Music Theory for Conworlders: Simple Scales from the Overtone Series

Previously I described the overtone series. Let's use it to build our first scale!

For ease of calculation, we'll start at 50hz: 

We notice that for each octave we go up, the number of tones increases:

50 100 ← first octave
100 150 200 ← second octave
200 250 300 350 400 ← third octave
400 450 500 550 600 650 700 750 800 ← fourth octave

the overtones of a 50hz tone, given in hertz, presented in successive octaves

1 2
2 3 4
4 5 6 7 8
8 9 10 11 12 13 14 15 16 

 the multiples that give the overtones, presented as successive octaves 

We pick the 400-800 (or 8-16) octave simply because it's the first octave in which there's 'enough' tones for a reasonable scale (four-note scales for some reason generally aren't considered much in scale design - it seems most scales you find in human cultures consist of five or more). We use that as a model for every octave from now on, so we end up having this:

50 56.25 62.5 ...

100 112.5 125 ...

200 225 250 ...

Since the difference now actually is a constant number of hertz, we get a scale represented by adding a constant to the enumerator of a ratio:

8/8 9/8 10/8 11/8 12/8 13/8 14/8 15/8 16/8

overtones from the eighth to the sixteenth, tuned down until they're in the first octave 

And the intervals between subsequent notes will shrink:

9/8, 10/9, 11/10, 12/11, 13/12 ...

In fact, the first step is almost twice the size of the last step. Here's a short two-voice composition in this scale, although from C to C rather than G400 to G800, 'Overtone ditty':

So, how would this kind of scale appear in real life? Consider the string or the tube of air in organs, flutes etc. If we construct frets linearly, so that there's one fret at roughly half the string, and the rest of the frets are equidistantly along the neck, you'll approximate overtone tuning fairly well. With tubes of air, the same roughly applies: half the length gives an octave, then make pipes that are equal divisions of that length. Overtone singers also easily can produce overtone scales, and overtone-instruments such as overtone flutes.

Of course, we needn't use 1/1 as our 'root', we can use some other pitch as our 'root': we might start from, say, 550 - and get 550 600 700 750 800 900 1000 ... in which case the 'steps' will not just shrink - we get a large step between 800 and 900, from which a new sequence of shrinkage appears for a while. The scale will start out as a series along the lines of 11/11 12/11 13/11 14/11 15/11 16/11 18/11 20/11 22/11.

Here we go on using Overtone Ditty, but I tune down the fundamental so that the root for the melody keeps being C. However, the starting point - the 8/8 - from which the overtone series is calculated is reduced by one each iteration (except the last that returns us to C as both the root and fundamental):

The first iteration was the same piece as previously; in the second part, whenever the first part went 8/8 9/8 10/8 ... the new one went 9/9 10/9 11/9; in the third you got 10/10 11/10 12/10; the fourth even went as far as 11/11, 12/11, 13/11 ...; It was a bit tedious to retune the parts, so I didn't go any further than that, I hope the different sounds of the four parts illustrate pretty well what's going on.

I might later compose a short ditty illustrating some of the next few ideas.

We could of course build a similar rational scale using any denominator, and not be restricted to exactly 8 (or even 2ⁿ, n ∈ Z) pitches - any number, really, could work. Let's look at the set we get with the series from 7 to 7*2:

7/7 8/7 9/7 10/7 11/7 12/7 13/7 14/7

This would correspond to starting out by halving the length of a string or pipe, and then placing frets or producing new pipes (or adding holes) at equal distances between the full length and the half-length.

For each such scale, we can also investigate and use the scales we get from using some other tone than the original N/N as our starting point and introducing a slight "discontinuity" in the sizes of the intervals, such as:

8/7  → (1/1)' (i.e. 8/7 is now the new 'standard' for the previously given scale; 'new' is marked by an apostrophe in the ad hoc notation I now am using)

... 7/8) 8/8 9/8 5/4 11/8 3/2 13/8 7/4 2/1

(You can see how the interval size consecutively shrinks until you hit the distance from 7/4 to 2/1. Repeated shifting along this procedure will of course move the discontinuity rightwards.)

A few examples then, would be 7/7 to 14/7, 5/5 to 10/5, 6/6 to 12/6. I've thrown in 16/16 to 32/16 as well to give an example of a rather crowded scale.

