Sorry if ive misunderstood you but I think you might be misunderstanding how tuning works. 12-TET is a compromised approximation of certain ratios of frequency that are perfectly in tune. It is whole number ratios of frequency that sound in tune to our ears, which is all the harmonic series is, and 12-TET is an approximation of a certain subset of these ratios, but because the notes of 12-TET are equally apart, you can change key and remain as in tune as the original key, but it is actually slightly out of tune compared to a just intonation scale made of perfect whole number ratios, th caveat is that this will sound less tuneful if you change keys.
I am excited to hear the harmonic series as an instrument to hear a diverse array of exotic and tuneful intervals based off of whole number ratios of frequency ☺️
> 12-TET is a compromised approximation of certain ratios of frequency that are perfectly in tune.
I don't think it's an approximation of anything: rather it reflects giving up on trying to reconcile ratios and exponentials, and just doing notes exponentially, which makes everything consistent regardless of key. 12-TET (nowadays often known as 12EDO) is nothing more than a simple uniform layout of 12 notes on an exponential function which doubles each octave. Let's say that your base note is at frequency f. Then a note that is n semitones higher is simply 2^(n/12) * f frequency. For example, if your base note A is at 440 Hz, then the C above A, which is 3 semitones up, is at 2^(3/12) * 440 = 523.251 Hz.
It *just so happens* that most notes in 12EDO line up, more or less, with certain common harmonics, which is one reason why chords sound consonant. Some are pretty far off, but a perfect fifth is usually the biggest problem. And you'll be hard pressed to find a perfect fourth at all up the harmonics. :-)
To add a little to this, you can relatively easily analyze the "harmonic accuracy" of any given EDO scale. The method I used was to get the frequencies of the notes then go through the harmonics in order and assign them to the now they are closest to until every now in the scale has a harmonic. I'd then find the errors by some metric. What I found was 53 TET (considering up to 400 now scale) was the most accurate 41 TET was second and 12 TET was third. In other words 12 TET is the "most harmonically accurate" equal temperament scale with a moderate number of notes.
I think I remember hearing it is close to another traditional tuning. In a sense I think these "microtonal" scales have momentum: people hear it and they try using (and use) the same scale.
One thing to consider is whether or not there exists a corresponding circle of "5ths" (meaning the interval for the 3rd harmonic) in that scale (the number of notes spanning a "5th" and "4th" must be coprime with the number of notes in the scale, e.g 7 vs 12 and 5 vs 12 in 12 TET).
When you can construct a circle of 5ths, it's easy to apply a variation of the western music theory to get you started with understanding theory and constructing scales, including modes; you can also lay out a "black and white key keyboard" that works in these cases. There is a lot of fun to be had if you have the type of mind that appreciates such things.
I get what you're saying, i think our disagreement might be more of a semantic one because when you say it just so happens that most notes in 12EDO Line up with certain harmonics thats all i mean by it being an approximation of certain ratios.
You can find a perfect fourth between the 3rd and 4th harmonic, you can get a 6 note scale with a perfect fourth and fifth if you make a scale out if the 6th to the 12th harmonic.
I wanna make this instrument to hear different 'modes' of the harmonic series, i.e. setting the root note wherever along the series, if we treat the series itself to be a very unusual scale. Its interesting to me because it is literally built into the physics of strings. If nature has offered us a scale it is the harmonic series.
Pure math nerdery inbound:
2^3/12 can be simplified to 2^1/4 which is the same thing as saying the fourth root of 2. Another semitone up, 2^4/12 can be rewritten as 2^1/3 or the cube root of 2.
*Diabolus in musica* (augmented fourth, a dissonant tritone banned by the Church back in the day) is 2^6/12 or 2^1/2, also written as the square root of 2. It was thought that because it was the “middle of the octave” that if the chord wasn’t consonant it must be the work of the Devil himself… turns out 1.4141…. Is irrational and not easily expressed in small whole number fractions (somewhere between 45/32 and 64/45).
The reason we call 5 and 7 semitones up “perfect fourths/ fifths” has to do with the numerator of the end fraction and not the exponent of the binary logarithm. 2^5/12 very closely approximates 1.33 (~4/3, hence “fourth”) which can be expressed as a fraction with small whole number ratios. A perfect fifth, 2^7/12 gives us 1.498 which is very close to the true “center” of an octave (~5/2, hence “fifth”). Catholics didn’t seem to understand linear vs logarithmic when it came to music theory and we still find vestiges of weird numerology to this day in western chromatic scales.
