Piano World Home Page Forums Our Most Popular Forums Piano Tuner-Technicians Forum Do we hear 'in tune' intervals by size or beats?

Hi Paul,

I think the inharmonic partials mentioned in the reference you gave arise from a mechanism (interference between the two different pipe diameters and lengths) separate from the the fundamental pitch generating mechanism and are not related to the fundamental pitch. Longitudinal waves on a piano string are an analogous situation. They can be described as inharmonicity, but are not directly related to the fundamental pitch of the string.

Edit: The point I am trying to make above is that in tuning a piano, one must accommodate the inharmonicity that is related to the fundamental pitch. The other inharmonic effects - longitudinal waves, hammer noise, woody sounds, etc., must be dealt with by other means prior to tuning.

Hi Robert,

You wrote: ..."...unlike in a piano where the irregularity of IH makes the calculations much more complicated, the mathematics of how pitches beat in a no-IH instrument like a pipe organ is simple. If you do observe any unevenness in the progression of tempered interval beat rates in a pipe organ tuned to 2:1 ET, it would be due to tuning error."...

This is where I do not agree. I appreciate that you address "tuning error" as one possible cause for uneven progression, but I was not not talking about that. I was talking about the 'mathematics'.

..."It is possible that the air pressure in the pipe chest might not be perfectly regulated, so the pitch of a single pipe could change slightly when another pipe is played along with it. This might give the appearance of imperfectly progressing intervals, but that would only be because of the imperfect control we have on the fundamental tuning."...

With that, are you saying (in between the lines) that you haven't managed to put the first 2:1 octaves ET theory into 'no iH' practice?

..."If the tuning could be guaranteed to stay at the pitches specified by 2:1 ET, the beat rates of the intervals can be guaranteed to be perfectly progressive in exactly the same ratios as the pitches are progressive."

As I said above, here I cannot agree. I cannot agree not because "the mathematics of how pitches beat in a no-IH instrument like a pipe organ is simple", but because 'that' mathematics is simply a convention. In other words, it was made a bit too 'simple'.

Using your words, "perfectly progressive..." could be garanteed only if 2 divided by 3 was a finite number, but in fact it is not, it gives 0.6666.....

For the same reason (2 and 3 being primes), 4 (that is 2*2) divided by 3 is not a finite number, it is an irriducible 1.3333.... (1.3^_ (period 1))

You will see now (and I hope you acknowledge) that 4/3 (the conventional fourth) multiplied by 3/2 (the conventional fifth) will not reach 2 (as in the 2:1 ratio). Leave all 'tuning errors' aside, that is why we cannot have 'perfectly' progressive intervals on the basis of 12th root of two.

I was taught in elementary school that 4/3 * 3/2 = 2.

Kees

Me too.

Alfredo, you must have a fundamental misunderstanding of how tuning is constructed.



The 4th and 5th are complimentary intervals, and create a perfectly just 2/1 octave when multiplied.

(1) 4/3 * 3/2 = 12/6 = 2/1

That aside, yes, we also certainly can have progressive intervals at an equal temperament based off a ratio of 2/1. As a matter of fact, we can have perfectly progressive intervals in any type of equal stretch, rational or irrational!

For any type of piano, an equal tuning can be calculated to have all intervals be progressive with any kind of stretch from near zero to infinity.. the limitation being the amount of tension the strings are designed to carry. The difference created by these stretches would only be in the alignment of intervals and thereby, the quality of the tuning.

The principle of primes is so commonly misused. A prime defines a power structure (geometric curvature) that is unique. In tuning, this is only relevant such that geometric series of ratios with different primes will never be able to align.

For all integer and irrational floating point values of 'x':

(2) 3/2^x != 2/1^[0 ... x]

(3) 4/3^x != 3/2^[0 ... x]

All this means for tuning in equal temperament is that a curvature must be chosen which will be a nice compromise for all geometric series that are of importance.

An imperfect compromise, always - but this has nothing to do with intervals being progressive, and everything to do with auditory alignment to maximize the beauty of the sound.

..."Alfredo, you must have a fundamental misunderstanding of how tuning is constructed."...

It must be, either me or you, Tunewerk.

Are you saying (in between the lines) that you have managed to put the first 2:1 octaves ET theory into 'no iH' practice?

..."As a matter of fact, we can have perfectly progressive intervals in any type of equal stretch, rational or irrational!"...

I look forward to hearing your demonstration. Untill then, please avoid superfluous posting.

...SNIP..."All this means for tuning in equal temperament is that a curvature must be chosen which will be a nice compromise for all geometric series that are of importance. An imperfect compromise, always - but this has nothing to do with intervals being progressive, and everything to do with auditory alignment to maximize the beauty of the sound."...

Yes, a '..nice compromise for all...', this rings the bell.

Ah... Alfredo, what I said above is self-evident to anyone who understands the theory correctly. A demonstration would be unnecessary and superfluous.



No. If you read my lines instead of inserting meaning in between them, it might save you time.

A demonstration would prove that what you refer to is a lame theory but, yes... it's getting late now.

Good night, a.c.
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Alfredo, what kind of demonstration would you accept, that does not require my traveling to Italy, or your traveling to the USA? Would you, for example, take my word for it if I said I tuned a rank of pipes with 2:1 ET and measured the beat rate of thirds and found them to be uniformly faster as I went up the scale?

So we've all been taught wrong? What is 4/3 * 3/2 according to you, if it is not equal to 2?

Kees

Hi Robert,

No need to come to Europe :-) I would take your word for it, perhaps adding a few ten-thousandths to the ratio you used.

What I think is that we approach 'exatitude' and what we may define 'perfect' in different ways. I was not addressing the thirds beat rate progression only, but all intervals within a theoretical 2:1 octave.