To understand what happens with our scale (given as a series of numbers), it's simply that we multiply each interval by the inverse of the number we want as our new root (so, if in the series from 7/7 to 14/7, we now want 8/7 as our root, we multiple each with 7/8), and those that now are less than 1/1, those that drop 'under' our new root, will reappear from an upper octave. Mathematically the same result is obtained by multiplying by two any interval that is now lower than one - 7/8 becomes 7/4, 8/9 becomes 16/9, and thus remains in the scale. We could do the same for 5/4 and the whole 7/7-to-14/7 series:

5/4  → (1/1)'

... 4/5 9/10) 5/5 11/10 6/5 13/10 7/5 8/5 18/10 10/5 

Another thing we can do is the omission of some notes. We could take our eight tones:

1/1 9/8 5/4 11/8 3/2 13/8 7/4 15/8 2/1

and pick some way of removing a few. Maybe we want to have a highest 'factor', i.e. any ratio involving larger primes is omitted. Omitting 11 and upwards would give

1/1 9/8 5/4 3/2 7/4 15/8 2/1

Which is almost our pentatonic scale - it has a somewhat sharp sixth, and a major seventh to boot that extend it, though. (Why I call those intervals by those names, I'll explain later).

We could of course do some other approach too: only pick primes and composite numbers which are square and composite numbers where at least one factor is 2:

1/1 9/8 5/4 11/8 3/2 13/8 14/8 2/1

This is a weird variety of mixolydian (and I'll get to that later). We needn't even be particularly logical about this - we can just exclude some intervals because we think they sound bad in combination with the others.

"Proper" overtone scales appear in throat singing, and part of the overtone series is close to parts of the western diatonic scale. Intervals do recur - if you go high enough up. But for each recurrence of an interval, there'll be twice as many new intervals appearing if you ascend the scale one note at a time.

Although scales based on the overtone series aren't necessarily what you'll want to use in your conculture, overtone series are a good thing to be familiar with. A lot of what we'll be doing with scales that are much more like the diatonic scale will be based on the overtone series in one way or another.

Music Theory for Conworlders: A Short Survey of Western Music

Western music has certain properties that western listeners tend to take for granted:

• harmony
• functional harmony

To a lesser extent, we can add

• modulation

A few features that are probably somewhat more universal are:

• repetition
• structured compositions
• some kind of 'tuning practice' of some form
• rhythm

What are these things?

Harmony is the use of multiple distinct notes simultaneously. Functional harmony is a bit more specific - most western music from the baroque to now has some variety of functional harmony. Essentially, the use of chords as 'functional entities'. In a song where the chords go something like 

C - F - G7 - C

the different chords have different levels of tension, and this 'tension' depends on a few factors. We perceive the first chord as fairly consonant. The second is in some type of relation to the first that makes us hear it as slightly less consonant, and the third one is really even less consonant. The dissonance of the final chord is resolved as the first chord reappears.

Functional harmony is not necessary for music with harmony - some early blues, modal jazz, some funk, renaissance and medieval music, and lots of folk and ethnic music have modal harmony instead. In fact, functional harmony seems to be unique to music that has been influenced by European common practice music, whereas harmony does not have any such restriction.

Not all music has harmony at all - there are pieces where only one tone is ever heard at a time. Some cultures only have such music. Tuning is relevant to such cultures as well, since it influences what kinds of melodic options exist. However, the tuning will be less restricted by the requirements of harmony in such a culture.

In most human music, it does seem as though there is a 'tonal center' to most pieces - a tone, that simply speaking, feels like 'home'. A tone - or set of tones - that does not or do not impart any tension when heard. This will depend on the key you're playing in and the way your melody and harmony are used, so it's not like there's necessarily one particular tone that functions as the 'tonal center'. Modulation is a thing in western music where one actually shifts the tonal center during the duration of a piece. Such a thing also brings with it some technical requirements for the scales that would be used with that music - i.e. there must be more than one tone from a which a relatively full, sufficiently well-in-tune scale can be found.

We will soon account for several of these things.

Music Theory for Conworlders: Modular Arithmetics and Some Tricks

We will do a slight diversion into a relevant topic to prepare ourselves for a thing that I'll soon get into here. The topic is modular arithmetics.