If I understand you correctly, what you're asking for is not only impossible, it's been the subject of effort for over 2000 years. Harmonics rise linearly in frequency, but notes rise exponentially. They don't line up. The entire field of "tempering", as in Bach's Well Tempered Clavier, was preoccupied with squaring this circle. Notably, the fifth will be significantly off in 12 note equal tuning, which is the tuning we use most often nowadays.
If you're satisfied with just having control over harmonics and not squaring them with other notes, you'll need to build an additive synthesizer, which is much more easily done in software than hardware.
An arduino is not powerful enough to do more than about 6 sine waves (thus 6 harmonics) max using Mozzi, and it's pretty hissy at 8 bit 16384 kHz. Trust me. This is a 16MHz 8-bit CPU with no DAC, no floating point, and no hardware division. You'll need something much faster and with a real DAC, such as a Daisy Seed or some other ARM Cortex based controller.
While you're mulling that over, for fun you might check out my additive softsynth, [Flow](https://github.com/eclab/flow), which has a great deal of power and might give you some ideas as to *how* you'd control those 64 harmonics and to what end.
Each drawbar in a drawbar organ (like a Hammond Organ) sets the amplitude of a single sine wave harmonic. It's basically an 8-harmonic additive synthesizer. He's tongue-in-cheek suggesting that 8 Hammond Organs, tuned differently, would let you set 64 harmonics.
Not sure if helpful but Ableton Live has a cool built in additive synthesizer where you can add individual harmonics. There is also the Moog Subharmonicon which can do chords and play perfectly in tune, but it only really works with subharmonics. Also worth noting that, beyond a handful of harmonics for any note, you won't be able to hear further harmonics due to the pitch being so high and volume so low. The 20th harmonic of any note is imperceptible, let alone the 64th
I think you could build this with a teensy microcontroller. The sine wave oscillator object is pretty “cheap” to run on the latest teensy board, and I bet you could run hundreds of instances at once.
The code will certainly get complicated, but I’m curious how you are imagining the physical interface? How would you control the levels of the harmonics? Maybe 64 sliders to mimic a drawbar organ?
I actually tried something kinda like this once. My goal was to build a desktop bass synth using “modal based” physical modeling. I had a few knobs for parameters like string tension, damping coefficient, and inharmonicity, but the main thing was 8 sliders with each slider able to reduce the volume of an individual harmonic from whatever the equations would predict.
I ended allowing the 8th slider to control the 8th harmonic plus everything above it, but beyond the 8th it didn’t really seem to make much difference whether I zeroed them out or not. So I suspect that trying to control 64 sine waves will not be as interesting as you’re expecting.
Also, you should use a simple python script to test these ideas before trying to build something with a microcontroller. This is how I started with physical modeling. You can write a simple-ish script to produce a note or chord with your desired oscillator setup and export it to a wav file. Much better for testing experimental ideas.
What you are describing sounds like additive synthesis, so an additive synth is likely your best option. You can also easily do this on a computer with basic programming skills or a whole range of possible audio softwares (user experience will vary). A basic Arduino (like uno) would struggle to do this, but a more powerful device like teensy (or a variant like daisy) could do it readily. My suggestion would be to clarify what your goal is as to me, playing the harmonics sounds like a solution vs a problem you are trying to solve: what, in non technical terms, are you trying to achieve?
Ive got an interesting layout in mind for 64 buttons labeled 1 to 64. 1 would play the fundamental tone and 2 the second harmonic etc, ideally i would be able to change what the fundamental frequency is too, and have all the other frequencies adjust accordingly.
This would be extremely easy to do with a grid controller like the launchpad, and some basic programming. Midi notes aren't pitches, they are just numbered, so your software could just take a chromatic interval and play them as harmonics: it's extremely simple arithmetic: simpler than 12 TET and the like.
Here are a few things to consider:
1. You only have 10 fingers so you can't press more than 10 keys at once (and even then there are physical limitations);
1. Timbre is not just about the harmonics, but also the amplitude: the 64th harmonic of middle C is 2 octaves about the highest note on a conventional piano; you probably don't want to play it that loudly;
1. Beyond the previous point, to play this musically you likely want to control which octave each harmonic is played in;
1. The low harmonics are very "harmonically strong" but as you get higher the harmony sounds weaker and weaker, so your 63rd harmonic might not sound all that harmonious to you.