What I can say is that there is only one way to combine fourths, fifths, thirds and octaves in an exponential scale and obtain a 'perfect' octave: that is when you combine 4 (the double octave) and 3 (the octave + fifth). In this case, the 'perfect' octave is not 2:1 but a little bit wider, a bit more than five ten-thousandths wider than 2:1.

As mentioned, you would probably hear those approximations only when you check all intervals, including 12ths. Did you?

Of course, I would be happy to check non-iH 2:1 octaves aurally, even if it was a simulation: progressive thirds, tenths and 17ths, and consistent fourths, fifths, octaves and 12ths.

In fact, I would look forward.

I think that decimal approximations of exact values are being mis-understood and mis-used to further a theory.

Alfredo, the value of pi, for example, is an exact quantity defined by the ratio of the circumference of a circle to its diameter. The decimal approximation of pi is not exact and not used in conceptual mathematics. It is used in technological applications.

Hi Alfredo,
Here is a link to a simulation of ET with no iH. It is just a portion of the full compass but shows that the intervals are progressive.

Thank you, Prout, I appreciate your effort.

Yes, I said 'simulation', but meaning a simulation of non-iH 'sounds', what do you call them, recorded sounds or notes in a scale, say like Pianoteq's, so that one can check them aurally?

Anyway, look at the values for A0, A1, A2, and A3, relative to M3, P4, P5 and M6, perhaps you will notice two facts: how fifths (P5) get divaricated, comparing them to M3, P4 and M6 (imagine what 12ths would sound like), and how decimal approximations (roundings?) are inconsistent: on A3, the M6 would slow down, as the M3 (1.09 - 2.18 - 4.37 - 8.73) is unstable.

Regards, a.c.
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Hi Alfredo,

Again, I think you are missing the fact that the numbers presented are approximations only, rounded to 2 decimal places. Had I chosen to round them to a larger number of decimal places, they could be as accurate as you could desire. What is important to note is that the beat rates of the P4 and P5 within the octave are exact when the number of decimal places is increased to an arbitrarily large number (common language used for 'arbitrarily large number' is 'infinity').

Hi Alfredo,

Here is a wav file of simulated M3s from F3A3 to C#4F4. They show a perfectly progressive increase in beat rate within the perfect 2:1 F3F4 octave. Hope this helps validate the math of no iH ET.

Regards.

Thank you, Prout, for addressing approximations.

For me what is important to note also, it is that even if we extend the number of decimal places to infinity, we are unable to remedy an 'arbitrary' model, as the first ET is.

Now I hope Robert will offer us a sound reproduction of a non-iH 12th root of two tuning.
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Edit: Sorry, I had not seen your latest post, ...arriving at home from work I just replied to your second last post. I will listen to that file and reply asap.

Prout, sure that wav file is OK? Are you yourself able to reproduce it? I can only hear two very low notes that go on chromatically. Perhaps there is something wrong with my LT?

Hi Alfredo,

I now realize that the file I posted is not correct for our discussion. Each M3 in the file is the result of two essentially pure sine waves interfering with each other. The resultant sound is interesting, but incorrect. For example, the F3A3 sound (first 5 seconds) produces a clearly heard set of harmonics based on about 8.6 (8.6, 17.2, 25.8, etc.) that are a result of the 174.61 Hz and 220.00 Hz. This is not the beat rate from the upper harmonics, since there are no upper harmonics.

Sorry, I will try to add some harmonic distortion to the mix.

Regards.

Hi Robert,

If you like, have a look at this (below), it is down this page:

http://www.pykett.org.uk/temperament_-_a_study_of_anachronism.htm

..."Appendix 2 – Arithmetical Precision required in Temperament Studies

The degree of precision required in numbers and arithmetic operations to do with temperament arises as follows.

Consider the process of tuning by beats. Let there be two flue pipes, one of which is already in tune and one which is to be tuned to it. Tuning becomes progressively more critical and difficult the higher the frequency because the beats, which are frequency differences, become faster as those frequencies increase for a given change in length of the pipes.

A verdict of “pretty well in tune” would probably be given if there was, say, one beat in around ten seconds for any pair of pipes. Although stricter criteria could be adopted there is no point in making things too difficult, partly because of the tuning drift which occurs naturally due to temperature variations etc after an organ has been carefully tuned. Therefore, using this criterion, the two pipes would have to be tuned until their fundamental frequencies did not differ by more than about 0.1 Hz, because one beat in ten seconds implies a frequency difference of 0.1 Hz.

Remembering that tuning is most critical in the upper reaches of the compass, consider the fifth C on the keyboard, i.e. the C below top C on a 61-note organ keyboard. Even higher notes could be chosen, but again we have to adopt reasonable parameters if the discussion is to remain sensible and practical. The fundamental frequency of this note on an 8 foot stop is 1046.5 Hz for an organ tuned to A = 440.00 Hz in Equal Temperament. For simplicity we shall use the approximate figure of 1000 Hz.

A frequency tolerance of about 0.1 Hz at a frequency of about 1000 Hz implies a tuning accuracy of the order of 0.0001 or 0.01%. This is therefore also the precision required in temperament calculations which have to deliver the frequencies of the notes in a particular temperament. But because there are usually several steps in the calculation of each frequency, it is necessary that the numerical precision of the numbers used in each step is greater than that required in the final answer, otherwise the answer will not be accurate enough owing to truncation or rounding errors. Therefore at least one more significant figure is required throughout the calculations, meaning that numbers must be represented to at least a precision of 0.00001 or 0.001%. This is the same as a precision of 1 part in 100,000, or 6 significant figures, as stated in the main body of this article."...

Is that what you where referring to, few posts ago, mentioning 'tuning errors'?

Regards, a.c.
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