You all probably know how the minutes wrap around at 60, and the hours at 12. These are examples of modular arithmetics. We can do modular arithmetics with any number, not just 12 and 60.

5 + 5 ^{mod 8}= 2, because 5+5 = 10, 10/8 leaves a remainder of 2.

Modulo twelve is worth considering as a first example, however, since it has certain properties that help illustrate some of the properties of modular arithmetics. I will map the tones of western tuning to the twelve numbers of modulo twelve:

0	1	2	3	4	5	6	7	8	9	10	11
A	A#	B	C	C#	D	D#	E	F	F#	G	G#

Note that we could just as well have picked A# or D or Db or whichever other tone as 0. Now, as it happens, these numbers encode two things. 1) They encode individual pitches. '5' = 'D'. However, we can also consider them to be representations of intervals - i.e. 5 also is the distance from 0 to 5. In such a notation, we can start combining intervals. If we call '2' the 'second', A to B forms a second, but so does C to D, or C# to D#. G# to A# as well, due to our resetting the counter at twelve.

Let us consider what happens if we pick a number from (1, 11), and repeatedly add it, i.e. 0, 0+2, 0+2+2, 0+2+2+2. Turns out we get 0,2,4,6,8,10,(12=0,14=2,16=4, ...) and we have a cycle. We'll never break free of that cycle just by repeatingly adding twos. What happens with 3? 0, 0+3, 0+3+3, ...= 0,3,6,9,12=0,... and with 4? 0,4,8,(12). 6? 0,6(,12). How about 8? (0,8,16=4,12), And 9? 0,9,18=6, 15=3, 12. Finally, 10 gives us the same result as 2, only 'backwards': 10, 8, 6, 4, 2, 0.

Turns out only numbers that are coprime with the period (i.e. the 'size' of the set) will crank out each possible number if you keep iteratively adding it. Obviously, 1 will do that for any size. But that gives a boring structure. In 12, the only other options are 11 (which gives 1 in reverse), or 5 or 7 (which are each other's reverses as well - this due, in fact to 5+7=12). This has some further consequences: for prime-number-sized sets, any integer will generate the whole set, for composite-number-sized sets, there will be smaller 'cycles' in them. The cycles are subgroups - one can in fact do modular arithmetic on them as well, i.e. the {0,2,4,6,8,10} set basically behaves exactly like a {0,1,2,3,4,5}, mod 6 set - i.e. we map each integer in the first to its halved version in the second, and cut the period in half as well.

When we iterate over the set using such a coprime 'generator', we may get some interesting structures, as we can see for arithmetic modulo twelve. We see something that is slightly regular, slightly irregular appear:

0 ∪ (0+7) = 7
0, 7 ∪ (7+7) = 14, mod 12→2
0, 2, 7 ∪ (2+7) = 9, mod 12→9
0, 2, 7, 9 ∪ (9+7) = 16, mod 12→4
0, 2, 4, 7, 9 ∪ (4 + 7) = 11, mod 12→11
0, 2, 4, 7, 9, 11 ∪ (11+7) = 18, ... 6
... 0, 2, 4, 6, 7, 9, 11, (12)

(We can go on, obtaining along our way things like 0,1,2,4,6,7,9,11; 0,1,2,4,6,7,8,9,11; 0,1,2,3,4,6,7,9,11; 0,1,2,3,4,6,7,8,9,10,11; and finally the full set)

We can see how the first part, 0,2,4,6 in fact is common with what we'd get if we went and took '2' as our generator. This is somewhat natural: from the vantage point of, say, 0, adding two sevens in succession will obtain a '2'; repeating it twice more will add a '2' to the previously obtained '2'. In some sense, 7 is half of 2! We also notice how 7, 9, 11 form a similar - albeit shorter structure, that is 'offset' by one from the 0,2,... series; if we were to start at 7 and do the 'add two' operation, we'd obtain exactly those values first. By adding sevens, we create two 'slighly offset' two-series. However, we could have stopped at another time: we could have stopped when we had 0, 2, 4, 7, 9. At this point, we have differences of two and three; Both two-series are now one member shorter, and they are separated at both ends by a larger gap.

