But unless the music also contains some 1Hz modulation (or a power-of-two multiple thereof) then the 432 base frequency isn't related to anything fundamental in musical terms. Speaking as a DJ, if you take a track and play it a little bit faster or slower it still sounds great or awful as at the default speed in most cases. 432Hz vs 440Hz is a ~2% difference, while DJ equipment commonly allows for +/-10% pitch variation so you can match the pace of different tracks smoothly while people are dancing. Only the very tiny number of people with perfect pitch find this disorienting to listen to. If there were really something special about the duration of the second and the base pitch relative to that, you'd have expected it to emerge from dancefloors years ago. In reality 432Hz is basically cargo cult numerology, something fun to think about when you are not having any success coming up with a kickass tune. And kickass tunes derive their quality from the relative rations of the note pitches, not from some absolute Magic Frequency.
Trust me on this. I really love numerology, sacred geometry and so on, and I try to integrate this into my artistic work regardless of medium. I would love for there to be some special key that would unlock the gate to cosmic/ biological/ quantum harmony and allow my artistic work to automatically echo the heartbeat of the universe. I'm a mystic by temperament and have been looking for such things my whole life. I would go so far as to say I have some religious faith in the significance of such things. But this ain't it.
Yup. Another example: in most of the world (where the television system is PAL or SECAM) movies are played on TV or released on DVD at 25 frame per second. But to achieve this frame rate, they take the theatrical film release at 24 fps and speed the video and audio up by 25/24th (4.17%). Of course almost nobody knows this because nobody notices it. https://en.wikipedia.org/wiki/576i#PAL_speed-up
For NTSC, the film release is sped down by a much smaller amount (0.1%) during the 3:2 pulldown to adjust from 30 to the exact 29.97 fps that NTSC needs.
This video claims that TBS is currently increasing the speed of Seinfeld episodes a whopping 9%, reducing a 25 minute episode to some 22.5 minutes.
Technology can work in your favor too!
Speed up: ] key
Speed down: [ key
Normal speed: = key
Also, some youtubers speak way too fast. Slowing them down helps comprehension a lot, I find.
Though, I can now instantly tell if I'm watching a PAL or NTSC version of Star Trek TNG based on the the first few bars of the opening song.
Which is weird, because I don't even have pitch perfect hearing and I'm not even that musically inclined.
I was actually thinking about a similar but less subtle example: I use a transit system that involves tagging on and off with an RFID card (Clipper on Caltrain). When you tag your card, it makes one of three different noises depending on what happen—and I couldn't possibly tell you which one corresponds to what. But if I'm actually using it and I make a mistake, I notice immediately because I'm so acclimated to the pattern it normally makes!
If I hadn't realized this, the UI for the system would have looked absolutely terrible. The different beeps are the auditory equivalent of "mystery meat navigation" but even worse because they don't carry any semantic meaning at all. But because I always use the system in the same way and the noises were consistent, it actually works really well even if I never consciously learned what noise corresponds to what.
(The fact that there are wrong ways to use the system is bad design, but it's a function of how the whole train system is set up, not the fault of Clipper's designers.)
I'd be really interested to read anything more concrete about audio affordances though, if anyone knows of links to further research, etc!
A fantastic book on how these are all related is "metaphors we live by". I'd expand on this but I'm typing on mobile and on a hurry. But seriously: read the book. It takes a long afternoon. It's great, short, illuminating, and not excessively dense or padded.
* One beep - successfully paid fare
* Two beeps (different) - successfully paid fare, and you're about to run out of money on your card
* Two beeps (equal) - tagged off (e.g. on Caltrain or San Francisco Bay Ferry)
* Three beeps - read error or insufficient value
There might be others. Indeed, they've introduced a quieter "here is a Clipper reader" beep to help visually impaired people locate them: https://vimeo.com/183916243
 e.g. http://www.sciencedirect.com/science/article/pii/S0010027715...
And in cinema they were better (I seem to remember Geoffrey Rush 'playing' Rachmaninov 3rd Piano Concerto in that film, at excellent pitch in the cinema), so I knew they were sometimes right. But i think the tape, TV and DVD formats I'll have rewound most often in perplexity.
