#mathemusic

[mathemusic-blog]

Entry 3: The Harmonic Series

Let’s start with knowing what it sounds like first this time, shall we?

After learning what intervals mean mathematically, we can finally learn what they sound like. The video shown above is the sound of the Harmonic Series, which is a pattern of intervals found in nature everywhere. To explain why the harmonic series is so important to the nature of sound and how humans hear it, I have to explain the basics of sound design. Feel free to skip this part if you’re already familiar with the harmonic series.

This is a SINE WAVE, a perfect version of the most common type of sound in nature (since, of course, no sound will be exactly like this, air pressure may be a few molecules off from perfect numbers). When playing this wave with a speaker, the y axis represents the position of the speaker and the x axis represents time, making a longitudinal wave like the one from last entry. When pushing and pulling something this is the pattern it tends to make, since you need to slow something down in order to reverse its direction.

So we know why sine waves are shaped the way they are, but why are they important? It’s important to know that when multiple different notes are played at the same time, they change the shape of the same wave of air. When two pitches, eg. 220 and 440 hz, the shape of the wave that it makes can be produced with the function y=sin(2pi220x)+sin(2pi440x), shown below.

–note– If you’re not familiar with math, the reason both of the sin waves have 2pi in them is because normal sine functions repeat themselves every 2pi values of x. We want 1 hz to repeat itself every 1 second, so all sine waves based on sound will contain 2pi in them so we can simply insert the hz value afterwards. The height of the sine wave (which represents the volume), however, is 1 by default. The volume can be changed by multiplying the sine wave by a number, eg. 2sin(2pi440x) gives you a sine wave with a volume of 2 that repeats itself 440 times per one space of x. –note end– As you can see, the addition of another tone changes the shape of the wave. This is important to know, because the math it comes with proves something very useful: every sound, natural or artificial, can be made by adding sine waves together. Every sound you hear, every leaf rustling and voice talking, is made up of a FUNDAMENTAL TONE, the loudest sine wave that defines the pitch of the entire sound, and the OVERTONES, the quieter tones that are layered above and below it in pitch to give the sound its shape. This will all sound very strange at first, since sounds of the fundamental-overtone type are combined into one whole sound by your brain and this concept isn’t meant to be completely understood by humans, so I will give a few videos to watch that will help show how sounds appear without some of their overtones.

Overtone singing | Additive synthesis | Wind instrument filtering

Now that what you thought of as sound has been completely destroyed and built back up, let’s go back to the harmonic series.

The HARMONIC SERIES is a set of all possible overtones that can be above the fundamental. Many different shapes of waves can be made just from different combinations of these tones at different volumes. This video shows how to make a few different sounds. The harmonic series, since it occurs so often, will be the core of our new system of music.

There are a few ways to identify the pitches in the harmonic series with math. The first and most obvious way is taking a fundamental tone, 82 hz in this case, and multiplying it by each number one at a time. The first pitch is 1*82=82, the second is 2*82=164, the third is 3*82=246, etc. This is the ABSOLUTE method of finding the harmonic series, meaning all pitches are produced from one fundamental pitch, the TONIC (which we can now define as the fundamental pitch all other pitches in a song are multiplied from). The second way is, starting with the tonic, each new pitch is obtained by taking the last pitch and multiplying it by an interval of the form (x+1)/x, eg. 82, 82*2/1=164, 164*3/2=246, etc. This is the RELATIVE method, meaning all pitches are produced from the last pitch calculated. These intervals taken from the relative harmonic series are the tools we will use to make a new system of music.

Name Interval Piano notes Octave 2/1 12 Fifth 3/2 7 Fourth 4/3 5 Major Third 5/4 4 Minor Third 6/5 3 (Unnamed) 7/6 between 2 & 3 (Unnamed) 8/7 between 2 & 3 Whole Tone 9/8 2 . . . Semitone 17/16 1 . . .

The table above is just like the one from entry 1, but I put the full list of intervals there. It is now obvious that this table is a list of the relative intervals in the harmonic series, with each new one becoming more and more dissonant compared to the one before.

Today we learned about the most pure type of sound, a SINE WAVE, and about how the FUNDAMENTAL tone that defines the pitch can havd OVERTONES added to it to shape the wave and make a completely new sound. We learned that the HARMONIC SERIES is a list of possible overtones to pick from, and that it can be calculated in an ABSOLUTE sense based on the fundamental TONIC tone or in a RELATIVE sense based on the last tone played.

So we have our intervals and we know what we're working with. Next time I will focus on how to use these intervals in what order, and how to position notes into chords. Until then, listen closely to what you hear, think beyond what you're conditioned to think, and thank you for reading.

[mathemusic-blog]

Entry 2: The Hertz System

If you’re the intuitive type, you’ll have a lot of questions from the previous post, especially: What about piano notes 6, 8, 9, 10, and 11? If you started on 7 and counted down, why did you skip 6? What does it mean for an interval to be in between piano notes? I plan to answer these questions and much more in this post. The focus, especially for visual learners, will be: Why isn’t music more symmetrical? If there are 12 notes in an octave, why aren’t 0, 3, 6, 9, and 12 the most perfect notes?

