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Hi,

I read somewhere that :

https://prnt.sc/1s4wj50

You can use a P4 and P5 intervals within the same octave to test for a 4:2 octave. According to the author: "The test note is a P4 above the lower note and a P5 below the upper note in the octave. Provided the P4 is wider than beatless, and it has a faster beat rate that P5, the octave is wider than 4:2."

(1) What is the rationale that it will give a wider than 4:2 octave, but not a 2:1 octave ?

(2) Does it also mean that if we tune to the effect that both intervals are beatless and therefore equal, it will be a perfect 4:2 octave ? And why ?

Please enlighten. Thank you.

Last edited by Fazioli-Yang; 09/14/21 02:20 AM.
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Look at the coincidental partials involved in the beating.
The P4 beats (primarily) between the the 4th partial of the lower octave note and the 3rd partial of the test note.
The P5 beats (primarily) between the 3rd partial of the test note and the 2nd partial of the upper test octave note.

Therefore, in both cases the 3rd partial of the test note forms the beat:
... against the 4th partial of the lower octave note and
... against the 2nd partial of the upper octave note.

This is therefore a direct comparison of the 4:2 partials in the octave. So, in answer to your questions:
1) Provided the test note is correctly tempered (wide P4, narrow P5), then it is indeed a test note for the 4:2 octave, not the 2:1.
2) Consequently, yes, if both test intervals are beatless (or generally speaking, equal), then the octave is indeed a perfect 4:2.

(Personally, I prefer the M3-M10 test, as I find it easier to compare two moderately beating FBIs RBIs than two SBIs. (Test note is a M3 below the lower octave note.)

Last edited by Mark R.; 09/14/21 03:06 AM. Reason: corrected "test" to "octave" and "FBI" to "RBI"

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Originally Posted by Mark R.
2) Consequently, yes, if both test intervals are beatless (or generally speaking, equal), then the octave is indeed a perfect 4:2.

Should they be equal in the context of Equal Temperament ? If P4 is wide and P5 is narrow, P4 should beat slightly faster than P5.

This is where I'm confused.

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Yes, using 4ths and 5ths to DETERMINE the size of the octave is very difficult to do accurately and consistently. Part of the reason for this is that the 5th has two audible partials 3:2 and 6:4 with the 6:4 being quite often louder than the 3:2. If the 6:4 is mistakenly tuned as if a 3:2 one will quickly be wildly off the mark.

Many of us analog tuners use the 4th/5th quick test simply to get us into the ball park. Then we use the 3rd/10th test to analyze the octave like a micrometer. We can set it very accurately this way.

Furthermore, to test a 5th at the 3:2 (and not make a mistake) we use the 6th/10th test also like a micrometer. This completely avoids mis-interpreting the 6:4 for the 3:2.

Peter Grey Piano Doctor


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I test on fourths and fifths which are almost beatless. Fifths are slightly narrow, and fourths are slightly wide. This carries over for several octaves, and the amount that they are off doubles every octave. The important thing is that the 11ths and 12ths, and the 18ths and 19ths do not beat too much. You can tell if the octave is sharp or flat by which one beats too much, and of course, you can test with 3rds, 10ths, and 17ths.


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Originally Posted by Fazioli-Yang
Originally Posted by Mark R.
2) Consequently, yes, if both test intervals are beatless (or generally speaking, equal), then the octave is indeed a perfect 4:2.

Should they be equal in the context of Equal Temperament ? If P4 is wide and P5 is narrow, P4 should beat slightly faster than P5.

This is where I'm confused.

If an exact 4:2 is indeed the octave type you want, the stacked P4 and P5 should simply have the same beat rate, e.g. both beating at 1 bps. Not qualified by some "context", but simply equal -provided, of course, you listen for the beat only at that one pitch (*), namely
[partial 4 of the lower note] = [partial 2 of the upper note] ~ [partial 3 of the test note].
Remember, by definition 4:2 means that partial 2 of the upper octave note is exactly matched with partial 4 of the lower octave note. So if they have exactly the same frequency, this means that partial 3 of the test note must produce exactly the same beat rate with both.