Another confounding: living in California a lot and not having incorrect pitch on local copies. And nowadays, when sources are digital, bit may have originated from analogue recordings, it's now very unclear what to expect.
No. For NTSC they just drop (throw away) 1 frame from every 1000 frames, so there's no speed-up or speed-down in audio track.
Let's say you have 100 seconds of film material with corresponding audio duration. That would be exactly 2400 frames, because film is 24 fps. When you do 2:3 pulldown (that's one of the methods used to do telecine, i.e. film conversion to television format) you'll get back exactly 3000 video frames. But in NTSC format, (if you'll leave audio untouched, i.e. it's duration didn't change, it's still is 100 seconds) for 100 seconds of video duration you must display 2997 video frames, not 3000. So you just throw away 1 frame from every 1000 frames to get the exact 29.97 fps.
Doing this, at worst case you'll get audio/video desynchronization of 33 ms, which I don't think is noticeable (I tend to visually notice desynchronization when it's at about 100 ms or more). But if you'll drop that 1 excessive frame in the middle (not at the end) of the sequence of 1000 frames, you'll get audio/video desynchronization which only varies from -16 ms to +16 ms. Which is not worth the trouble of fixing this by going the other way around it: leaving video untouched and slowing down the audio by 0.1%.
A friend of mine bought the DVD of 'Rent' when it first came out in our region. A month or two after it came out, the DVD got re-issued because it's a musical, and people had been complaining that the conversion had been messed up. I think she got the re-release for free.
I could see making a case that Planck time is special enough for 440 vs 432 to matter, but it is so far off the scale of human hearing that its even integer points would form an effectively continuous scale when it comes to audio.
Hate to be a stickler, but: "...conductors of negligible cross-section and of infinite length." It's still one of my least favorite SI definitions.
- 1 Liter of pure water at 4 Celsius is 1 Kg.
- 1 Liter is a cube with 10 cm on each side.
All our music would sound wrong.
 of 4 measurements (761, 767, 771, 780) — the exact mean being 769.75
60 Hz is everywhere, often at a subliminal level, sometimes at an obvious/conscious level like when you walk underneath powerlines or use a garage-door opener. In any case I suspect it's a lot more obvious and perceptible to most people than any cosmic vibrations. I bet fixing your music to resonate with THAT, would (perhaps sadly) feel a lot more "cosmically in-tune" than what they're proposing with 432 Hz.
Then again, maybe it's just a number and our brains couldn't care less either way.
Meaningless in practice, but it makes it convenient to describe the tuning of, for example, a stringed instrument.
Main> map (\(x,y) -> (432.0 * (fromIntegral x)) / (fromIntegral y)) majscale
That's pretty cool. The whole major scale comes out as whole numbers in just intonation. (In 12-tone equal temperament it doesn't, because 12TET uses the 12th root of 2 as a uniform division of the octave.) This looks like your explanation isn't just a neat trick, it's likely to be the reason why 432hz was proposed as a standard in the first place.
Main> let scale = [(1,1),(16,15),(10,9),(9,8),(6,5),(5,4),(4,3),(45,32),(3,2),(8,5),(5,3),(15,8),(9,5),(15,8),(2,1)]
Main> map (\(x,y) -> (432.0 * (fromIntegral x)) / (fromIntegral y)) scale
..and if we construct a more chromatic scale, not all of them come out quite as whole numbers, but we don't have any repeating decimals or anything like that (though we would if we included weirder intervals like 8/7). Nice.
edit: This works for a just major scale constructed starting from A. If you construct a scale from a different root note, it might not work out quite as cleanly.
But really, the moment you decide thirds and fifths both matter, and you want to build them on more than one note, you have to either continuously re-tune or give up on JI and worry about temperament instead. The numbers work great in a Pythagorean tuning, where only fifths matter: that tuning gets you both A=432 and C=512 (middle C = 256).
Just to pick two:
45360/100 is not that far away either
55440 (a superior highly composite number, 2ˆ4 ⋅ 3ˆ2 ⋅ 5 ⋅ 7 ⋅ 11 ) would have a built in Base-10 mnemonic
Even introducing the factors of 5 creates centuries of complications (temperament) that caused us to finally give up and throw out all the integer factors, replacing them with the twelfth root of 2.
