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Link please.

I see the beauty but also a flaw. F3D4 is closer to G#3C4 than to G3B3 even on low IH instruments and more so on high IH instruments.

You are correct in stating
F3A3 < F3D4 < A3C#4
if the "<" mean less than, but if this equation means that the beatrate of F3D4 should be the average of the beatrates of F3A3 and A3C#4, it is not true. It should be closer to the beat rate of A3C#4 than to the beatrate of F3A3, not in the middle.

Theoretically speaking, if you ignore this subtlety you will end up with non progressive M3's. That's OK because no-one can tune progressive M3's aurally anyways, but something to consider.

Another thing that comes to mind here is that you have stated a number of times that you have measured human abilities (tolerances) for beat rate progressions detection, which you have. However you measured the ability to detect beat rate differences for beating sine waves, not real piano tones. Clearly the real accuracy will be lower, possibly by a large factor.

I don't intend to "shoot you down" and knowing beat rate discriminatation for beating sine waves is obvious better than knowing nothing at all, but if you claim you have "proved" beat rate discrimination limits and don't mention these caveats you should expect to be challenged on this.

The value I see in your pursuit of determining "objectively best octave size in the temperament octave" is as follows. On a good day you just listen and find the best octave. On a bad day you can't decide and would like to have some method to do what you'd do on a good day. If you had an ETD you'd just measure IH and let the software figure it out. It usually does a good job, which is why 90% of PTG member tune electronically

If not what you can do is to listen to beat rates of m3's and M3's and see how much off they are from what would be predicted from a simple no-inharmonicity model using a twelfth root of 2 relation between pitches which Alfredo incorrectly thinks people use.

By listening to this you can get an idea of how much IH you have under your fingers, and then, using the data collected from your survey you can make a recommendation for a 4:2, 4:2/6:3, or even a 2:1/4:2 octave. For BDB: the symbol ":" does not indicate division here as you learned in elementary school, it has a different meaning.

Kees

I'm sorry you do not value the peer review approach. This is all the garbage before the final product. For anyone else reading these posts and threads with the same frustration, please keep this in mind.

As for garbage piano - they're what we tune. This are my typical day to day pianos and I still want to tune them the best I can.

Microphone - does not change partial frequency, only strength.

Unisons - yes, that is a valid concern and a requirement for the next test.

I don't see this relationship as flawed. It is only stating a relationship between beat speeds with no implications about scaling and ratios, and using a "window" to constrain the choice of beat speeds to tune another interval.

Going on however, there seems to be too much emphasis on the use of RBI checks as a means of tuning. I think we should not diminish the importance of listening to intervals themselves for purity or otherwise and then use checks as confirmation.

The only pure intervals on a piano are unisons which does not help much to set the temperament.

The relationship is flawed or not depending on the meaning of the "<" symbol as I tried to make clear.

Kees

Yes I see the beauty. In fact I'm curious and I'll give it a try. The success of this sequence will depend on the ability to tune congruent octaves, M3s and M6s.

To avoid going too high, where M3s beat too fast, I will tune C#3 in addition of the usual CM3s. This means this sequence may be apropriate to tune pianos with the bridge break and top most wound string below C#3, to avoid jumps in iH.

Here is the sequence:

A4, A3, F3, F4, C#4 as usual, progressive CM3s,

then tune:

C#3 to C#4 as the best sounding octave possible, check C#3F3 fits in the progression of CM3s.

A#3 to C#3, C#3F3 < C#3A#3 < F3A3, check the fifth A#3F4: C#3A#3 > C#3F4

D4 to F3, F3A3 < F3D4 < A3C#4, check the fifth A3D4: F3D4 > F3A4

F#4 to A3, A3C#4 < A3F#4 < C#4F4

F#3 to F#4 as the best sounding octave possible, check F#3A#3 fits in the progression of contiguous and chromatic M3s, check the fifth F#3C#4: A2F#3 > A2C#4

D3 to D4 as the best sounding octave possible, check D3F#3 fits in the progression of contiguous and chromatic M3s, check the fifth D3A3: F2D3 > F2A3

B3 to D3, D3F#3 < D3B3 < F#3A#3, check the fifth B3F#4: D3B3 > D3F#34

D#4 to F#3, F#3A#3 < F#3D#4 < A#3D4

G4 to A#3, A#3D4 < A#3 G4 < D4F#4

G3 to G4 as the best sounding octave possible, check G3B3 fits in the progression of contiguous and chromatic M3s, check the fifth G3D4: A#2G3 > A#2D4

D#3 to D#4 as the best sounding octave possible, check D#3G3 fits in the progression of contiguous and chromatic M3s, check the fifth D#3A#3: F#2D#3 > F#2A#3

C4 to D#3, D#3G3 < D#3C4 < G3B3, check the fifth F3C4: G#2F3 > G#2C4, check the fifth C4G4: D#3C4 > D#3G4

E4 to G3, G3B3 < G3E4 < B3D#4, check the fifth A3E4: C2A3 > C2E4

G#4 to B3, B3D#4 < B3G#4 < D#4G4, check the fifth C#4G#4: E3C#4 > E3G#4

G#3 to G#4 as the best sounding octave possible, check G#3C4 fits in the progression of contiguous and chromatic M3s, check the fifth C#3G#3: E2C#3 > E2G#3, check the fifth G#3D#4: B2G#3 > B2D#4

E3 to E4 as the best sounding octave possible, check E3G#3 fits in the progression of contiguous and chromatic M3s, check the fifth E3B3: G2E3 > G2B3


The size or stretching of the octaves must be uniform. Each time we tune an octave we must check with the apropriate tests M3M10, P4P5, m3M6, M10M17, etc. that we are tuning coherent or congruent octaves.

