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Folks, changing the definition of cent is not going to change any tuning any more than measuring a string length in inches instead of centimeters alters its pitch.

Kees

As long as you agree that a spreadsheet cannot tell how a tuning sound, I will agree with you. !

did you check the pitch difference heard between a plucked string and the same played by the hammer ? which is the real pitch ?

The difference doesn´t matter per se. What is important is the concept.

Before iH was discovered, tuners were able to tune correct octaves, in the low bass and in the high treble. To speak of tempered octaves was nonsense.

Now that we are aware of iH we tune the same correct octaves but we can better understand the behavior of octaves in the high treble compared to the low bass. And now speaking of tempered octaves makes sense.

We can no more believe we tune Pithagoras's octaves where the fundamentals are in a 2:1 ratio, that model doesn't work for pianos.

So the model you choose makes a difference!

I kind of agree with both of you. Kees is right on his idea of scaling/measurement. The problem is that it's a slippery son of a… target.

It touches something I've thought quite a bit about. Just where does the need for stretch come into play? We like to measure it by the octave, but guess what - that octave isn't really there. Or, at least, it can be measured from 12 different angles.

As long as we are limited to any known scale (yes, even a logarithmical one - they are hopelessly symmetrical, too) the best we can do is to establish it, and give offsets/deviations. This is what any ETD will be capable of, and even though it is a crippled reading, it is all we've got for now.

It gets to what I was asking: What is the frequency of a piano note? Do you measure it from peak to peak or from zero to zero? The wave put out by a piano is not periodic, and the two measurements are different.

Remember the old joke about the mathematician, the physicist, and the engineer, where they are put at one end of the room and a pile of money is at the other end of the room? They are told that they can have the money if they get to it obeying one rule: After they go half the distance to the money, they need to stop, wait a minute, and repeat until they get to the money. The mathematician thinks about it and walks away. The physicist walks halfway to the money, waits a minute, walks halfway again, waits a minute, thinks about it and walks away. The engineer walks halfway to the money, waits a minute, walks halfway again, waits a minute, walks halfway again, waits a minute, walks halfway again, waits a minute, walks halfway again, waits a minute, walks halfway again, waits a minute, walks halfway again, waits a minute, walks halfway again, waits a minute, says "That is close enough!" and grabs the money and leaves.

I prefer something which is between a 3:2 and 6:4 fifth. It has nothing to do with light, it just minimizes the maximum of the 3:2 and 6:4 beat rates.

Kees

Right!

The same is true for Kamin. The good sounding octave won't be 3:2 nor 6:4.

There is an interference between the two of them. If they are close enough they act on the other to produce a one unique sound.

Like unisons. When the frequence of one string approaches the other there is a range in which it acts on the other string and makes it vibrate at the same frequence and they fall into the unison.

Neither, there are various algorithms to measure the fundamental, all very accurate. More accurate than anyone can tune.

Kees

Rafael, Kees is making a point. If anybody want a new scale instead of cents, they have to implement it. But why? The octave will be whatever you want it to be. Why not stick to cents and play around with F' ?

What are those algorithms? What does "accurate" mean in this case? What definition of frequency are you using?

I liked very much the joke refered by BDB. "Close enough to grab the money"

Close enough to achieve a good tuning.

Here is an extract of the book by C.Chang. "Fundamnetals of piano practice"

d. Sympathetic Vibrations. The accuracy required to bring two strings into perfect tune is so high that it is a nearly impossible job. It turns out that, in practice, this is made easier because when the frequencies approach within a certain interval called the "sympathetic vibration range", the two strings change their frequencies towards each other so that they vibrate with the same frequency. This happens because the two strings are not independent, but are coupled to each other at the bridge. When coupled, the string vibrating at the higher frequency will drive the slower string to vibrate at a slightly higher frequency, and vice versa. The net effect is to drive both frequencies towards the average frequency of the two. Thus when you tune 1 and 2 unison, you have no idea whether they are in perfect tune or merely within the sympathetic vibration range (unless you are an experienced tuner). In the beginning, you will most likely not be in perfect tune.
Now if you were to try to tune a third string to the two strings in sympathetic vibration, the third string will bring the string closest to it in frequency into sympathetic vibration. But the other string may be too far off in frequency. It will break off the sympathetic vibration, and will sound dissonant. The result is that no matter where you are, you will always hear beats -- the tuning point disappears! It might appear that if the third string were tuned to the average frequency of the two strings in sympathetic vibration, all three should go into sympathetic vibration. This does not appear to be the case unless all three frequencies are in perfect tune. If the first two strings are sufficiently off, a complex transfer of energy takes place among the three strings. Even when the first two are close, there will be higher harmonics that will prevent all beats from disappearing when a third string is introduced. In addition, there are frequent cases in which you cannot totally eliminate all beats because the two strings are not identical. Therefore, a beginner will become totally lost, if he were to try to tune a third string to a pair of strings. Until you become proficient at detecting the sympathetic vibration range, always tune one string to one; never one to two. In addition, just because you tuned 1 to 2 and 3 to 2, it does not mean that the three strings will sound "clean" together. Always check; if it is not completely "clean", you will need to find the offending string and try again.
Note the use of the term "clean". With enough practice, you will soon get away from listening to beats, but instead, you will be looking for a pure sound that results somewhere within the sympathetic vibration range. This point will depend on what types of harmonics each string produces. In principle, when tuning unisons, you are trying to match the fundamentals. In practice, a slight error in the fundamentals is inaudible compared to the same error in a high harmonic. Unfortunately, these high harmonics are generally not exact harmonics but vary from string to string. Thus, when the fundamentals are matched, these high harmonics create high frequency beats that make the note "muddy" or "tinny". When the fundamentals are de-tuned ever so slightly so that the harmonics do not beat, the note "cleans up". Reality is even more complicated because some strings, especially for the lower quality pianos, will have extraneous resonances of their own, making it impossible to completely eliminate certain beats. These beats become very troublesome if you need to use this note to tune another one.