(Anyone who wants to reply saying that factors of 7 explain "blue notes", please be specific about how this works. I think this is a fictitious idea in music theory that propagates itself because it would be really cool if it were true, but the usual explanations produce false predictions.)
We know our bodies do follow circadian rhythms, so you have the initial link to length of day. Then everything else is split into sets of 12, a number with exceptional properties that natural selection may optimize towards:
"The number twelve, a superior highly composite number, is the smallest number with four non-trivial factors (2, 3, 4, 6), and the smallest to include as factors all four numbers (1 to 4) within the subitizing range. As a result of this increased factorability of the radix and its divisibility by a wide range of the most elemental numbers (whereas ten has only two non-trivial factors: 2 and 5, and not 3, 4, or 6), duodecimal representations fit more easily than decimal ones into many common patterns, as evidenced by the higher regularity observable in the duodecimal multiplication table. As a result, duodecimal has been described as the optimal number system."
Another thing is that our current system of setting the duration of a second as a constant may be non-optimal for some purposes. Other systems let the length of time units vary with the seasons. It is interesting to think that under these systems, by maintaining the same base frequency, the tone of the note would change over the course of the year and with latitude.
That's exactly why, actually. This is what the second was once defined as, but the problem is that the length of a day varies slightly - up to ~8s depending on the time of year and other seconds. As the article mentions, the amount of time that is defined as one second is now taken as an arbitrary fraction of the decay time for Cs-137 that is close the mean value of the length of the second defined relative to the time it takes the planet to rotate.
In terms of keeping time in our common lives it's not really a problem for there to be a little bit of ambiguity in how long a day/minute/second is, but this is important for things that rely on precision timing - like tuning in music. It'd also be crazy hard to keep track of how long a second is at any given moment, since some of the variation is random from tidal forces.
I think you could use something like this, which may be accurate enough:
It may be more accurate to use something like the JPL ephemerides to convert between the two though:
Finally, the current definition of a second is meant to approximate the previous definition of a second. I am saying the previous one may have some correlation with biological activity.
edit: Or perhaps you meant to tune variably, and not redefine the second. That sounds more sensible, but not as practical.
A day is defined as sunrise to sundown.
A night is defined as sundown to sunrise.
Every day consists of 12 hours.
Every night consists of 12 hours.
Each hour consists of 60 minutes.
Each minute consists of 60 seconds.
There are always 60*60*12 = 43,200 seconds each day and night.
Within a given day or night every second is of equal length.*
On the equinox (length of day = length of night) we will have a "base unit" s0.
At times/places when there is more daylight than that, the duration of a second
during the day s_d > s0 and during the night s_n < s0.
*It may be better to continuously vary the duration of a second so there
is not a sudden change at sundown/sunup. It can perhaps be linked to
https://en.wikipedia.org/wiki/Solar_zenith_angle. I think the Egyptians used
to use Sirius at night for this, but don't remember where I heard that.
Anyway now we can always be monitoring the position of the sun even when we
cannot see it at the moment.
Move to the equator? J/k, I would not think this system would be law, or used for all purposes. Despite the huge pain that dealing with timestamps from various locations can sometimes be today, I suspect my scheme would make it more difficult to synchronize long distance activities.
It would more be a chill, get back to nature, type of timekeeping. Used at resorts and rehab centers.
The only free parameters are:
F0 = the pitch of A4 on the when there are 12 hours of day/night
k = a scaling factor that determines how much to change the pitch based on duration of the variable second
I am saying the second was originally linked to length of day, so the calibrated clock should be used. Also, at some point the differences due to latitude changes, etc will be biologically indistinguishable from stuff like the effect of clouds.
How precisely is that 24.2 known I wonder..
By recording the daily rhythms of hormones and body temperatures in 24 healthy young and old men and women over a one-month period, the researchers conclude that our internal clocks run on a daily cycle of 24 hours, 11 minutes.
Researchers previously reported a range of 13 to 65 hours, with a median of 25 hours, 12 minutes. The variation between our subjects, with a 95 percent level of confidence, was no more than plus or minus 16 minutes, a remarkably small range.
The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.
Not sure of the numerological significance of 9192631770.
It has none. It just happens to make the new cesium-based definition of a second roughly equal to the historical definition of a second as 1/86400 of a mean solar day.
