Near-Equal Temperament

[andonoff]

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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 it's 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 share with me your thoughts.

https://nearequaltemperament.com/ (https://nearequaltemperament.com/)

[BertKoor]

[...] the error is 0.00128 cents.

Nice theory, but not applicable in practice. Do you have any idea to what precision real-life equal temperament instruments (such as a piano) are tuned?




Spoiler: two octaves from middle C the deviation from ideal 12TET is already 5 cents.
https://en.wikipedia.org/wiki/Piano_tuning#Stretch

[jancivil]

Yes, a 'perfectly' equal-tempered piano doesn't sound right, for reasons. IE: if you designed a machine to just nail it, just no. Piano tuning is an art, the science doesn't actually out.

[BertKoor]

Btw 0.00128 cents the difference between 1000.0 and 1000.00073938 Hz. It "beats" at 22 minutes.

[jancivil]

close enough for jazz

[BertKoor]

Close enough for Rock'nRoll

[BertKoor]

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

If the idea is to let humans stack the intervals, there will be a small error in each of these intervals which gets accumulated. The result of stacking a total of 8 intervals has accumulated 8 errors and will be quite imperfect.

Quote from the website regarding the JND (Just Noticable Difference) :

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

The most common method we use for comparing pitches is to use a 440Hz pitch fork and play the same A with the instrument. If two close frequencies are added, you hear the difference as a beating frequency. This is a natural phenomena that is easy to observe, no training or equipment is required.

I estimate our tolerance is a 10 second beating period which is a difference of 0.1 Hz. The difference between 440.0 Hz and 440.1 Hz is about 0.4 cents. To get an idea of how many significant digits you need, 440.09 Hz is 0.35 cents off and 440.11 Hz is 0.43 Hz off. Maybe I overestimated our tolerance and it's actually 5 seconds, then 440.2 Hz is what you get which is about 0.8 cents off.

Still close enough for Rock'nRoll (or jaath)

[askoan]

Noob here. I consulted the forward technique process. I'm not sure i understood the practical point.
Does the main application of the concept consists on which order an experimented human without electronic tool has to tune an instrument in order to get more accurate results ?

[andonoff]

Forward method is to understand the conception of 7 P5s + M3. The inverse one is much better since the overhead is reduced.
Inversed P4 = P5 + octave down.
Expressed by ratios:
3/4 = 3/2 * 1/2
Instead of doing P5 and then lower the octave, just tune the next "jump" tone by 3/4.
I believe the manual tuning is only for professional. I don't know sh*t about it.
Not sure about electronic equipment, may lead to intolerable discrepancy.

[andonoff]

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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/ (https://nearequaltemperament.com/small-scale/)

Since the intervals are dissonant, is this possible?

[BertKoor]

Just watch a piano tuner at work. I think you're trying to solve a non-existent problem.

[jancivil]

"if the pitch changes with less than 3.5 cents, we won’t be able to notice a difference."
speak for yourself

[jancivil]

For example:
9/8 (3/2 * 2) and 15/16 (3/2 * 5/8)
3 major seconds + 1 inversed minor second"
this goes back to antiquity
'Pythagorean tuning'

inverse of a minor second is a major seventh btw.

here's an example of a system which uses a small interval known as the syntonic comma (81:80):
...
forget typing, I'll post a picture:

22 sruti sys.jpg

this is one assessment of the 22 Srutis in Indian Classical.
Alain Danielou, who also assessed that within an octave are 53 useful intervals
(many of which are corrective ie., commas.)
again based in ancient knowledge.

Many things are possible, this "dissonant interval" notion notwithstanding

[DrEntropy]

This is really quite interesting, although I suspect not practically useful.

According to https://en.wikipedia.org/wiki/Schisma something like this was known to Kirnberger but not sure if it was used for tuning as it does not seem practical (tuning several 'auxiliary' fifths seems
more complicated then directly tuning flat fifths using beat counting)

The reason why the schisma might be involved is that going up 7 just fifths and 4 just thirds is equivalent (modulo octaves) to going up 11 fifths and down one syntonic comma. However, the Pythagorean comma is what we temper out when doing equal temperament, not the syntonic. The difference between the syntonic and the pythagorean is the schisma, only about 1.95 cents. Tempering out the entire comma for 11 fifths is too much, so we have to correct by the 12th root of the Pythagorean comma to get to ET, and that also turns out to be about 1.95 cents! These nearly cancel. Fascinating.

EDIT: https://arxiv.org/pdf/0907.5249. A paper that discusses this ....

[anomandaris1]

The reason why the schisma might be involved is that going up 7 just fifths and 4 just thirds is equivalent (modulo octaves) to going up 11 fifths and down one syntonic comma. However, the Pythagorean comma is what we temper out when doing equal temperament, not the syntonic. `.

Standard music theory and staff notation assumes infinite chain of fifths in syntonic temperament, not 12 equal or schismic. If we assume 12 equal or syntonic+schismic we get tons of different "enharmonic" equivalences and chord progressions not related to diatonic music.
29 equal is interesting, the error is similar to 12 equal, but in the opposite way -> major thirds are flat, minor thirds are sharp. It is schismic, so major third is not equal to two whole tones (two whole tones give a dissonant "Pythagorean ditone"). The error of fifths is around 1.49328 cents instead of 1.955 (as in 12 equal).