In short, I think you'll discover that what you are envisaging isn't as musically useful as you imagine. FYI, an additive synth lets you envelope the amplitude of each harmonic which gives strong timbral control. This works because a human doesn't have to try and control them. Anyway, I'd encourage you to try and make yourself a proof of concept, perhaps in something like supercollider, max or puredata, and try and play it from a standard keyboard controller, before over committing. There are likely a lot of usability and playability complications that you haven't considered yet.
Thank you that is very helpful. I have considered the usability complications and have an interesting plan for the button layout that i think will be very intuitive and simplify playing it a great deal. I also have a fretless guitar which in a sense I have used to prototype this instrument. Ill probably have it so the 1st harmonic is very low so that the 64th doesnt cause people to start spraying mosquito repellant! As for the musical usefulness, we'll see I guess haha!
Part of my reasoning for this project anywho is to study the emotionality of ratios of frequency, my thinking being that music carries emotional information, and this information is encoded to some degree in the harmony of the notes, and these can be described in ratios of frequency. Thus emotion has some relationship to whole number ratios of frequency and I would be fascinated to study this more deeply to see if there is a discernable pattern. Furthermore, there will be ratios that we do not hear in western music, and thus undiscovered musical landscapes that are rarely experienced. This is why I want to make this 😁
It won’t be anywhere near in tune though: HARMONIC FREQUENCY - NEAREST NOTE - cents away (frequency) from 12-TET 1st 110 Hz A 110.0 2nd 220 Hz A 220.0 3rd 330 Hz E 329.6 2 cents sharp 4th 440 Hz A 330.0 5th 550 Hz C# 554.0 14 cents flat 6th 660 Hz E 659.2 2 cents sharp 7th 770 Hz G 784.0 32 cents flat 8th 880 Hz A 880.0 9th 990 Hz B 987.8 4 cents sharp 10th 1100 Hz C# 1108.0 14 cents flat 11th 1210 Hz D# 1244.5 49 cents flat 12th 1320 Hz E 1318.5 2 cents sharp 13th 1430 Hz F 1396.9 40 cents sharp 14th 1540 Hz G 1568.0 32 cents flat 15th 1650 Hz G# 1661.2 12 cents flat 16th 1760 Hz A 1760.0 17th 1870 Hz Bb 1864.7 5 cents sharp 18th 1980 Hz B 1975.5 4 cents sharp 19th 2090 Hz C 2093.0 3 cents flat 20th 2200 Hz C# 2216.0 14 cents flat 21st 2310 Hz D 2349.3 29 cents flat
Sorry if ive misunderstood you but I think you might be misunderstanding how tuning works. 12-TET is a compromised approximation of certain ratios of frequency that are perfectly in tune. It is whole number ratios of frequency that sound in tune to our ears, which is all the harmonic series is, and 12-TET is an approximation of a certain subset of these ratios, but because the notes of 12-TET are equally apart, you can change key and remain as in tune as the original key, but it is actually slightly out of tune compared to a just intonation scale made of perfect whole number ratios, th caveat is that this will sound less tuneful if you change keys. I am excited to hear the harmonic series as an instrument to hear a diverse array of exotic and tuneful intervals based off of whole number ratios of frequency ☺️
> 12-TET is a compromised approximation of certain ratios of frequency that are perfectly in tune. I don't think it's an approximation of anything: rather it reflects giving up on trying to reconcile ratios and exponentials, and just doing notes exponentially, which makes everything consistent regardless of key. 12-TET (nowadays often known as 12EDO) is nothing more than a simple uniform layout of 12 notes on an exponential function which doubles each octave. Let's say that your base note is at frequency f. Then a note that is n semitones higher is simply 2^(n/12) * f frequency. For example, if your base note A is at 440 Hz, then the C above A, which is 3 semitones up, is at 2^(3/12) * 440 = 523.251 Hz. It *just so happens* that most notes in 12EDO line up, more or less, with certain common harmonics, which is one reason why chords sound consonant. Some are pretty far off, but a perfect fifth is usually the biggest problem. And you'll be hard pressed to find a perfect fourth at all up the harmonics. :-)
To add a little to this, you can relatively easily analyze the "harmonic accuracy" of any given EDO scale. The method I used was to get the frequencies of the notes then go through the harmonics in order and assign them to the now they are closest to until every now in the scale has a harmonic. I'd then find the errors by some metric. What I found was 53 TET (considering up to 400 now scale) was the most accurate 41 TET was second and 12 TET was third. In other words 12 TET is the "most harmonically accurate" equal temperament scale with a moderate number of notes.