We could of course consider something like '4' or '3'; these divide twelve, but they can have their own internal structures that we simply repeat in the octave

0, 2, (3). → 0,2, 3, 5, 6, 8, 9, 10, 11, 12

0, 3, 4 → 0, 3, 4, 7, 8, 11, 12 

Since the period is 3, what we really have is 0, 2,3. We could fill these out all the way to twelve: 0, 2,3 5,6 8,9 11,12. In essence, we make a stencil: x-x, where x=in the set, - = not in the set, and we line these stencils up one after the other, getting x-xx-xx-xx-x. We can do the same for four, although now we have two options: x--x, x-xx. (I omit the full options xxx, xxxx since they're not all that interesting.)

Now, these are methods for scale building that more or less actually happen in western art music (the x-x scale is the dim scale much used in some styles of jazz, the x-xx scale is somewhat more difficult to find examples of, but Messiaen probably has some works. However, the stack of sevens that I elaborated a bit on above, that actually gives the lydian mode, and starting from the fifth tone of that (the one at index '7'), we obtain the regular ionian major scale, i.e. the vanilla major scale.

If the number of tones in our octave is not twelve, we may find other patterns that can be constructed like this, and we may even find that several different 'generators' generate multiple sets of interesting patterns. More of that later, however.

Music Theory for Conworlders: Scales and Culture

We're going to take a short break from the mathsy stuff, and consider some other things with regards to scales and music. The approach to scales I am taking is very much a western approach - although probably not unique to the west, it lacks things that are not present in the concept of scales in western music. It might be informative to look into what other cultures subsume into their concept of musical scales.

Many cultures consider scales not just a set of pitches, but a tuplet consisting of a set of pitches and a basic rhythm. Melodies or improvizations in a scale are expected to use that rhythm. The same pitch set can appear with another rhythm, but is then considered another scale. There's no guarantee that all rhythms are represented with all pitch sets. Examples of such systems are Indian ragas, Arabic maqamat and the ancient Greek scales.

Some Indian traditions also associate scales with time of day and other things - songs in some scales are supposed to be played in the morning, songs in others in the afternoon, etc.

In Europe, we tend to associate chords and progressions with keys, but since our keys are all identical copies that are transposed, the same chord progression can work in any key, provided it's been properly transposed. This is not necessarily the case in scales everywhere - even some renaissance instrument makers used temperaments that omitted parts of the gamut, and you can find some modern instruments even that lack several of the twelve tones usually expected - some harmonicas, for instance, or autoharps.

Some systems go further than that and approach their entire music's tuning like our harmonicas or autoharps - just pick one key, and don't bother making any other keys playable (duly note that the renaissance instrument makers that restricted the number of available keys still usually made their instruments have the full scales of C, F, G and a few others available). Indonesian Gamelan, to some extent, is an example of curtailing the available intervals in such a way. The Gamelan, however, further complicates things by having two entirely unrelated scales - one essentially five equal steps to the octave, the other close to a seven-note subset out of nine equal steps to the octave. Even then, variation between one village's gamelan and another may be drastic, and their tuning is thoroughly an art, and not an industrial standard.

Further, a scale may be somewhat flexible - lots of Chinese and Japanese music, as well as the American blues flex their tones' pitch considerable during the course of a song. This seems more common the smaller the size of the used tonal palette. Slightly comparable may be how the melodic minor is different when ascending from when it's descending. (In practice, the pitches that are altered - in Cminor, they'd be A and Ab, B and Bb - are not just altered when descending or ascending in actual compositions - it depends on chord choices, etc.)

Western music has been rather chord based for quite a while. Ancient Greek, Arab, Turkish and Persian music are rather more 'tetrachord'-based - a tetrachord is a span of subsequent tones , and the melody tends to weave melodies out of those four notes for quite a while, until switching to another tetrachord. From what I gather, they tend to switch tetrachords either so that the first scale and the last scale share the middle tetrachord - i.e. like going from GABC to CDEF to GABbC - CDEFGABc is C major, FGABbcdef is F major - CDEF is in both, or just switching between two tetrachords with the same end-points: GABC to GABbc. Playing around with such concepts (and extending them or retracting them), may provide some ways of making very 'culture-specific' music.

Unlike tetrachords, our chords tend to have the tones played simultaneously (although the practice of arpeggiation - sounding one chord tone at a time - can be played in a way that deviates from that), but they also form a very important melodic backbone; lots of melodies mostly consist of the tones of whichever chord is currently playing with the occasional tone outside of that set. We can of course come up with rather drastically different chords which might be used in similar ways.