The first and absolutely the most hard concept is the one we’ll need to go through first, and it helps very much if you know how semi-log graphs work. A video tutorial will be linked here.

In this post, we’ll be introducing an different way of measuring music than piano notes: HERTZ, abbreviated Hz. This will sound a bit familiar if you’ve been in sound physics or even electrical engineering before. The Hz system counts the amount of times something repeats in a single second, in this case air waves. Sound, physically, is the difference in air pressure from the air that surrounds it. These sound “waves” go up and down in pressure a certain number of times per second, which is what the Hertz measures. If the air in a certain place, like your ear, goes up and down 400 times in one second, you hear the tone 400hz. The higher the hertz is, the higher pitched the note sounds. But, there’s a catch. –note– For those curious, the amount the pressure changes is the volume of the sound, but we won’t focus on that right now. Go here to find out more about sound waves (pardon the animation and potential nightmares). –end note–

The human ear hears sound in a unique way: EXPONENTIALLY. You can’t add 200 to the hz value and expect to go up an octave each time, eg. 200=c4, 400=c5, 600=c6. Instead, you have to multiply. To make an octave interval, you must play the starting tone eg. 400hz, and multiply it by 2, making 800hz. This means that an increase from 200 to 400 hz will sound like the same interval from 400 to 800 hz, only both will be higher. This may seem like a lot of information at once, so let’s look at a graph to see what’s going on.

Since the original note (A) followed by a number is western terminology to show which octave you're in, the higher the hz values (shown on x axis) go, the closer together they hear; an interval from 40 to 80 hz will sound the same as an interval from 10000 to 20000 hz. In other words, the higher the hz value rises, the slower the pitch seems to rise. –note– For those familiar with math, the growth is the interval to the power of x, or y=tonic*int^x, using the octave interval 2 and the tonic 200 as an example to make y=200*2^x. This will make values on the x axis equal to the amount of intervals (in this case octaves) higher (positive) or lower (negative) from the original (in this case 200), and values on the y axis equal to the hertz value needed to play at the same time. –end note–

The intervals from the last post play into this. Each interval is what you need to multiply your current tone by to obtain the hz value for that interval. The perfect fifth, for example, is 3/2. To get the interval a perfect fifth above the tone 440, we only need to multiply it by the interval, 440*3/2=660hz, to get the interval’s hertz value. The dissonance plays into this as well; it is simply equal to the denominator (underside of the fraction) of the interval. The dissonance of 3/2 would be 2, the dissonance of 9/6 would be 6, etc. –note– To learn more about tuning, I recommend this video. –end note–

The piano keyboard is a sort of SEMI-LOG GRAPH, making it look like the linear numbers are moved up so that how they look and how they’re spaced apart is more like how the human ear hears them. This means, though, that the intervals themselves are no longer perfectly spaced apart from each other. The piano’s intervals are not symmetrical. Let’s play a CHORD, notes on different intervals all played at once, and see how it looks on a piano as opposed to a hertz number line, showing where all the numbers are in relation to each other. The chord’s intervals will be 1 - 5/4 - 3/2 - 7/4 - 2 (which you should notice is the same as 4/4 - 5/4 - 6/4 - 7/4 - 8/4, I’m only counting up quarters in an octave), and the tonic will be 275hz (or C3 on a piano, the third C note), meaning the chord played will be 275, 343.75, 412.5, 481.25, and 550 hz all at once. Notice that for each hz value I’m only multiplying the intervals in order by 275. Let’s see what this looks like on a piano:

| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |___| |___| | |___| |___| |___| | |_ | | | | | | | | | | | | | | | | | | | | | | | | | | | |_____|_______|_____|_____|_______|_______|_____|_____| Notes Away 0 1 2 3 4 5 6 7 8 9 10 11 12 Scale | - | - | - | - | - | - | - | - | - | - | - | - | | | | | | Interval 1 5/4 3/2 7/4 2 Numberline | | | | | Heard as | | | | |

As you can see from the graph, it follows the same pattern as a semi-log graph. Intervals the same space apart will sound closer together to the human ear when pitch gets higher, but they will sound farther apart when the pitch gets lower. The hertz system SCALES (or adds numbers) LINEARLY, each value adding to the pitch so that intervals are evenly spaced apart from each other, while a system like piano notes scales exponentially so the human ear hears all pitches as the same distance apart, even though it makes symmetry sound very dissonant.

In this entry, we learned about the HERTZ system and how it SCALES LINEARLY, or adds numbers so that all numbers are the same distance apart from each other, as opposed to the piano which scales EXPONENTIALLY so the human ear can hear each note exactly the same distance apart from each other. A SEMI-LOG GRAPH is distorted so that numbers from a linear system appear higher up, showing what they will sound like to the human ear. A CHORD, introduced in the middle of the entry as an extra, is a bunch of notes played all at the same time.