Thanks for all the comments. I will look through them all more deeply.

Uh, I think that was a peer review.

"Does anyone else see the beauty of this approach? - We have tuned a wide P4 by exactly the amount that iH requires in order to have progressive RBI."

I don't think this is anything new. I had noticed this long ago and found it inadequate. As you continue through what can be described as a "down a third, down a third, up a fourth" sequence, you find a point where the newly tuned notes produce intervals with the first notes that were tuned. At this point (the ninth note...?) you find out that what you thought produced a tempering of the SBIs of "exactly the amount that iH requires in order to have progressive RBI" was just wishful thinking.

If you want to prove otherwise, post a video of this sequence for "peer review."

Today I tuned a Wendel & Lung grand, model 178 (5'11"). A nice piano. I tried the sequence I posted above. After 25 minutes fighting the piano to set the temperament I gave up and used the old good ET Via Marpurg of Bill Bremmer. In less than 10 minutes I've got a nice ET.
It's difficult to only use octaves, M3s and M6s avoiding tuning fourths and fiths! I got lost when I found the first bad sounding fifth and tried to correct it by retuning the last tuned M6. A horrible mess!

I tried something similar earlier this year. I figured if progressive M3s and M6s is a barely achievable, but still clinical, definition of ET, why not tune and listen to only these two intervals? I got the temperament as close as I could and the SBIs were horrid!

It was worth trying, and a lesson I won't forget!

Because we are human, we can "think" a M3 is faster or slower than a neighbour, or in my sequence, within a beat speed window, but our ears can be fooled. The SBI provide a safeguard - a second, and more importantly, different way to listen.

True!

In my sequence I guess the problem is the compounded error as I tune each note to the previous tuned note.

Compound errors with an RBI sequence??? How could it be??? SAY IT AIN'T SO, JOE!

I mean I can tune a good set of CM3s C#3, F3, A3, C#4, F4, A4

And then another good CM3s chain D3, F#3, A#3, D4, F#4

But the windows C#3F3 < C#3A#3 < F3A3 and F3A3 < F3D4 < A3C#4 are insufficient to guarant the second chain is "in tune" with the first one.

The same happens with the 3rd and 4rth chains os CM3s.

The Sanderson/Baldassin solves this by directly tuning fourths and correcting their tempering until they fit in the set of CM3s and octaves previously tuned, constructing a nine note mini temperament.

ET Via Marpurg avoids this error compounding by tuning temporary pure fourths and fifths that put the ball in the park and then refining their tempering with equal beating techniques used in P4s and P5s.

Both sequences are "self correcting" opposed to "error compounding".

Another aspect concerning accuracy when estimating different beat rates is that for me it's really easy to detect a mistuned interval if I play a run of chromatic alike intervals. That's what I do to refine my tunings, I play a run of chromatic fifths or fourths or octaves or M3s or M6s or M10s or P12ths or M17s and I can discern very small differences between them. I do this to check the quality of my unisons also! I listen to the center string right string unison and rhen compare with the center/left strings unison. They must sound the same. It's really easy to detect the smallest deviation. I also play a run of chromatic notes looking for unisons that sound "different" and stick out like a red light.

That's how the nine note mini temperament in the Sanderson/Baldassin sequence is tuned: you play a run of 5 chromatic M3s, a run of 4 chromatic P4s and you have also 2 chromatic P5s. It's really easy to detect an interval which sounds different.

I am just not able to achieve that level of accuracy when hearing/tuning at a M3/M6/CM3 "window". The partials involved are different and distant. For the first M3 we listen to the 5th partial of the lower note beating against the 4th partial of the upper note, then against the 3rd partial of the note a M6th above and then we have to listen at a different and distant pair of 5th and 4th partials for the CM3. I believe this is the main problem with this "window" technique.

I am glad you posted this, Rafael. For some reason the phrases "compounded errors" and "self-correcting" always rankle me. I now know why. They are misleading terms. The errors do not compound, they accumulate. And intervals are not self-correcting, there is just a maximum error that they can be off and pass a test.

It is like interest on a loan. If it compounds you owe interest on the interest. If it accumulates, interest is only charged on the principle. Likewise, if when tuning interval after interval, say with a 1 cent error, after 5 intervals the error totals 5.0 cents, not 5.1 cents.

And CM3s do not self-correct to 0.0 cents. They can be an entire cent off and still be definitely progressive. Someone could then counter with the argument that these errors then "compound", but really they, too, only accumulate.

Self correcting comes when you get informed you are wrong, by how much and in what direction.

In a P4/P5 sequence you must tune all and every note to see if the circle closes and you have no info about what you did! Even if the circle closes you are not sure your intervals are good. Errors can not only add but cancel out previous errors. Such a sequence is not self correcting.

In a CM3s sequence you start by tuning an octave that is presumed correct. And you have to tune only two more notes to close the "circle". If you tune a too wide third another third will be too narrow! It's immediate. You get informed at the next step. It is telling you you are wrong, by how much and in what direction. You can correct the tempering of these CM3s until they fit in the octave. In the Sanderson sequence you make a similar self correcting tempering for the fourths if your 4ths are too wide or to narrow you'll get informed in the next step and you'll know how to correct them.

In the ET Via Marpurg sequence, errors can not add/cancel. As soon as you tune a bad interval another interval shows you your error.