432 tuning fork

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Permalink Reply by Jens Mügge on September 25, 2010 at 6:20pm

Thank you Marco, for posting this topic here. I have heard that some instruments should be tuned in 432 hz - but you can tell me everything about music theory. Maybe Roberto Laneri or other composer and musicians would like to share their opinion. Interesting to read what they think about it.

What I think: I think it is nonsense to use a frequency of 440hz - but I can't explain it. I feel only that 440hz is not the frequency of our time anymore. Is that right that this frequency changed from time to time and that Bach composed on another basic frequency?

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Permalink Reply by Skye Løfvander on September 26, 2010 at 8:00am

To be on common ground it may be good idea to call for Wikipedia on Concert Pitch.
The amount of deviance in between the two tones 440 and 432 Hz is about one third of a semitone. Christopher informs us that it is about 1,8% which is correct as semitone step is a little less than 6%, but in music theoretic terms it would be more correct to express it in the cent system where the semitone value is 100 cent (the octave = 1200 cent). So we are talking about a microtonal difference of 31.8 cent (log(440/432)x1200/log2). Since this is a forum for overtone entusiasts it should cast some light also to mention that 54:55 is the proportion in question. The Pythgorean comma is close to 74:73.
Pythagorean tuning may be used for dronal and modal music but as soon as you move in to the field of polyphony and chord based music (roughly speaking the last 500 years of west european music) it cannot be used because of the bad dissonances in between some of the chords - which is due to the mechanism behind the above mentioned Pythagorean comma - 12 perfect fifths in succession (Pythagorean tuning) deviates from 7 octaves from the same basis with this value.
2^7 ~ (3/2)^12.
This disturbance of course occurs due to beating notes, so even though the Pythagorean comma is smaller than the concert pitch deviation, you have to consider that there is a difference in circumstance because there are other tones activated for reference. So 55:54 is not a very drastic deviance when dealing with a fixed point for concert pitch but would be a terrible dissonance to deal with inside a tonal system- as you can read in the Wiki-article history has seen much bigger differences in concert pitch definition.
Personally I prefer to use c'=256 Hz, but that is more from a philosophical point of view as this is eight octaves above the time unit 1 second. But like Marco points out you may question whether a time unit which is defined as 1/24x60x60 of a day has a very strong foundation.

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Permalink Reply by Marco Tonini on September 26, 2010 at 12:37pm

Thanks to all,
we know our body/mind is more affected by some frequencies rather than others, but in my opinion the problem is to establish the mathematical meaning of a pitch and the relation between number and time unit.
We can assign a lot of meanings on number 432, but we know 432 Hz and 432 MegaHertz are not the same thing, 432 meters and 432 kilometers are different sizes, likewise 432 byte and 432 Gb, or 432 celsius/fahrenheit/kelvin degrees, and so on...
So I think the question is: why does world use the second as unit of measurement? Is it a conventional unit?

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Permalink Reply by Skye Løfvander on September 26, 2010 at 1:28pm

Well, actually the answer to that last question has got to do with music.

When the Mesoptanians divided the circle in 360 degrees it was because their mathematics had a sexigesimal (60 digit) basis. Many of us still carry a remembrance of that advanced culture on our left wrist where the hands of the watch takes turns of 60 seconds, 60 minutes and 12 hours.
I call it great fortune that the circular display of time survived the more linear digital (13:09:43). Not only is it important to emphasize the cyclical aspects of time. The clock also gives us an intuitive knowledge of how to divide a circle into primary geometrical shapes:

Trigon: 0-20-40 minutes; Square: 0-15-30-45; Pentagon: 0-12-24-36-48,... etc.

60 has more factors (divisors) than any other number of that size... and if we go into it we realize that the old boys constructed a frame of understanding which also included music. The names of the individual lyre strings were related to the gods:

60: Creator/ Sky God Anu
50: Mountain God Enki
40: God of the fresh waters Ea/Enlil
30: Moon God Sin
20: Sun God Shamash
15: Venus/Ishtar
12: Mars/ Underworld God, Nergal
10: Jupiter, Bel/Marduk

If you take the length proportions on the basis of a fundamental tone 60 units, you will find the harmonic series 2-3-4-5-6 as lengths 30-20-15-12-10. And 60-50-40 is a minor trichord
Furthermore the octave 30:60 contains a heptatonic scale expressed in natural whole numbers composed by the three fundamental musical functions octave (2), perfect fifth (3) and just major third (5).

30-32-36-40-45-48- (50)-54-60 (2x3x5- 2^5- 2^2x3^2- 2^3x5- 3^2x5- 2^4x3- (2x5^2)- 2x3^3- 2^2x3x5)

Interpreted as lengths with 60 units as the fundamental that is the major scale, which we use today although in a different tuning system.

do-ti-la-so-fa-mi-(mib)-re-do

Conventional?
Convenient!