Finally, other practices may be associated with different scales - working songs to ensure the right working rhythm, different scales for different religious uses, etc. There's a world of possibilities.

Music Theory for Conworlders: Dissonance and Consonance

In monophony - music where only one voice is heard, or all voices do the exact same thing - dissonance and consonance mainly seem to relate to whether a note feels at rest or not with regards to the melody. Thus, a melody like C E D B' C, B' is likely not to be at rest, whereas C is quite restful. Why this is so is not the topic right now, but we'll go on to a more easily understood situation - that of consonance and dissonance in a situation where voices do not do the same thing.

A few centuries ago, the main hypothesis as to what causes dissonance is 'complexity of ratios'. Basically, a simple ratio such as 3/2 will regularly and often have waveforms 'coincide' - the tops (or troughs) of the waves will recur in the same position relative to each other often. With more complicated ratios they will recur less often, and therefore 'keep the ear in torment'. Thus 2/1 will be more consonant than 3/2, 3/2 more consonant than 5/4, 5/4 more so than 9/8, 9/8 more so than 11/8, and so on. The exact way of measuring 'complexity' of a ratio differed between scholars that held that hypothesis, but higher prime factors were more complex than lower prime factors; however, 25/16 probably was comparable to, say, 13/8, despite 25/16 having no factors higher than 5, and 13 having 13 and 2 as its only factors. (25/16 = 5²/2⁴, 13/8 = 13/2³).

Nowadays, the main hypothesis deals with overtones. Ultimately, for regular overtone series, the results will be similar to the previous hypothesis - simple ratios will mostly be more consonant than complex ones. We noted previously that tones tend to consist of a whole stack of frequencies.

If you were to generate sinusoidal tones (i.e. tones lacking all overtones), you'd find that only when they're fairly close to each other do they cause dissonance - anything wider than (roughly) a minor third never sounds dissonant, even solidly dissonant intervals like the tritone and the major seventh sound kind of neutral and uninteresting.

Music File Hosting - Audio Hosting - differencetone_2sources

an example of two sinusoidal tones a bit too close to each other to be dissonant

Upload Music - Play Audio - Sinusoidal Dissonance

two sinusoidal tones that are further apart yet close enough to be dissonant

We introduce the concept of 'critical bandwidth', which is a range of intervals that by their very nature are dissonant - roughly, this corresponds to 'about ten hertz off' to 'about 6/5' (roughly A-C, the minor third)*. So, for A440, any frequency in the rough area of 450hz to 528hz will be dissonant. But any musician knows that there are intervals wider than the minor third that are dissonant - for instance, the major seventh (A-G#) and the tritone (A-D#).

The reason for these is dissonance between overtones. The dissonance of two notes played together is basically the sum of the dissonances between the overtones. Thus, tones whose overtones often are within the critical bandwidth of each other are very dissonant, tones whose overtones often are closer than 10hz from each other or further than 6/5 from each other will be consonant.
Let us compare the overtones of A440, E660 and D#622. Each of these tones gets a column, so A440 and its overtones are under A, E660 and its overtones under E, etc. The overtones are ordered wrt pitch:

	A	E	D#
A	440
D#			622
E		660
A	880
d#			1244
e	1320	1320
a	1760
a#			1866
b		1980
c	2200
d#			2488
e	2640	2640
g*	3080		3110
g#		3300
		.
		.
		.

A sample of how overtones serve to 'create' the timbre of a tone may serve to help at this point:

Play Music - Audio Hosting - Overtones

Looking at the table, we find that d# has way more "conflicts" with A than E does. Certainly, E will start having dissonances with A - heck, after a certain point, overtones will start clashing internally (which is why a note with very strong overtones far up the series will start coming off as noise!)

We would end up with something like this:

Music File Hosting - Listen Audio Files - Tritonus by overtones

vs. this:

Upload Music - Audio Hosting - Fifth in Overtones

However, this also explains why the very complex ratios that lightly detuned intervals produce do not strike us as very dissonant. Clearly 659/440 is more complex than 3/2, (659 is a rather big prime compared to 3 and 2!) yet it's quite a tolerable detuning. The reason is that the overtones now come within the tolerable range just before the critical bandwidth, where they previously perfectly synched. Of course, the volume of each overtone has an impact on the perceived dissonance.

