Here is a question. It's something my tuner showed me, but I can't remember it all, plus I'm puzzled about why it works.

He showed me 6:3, 4:2, and 2:1 octaves. I understand the matching at the 6th and 3rd partial; the 4th and 2nd partial; and the 2nd and 1st partial.

He also showed me an equal beating test for them. Say you're tuning A2 and A3 to a 6:3 octave. Then you play the minor third A2 C3, followed by the major sixth C3 A3. These two intervals should have equal beats. This is what I don't understand: what pitches are you hearing beating (that is, are you hearing fundamentals beating, or partials?), and why does it work?

He also showed me a similar equal beating interval test for 4:2 octaves, but I can't remember what it is. Can someone enlighten me?

Thank you for any help you can provide in filling in these gaps for me. I'm not expecting ever to do any tuning myself, but I'm curious about how it works.

The test note in your case would not have been E3, but C3.

When playing A2 and C3, a beat is heard at E5. When you look at the partial series of A2 and C3, you see that E5 is is the first common partial. More specifically, it is the 6th partial of A2 (keep this in mind), and the 5th partial of C3.

Similarly, when playing C3 and A3, a beat is heard, again at E5. Again, E5 is the first common partial of C3 and A3, and more specifically, it is the 5th partial of C3 and the 3rd partial of A3.

Therefore, the 6th partial of A2 and the 3rd partial of A3 are both tested against the 5th partial of the test note, C3. You listen to the two beat rates at E5. When they are equal, you have a 6:3 octave.

This is the nature of any such octave test: choose one test note, of which one specific partial is tested against both the octave notes. It's good to choose a test note such that its 5th partial is used, because the 5th partial is quite prominent, and the only partial it could be confused with, is the 10th, which is normally quieter than the 5th in the mid-range of the piano. Even if it is audible, one can concentrate on the 5th.

For 4:2, one would use the M3 below the lower note as a test note. In this case: F2. The beat would be heard at A4, which is
... the 5th partial of the test note,
... the 4th partial of A2, and
... the 2nd partial of A3.

I hope I explained it sufficiently, and I hope you get the pattern.

Mark R., thank you, that's fantastic!

I changed my post to say C3, not E3 -- that was my typo, not an error from my tuner.

Why is the fifth partial particularly prominent?

Are there similar test notes for testing other intervals? Is there an online guide somewhere for how to tune a temperament octave, that describes this kind of thing?

Online resources: I just found several articles at Bill Bremmer's webpage. Thank you, Bill!

Thase octaves refernces and checks are convenient and used in the angloi saxon method to control the "stretch" or "size" of the octave.
Unfortunalmey ther is a defect in that process :

The main one is that the partials in pianos are very often uneven in pitch so the accuracy is not guaranteed 100%

The other one is that the tuners tend to forget that an octave can be listened to without any particular focus on a partial.

What we hear is a mix, indeed we can train to focus on apart of the tone or at a certain level and that is very useful, but a more global listening is more useful in the end.

The other big problem with the comparison between fast beating intervals is that their speed is not as even as one think. due to the reinforcement caused by partials in higer ranges, (and possibly other factors) the speed of the intervals sound like accelerating, then slowing, the,n accelerating, making the comparison not always as easy and clear as one wish.

SO I don't reject the value of those tests, but I don't take them as seriously as I did.

I dont know if it is only possible after so many years of tuning, but direct tuning of the intervals is possible, even for the slow ones. Any method that allow to compare beat progression from intervals that follow each other is also good and easier than the ladder of thirds or a precise speed to be attained.

Generally speaking if one want to have a facilitated progression of FBI, the enlarging of the first octave is the way to go.

A very neat trick is also to trigger the beats listening by playing the first note then the second once the tone of the first is cleaner. the samples of intervals from Weiyan show that very well.

A musician/pianist that is learning to tune would be ready to directly tune in a musical tone. the use of "standards" for octaves is not always as musical as I wished.


Mark provided perfect answer to the question you where asking, however.

Kamin

Can you describe what you listen for when tuning octaves? Have you assessed how your results compare with Anglo-Saxon ratios, and how much they vary from piano to piano?