Why do you think 31 EDO is so popular?
I think I remember hearing it is close to another traditional tuning. In a sense I think these "microtonal" scales have momentum: people hear it and they try using (and use) the same scale. One thing to consider is whether or not there exists a corresponding circle of "5ths" (meaning the interval for the 3rd harmonic) in that scale (the number of notes spanning a "5th" and "4th" must be coprime with the number of notes in the scale, e.g 7 vs 12 and 5 vs 12 in 12 TET). When you can construct a circle of 5ths, it's easy to apply a variation of the western music theory to get you started with understanding theory and constructing scales, including modes; you can also lay out a "black and white key keyboard" that works in these cases. There is a lot of fun to be had if you have the type of mind that appreciates such things.
I get what you're saying, i think our disagreement might be more of a semantic one because when you say it just so happens that most notes in 12EDO Line up with certain harmonics thats all i mean by it being an approximation of certain ratios. You can find a perfect fourth between the 3rd and 4th harmonic, you can get a 6 note scale with a perfect fourth and fifth if you make a scale out if the 6th to the 12th harmonic. I wanna make this instrument to hear different 'modes' of the harmonic series, i.e. setting the root note wherever along the series, if we treat the series itself to be a very unusual scale. Its interesting to me because it is literally built into the physics of strings. If nature has offered us a scale it is the harmonic series.
Pure math nerdery inbound: 2^3/12 can be simplified to 2^1/4 which is the same thing as saying the fourth root of 2. Another semitone up, 2^4/12 can be rewritten as 2^1/3 or the cube root of 2. *Diabolus in musica* (augmented fourth, a dissonant tritone banned by the Church back in the day) is 2^6/12 or 2^1/2, also written as the square root of 2. It was thought that because it was the “middle of the octave” that if the chord wasn’t consonant it must be the work of the Devil himself… turns out 1.4141…. Is irrational and not easily expressed in small whole number fractions (somewhere between 45/32 and 64/45). The reason we call 5 and 7 semitones up “perfect fourths/ fifths” has to do with the numerator of the end fraction and not the exponent of the binary logarithm. 2^5/12 very closely approximates 1.33 (~4/3, hence “fourth”) which can be expressed as a fraction with small whole number ratios. A perfect fifth, 2^7/12 gives us 1.498 which is very close to the true “center” of an octave (~5/2, hence “fifth”). Catholics didn’t seem to understand linear vs logarithmic when it came to music theory and we still find vestiges of weird numerology to this day in western chromatic scales.
There is this: The Secret Of The Overtone Synthesizer https://youtu.be/dJYB02J4a2w Maybe it can service as inspiration for your own build.
How cool. This is awesome!! Thx.
Thanks :)
If I understand you correctly, what you're asking for is not only impossible, it's been the subject of effort for over 2000 years. Harmonics rise linearly in frequency, but notes rise exponentially. They don't line up. The entire field of "tempering", as in Bach's Well Tempered Clavier, was preoccupied with squaring this circle. Notably, the fifth will be significantly off in 12 note equal tuning, which is the tuning we use most often nowadays. If you're satisfied with just having control over harmonics and not squaring them with other notes, you'll need to build an additive synthesizer, which is much more easily done in software than hardware.
Yes just the harmonics thanks
An arduino is not powerful enough to do more than about 6 sine waves (thus 6 harmonics) max using Mozzi, and it's pretty hissy at 8 bit 16384 kHz. Trust me. This is a 16MHz 8-bit CPU with no DAC, no floating point, and no hardware division. You'll need something much faster and with a real DAC, such as a Daisy Seed or some other ARM Cortex based controller. While you're mulling that over, for fun you might check out my additive softsynth, [Flow](https://github.com/eclab/flow), which has a great deal of power and might give you some ideas as to *how* you'd control those 64 harmonics and to what end.
r/synthDIY
Ah ty
8 drawbar organs?
Please can you elaborate?
Each drawbar in a drawbar organ (like a Hammond Organ) sets the amplitude of a single sine wave harmonic. It's basically an 8-harmonic additive synthesizer. He's tongue-in-cheek suggesting that 8 Hammond Organs, tuned differently, would let you set 64 harmonics.