I wonder if the same phenomenom is present when tuning near pure intervals,other than unisons, I mean octaves, 12ths and 5ths.

That would become too technical. "Pitch detection" and google/wikipedia should get you going here if you're interested.

The best method that I know of to find all partials and their decay rates is described in this paper

Kees

Kamin:

In Cordier tuning do you get enough stretch in the bass? Are the pure fifths (3:2/6:4 equal beating I think) continued all the way to the bottom?

Kees

Here are some plots of the sizes of various intervals for Cordier tuning, which I interpret as equal beating 3:2/6:4 fifths (i.e., minimizing the maximum bps for those two partials). The first plot is for my Heintzmann upright, the second for a Steinway D. I used tunelabs modification of Young's model for inharmonicity.


Heintzmann


Steinway

For the upright Cordier seems about equivalent to 6:3 octaves throughout.

For the Steinway everything is stretched. (Note the P5 data are based on 3:2.)

So that answers my question to Kamin from a theoretical point of view. Kamin, do you observe this also in practice, i.e., Cordier on uprights in about the same as tuning 6:3 octaves, but on grands it stretches way more?

Kees

Well, there comes a point where you have to define the pitch of a piano note well enough so that you can distinguish between 440*2^(1/12) = 466.163761518 and 440*3^(1/19) = 466.191468485 on a piano, otherwise there is no point in discussing it at all. I am not certain it can be done. I do not believe that a piano note's pitch is that well defined.

That would be a beat rate of 2/min, which can not be distinguished electronically or aurally.

However your premise that "you have to define the pitch of a piano note well enough so that you can distinguish ..." is unclear to me. Why would you come to this point? Maybe you should argue with Alfredo instead of me.

Kees

Kees,

your graphs look the way I hear it. Have you input (and posted) a similar graph for Stoppers pure 12th tuning? If you have so, please forgive me, I had summer vacation and the weather is unusually hot here...

And (withdrawn from the same vacation account) - which program did you use for the graphs?

Here it is for 3:1 12ths across the board:


Heintzmann


Steinway

I used my own program, described in the "tuning math" thread.
(Again P20 should read P22, forgot to fix it.)

Kees

Great stuff, and very useful and logical layout! Thanks, I'll check into the math thread.

All:

My, it was a busy evening!

Cents are very handy as a measuring tool because it always means cents from something else. But contrary to Gadzar’s belief, I do not calculate in cents, I calculate and analyze in the aural tuners measuring device of bps which results in Hz. Then where appropriate, I display results in cents deviation from theoretical pitches. Unlike Kees’ fine graphs, I prefer the y-axis to be in bps. That has more meaning to me.

Looking at Kees’ graphs solidifies something that has been developing in my mind. How do we determine what is considered a high, medium or low iH piano? Even with an ETD taking the iH reading of one note does not mean much. The slope of the iH curve is important, too.

On the graphs, any curve could be straightened into a line resulting in the other curves changing their shape. The ones that interest me most, lately, are the 12th and the 4:2 octave. Particularly at what point going up the scale does the 4:2 give the same stretch as the 12ths, or in other words, where 4:2 octaves produce pure 12ths.

For an aural tuner, this is one way to determine the iH (high, med, low) of a piano. I would say high is when this happens below the temperament, medium in the temperament, and low above the temperament. So looking at Kees’ fine graphs and noting where the 12ths line crosses the 4:2 curve the Hientzman could be considered medium iH and the Stienway D low iH.

So taking what Gadzar said about real vs simulated pianos to heart and keeping in mind what Pat correctly mentioned about being stubborn (my choice of word) during a discussion, I meticulously tuned my Charles Walter Console (the scaling is more of a Studio with #1 being 48 inches) with 4:2 octaves until pure 12ths were produced and continuing with twelfths from there. My simulator predicted that pure 12ths would be produced at the C4-G5 12th, but in reality this happened about an octave higher. I have no explanation just an observation. Perhaps the iH curve that I use is not the same as my particular piano.

What I am going to be playing with is the relationship between a tempered F3-C4 fifth, a 4:2 C4-C5 octave, and an F3-C5 12th to get an idea of the iH of pianos as I tune them.