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I really don't know. I never play organ, I've never tuned one. I guess if you ask Mr. Capurso he will say CHAS is better.

For me all I can say is that if I tune the high treble of a piano with clean 2:1 octaves, it sounds flat to me, I wonder if an organ will have the same flatness when tuned with exact theoretical harmonic frequences.

Those are not from a piano; those are just some sounds you have put together.

In order to calculate the beat rates you must first know the inharmonicity constants for every note for the particular piano. So, how do you know what inharmonicity to use? If you use a zero-inharmonicity 42-ET model with the hope that the beat rates are correct, it will only be correct for one fictitious piano. How do you know it will work on both a grand piano and a spinet?

Also, lets suppose that you have a calculated beat rate spreadsheet. How do you then tune a note so that the beat rate is correct? What technique do you have to say that your beat rate is precise?

Like Gadzar, I don't believe that all ET semitones for a real piano are the same across the range of a real piano tuned either aurally or with an ETD.

Kent, you need to be more precise with real techniques and evidence if I am to take you seriously.

Thanks, this has been fun. This has gone about how I suspected it would. 8^)

In a world where our understanding of inharmonicity is complete, and we actually have the ability to measure inharmonicity for each note and calculate a tuning that takes into consideration each note's individual inharmonicity, then it is obvious that equal temperament is impossible.

And yet, I am employed every day to tune equal temperament, and other techs have been so employed at least since Montal a century and a half ago.

The concept that Virgil Smith called the "whole sound", and Brian Capleton calls the whole "soundscape" includes some effects that are not taken into consideration when one looks only at individual partials of individual notes.

I have spent the last year studying the tuning literature to see if I could deepen my understanding of tuning. I will present what I have found in a class in July. Presenting a class such as this is the only way I have after 35 years of tuning for me to have any chance of learning something new about the art/craft of tuning.

I would have thought that one event here would have been a tip-off that there is more to tuning than can be accounted for with individual partials: Kees wrote, "As it turned out nobody was able to actually tune a temperament octave with strictly progressive M3's, with the exception of Bernhard Stopper..." I doubt that it was an accident that Bernhard Stopper was able to do this.

I can hear people now, "How can there be 'more than individual partials'? That is impossible!"

Or maybe we just have more to learn about tuning. I gotta go work on my class, and then go tune some pianos in equal temperament!

I like that. partials only are ( often ?) a little off what we want.

We look for an optimal bloom in octaves, that mean depending where in the piano , different "sizes" will be used, but no partial is left aside doing so, we try to have all them right and do not cry if the octave have some activity, as long it sound as a pure interval.
I seem to use 2:1 and 4:2 more than a directly tuned 6:3 , that sound off to me.

2:1 , 4:2 6:3 are all tuneable directly by ear.

Checking with beat rates of different intervals is not as precise I thought for a long time, they are tools, mostly learning tools or control tools. Due to the fluctuation in beats strength and speed, how can I compare them precisely ? I just persuade myself they beat the same, a little faster:slower can be heard but not very little speed differences.

All:

I think we are missing the forest for the trees.

Pick an ET model, see what pure or equal beating intervals define it, apply those intervals to a tuning of an instrument with iH and the result will be a certain amount of stretch. For instance:

12th root of two is understood to be 2:1 octaves.

19th root of three is understood to be 3:1 twelfths.

24th root of 4 is understood to be 4:1 double-octaves.

and so on.

A mathematical model when applied to a real piano is just a way of saying what intervals are pure or equal beating.

I don't quite agree with Mr. Garlick. A piano can be tuned with pure 2:1 octaves or pure 4:2 octaves if it is recognized that these are actually two different intervals. And I think we must view multiple partial matches as different intervals, otherwise there is the possibility of an interval both increasing and decreasing in beatspeed at the same time. (Or being both wide and narrow at the same time. It would not always be possible to say whether multi-partial intervals are progressive or not.)

And in cases where a pure interval is tuned, an optimum way of dealing with jumps in iH is offered. For instance if a 4:2 pure octave definition of ET is used, which also means that stacked 4ths and 5ths beat equally, then 4ths and 5ths should be tuned to be as progressive as possible across a break. Without going into detail, this only holds true for test intervals that are wholly within the defined interval. The M3-M10 test for 4:2 octaves wouldn't work. "Making the M3s and M10s as progressive as possible across a break wouldn't work as well."

Do you mean the spreadsheet shows the odd interval beating slower as you ascend chromatically?

Yes. Here is my M&H BB tuning prediction based on its own iH.
Note that the M3s are progressive, but the P5s and P4s are not.
I should also note that the octaves from C3-E4 are basically 4:2 transitioning to 2:1 at C5. This is a test only. The stretch at C5 is +0.6 cents.

An interval is produced when we play simultaneously 2 keys in the piano. For example if we play C4G4 the intervel is a P5. In this interval we have two distinct beat rates namely 3:2 and 6:4. Some may say these are two different intervals but the truth is that there is only one interval with two different beat rates coming from two pairs of almost coincident partials. There is no way to play a 6:4 fifth without its corresponding 3:2 fifth.

When ascending chromatically we can ttune in a way to have one beat rate progressing evenly faster but we have no control over the secpnd beat rate. In a well scaled piano normally the second will follow the first one but when there are jumps in the iH then disruptions in the progression can not be avoided.

Plus a beat created by the rubbing between partials.

I have a nice table of those "phantom beats" they obviously interfere with the most prominent ones.

a 3!2 or a 6:4 fifth is not something with a meaning, to me, this is not even a 5th . A fifth is an interval with a slow beat.

A beat is periodic energy fluctuation whatever the number of composites it contains.

To hear them on a piano, go to a piano, tune a 2;1 octave (or hire somebody else if you don't know how), then listen (with your ears). For the 4:2 and other octave follow the same procedure.

Kees

2^(1/12) = 4^(1/24).

Kees

I don't follow you. If you tune pure 4:2 octaves then both tests P4P5 and M3M10 will show the 4:2 octaves are pure.

Maybe you mean that P4 will progress evenly while there will be a jump in the progression of M3s, don't you?

Good demo with very audible beats, I believe they can be hidden in a sort of bloom. for instance 4:2 2:1 balance gives a sort of platform where the 6:3 can be hidden (I believe it is an energy question)

Yep, I know. The one implies, and should be understood as, the octave being what is divided up into 12 parts, the other the double octave being divided up into 24 parts. that is how we can go from a mathematical, non-iH, model to an actual tuning on a real piano.

Yes, thank you, I my post was confusing. I should have said "Making the M3s and M10s as progressive as possible across a break wouldn't work as well." I will correct it.

(2^(22/3))^(1/88) then?

Kees

Human ear :

distortion approx : plus 1/4 tone on a 3 octave span, in soprano region.

There iH is helping us to hear that a piano is in tune

No. Please note that the 12th root of 2 and the 24th root of 4 are the same number, so they cannot be used to differentiate between 2:1 and 4:1.