of intervals. The wide RBIs less than double each octave while the narrow RBIs more than double. (Ever wonder why you can use 10th and 17th so high into the treble?) The SBIs act differently depending on the iH slope and also on the stretch scheme. The 12ths are a powerful tool however they might used.

Re: pure 12ths. I used to tune pure 12ths for that reason, but recently have experimented with tempering the 12ths in favour of a cleaner double octave. Pure 12ths really do sacrifice the double octave. Also, the 12th is not the only interval that will set up sympathetic vibration. The compound octaves do as well so that is why I favour the tempered 12th; so the double octave is cleaner and rings as well.

May I suggest that we post videos or recordings to prove our thesis as opposed to mathematical treatese? It may be more palatable to the larger audience and also more convincing. I'll see what I can do.

Mark:

Sorry, I have no AV equipment.

Ah, I found the 2-1/2 year old Topic that has what I need to get this started:

http://www.pianoworld.com/forum/ubbthreads.php/topics/1445359/1.html

Like I said, give me some time and I will start a new Topic. I am tempted to create an executable program for this and make it available (with a text file of the vb code). We'll see...

Hi Jeff,

Thanks for taking the time to find your old posts.

I read through some of them. I have to be honest. As a mechanical engineer and professional musician and piano tuner, I have delved deeply into mathematics and also holistic practical performance. I have always found mathematical analysis to come up short of high level practical performance execution.

For example, your analysis uses 4:2 octaves as assumptions. Regardless of what the graphs show, 4:2 octaves are just not good enough.

Also, your assertion of always beating partials somewhere in the octave is argued by many experienced tuners, including myself, claiming that octaves can be tuned beatless. How do you explain that? Remember, I'm an engineer, I know the math, and I know the math says it isn't possible. However, as a human piano tuner, I do it every day.

My claim of being able to create in tune single, double, and triple octaves can only be proven in practice; I have to post a video proving my technique. I think then you will hear what I am talking about. Until then...

+1

+1

Mathematics is more than numbers, so math can be used in tuning. However, there is a lot of the physics that is not very clear. I talk about it a bit with a friend from my college days who went on to work at Lawrence Berkeley Lab as a mathematician, and he agrees that we both use a lot of math, but in different ways, and it is not certain who does the most useful work.

I pointed out once before that we do not even have a real accurate definition of the pitch number of a piano tone because the vibrations are not strictly periodic. That is why I said that "in tune" is a vague term.

Mark:

I am mistaken. You do not show a "true interest." It seems that what you think you hear must be what everybody actually hears, but are sometimes mistaken. And since you have "delved deeply into mathematics" you do not need me to show you the proof. You can come up with your own proof. But why bother? If it does not agree with what you think you hear, the math is obviously wrong.

As I said:


Let's leave it at that. I am not interested in some "Schwartz" comparison.

Hi Jeff,

With your permission I will still post the video if I can get around to it.

With my permission?

Folks, am I missing something here? It wouldn't be the first time.

Hi Jeff,

I tried to send you a PM but your account doesn't accept them.

Yes, you are missing something here. I asked your permission as a matter of courtesy since you feel I do not show a "true interest" in the subject. I found your post to be a bit defensive and I didn't want to agrivate you any further.

I also feel that your 'true interest' is in the playing with of numbers and not the practical execution of the art of piano tuning. Please don't get me wrong. I do not feel that an interest in the purely mathematical aspect of piano tuning is inferior. I admire anyone who can "delve deeply" into that and come up with some interesting proofs. The proofs I am interested in are more practical and similar to the ones I've already posted. My critiques of your approach were from that perspective, and not from that of the validity of the mathematical analysis.

I hope you are ok with that and do not take the posts too personally. I have more than a true interest in this subject. It is my passion, as I know it is yours. I hope we can see the validity in each other's perspective.

Peace,

With regard to “proofs” here is our former Prime Minister on the subject of proofs that need to be proven. What he actually says is below the video in text.

A proof is a proof

Mark:

I am frustrated with myself for jumping to a conclusion. I am not going to say just what it was.

Here's the thing: Through the years I have read that people hear things different than I do. And I have read that the math says one thing when I have found that it says another. And I have found that what I hear and the math that I use agree with each other.

OK, so you hear something different than my ears hear, and perhaps your math says something different than my math says, and perhaps what you hear and what your math says do not agree with each other.

Post what you like. I am confused by your courtesy.

Hey Jeff,

No worries. I suggest a Skype meeting where we can demonstrate each others techniques. Send me a PM.

Mark

No thank you.

From A#4, I tune 4:2 or 4:2+ octaves up. All depending, from about F5 >, I start tuning & checking my 3rd and 17th, for example, F3/A3 and F3/A5 etc...and continue up as far into the treble/high treble as I can on any particular piano. Then it is single octaves up to the top, either melodic or harmonic or both.

Sometimes, I tune M6/M17ths up through the treble.

Going down from F3, I tune a 5th down, check my 4th and octave and move on down as far as the piano will allow. For single bass strings, I use a variety of techniques, once again, all depending on the piano. It may be ghosting or it may be making the notes sound as good as I can get them to sound with octave, 10th, 17th, 12th and double octave.

Ok, I admit, I use a lot of checks as I tune, I actually, as I wrote earlier, tune with my checks and check with my tune!?!

For the high treble, if it is a good quality piano, and I am tuning for a discerning client, I have begun using the chord of nature, which I think Kent Swafford is the author of.

In general, I tune whole tone tuning. I strive for as clean sounding tuning as possible, smoothing out the 12th and D8ve. I used to stretch my tunings quiet abit but now prefer as clean sounding/just tuning as the piano will allow me.