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Permalink Reply by Skye Løfvander on September 26, 2010 at 2:05pm

If you want a C-E-G harmonic trichord you should obviously not go for Pythagorean but for Just tuning where the major third is ... just!
The Pythagorean major third is defined as 81:64 whereas the just major third is 80:64 - or more simply and convenient (this is an overtone forum): 5:4.
So the just harmonic trichord is defined as 4:5:6 - do-mi-so. (the Pythagorean: 2: 162/64: 3)
The Pythagorean major third is 'way out of tune', 16.7 cent, one third of a semitone, and if the other tuning forks are played within the same session the deviation will be grave.
The deviation between Pythaghorean tuning and 12 TET is much smaller as long as we stay by the fundamental major tricord (tonic). The deviation of the fifth is only 2cent.
The problem of those two tuning systems is that the just (pure/ harmonic) third has been sacrificed - like to some degree also the sixths.

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Permalink Reply by Skye Løfvander on September 26, 2010 at 2:39pm

I forgot to give some references:
Ernest McClain:
http://www.ernestmcclain.net/
Richard Dumbrill:
http://mustradme.weblog.leidenuniv.nl/#Dumbrill

And then to go back to the original question about concert pitch which has becomed linked to the frequency of a':
Maybe overtone singers should not advocate the Pythagorean major sixth, 27/16, but the interval which is pure and comes beautifully to expression in the harmonic series as singable with overtones (which 27/16 is not):
So, let's make a new fraction for a'= 426,667 (the just/pure major sixth, 5:3, from fundamental c'=256 Hz
;-)

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Permalink Reply by Skye Løfvander on September 26, 2010 at 3:22pm

More Richard Dumbrill

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Permalink Reply by Skye Løfvander on September 26, 2010 at 4:25pm

I am not sure that I understand what piece of the puzzle you think is missing.
Maybe some Wiki pages about Mesopotamian sexigesial systems and measurements (?):
http://en.wikipedia.org/wiki/Ancient_Mesopotamian_units_of_measurement
http://en.wikipedia.org/wiki/Sexagesimal

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Permalink Reply by Skye Løfvander on September 26, 2010 at 4:26pm

I am a singer who occasionally perform for others, I am not into tuning forks but I suppose most of the following thoughts may apply to fork situations as well:
So when I tune my instrument, for instance my monochord (whose strings all have the same pitch so that is easy), what would I think of? Planetary frequencies?

It is a music instrument so of course what matters is the tension, material and diameter of the strings and how their resonance with the sound body with its proportions and material characteristics is facilitated. To put it shortly: I tune my instrument so that it sounds good!
Other considerations comes into question if I intend to sing a mode or scale where it does matter what function the drone has – so I also need to adjust the pitch of the instrument to that.

But the resonance of my instrument is only one level of a complex picture. I also need to consider at least two more levels of resonance involved:

1) That of human body resonances – being aware that body is obviously also a music instrument, a very complex one.
This perspective may again be divided in two:
- As I am a singer it is not enough that the instrument sounds good in itself it should also match the resonances of my voice optimally. This is not stagnant – my voice changes from day to day and depending on environment and situation.
- Furthermore I should somehow be aware of what you may call the resonance of the audience.
Both these aspects may be interpreted as having a physical perspective and a feeling 'soul' perspective.

2) Finally I should really consider that the room which lends its frames to my performance – however small it may be – is also an instrument with resonance characteristics. So actually I would say it is much more important to tune according to the acoustic space than any speculation about planetary resonances.

Planetary perspectives are great, specially when we acknowledge our own fundament: Planet earth!

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Permalink Reply by Marco Tonini on September 27, 2010 at 1:49pm

Sorry Christopher,
I didn't know you posted this: http://www.overtone.cc/forum/topics/time-and-frequency, it seems to me the same question...

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Permalink Reply by Skye Løfvander on September 27, 2010 at 2:58pm

Well, now I think I got what you are pointing at!
You are on a quest for some measure reference which may be easier to relate to for shorter time intervals than a day.
The metric system is somewhat comical as every time I need to weigh flour for a dough - a very common thing - I need to relate to the size of the earth (because one meter was defined as 1/40.000.000 of the circumference of the globe, and one kilogram was originally defined as the weight of one cubic decimeter of water... and they didn't even get the earth diameter right!).
But obviously all measures - time and space - in nature are subject to variances. The ancient measure systems got very confusing... so many versions of an inch because peoples thumbs are different.
I have also been on the quest and found some measures in the human body which hardly vary at all from individual to individual and during evolution. They are lengths, so you may convert it to wawelength and further to frequency.
You may find a presentation about this (in Danish, sorry!) here:
https://docs.google.com/present/edit?id=0Ae299qtf5h57ZDNtc3g4dF8yMj...
If you find some time unit which would be suitable for a common fix point - preferably rooted in nature I am very curious to hear about it!

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Permalink Reply by Marco Tonini on September 27, 2010 at 8:20pm

I ask again: why one second time unit is exactly one second and not a second plus or minus some istants? I don't find a cosmic/universal/heavenly reason. So in my opinion it is a nonsense to try to call a pitch with a number. Obviously it is necessary to compare a pitch with another, for example 100 Hz is higher than 101 Hz and so on, but how can I say 432 Hz is better than 440 Hz only thanks to mathematical arguments?
I can say 432 is a magic number because .... or it is a perfect number because .... or it is a divine number because ... but in this case we talk about Hz, that is in relation with an other unit of measure, the second.
All of these considerations are useless in comparison for example with the sectio aurea and others, ratio between numbers.

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