Frequency choices even then were driven by cost/complexity to the extent that e.g. rather than trying to synchronise things in other ways or decoupling various clock rates, the CPU clock rate of many computer models were chosen based on how other things were clocked.
The Commodore 64's and Amiga's at least were both clocked slightly differently in the US and Europe because the CPU clock affected bus timings, and bus timings affected the custom chips for graphics and sound, and it was most convenient to time it all to match the video refresh rate of NTSC and PAL respectively.
Given that, I'm not at all surprised if the BBC decades earlier found it more complicated to handle 439Hz given whatever clock signals they had available to create it from and/or wanted to be able to match it against.
(Technically nonsense, of course.)
I would take it a step further. I'd make up my own units based on seemingly natural things and then tune to numerically interesting values of those.
"This next song's beat is 432 times faster than a cricket chirping at 50 degrees C."
Edit: An exception would be trying to 'fatten' the sound of an acoustic guitar and using a 25-35ms delay, which puts it close enough that it's not perceivable as a delay, but will provide a noticeable effect on the tone of the guitar.
Presumably the music's timing is based on seconds, though, right? That might conceivably mean that primeness could potentially matter. I don't really think that it does, mind. But I'm not so quick to think that it doesn't, either.
I recognize that this bit is entirely personal, but I think that can make otherwise "simple" music more enjoyable. But my enjoyment of music is almost entirely dependent on the careful management of the "Wall of Sound" effect and by how much noise (in the form of timbre, attacks, number of different voices) is introduced or removed from the track. And I think that's the key difference between, say: a library piece, or a movie score; and pieces meant to be listened on their own.
Also, the expression "cargo cult numerology" pretty much made my day.
Tighter differences between each notes' frequencies??
Please explain this thought.
For instance, a half-tone from 432Hz is 457.69Hz, which gives out a difference of 25.69Hz; but a half-tone from 440Hz is 446.16Hz, which gives out a difference of 26.16Hz. Between 432Hz and 864Hz we can fit a whole scale, while in the same frequency range we'd be missing a note with 440Hz tuning.
Of course, this is a slight difference and it might not be noticeable at all. I personally don't hear much difference between the two tuning. Instead, I'm playing devil's advocate.
Then most songs would sound 'better' a half-step down, etc.
The boiling point of water in Celsius is 100 degrees, and the freezing point is 0 degrees; in Fahrenheit, 212 and 32, respectively. What's significant is the scale of each unit. 99C is colder than 211 Fahrenheit.
In the case of music, which is all about relative frequencies, establishing A determines the size of the steps. The above poster is proposing that A=432 yields more interesting frequency relationships than A=440. I have no idea.
Consider the beggining of Bach's "Little" Fugue in G Minor (BWV 578), where the subject of the fugue is first introduced by a soprano voice and then repeated by an alto voice, while the soprano voice does the coutersubject. I'd say that, played in isolation, the melody sounds better when sung by the alto voice. But lowering the whole Fugue by a half-step wouldn't make it much better.
> A=432 Hz, known as Verdi’s ‘A’ is an alternative tuning that is mathematically consistent with the universe. Music based on 432Hz transmits beneficial healing energy, because it is a pure tone of math fundamental to nature https://attunedvibrations.com/432hz/
Oh new age pseudo science. My mom loves this stuff and I never fully understood why. My sister gets angry at me when I point things out or level any criticism because "it makes her happy". This apathy to the issue from people who should know better is likely why nonsense like this keeps spreading in an age of Snopes and Google.
If family members get a positive placebo effect from listening to "5 Hours High Quality BEST 432hz Meditation Healing Chakra" on YouTube...Who cares?
It doesn't bother you when friends and family are sold on something that is baseless and are completely opposed to actually researching it and at the very least becoming more skeptical about it?
How harmful is it to meditate to 432Hz compared to... well maybe without that superstition they wouldn't even meditate at all. It's probably advantageous, even though it's bullshit.
As long as they aren't engaging in harmful behaviors (e.g. anti-vax), best to let it go.
It can also make them stronger, I'm not proposing you always ignore superstitions. Just ignore them when there's a cost and minimal payoff.
(Not something I worry about personally, parents long gone).