I am trying to get a mental picture of how you aim to match up, or should I say mix up, the partials as you stretch the octaves across the keyboard.

Hi Ian,

I listen to the global activity and level of consonance, I did not try to really compare with the usual tests but I will.

It may depend of the age of the strings (they raise in iH with age) with voicing probably.

I try not to focus on beats at a particular level as long they are not easily noticed, but I suppose that if I where to focus on 4:2 or 6:3 I would hear beats, while if I listen more "globally" the acceptation is larger, I believe that when I aim for "the beginning of beat" it may hide in the slope of the sustain, but it may be located at the lowest level( 2:1) .

I will check that and let you know may be the octave type will be similar, due to the more or less strong presence of some partials along the scale.

Really I tried when I first begun to use that approach , to test a little and I was expecting very stretched octaves as large 6:3 (due to that "beginning of a beat) , I was surprised to find very little raise in bps between 3ds and tenths for instance.

Afterthat I find it too tiring to use those comparative tests that imply 2 different intervals so I rarely use them.

You are hearing partials beating, not the fundamental.
Have your tech tune A-2 to A-3 as a 2-1 octave, then a 4-2 and then a 6-3.
Take a close listen and decide what sounds best to you.
Equal beating only is not real accurate by itself. It should be suplemented by using other tests. Say you tune your A-2/A-3 as a 4-2 octave, the 2-1 octave test should test wide and the 6-3 octave test should test narrow. You can also listen to chromatic fast beating intervals like tenths or sixths or thirds to help confirm if it fits or not.
You will likely find that a few compromises are made to get the best sound.

It isn't particularly, in itself. What I meant is this: to choose the test note such that its 5th partial is used, is a good choice, because the only partial it could be confused with, is the 10th - which is normally much less prominent than the 5th. Hence, the 5th is usually easy to identify and focus on, which makes the test less ambiguous.

Try it on your piano if you want. When you play A2-C3 and C3-A3, you will hear a beat at E5 (5th partial of test note) and you should also hear, probably more faintly, a beat at E6 (10th partial). If I understand correctly, it should be faster than the one at E5.

If the two intervals' beat rates at E5 are equal, that's a 6:3 octave. If the beat rates at E6 are equal, it's a 12:6 octave.

I noticed more than once that it was not always the case. They "should" but I lately heard that the partial series is too inconsistent to be used as a primary mean to regulate the octave width. Possibly the 2:1 test is the most efficient, then when tuning in the basses the consonance and progressiveness of other intervals are enough to me.

Kamin

Thank you for your observations.

I had been thinking about them (and ET v UT) when I came across the reference to Haye Hinrichsen's paper about entropy based tuning; the one I posted in the ETD v Aural thread yesterday.


The paper itself is very interesting. It is all about the compromises Gene mentions and I think it also supports much of what you and Alfredo say. I'll start a thread about it.

Entropy-based tuning of musical instruments

there are good comments on that MIT (?) web site. I was surprised it may be due that the people interested in piano tuning are not so much on the present forum.

The developer of the software I tested recently , Dirk, told me he use an entropy based algorithm too, he also state that the article is missing a few points from the puzzle (and that presenting that as "the end of the piano tuner" is not giving much credibility to the author ( )

I will edit that file I am uploading , where the way I listen to octaves or other intervals is heard. (mostly in the basses corrections) the file is 1 hour long so it may be too long to load it...
http://soundcloud.com/olek-4/vertical-piano-atlas-unisoning

To put the record straight, that sensationlistic comment about "the end of the piano tuner" came from the journalists, not the author.

Haye Hinrichsen, the author, added this comment below the article:
Thank you for reading the article and discussing it in here on the [MIT magazine] forum. I was not informed by the writer about the existence of this blog. In fact, I do not spell the end of professional tuning, instead my paper is primarily of academic interest. I wanted to understand why professional aural tuners, who produce 'dirty-looking' tuning curves with a lot of fluctuations, are often considered to be better than advanced electronic devices, which produce smooth tuning curves with the correct stretch.

Yes it was supposed, thanks for clearing that.