This model also predicts that instruments with unusual overtone series may cause dissonance in other intervals than more regular instruments do, and we find that this in fact is the case. One can even make intervals like the octave come out as very dissonant by picking weird timbres.

I have no methods for efficiently finding the least dissonant intervals for two tones with arbitrary weird overtones, so I won't give you that kind of mathematicking right now.

In addition, we must note that consonance and dissonance are somewhat subjective - we perceive them in part due to conditioning. However, it does seem the main thing we condition is 'where' the line is drawn - how many conflicting overtones we can tolerate. We also seem to fill in some missing overtones - so functionally a tritone will sound somewhat dissonant even if the overtones that would usually cause the dissonance now are missing. The dissonance will be less acute, however.

Finally, most human musics have both dissonant and consonant intervals - the interplay between consonance and dissonance, or tension and release - seems to be fairly common. However, due to the way intervals work, we easily get dissonant intervals as a byproduct once we have a sufficient number of consonant intervals - the cracks in which consonances can fit without trampling into other tones' critical bandwidths quickly shrink.

* A more accurate description would be "6/5 or 50hz, whichever is bigger"; it seems the critical bandwidth is wider in the lower registers. In fact, go low enough and octaves and the like get dissonant. This is why most music with more detailed timbres and such tend to leave a lot of space around the bass - you don't want to crowd the lower registers.

Music Theory for Conworlders: Intervals

For most humans, the important thing when listening to music is not the absolute pitches. A song with A440hz, C528hz, D586hz, E660hz and G782hz is not identifiable on account of those pitches being at those particular hertz. A melody that goes ACACDED ACACDEGa aGaGEDCA is not recognizable for the reason that it plays those particular frequencies in that particular order, it is recognizable due to other reasons.

The thing that makes it recognizable is the relations between the notes. (Music is a bit tricky though, and it seems the rhythm of a melody is even more important for recognition - a piece of advice an old fiddler once gave me was that it's more okay to play a wrong tone at the correct time than the right tone too early or late.) Of course, our hearing has some leeway, but roughly speaking, the ratio between the involved pitches is the interesting thing. Not the absolute difference in frequency - a difference of a hundred hertz will sound very different in a motif that goes 100hz - 200hz - 100hz - 200hz (basically that's a disco bass octave thingy), or one that goes 800hz - 900hz - 800hz - 900hz (that's basically one part of the Rudolph the Red-Nosed Reindeer's main melody, although the sheet music I found for that puts it closer to ~330 - 371 - 330 - 371).

So, the distance between 100hz and 200hz is comparable to the distance between 800hz and 1600hz. The distance between 800hz and 900hz is comparable to that between 100hz and 112.5hz.

The scale I provided in the first paragraph - a version of the pentatonic minor scale - is basically this shape:

A		C		D		E		G
1/1		6/5		4/3		3/2		16/9
	6/5		10/9		9/8		32/27		9/8

The last ratio in the lower line is the remainder needed to get to the next 'a' at 2 times the frequency of the previous A.

Here, the upper series gives each note as its relation to our starting point, A. We could basically pick any frequency we like for A, at this point that does not matter at all. The second list is the intervals between each neighbouring pair of notes.

How do we combine the distance from A to C with the distance from C to D to calculate the distance from A to C? We multiply them! A/B * C/D = (A*C)/(B*D), and as it turns out, 6*10/5*9 = 60/45 = 4/3.

Multiplication is somewhat complicated, and it gets difficult to compare ratios at a glance - 32/27 and 6/5 are actually pretty similar intervals, but this is hard to spot. For this reason, the unit 'cents' has been invented. One cent is a hundredth of a semitone, but that doesn't tell us much of its mathematical properties. The cent is 1200 * log₂A, where A is the interval we're looking at.

If you don't know logarithms, logarithms basically 'shift the gear' of the numbers we're considering in such a way that log(A*B) = log(A) + log(B). In log₂ , if we deal with a doubling, we just add 1; in 1200*log₂ we add 1200. log₂(5/2) = log₂(5/4) + 1; since we're dealing with the weird situation where we have a factor of 1200 everywhere, 1200log₂(5/2) = 1200log₂(5/4) + 1200. Stated simply, logarithms change gears so that multiplication turns into addition.