Personal example: As a musician, A=432 doesn't sound that different to me, not enough to make that much of a difference in tone. However I do like A=425, an arbitrary "in between" tuning on the flat side of the middle between A and Ab in A=440 tuning. It adds an entire new mood set for each key that is rather different than A=440, distinct enough to me where the moods don't overlap (whereas with A=432 it does).
There is no science that can explain this sort of thing that I know of, and I fully expect that others would have different interpretations.
Tuning is arbitrary. I've listened to some comparisons between A=432 vs 440 and I think the 440 actually sounds better. But then again, like this article even states, it depends so much on the piece. If you're listening to stuff from the 1700s/1800s, it would make sense to try to use the tuning they used at the time, if only to try to get the sense of emotion intended by the originally composers/arrangers/musicians (even then it's a best guess).
But yes, it is pretty arbitrary. It's the tuning of a note depended on the number of cycles per some arbitrary time unit we defined.
If you're listening to a song today, with all our sound precision technology, you're hearing it the way the original author/musicians wanted it to sound (depending on your equalizers, speakers and various other audio setup).
I'm curious how you can tell. A double-blind test spaced a few minutes apart (enough for you to forget the frequency so you couldn't tell if what you were hearing was tuned higher or lower than what you heard on the previous trial) would be interesting.
I play a lot at A=440, but I also play a lot at A=415 - mostly on the viola da gamba. A=415 is roughly one semitone lower, and it changes the way the instruments resonate. My viol happens to sound good at either pitch, but as an ensemble we find that when we're playing at A=440 the whole sound doesn't blend in quite the same satisfying way. This is due to the construction of the instruments themselves, but also just how those slightly lower pitches sound.
Oh and recorders built to A=415 just sound great. I think that slightly bigger body gives them the chance to produce a richer resonance, but then again you also don't find cheap plastic recorders (or cheap wooden ones) built at A=415 because only "serious" players go looking for those instruments.
One nitpick on the article though - historical tunings don't range from A=415, my recorder teacher has two flutes at what is sometimes called "French baroque pitch" - that's A=392!
The article had a pretty good explanation: that strings sound different at different tensions. Musical instruments tuned lower would sound darker, mellower, and "looser" than those tuned higher. The instruments would probably feel a little different to the musicians, who would probably play differently, especially if they'd just heard somebody expound on the mystical properties of a certain tuning. I'm not at all surprised that it would sound different.
On an FFT graph, you'll have the same fundamental and a very similar next few partials, but eventually you'll see (and hear) the results of the different tensions. Those differences represent what people like and don't like about different tunings -- nothing to do with the slightly different pitch between 440/442/438/432/etc!
Nonsense spreads not just because of apathy.
The problem with nonsense that purports to be true is that there's usually a scam attached to it, to separate people from their money. Fictional stories don't have this: everyone knows they're fictional before they spend their money on it, and they know what to expect from it (i.e. pure entertainment, not a cure for whatever ails you).
We understand Star Wars is space wizards. New-agers do not understand that their beliefs are mostly bogus.
Science is very good at describing things and why they're interesting, but science can't tell you what's beautiful or how to live. I understand your desire to point out the meaningless nature of setting up 432hz as some gold standard and agree with you that it's nonsense, but a spoonful of sugar helps the medicine go down' you could invest more time in exclaiming over the beauty and meaning of the relative pitch relationships in music, the most popular of which happen to nicely (albeit slightly inaccurately) mirror the ratios of planetary orbital periods. Your sister has the right idea in the wrong context, and you're missing out on some aesthetic insights by overlooking this.
Yes it can. If you feel otherwise, it is because you're ignorant of how science actually works and how general its base principles are.
Decisions made by the J.C. Deagan mallet instrument company at 1770 W. Berteau in Chicago are the sole reasons why A=440. I once worked in this building, still called the Deagan building.
The company made chimes and xylophones and instruments. They had customers all over the US. Each symphony in each city used a slightly different reference tuning. Philadelphia used 442, New York 438, etc.
This fact forced Deagan to retool their machinery every time a different order came in. This cut into profits. The owner, J.C. Deagan urged his customers to standardize on one reference tuning. One reference tuning meant one set of machine tools for his machines and no time or money spent retooling.