(*) For my part, listening for a slow roll only at this partial is really difficult. I admire SBI-tuners for this skill. I've experienced exactly what Peter has described above. On P5s I often hear the 6:3 beat much more strongly than the 3:2, and I really cannot quantify how fast the 3:2 is beating (without an explicit M6-M10 test). So, my go-to test note is the M3 below the octave, which actually tests both the 4:2 octave and the P5 below the upper octave note, e.g. when tuning F3-F4 and then placing A#3 inside that octave.


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As you may very well know, the physical basis for ET is the "slightly" contracted 5th. This RESULTS in expanded 4ths, expanded M3rds, contracted m3rds, and expanded M6ths, etc.

Generally speaking, our current DEFINITION (or proof) of a "true" ET is gradually and evenly accelerating or decelerating M3rds and M6ths throughout the scale. If this is exhibited very well we call it ET. The ET requirements really do not specify octave widths, therefore one could theoretically have numerous "ET's" with varying octave widths (consistent of course) all resulting in smoothly rising 3rds and 6ths, yet anywhere from fairly noisy 5ths to actually perfect beatless 5ths and still call it "ET". The "equal" aspect refers to the basic fact that no musical keys are favored one over another (as in WT).

Most of us strike a balance where the 3rds and 6ths ascend evenly, and the 4ths and 5ths are pretty clean (although adhering to wide and narrow respectively) but they vary slightly, and octaves are pretty clean at either 4:2 and/or 6:3 depending on the scale and other issues. In my experience one cannot have perfectly rising and falling of ALL interval types simultaneously. Something is going to vary a little. It's up to us to decide what and where within certain good musical constraints.

Peter Grey Piano Doctor


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I think Peter describes the situation very accurately. "True ET" on an acoustic piano is not physically possible in regards to ideal beat rate progressions.

Even the 4th/5th test does not always reconcile with the 3rd/10th to set a "pure 4:2" temperament (for me this is usually F3-F4) octave. This means a "pure" octave set using the 3rd/10th test won't be the same "pure" octave set using only the 4th/5th test.

Some pianos seem to like a slightly wide 4:2 temperament octave, but some seem to prefer a pure 4:2 in order for the octave to sound clean. I believe this is due to the difference in in harmonicity between the strings of F2 and those of F3. F2 is close to the bass tenor break on small pianos (or even beyond it on some spinets and consoles) and may sometimes be the lowest plain wire. This means it is pretty slack with sometimes wildly sharp harmonics.

This lowest plain wire is probably the lowest tension string on the piano and this can really create issues when tuning temperament: As you descend to the lowest intervals thirds get slower than you would expect and there can be large discrepancies between the octave beats at the 2:1, 4:2, and 6:3.

Similarly, at the bottom of the temperament you can have a large discrepancy between the 3:2 and the 6:4 beats in the 5th. The 3:2 may be pure with the 6:4 being annoyingly busy. These are the main reasons people get so frustrated tuning spinets, but once you understand the phenomenon, you can make sense of them and get decent results.

In regards to the 4th/5th,, if you consider that the difference in inharmonicity between A3 and D4 is different than between D4 and A4 you can see that you may get slightly different octaves using the 4th/5th vs the 3rd/10th. I had noticed this sometimes when tuning but it took me a while to wrap my brain around what I was hearing.

In other words, you could have a 3rd/10th test proving a pure 4:2 octave, but that doesn't necessarily mean the 4th and 5th will beat the same. I've often detuned the A3-D4 4th to beat at 2 or 3 beats a second (to make it very easy to hear) and then set the D4-A4 5th to beat as close to that as I can and then check the 3rd/10th and it doesn't always come out the same!

So, back to what Peter is saying - you really need to listen from both perspectives and find the best compromise. Tuning is all about compromise, and it's one of the biggest weaknesses of ETDs is that they are generally uncompromising.


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Originally Posted by rysowers
So, back to what Peter is saying - you really need to listen from both perspectives and find the best compromise. Tuning is all about compromise, and it's one of the biggest weaknesses of ETDs is that they are generally uncompromising.

As I understand it, Pianometer is all about compromise. Here's what it says on the website:

Multiple intervals — Unlike other tuning platforms that set a “temperament” and then calculate the tuning outward with stretched octaves, PianoMeter independently calculates the ideal tuning of all 87 notes based on multiple interval relationships with other notes. Our algorithm finds the tuning based on minimizing the beat rates of:

• octaves (2:1 through 10:5)
• twelfths (3:1 and 6:2)
• fifths (3:2 and 6:4)
• fourths (4:3 and 8:6)
• double octaves (4:1 and 8:2)
• triple octaves (8:1)

Maybe Anthony Willey can say more?