Next time, we will take a rest after the massive hurdle we just jumped over and explain the intervals more thoroughly: what they mean, and what they sound like. The third entry will be the first one with actual sound; a bit rare for a tutorial about music. Oops. Until then, thank you for reading.

[mathemusic-blog]

Entry I: Introduction

Welcome.

This blog is meant to be a guide to Microtonal Music Theory, and not Western Music Theory (usually taught in modern music classes around the world, and basically every music class in Europe and the Americas). The western system has existed for a very long time, developing since the 1100’s when Roman Catholic churches began growing accustomed to thirds and fifths and becoming standard in the 1600s baroque era when harmonic major and minor tonality started to take over all other musical systems (see here). This 12-tone system is fine, but I think every artist should realize that the number 12 was picked from a list of possible other numbers. There are literally infinite systems, such as 66-tone (Arabic Maqam) or 6-tone (Indonesian traditional).

There is, however, one single system that ties them all together. One system that knows infinity and uses it. If music is composed in this system, a bit of math can average out all the numbers onto a system of any tones, including 12-tone. –Note– Alternatively, we can do today what the renaissance-era composers could not and make perfect music with this infinity-system with a tool only recently invented: computers. There are not very many computer programs I’m aware of that help create non-western music, except for a company known as H-Pi. Check them out if you’re interested and let me know if there are non-western programs you find somewhere else. –Note end–

This guide will teach how this infinite-tone system works, whether or not you have a background in music. Every term will be explained, and if necessary new terms will be made to explain more advanced topics. Beware, though. This system is as hard to understand as trigonometry or polynomials, so if you have a very hard time with math, you may get stuck. Let’s begin.

First, let me explain what 12-tone or 66-tone is. The western system, as shown on the average piano, has 12 notes all the exact same distance apart from each other. If you start on any given note, you can play up a scale (to the right on a piano, using the black and white keys) and end up playing 12 notes before you end up on a note that sounds almost the same as the one started on. This note is an OCTAVE away from the original note. It is known as an octave only because traditional western scales have 8 notes rather than 12 (on a piano, playing only white keys). On a 66-tone system, the octave will sound the exact same, but 66 notes will have to be played in order before the octave is reached.

An octave is an example of an INTERVAL, a note in relation to another note. Any note on a piano, for example, has a different note that is an octave away. This note can be found on a piano by counting up 12 white and black notes. An octave up from that note can be found by counting up another 12 white and black notes.

An octave is not the only interval, but it is the least DISSONANT. Dissonance is hard to explain at this point, but generally the more dissonant a note is, the more tension it holds. If you hear a song only use dissonant intervals it might sound very scary or exciting, or if you hear a song build up by using more and more dissonant intervals, you might feel a source of tension that is only relieved when the song goes back to less dissonant intervals. There is an infinitely long list of dissonant intervals, with the least dissonant being an octave and intervals increasing in dissonance more and more until the human ear can no longer hear the difference. Below is a table on the intervals, their DISSONANCE VALUE (higher numbers being more dissonant), and how many notes away -higher or lower- from the TONIC the interval is. (A tonic is the root note of a song, the note the song generally comes back to. In this case, the tonic is the note you start on.) Please keep in mind the piano notes are not exact. The only interval that can be perfectly played on a piano is an octave, and an instrument with a string rather than notes (such as a violin) may be more suited for exact intervals.

Name Dissonance Piano notes Octave 1 12 Fifth 2 7 Fourth 3 5 Major Third 4 4 Minor Third 5 3 (Unnamed) 6 between 2 & 3 (Unnamed) 7 between 2 & 3 Whole Tone 8 2 . . . Semitone 16 1 . . .

This may seem like gibberish at the moment, but if you have a piano or guitar, try playing the intervals and seeing how they sound (use frets instead of notes for a guitar, ignore keys being black and white). Get familiar with how the various intervals sound, and next tutorial will be much easier.

In this lesson, we learned about INTERVALS (including OCTAVES) or the relationship between two notes, generally the TONIC (the first note played) and a note above it. we learned about DISSONANCE and how more dissonant intervals can create tension, and we learned that every INTERVAL has a DISSONANCE VALUE. Thank you for reading, and I hope to see you again soon.

[swircle987]

(via https://soundcloud.com/polypulse/30030-whole-tone-scale-ambient-sound-through-platonopia?utm_source=soundcloud&utm_campaign=share&utm_medium=tumblr)

[swircle987]

This piece is a sonification of all of the possible filled Tic Tac Toe games, whether possible by the rules or not. In other words, you are hearing a particular sonification of every way to fill a Tic Tac Toe grid with X's and 0"s, assuming X always goes first. While teaching Algebra II one summer, a student asked if we could just play Tic Tac Toe all day. Of course, I said "No," but I did offer for us to "analyze the game of Tic Tac Toe all day!" As it turned out, we had to cover some mathematical probability anyway, so it turned into a really cool two-day lesson in probability - all focused on the game of Tic Tac Toe. As to the precise way in which I sonified these arrangements, I'll have to get back to you on that! It was several years ago at this point...but suffice it to say it had a whole lot to do with first converting every Tic Tac Toe arrangement into binary.