Deagan got results only when he contacted James Petrillo, the President of the AFM - the national musicians' union. Petrillo made it a work rule that A=440, overriding the conductors local preferences.
This was in 1910. Therefore there are no nazis involved.
Hey internet: don't be so goddamn gullible all the time, okay?
"Why do we use A = 440 Hz? (spoiler: no Nazis)"
For another example: http://www.mmdigest.com/Archives/Digests/199804/1998.04.05.0...
(And by the way, the Deagan building is awesome! Glad the whole Ravenswood corridor is getting redone and readapted.)
> Why do we use A = 440 Hz? (spoiler: no Nazis)
> I have no idea why this laughable article/idea persists
> Its central sin is that it obscures the mundane, industrial-based reasons for the tuning standard's emergence.
The article claims that tuning standards emerged because pitch inflation lead to problems for singers, is that inaccurate?
And yes, the tuning standard emerged from the wishes of the owner of a single mallet instrument factory in Chicago named Deagan.
I can say as someone who performs a lot of Baroque music (c. 1600 - 1750) that there is a relatively common standard of A=415 for music of this period. A=415 is often referred to as "Baroque pitch."
This mainly applies to ensembles that specialize in Baroque music, and use historical instruments. If you see your local symphony orchestra play Bach on modern instruments, they will probably play at A=440.
I have never performed at a pitch higher than 440. I'm sure it happens sometimes, but I think it's more of a niche thing.
One thing that gets opera singers (like me) up in a knot about this, is that the human voice is not a tune-able instrument. There are registration issues in fixed frequency ranges, which we can't change. At A440, a register shift happens a quarter tone lower than where Mozart and Verdi expected it. That creates a sound difference that an audience can hear, and a big difference technically.
If memory serves, a number of opera singers, including Pavoratti, were lobbying for a change of the international standard to 432. I thought this was a bit foolish, as the problem is better addressed by simply agreeing to adopt an alternate tuning for those pieces that would benefit from it.
Personally I prefer singing at A=415 since my voice is lower set. :)
This said I think it's more fun to talk about temperaments than absolute pitch, after all unless you have absolute pitch I don't think you can hear much of a difference between something played at 432 or 440, but going from equal temperament to something else is a lot more noticeable and makes a lot more impact to the music if it's written taking that into account.
In addition to varying by era of music, as you mention, this also varies by country. My violin teacher had done a lot of traveling and performing in different countries. She told me that, in one country, she was given criticism that her tuning was off in her solos. She realized that they were using A=445, and their ears were trained on that, so everything she played solo sounded flat to them. Once she discovered this, it was an easy "problem" to fix!
 I wish I remember which countries - I think Austria and Germany? I can't be sure. I'm almost postive it's not Russia, because I think A=435/438 is more common there - or at least was at the time.
 It's only really relevant for solos, because for a trained violinist playing in an orchestra, tuning is done by ear. So a professional violinist would adjust to the tuning used by the group without really thinking about it.
My father plays in a baroque orchestra (the Amsterdam Baroque Orchestra), and I know that when they play a repertory, e.g. Dieterich Buxtehude, the orchestra tunes to 465 meantone.
The regional variations depends a lot on orchestra politics. For example, when I played the munich phil in the late 90s they started at 443, but concert heat be damned, if the last chord wasn't 443 there were some yelling at the first violins and brass section for striving to much upwards.
The Swedish radio decided to play 441 for political reasons (it was a compromise that stuck), and I think still does.
(As a "bonus track" on the album, she included a realization of "Toccata and Fugue in D Minor." Especially appropriate for the season!)
Tuning is just a block of data on a digital synthesizer, but on an analog synthesizer it's not. If you use a keyboard controller, it's generating pitch control voltages with a fixed 2^(1/12) ratio between notes (or logarithm thereof). It would be challenging to convert this into a just intonation.
On the other hand, if the pitch control voltages are generated by potentiometers on an analog sequencer  and the musician is setting those potentiometers by ear, the musician will probably end up tuning to a just intonation because it sounds more correct to the ear.
Hideaway – Jacob Collier
I did not explicitly notice this the first time hearing it, but I certainly felt it. When I discovered the change it made sense to me. The mood is lifted over the course of the song as the tuning is also lifted.
