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#3143385 08/04/21 02:23 PM
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I have developed a new method for acoustic tuning to improve the harmony of Equal Temperament. The research is based on just intervals and it is still theoretical. The kernel of my theory is the stack of just interval 7 perfect fifths (7 * 3/2) and 1 major third (5/4). The result is perfect fourth and its ratio (1.3348388671875) is close more than enough the Equal Temperament. The mathematical precision is till 5th digit after the decimal point, the error is 0.00128 cents. JND is usually between 2-5 cents depending on professional experience. I'm suggesting a method to get equally tempered tones with unconditional accuracy. Repeating same stack 11 times gives a final error 11 * 0.00128 = 0.01408 cents. This is over 200 times better than any known tuning process. The full article and procedures how to achieve it is on following website. I would be grateful to share with me your thoughts.

https://nearequaltemperament.com/

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Boris,

Good that you've done a lot of math and thinking here. I trust you have already thoroughly investigated Well Temperaments as well as "the New Tuning", pure 12th tempering, etc. The important thing is how the piano sounds in a variety of styles of music. If you go ahead and tune a piano according to the rules you have set forth and play some music on it, record it and post it here, I'm sure you will get various comments from other experienced tuners (as well as musicians).

As many of us who have been doing this a while know, there is theory, and then there is practice. I look forward to hearing some of your work.

BTW, David Pinnegar in the UK also has a method based on 7 pure 5ths. Very interesting too.

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Could we see a video of you tuning this sequence, and then have you play various test and progressive intervals? Perhaps also a modulating piece of music?


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From the article:

"In other words, if the pitch changes with less than 3.5 cents, we won’t be able to notice a difference."

Hmmm.. Perhaps for tones played alone, such as a test for pitch memory? Those of us that tune know that approaching fine-tuning requires comparing intervals, getting to within at least 1/3 of a cent for each note of the expected goal.

Starting off your theory there with a bunch of piano techs probably won't get much support!

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Unfortunately I'm not professional musician and I can't tune any acoustic instrument.
The best I have done is creating a script based synthesizer for demonstration purpose.

https://ps-solutions.net/repository/ps-synth/ps-synth-12net_v1.31.zip

https://ps-solutions.net/projects/ps-synth/ - PS-Synth Project

The method I'm offering is just theoretical. In ideal world may be applicable because in terms of math it's fine. In practice may be not. I'm aware of all potential problems that could arise - resonance, inharmonicity, octave stretching, etc. - these are the biggest problems.

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A couple of points:

The "just noticeable difference" is defined for pitches played in succession. For two tones sounded simultaneously the human ear is far more sensitive. This includes not only unisons but also coincident harmonics of the two tones - hence octaves, fourths, fifths and even major thirds can be tuned to just intervals very accurately. Equal temperament intervals are set by recognising the beat frequencies in these intervals.

If I understand correctly, the basis of your method is the observation that the sum of 7 just fifths plus a just major third is extremely close to an equal temperament fourth.

Your methods for setting the scale using this fact are ingenious but I don't think really suitable for tuning a real piano. Tuning all those notes to be pure and then later retuning them to be (near) equal temperament wouldn't do anything for the stability of the tuning. Plus you are tuning a heck of a lot more notes than using traditional techniques. Using the fast stack you seem to need to accurately tune a stack of about nine just fourths, fifths and a third to set each note in the scale.

A little error correction - you state "See both numbers closely. They give an impression that the error of 5 is 7 times M3." Actually it's the other way round the deviation from just intonation is far greater the major third than for the perfect fifth.

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Originally Posted by Andonoff
Unfortunately I'm not professional musician and I can't tune any acoustic instrument.
The best I have done is creating a script based synthesizer for demonstration purpose.

https://ps-solutions.net/repository/ps-synth/ps-synth-12net_v1.31.zip

https://ps-solutions.net/projects/ps-synth/ - PS-Synth Project

The method I'm offering is just theoretical. In ideal world may be applicable because in terms of math it's fine. In practice may be not. I'm aware of all potential problems that could arise - resonance, inharmonicity, octave stretching, etc. - these are the biggest problems.

I read this after my previous reply. Yes, I totally agree. The theory is sound and it is theoretically possible to set a very accurate equal temperament scale using this scheme but unfortunately, in practice, probably not.

Last edited by Mr Dibbs; 08/05/21 05:41 AM. Reason: slight change of wording to be more friendly
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Just came up with another idea how to reduce the number of steps to 4. What about stacking these ratios 2 by 2?

For example:
9/8 (3/2 * 2) and 15/16 (3/2 * 5/8)
3 major seconds + 1 inversed minor second

https://nearequaltemperament.com/small-scale

Since the intervals are dissonant, is this possible?

Last edited by Andonoff; 08/16/21 05:07 AM.
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It is a remarkable result, that ET can be tuned using only just intervals and without counting beats. However when compared with modern methods, it is not very practical since it requires many more steps, and any errors will accumulate when tuning intervals sequentially.

I tried the fast inverse stack using the "Piano Tuning Game" simulator. You may find this app useful for testing out your theories in practice.

It turns out that you have independently rediscovered a tuning system proposed by Kirnberger (1766) and Lambert (1774). They were aware that (10935/8192) and (16384/10935) are extremely close rational approximations of the P4 and P5 in equal temperament.

https://web.archive.org/web/2021082.../mills_txt/_8000_onwards/msg____8437.txt

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Here is another mention of this tuning sequence in these forums.

https://forum.pianoworld.com//ubbth...retical-tuning-sequence.html#Post2301778

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Any system that claims accuracy of greater than about .2 cents is not in the "real world". I've studied this quite a bit, and it's even difficult to measure piano tone to .1 cent tolerance. Move your measuring device 6 inches one way or another and you can read .5 cents difference.

Don't build a chicken coop with a micrometer.

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Originally Posted by RonTuner
Starting off your theory there with a bunch of piano techs probably won't get much support!

Ron Koval

Yea. As a technician, after you've heard a couple dozen "latest and greatest" tuning and temperament theories you loose interest. They are never all that impressive and the time and effort spent to learn new tuning procedures is never worth it.


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Originally Posted by rysowers
Don't build a chicken coop with a micrometer.

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'Perfect!


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I think your approach is a theoretical one. But practically speaking, it is too long and too complicated. And you have to go through so many retuning, that eventually the errors you have created will be greater than what you are trying to correct.

Also, in your approach, you take the approach that tuning errors all adds up, but in practice it is not the case, as tuners adjusts to balance as necessary.

BTW equal temperament is not an harmonic objective. It is just a convenient way to balance several constraints.

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I disagree with this statement made in the article:

"When it comes to acoustic tuning, musicians rely only on their trained hearing because digital devices are not precise enough to get the right pitch."

This might have been the case 30-40 years ago, but it is no longer true, which is why most (2/3rds according to polls of PTG members) are now using tuning software, and the top programs such as Cybertuner and Veritune have been accepted as extremely accurate, especially in the outer octaves. The best tuners are probably capable of besting software, but most of us are not concert-level tuners.

The margin of error in the article is most likely exceeded by the false beating found in most pianos. It's also exceeded by the mechanical limitations of many pianos, especially those that suffer from high friction in the path of the string.


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