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I'm sure the tunings that Anthony's software creates are fine and represent a good compromise between the various intervals. cool

HOWEVER...

IMHO ETDs operate on the fallacy that there IS one ideal pitch for every note on the piano. There are many subtle variations on a beautiful tuning and it's easy to let an ETD waste a lot of your valuable time tuning pianos that may not need a complete tuning.


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Ryan,

I'm trying to wrap my brain around what you've written about M3-M10 vs. P4-P5, in that they may give different results. If I understood you correctly, you said that it has to do with the inharmonicity of the test note.

Would you check my thinking, please?

Take the example of the F3-F4 octave. If you are tuning it as a 4:2 octave, you are adjusting one of the two notes, so that the two partial envelopes give a match (overlay) at F5. So, you need some independent source of F5, to beat against
(a) F5 as the 4th partial of F3
(b) F5 as the 2nd partial of F4.

If you use C#3 as the test note (M3-M10), you are only using its 5th partial, regardless of where its other partials are sitting.
If you use A#3 as the test note (P4-P5), you are only using its 3rd partial, regardless of where its other partials are sitting.
Seeing that you are only listening at F5, why would the inharmonicity of the octave notes or the test note matter?

In essence: if the F5 originating from F3 and the F5 originating from F4 show the same beat rate against one source, that indicates a pure 4:2 octave. How could they then show differing beat rates against another source of F5?

Last edited by Mark R.; 09/16/21 03:06 AM. Reason: minor correction

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Mark, there is nothing wrong with your thinking.

The only problem with 4th/5th test is that, - as P W Grey has described - there is the potential of hearing the 6:4 partials instead of the 3:2 when adjusting the 5th.

So the 3rd/10th test seems to be the safer bet.

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Hi,

I find it difficult to reconcile these 2 :

(1) In ET, P4 is narrow and P5 is wide, and therefore P4 must beat slightly faster than P5

(2) Using P4 and P5 to achieve a pure 4:2 octave, P4 and P5 with a common note between them within the same octave must beat equally.

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In ET, P4 is wide, P5 is narrow.

Last edited by Hakki; Yesterday at 04:29 AM.
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Originally Posted by Fazioli-Yang
Hi,

I find it difficult to reconcile these 2 :

(1) In ET, P4 is narrow and P5 is wide, and therefore P4 must beat slightly faster than P5

(2) Using P4 and P5 to achieve a pure 4:2 octave, P4 and P5 with a common note between them within the same octave must beat equally.

(1) is incorrect.
Firstly, like Hakki pointed out, in ET, the P4 is wide and the P5 is narrow. That is a given. If you want to know why, read up on the Pythagorean Comma.
Secondly, whether the P4 "must" beat slightly faster than the P5, is a different matter. It depends on the chosen width of the octave. If (if!!) a pure 4:2 octave is chosen, then the 4:3 P4 should not beat faster than the 3:2 P5. (I've tried it myself, but in most instances I find the 3:2 P5 too difficult to hear and adjust unambiguously anyway.)

The width of the temperament octave could be seen as a matter of taste, but can be chosen deliberately to best fit the inharmonicity of the piano. Often the temperament octave is chosen somewhat wider than a pure 4:2. In this case, the P4 would indeed beat faster than the P5.


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Mark,

In my experience it is not terribly useful to stress over just how much faster the 4ths beats over the 5ths. I am simply and only concerned with whether the 4th is in fact wide and the 5th is in fact narrow. Precisely how much does not concern me because it can vary for numerous reasons I have little control over, and my final objective is to achieve smoothly ascending 3rds and 6ths (not smoothly ascending 4ths and 5ths). My hat is off to anyone who can achieve ALL of that consistently. (If one works only on very nice large pianos [like Virgil Smith used to] then that may in fact be possible).

I trust you are familiar with the 6th/10th test to check the 5th at the 3:2 partial level? This is a very clear and accurate test, and in conjunction with the 3rd/6th test for the 4th you can "rarely" go wrong. The beauty of these tests is that they each share the important partial for that interval and there is no guesswork.

Peter Grey Piano Doctor


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