There are so many elements of music that are routinely varied within a song — dynamics, texture, timbre, rhythm, key, and tempo (though less frequently during this age of the click track). I don't think I'd noticed tuning used prior to this.
I followed, until this. Earlier he debunks the significance of 432 per second (including that it's a sum of four consecutive primes), because a second is an arbitrary length of time. But now he says that 439 per second is difficult to generate electronically, because it's prime.
I'm not an EE. I'm willing to believe, but could a knowledgeable someone help us out here?
> The B.B.C. tuning-note is derived from an oscillator controlled by a piezo-electric crystal
that vibrates with a frequency of one million Hz. This is reduced to a frequency of 1,000
Hz by electronic dividers; it is then multiplied eleven times and divided by twenty-five, so
producing the required frequency of 440 Hz. As 439 Hz is a prime number a frequency of
439 Hz could not be broadcast by such means as this
Since 1000 Hz also was (is?) a broadcasted test tone, it makes sense that it is much easier to get perfectly matching 440 and 1000 Hz from the same high-frequency source than it is to get 439 and 1000 Hz. A more precise way of describing the problem is then that it's hard to generate since it doesn't share any divisors with other desired test frequencies.
If they only required 439 Hz they could have just used a slightly differently tuned source oscillator, yes.
If the choice is between a custom crystal and two custom boards, the custom crystal is going to be the far less painful choice.
Having all the broadcast frequencies controlled by a single reference crystal would have some obvious advantages (and a few disadvantages too, of course).
tl;dr: The BBC reference pitch was generated by dividing and multiplying a 1MHz crystal oscillator. A440 was achievable by this means; A439 was not.
However, when you are trying to build a device that oscillates at x Hz, there are engineering-ly sigificant values. For example (as already mentioned) a 1kHz oscillator might be easier to come by. As an additional example, the AC power in the UK is at 50 Hz, a factor of 440 Hz. So there is a free 50 Hz reference available everywhere, that can be multiplied (with a PLL, which is more robust to, say, temperature differences, than say, building a 50 Hz or 440 Hz reference yourself).
(50 Hz is just an example. AC line frequency probably is not stable enough in the short term to be a useful pitch reference)
Later, Hammond famously produced electro-mechanical organs referenced to line frequency.
Well, what do you expect from somebody simply following in the creative footsteps of Leonardo DaVinci & Thos. Edison?
I don't think it's a particularly stable reference in the short term. It's true that the power companies adjust it to hit exactly the nominal frequency over a period of many hours, but not at any one instant.
In fact: https://en.wikipedia.org/wiki/Utility_frequency#Time_error_c...
So, nope his claim doesn't make sense.
I wonder if there is some easy way of figuring out the simplest base number-multiplier-divisor chains for getting certain numbers?
In a 1985 letter, Dave [Packard] described Hewlett's audio oscillator as "the foundation on which Hewlett-Packard Company was able to grow into the largest manufacturer of electronic instruments in the world, the keystone that allowed four and one-half decades of major contributions to electronic measurement technology and equipment."
Or perhaps seconds are relevant in the way sound was broadcast.
An instrument will go ever so slightly out of tune by the time the orchestra is done playing. Temperature and humidity play a large role in how an instrument is tuned.
In the digital domain it's not an issue at all.
It is electronic gating to bring down a 1khz signal. Division and multiplication to a prime number is pretty hard...
I discovered that they were C, E, and G, where the A would have been 432 Hz (or very close to it; certainly much closer than to 440). Some internet searching turned up this phenomenon. Apparently ancient Hindus and Buddhists heard similar tones during meditation, and related them to various points in the body.
Maybe there is something to the notion that it's related to our neurology or physiology in some way.
What kind of code does one write to get the exact frequency of a sound in your head?
> and I think that if 432 Hz were some kind of a sweet spot, someone would have noticed by now.
because that's exactly what people who talk about 432 Hz are saying, that it's a sweet spot!
I dated an symphony oboe player for a few years. Oboes players play the note used to tune the orchestra before performances because the oboe has a penetrating sound and has very limited tuning capability (due to their construction) so other instruments are tuned to the oboe.
She said that the string players often wanted a higher "brighter" pitch than the true pitch for middle C. She had to carry an electronic tuner (about the size of a smart phone) to verify her pitch and resolve disputes with the some of the string players.
