# Normal Distribution

A question the other day by @espiegel123 on how to create a set of random values clustered around a mean led to this implementation of the Box-Muller algorithm. It generates a normal distribution (bell curve).

## I/O

Input Signal Range Notes
clk `0-1 gate` clock signal - unit produces a new value for each clock
mean `any number` mean value
std_dev `any positive number` standard deviation

Output Signal Range Notes
normally distributed random value `depends on mean and standard deviation` note that there is some truncation of tails due to 32 bit calculations

## Version History

Revision File Date Notes
1.0 Box-Muller Normal Distribution V1.0.audulus (18.5 KB) 02/25/2019 initial upload
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Thanks for this! What does standard deviation mean in this context though?

The standard deviation is a term from statistics that describes the width of the probability curve, or how widely distributed the random values are. A small standard deviation means the values are tightly clustered around the mean and a large one means the values are widely scattered. For a normal distribution, 68% of the values fall within 1 standard deviation, 95% fall within 2 standard deviations and 99.7 fall within 3 standard deviations. This is commonly referred to as the 68ā95ā99.7 rule. For other distributions the values will be different. With a standard deviation of 0.33 and a mean of 0, 99.7% of the values will fall between -1 and 1. With a mean of 0.5 and a standard deviation of 0.166, you will get 99.7% of the values between 0 and 1 with the curve centered on 0.5.

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Awesome, so itās almost like bandwidth of a BPF - great module to add to the utility collection!

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Itās basically the same idea as bandwidth. Like a filter, a normal distribution doesnāt ever completely go to zero. In the ideal case the possible output values actually go from -infinity to +infinity, but because we are using 32 bit math the width of the distribution is actually limited to 6.66 standard deviations from the mean. If you want to prevent the 0.3% of outliers > 3 standard deviations, you can always use a clamp() to truncate the tails early.

Yeah, the sqrt() expression is going to make the distribution asymptotic in either direction. But we can use this to remake the stored/quantized random voltages of the 266e Buchla Module?

Iām not sure exactly how the stored random voltages work in the Buchla unit. Iām debugging a counter for Mark at the moment, but Iāll check out the docs when I get a chance. The QRST module I posted has similar controls to the Buchla stored randoms. The QRST uses a biased random walk instead of a normal distribution so the value of the next step is influenced by the current step instead of being independent.

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Thx, interesting podcast

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I had trouble keeping up with the Hordijk stuff that was all laid out by @RudigerMeyer. Here we almost have a perfect introduction to westcoast synthesis, but not through Make Noise. You left us with no instructions and no example patch. This must be an homage to Don Buchla. I hope some smart people around here will wire up and share some patches that make use of the modules you offered in your āAn Uncertain Futureā bento box.

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I didnāt intend to be obscure. Like the original 266e this a source of randomness that you can potentially use to modulate just about anything. With the exception of the audio noise sources, which I included for convenience, all the outputs produce various sorts of random modulation signals. You could use them to modulate an oscillatorās pitch or a filter cutoff or anything else you would like. Iām not sure what you mean when you say:

The modules are fully documented. If there is anything in the moduleās documentation thatās unclear let me know and Iāll be glad to elaborate. I didnāt include a separate demo because I thought the clock and waveform displays were enough to demonstrate how each output behaved. I can put together a demo if you fell it would be helpful.
Even though itās modeled after one of Buchlaās designs, Iām not sure that anything in this particular set of modules could be said to be particularly āwest coastā since itās just a random source and west coast is really more about additive rather than subtractive synthesis,

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I added a demo patch to show some of the possible uses for the modulation signals but really they can be used to modulate anything that can be modulated. Think about the type of modulation you want. Slow and smooth or stepped? If you want stepped modulation do you want it quantized or not?

The quantized modulation divides the 0 - 1 range into between 2 and 24 steps and will only produce those specific output levels. Time correlation controls how far apart each output value can be from the last. At none each value is completely independent from every other one, at max each value can only be one step away from the last. The distribution controls the probability that a value will be towards the center (0.5) or edges ( 0 or 1). At flat all values are equally likely, at bell the distribution follows a bell curve with the center at 0.5 so a value is more likely to be near the middle than the edges.

The stored random output isnāt quantized. It can be any value from 0 to 1. When it receives a clock pulse it reads the skew knob and then adds or subtracts a random amount. The size of the random amount is controlled by the degree knob. Larger settings of the knob produce larger random modification ranges. The chaos knob control the distribution of random modifications. When chaos is set to low the random fluctuations around the skew value have a bell shaped distribution and when it is set to sign the distribution is flat (or completely random). Why Buchla chose those particular labels I have no idea, but I thought I would mimic his design since this is a clone of sorts. Hope that helps. Let me know if anything isnāt clear.

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Thats a great demo patch.

Well thatās was what I was after. I just meant that in using Maths, for example, there must be some interesting synthesis techniques other than modulating a cutoff. There is all this mystique around Buchla and whatnot so I thought you were about to uncover the the way in, so to speak.

• But if you read what you wrote above it is evident that there are already some complexities.