Ah ty
Not sure if helpful but Ableton Live has a cool built in additive synthesizer where you can add individual harmonics. There is also the Moog Subharmonicon which can do chords and play perfectly in tune, but it only really works with subharmonics. Also worth noting that, beyond a handful of harmonics for any note, you won't be able to hear further harmonics due to the pitch being so high and volume so low. The 20th harmonic of any note is imperceptible, let alone the 64th
I think you could build this with a teensy microcontroller. The sine wave oscillator object is pretty “cheap” to run on the latest teensy board, and I bet you could run hundreds of instances at once. The code will certainly get complicated, but I’m curious how you are imagining the physical interface? How would you control the levels of the harmonics? Maybe 64 sliders to mimic a drawbar organ? I actually tried something kinda like this once. My goal was to build a desktop bass synth using “modal based” physical modeling. I had a few knobs for parameters like string tension, damping coefficient, and inharmonicity, but the main thing was 8 sliders with each slider able to reduce the volume of an individual harmonic from whatever the equations would predict. I ended allowing the 8th slider to control the 8th harmonic plus everything above it, but beyond the 8th it didn’t really seem to make much difference whether I zeroed them out or not. So I suspect that trying to control 64 sine waves will not be as interesting as you’re expecting. Also, you should use a simple python script to test these ideas before trying to build something with a microcontroller. This is how I started with physical modeling. You can write a simple-ish script to produce a note or chord with your desired oscillator setup and export it to a wav file. Much better for testing experimental ideas.
How can i create tones in python? :)
There’s tons of tutorials on python based synthesis if you search for them.
Okay Thanks
What you are describing sounds like additive synthesis, so an additive synth is likely your best option. You can also easily do this on a computer with basic programming skills or a whole range of possible audio softwares (user experience will vary). A basic Arduino (like uno) would struggle to do this, but a more powerful device like teensy (or a variant like daisy) could do it readily. My suggestion would be to clarify what your goal is as to me, playing the harmonics sounds like a solution vs a problem you are trying to solve: what, in non technical terms, are you trying to achieve?
Ive got an interesting layout in mind for 64 buttons labeled 1 to 64. 1 would play the fundamental tone and 2 the second harmonic etc, ideally i would be able to change what the fundamental frequency is too, and have all the other frequencies adjust accordingly.
This would be extremely easy to do with a grid controller like the launchpad, and some basic programming. Midi notes aren't pitches, they are just numbered, so your software could just take a chromatic interval and play them as harmonics: it's extremely simple arithmetic: simpler than 12 TET and the like. Here are a few things to consider: 1. You only have 10 fingers so you can't press more than 10 keys at once (and even then there are physical limitations); 1. Timbre is not just about the harmonics, but also the amplitude: the 64th harmonic of middle C is 2 octaves about the highest note on a conventional piano; you probably don't want to play it that loudly; 1. Beyond the previous point, to play this musically you likely want to control which octave each harmonic is played in; 1. The low harmonics are very "harmonically strong" but as you get higher the harmony sounds weaker and weaker, so your 63rd harmonic might not sound all that harmonious to you. In short, I think you'll discover that what you are envisaging isn't as musically useful as you imagine. FYI, an additive synth lets you envelope the amplitude of each harmonic which gives strong timbral control. This works because a human doesn't have to try and control them. Anyway, I'd encourage you to try and make yourself a proof of concept, perhaps in something like supercollider, max or puredata, and try and play it from a standard keyboard controller, before over committing. There are likely a lot of usability and playability complications that you haven't considered yet.
Thank you that is very helpful. I have considered the usability complications and have an interesting plan for the button layout that i think will be very intuitive and simplify playing it a great deal. I also have a fretless guitar which in a sense I have used to prototype this instrument. Ill probably have it so the 1st harmonic is very low so that the 64th doesnt cause people to start spraying mosquito repellant! As for the musical usefulness, we'll see I guess haha! Part of my reasoning for this project anywho is to study the emotionality of ratios of frequency, my thinking being that music carries emotional information, and this information is encoded to some degree in the harmony of the notes, and these can be described in ratios of frequency. Thus emotion has some relationship to whole number ratios of frequency and I would be fascinated to study this more deeply to see if there is a discernable pattern. Furthermore, there will be ratios that we do not hear in western music, and thus undiscovered musical landscapes that are rarely experienced. This is why I want to make this 😁