An equilateral triangle whose area and perimeter are
equal has the area of exactly the square root of 432.
3x = sqrt(3) * x^2 / 4
3 = sqrt(3) * x / 4
3*4/sqrt(3) = x
12/sqrt(3) = x
sqrt(3) * (12/sqrt(3))^2 / 4
sqrt(3) * 144 / 12
sqrt(3) * 12
sqrt(3) * sqrt(144)
I've always wanted to play heavy strings, so I also explored tuning down. Imagine my surprise seeing what Dimebag Darrell was tuning to for "Floods" on 'Great Southern Trendkill' - C# F# B E G# C. I had trouble getting intonation that worked well for me, so I went ahead and got a Les Paul style Baritone. Definitely a workout, and the opposite of going lighter, but I really dig the tone it puts out. Worth the hand cramps to practice on the 27" scale.
Unfortunately they sound terrible arco so now I have to play cello when I play in my wife's orchestra. :(
 it occurs to me that you're talking about electric bass... in which case nylon strings would be a bit strange ;) sorry about that.
Not at all, there are nylon wrapped strings for electric now too! 
I'd definitely be more inclined to trust in his musical intuition than his scientific aptitude though.
It's not legal to "convert" Hertz to other units willy-nilly, especially because Hertz is tagged with an attribute that says "this relies on arbitrary constants", meaning you would have to multiply 432 by some conversion factor before you can compare it to pure numbers.
Thinking about things in terms of types (in a fuzzy, intuitive way) like this is a very powerful mental shortcut that is useful surprisingly often.
By nature of the instrument, we had a lot of relatively mystical-minded customers, and a solid subset inquired about the availability of instruments tuned to a 432 Hz reference.
What was interesting is that there is indeed a difference in 'feel' (character, vibe...) for most acoustic instruments when you tune them down e.g. to an alternative, lower, pitch reference. (Because most aspects of the system are either non-linear e.g. our sensitivity to various pitches, and maybe some sort of psychoacoustic quasi-significance assignment... ... or because simply they are changing one term in a function, e.g. given a fixed thickness of vibrating membrane, tuning it to a different pitch yields a different relationship to the many 'modes' in the system akin to standing waves... etc etc etc)
There's nothing mystical about it all of course... but the vibe changes, and many people prefer a slightly lower pitch, and now it is 'carried' by the instruments.
Then you get fuzzy thinking... I like this better, it feels better, QED an alternative pitch reference is the key to life itself.
What's funny in my experience is that our specific instrument did not have a fixed equal tempered tuning anyway, for technical reasons... so even with a 432 Hz A reference, the pitch values for say C# or F# were not === what they 'should' be according to 432 Hz adherents.
But try explaining that gently to an adherent... especially in the context of kindly debunking their mysticism around this topic... <wince>
Worth noting that today's strings are made from steel and nylon, as opposed to gut.
While I appreciate Jakub’s breakdown for the reader to get to explaining why the 432 number itself is effectively fabricated for convenience, he eventually states: “_The 432 Hz tuning, the divine tuning of nature itself, is ultimately defined as one vibration per 21279240.2083 periods of radiation of an uncommon chemical element_” however he’s missing the critical argument.
What is missing from the argument is the irrelevance of the number. For example, if 432 Hz resonates with I don’t know, the neocortex, there is no relevance what the number is or that we’ve associated with what we humans currently call a time second — that’s just a standard way of identifying the number.
What needs to be done is validation or invalidation of this frequency in terms of “healing and soothing properties.” It’s also important to not do so defensively, otherwise you’re arguing against organized religion or Santa Claus or Unicorns — it’s a useless effort to argue against something that has no scientific evidence. If someone is making a claim without evidence in the first place, you’re arguing irrationally already and you’ve already lost. Reason requires logic.
“Here is my repeatable scientific evidence why I feel the world is flat” is different than “I have a feeling the reason the sky is blue is from unicorn tears — prove to me it’s not”.
It is my suggestion that if someone feels that 432 Hz has healing and soothing properties, they take this hypothesis and run it through the scientific method. It will be these published results that can be responded to, directly. A quick search on plosone doesn’t show that there’s any documented research in this area currently, a new opportunity for those interested.