I wish! I was looking at a 259e ātwistedā waveform generator yesterday and was floored to see it cost \$1000 for just the one module. A 200e system 7 is almost \$30,000. The modules are undoubtedly well made, but the 259e is a pair of digital wavetable oscillators configured so one can modulate the other in a number of ways. Like many of his designs, itās an interesting idea but itās hard to see how its worth that kind of money. He has a different point of view for sure. Obviously some people think his stuff is worth the cost but I think you could come pretty close to the same sound with Eurorack modules that cost a whole lot less. I tried a Softube Modular emulation of the 259e, that is built in collaboration with Buchla and presumably runs similar code, and I canāt say I was particularly impressed. I ended up buying a āvirtualā set of modules from Mutable Instruments, 4MS and Intellijel instead. Much better value for money even in the virtual world. Softube offers emulations of the 259e and the 296e Spectral processor and like their real world counterparts they are two of the more expensive modules available

The thing about Buchla oscillator modules that makes them special to many is the special character of the waveshaping and the not-quite-sine. \$1000 for a 259e sounds like a bargain compared to what they have run in the past ā and I think the non-e 259 (which was fully analog and doesnāt have the wavetables) modules that are out there are even more expensive.

If one is enamored of Buchlaās waveshaping algorithms, the modules are really special. If one isnāt then there are probably better ways to spend oneās money.

I am but have to admire them from afar. Buchlaās oscillators take a serious investment in time to get a feel for them. A friend of mine who now has a nice Buchla setup thinks highly of the MakeNoise systems for getting into the same territory. At one time (before he invested in a Buchla), he put together a MakeNoise suitcase along the lines of the Music Easel and has great things to say about it.

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I have no doubt that Buchla makes a fine product, and in the final analysis, theyāre worth what someone is willing to pay. They seem overpriced to me, but then so does a Rolls-Royce. At the end of the day, no matter how finely itās crafted, itās just a car, I think there comes a point of diminishing returns when considering price versus performance, and IMHO, often very expensive products are desired because they are expensive, not because they are inherently better than less expensive alternatives. A digital module is a dsp processor, some DACs and ADCs, op+amps and some code. I honestly donāt see enough difference to justify the difference in price. Still, Itās an individual decision, and I suppose if youāre satisfied with your purchase, I guess it was a good one, I should probably add that Iāve never had the opportunity to demo a Buchla system in person, and living in the great synth wasteland of Kentucky, Iām not likely to, so Iām speaking on general principles rather than from personal knowledge and Iām not trying to be critical of Buchla systems or their owners

I think the Rolls-Royce analogy isnāt really quite applicable.

And, as I tried to make clear in my comment, some people find the unique sound of Buchla oscillators compelling. Others donāt. It isnāt a difference in spec or features. It is a question of whether one feels the sonic characteristics unique to Buchla designs compelling or not. Clearly, the unique thing about Buchla territory is not all that compelling to a lot of people. When I listen to some of the works by Alessandro Cortini or Todd Barton or Charles Cohen there are sonics that I find unique and compelling: crazily electronic yet with an organic character and which no other synth really has. It is a sound that really speaks to me. Iāve spent a fair amount of time with the Arturia Buchla soft-synth is pretty good but it doesnāt quite capture the waveshaping algorithm.

There would be no reason to buy a Buchla if that timbre-space doesnāt attract one. If it just sounds like another synth to someone (which seems to be the case with most people) then there wouldnāt be any reason to spend money on it.

(Btw, I donāt own a Buchla. I wish I could afford one. One of my main interests in Audulus is figuring out how to emulate the characteristics that really draw me to the timbres I find unique to Buchla systems)

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After your previous comment, I decided to re-evaluate my earlier opinion. I took another look at the Softube Buchla 259e which should be a fairly accurate reproduction of the hardware version, although Iāll grant you that there are undoubtedly subtle differences in the sound character. Itās probably as close as Iāll get to a real unit since theyāre way out of my price range as well. You were right when you stated that there was more depth to the oscillator than would appear from a cursory inspection. After spending more time with the module as well as modulating that which can be modulated in various ways, I was able to coax out some much more intriguing sounds than I had before. Fortunately the Softube version, while expensive relative to some of the other emulated modules, is still way more affordable than any hardware alternative. (at least if you donāt consider the cost of the Mac )
I still believe that his hardware is overpriced given his component costs, but as I said earlier, a fair price is really a matter of opinion. I saw where someone recently paid 3 million for a 59 Strat. Iām sure that reflects its collectorās value rather than any intrinsic quality as an instrument, but still it was obviously worth that to the purchaser.
As fas as better or worse is concerned, I try to avoid making too many value judgments when it comes to sound. Music is such an individual experience, that I think itās impossible to rank things other than from a purely personal perspective. As you say, āDo you find the sound compelling?ā The primary reason that drew me to Audulus was the ability to emulate the classic synths that Iāve admired since they were first introduced but was never able to afford. What keeps me interested is everything Iāve been learning about digital sound and most especially the conversations with this community of fellow users. While I have many local musician friends, none are particularly interested in electronic music, and itās really nice to have others with similar interests to talk to.

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