This gives us one further way of representing the scale given above (here rounded to integer cents):

A		C		D		E		G
0		316		498		702		996
	316		182		204		294		204

It is now easier to compare the sizes of the intervals. The size of the cent is picked to reflect the 12 tone equal temperament in a clear and simple way - each semitone is a hundred cents, and each tone of the western scale therefore is an integer multiple thereof. It is a relative measure, so it doesn't make sense to say that any particular tone is 0 cents - although we can decide for some context to use a certain tone as the starting point.

Another convenient fact is that an equal temperament of N tones to the octave will have steps that are 1200/N cents. This simplifies calculations a lot - if we live in the regular, non-logarithmic world, we need to take N:th roots, which is way more cumbersome.

Finally, any positive number is an interval: 11 is an interval, as is e⁵ or 36π/32. Human hearing stretches from about 20hz to somewhere a bit shy of 20 000hz, so intervals wider than 10 000 probably are not all that useful, since even if we pick the lowest possible point as one end of the interval, the other end will be outside our hearing range. Our hearing is not too precise, so differences in intervals of much less than a cent are probably not very useful either and therefore it might not make sense to distinguish 5 and 5 + 10^-12. And finally, due to the octave equivalence we've previously seen, we will probably only really want to deal with intervals "inside" of the range of [1, 2] and use those to fill out a reasonable chunk of the audible space. We will, however, look at some other approaches as well.

Next installment: more on intervals, Pythagorean tuning

Music Theory for Conworlders: Introducing Overtones and the Octave

As an introduction, we'll look at the basic concepts of western music theory. Most people know there's seven distinct tones in the major or minor keys, and there's twelve distinct tones in total per octave. People might also know of chords.

But what do these things even mean, and what kind of structure is there to them? Why twelve? What is an octave? What are tones, even? [Assuming most readers come here from a facebook discussion, this will basically have been covered already. If people come here from other places, I might add a more detailed definition somewhere later.

Now, I will probably return to the physics of tones a bit later, but for now, we'll observe that a tone has a frequency. What we mean to say when we say that in the ISO16 tuning standard A₄ is 440hz we mean to say that the sound wave is characterized by something that repeats 440 times per second.

We could make a wheel with equally spaced spikes that beat against a flexible piece of plastic, and have 440 of those spikes. If it rotates a whole turn in one second, the sound generated by those impacts would sound like a tone of 440hz to us.

There is nothing intrinsically special about 440hz, it is an arbitrarily picked standard. Baroque organs in Germany have A varying from roughly under 400 to about 500 hz. It's only fairly recently A440 has been standardized (basically sometime during the 20th century), and we find music that deviates from it as well.

Almost all instruments have a secondary important frequency-related fact going on - overtones. Most things will not just have one frequency, they will emit sounds of several frequencies simultaneously. For instruments, it's quite common for these overtones to be integer multiples of the lowest frequency, the 'fundamental'. (And these will later on be important when we look at concepts like consonance and dissonance, and therefore also when we look at chords and scales.)

So, when you play A440 on a guitar, you also cause generate the frequencies 880hz, 1320hz, 1760hz, 2200hz, ... . These contribute to the sound of the instrument - in fact, it's one of the things that distinguishes the sound of an organ from that of a guitar or that of a saxophone from that of a trumpet. We will start out by looking at instruments with this integer-multiples structure to their overtones. Other possibilities exist - pianos have near-integer overtones that tend to be slightly too large for integers (1, 2.01, 3.009, ...), and bells and gamelans have weird non-integers where multiple numbers close together may appear combined with large gaps as well.

To hear a synthesized version of this phenomenon, listen to the sample here below:
Play Music - Audio Hosting - Overtones

The very first overtone, the doubling of the frequency, has a rather special role in music. It corresponds to the octave, and it is nearly an universal interval in human music. Let us consider what happens with the overtones when you play two tones, an octave apart.
We have two tones, frequencies X and 2X. Each of these further is multiplied by a series of integers:
X, 2X, 3X, 4X, 5X, 6X, 7X, 8X, ...
2X, 4X, 6X, 8X, 10X, ...

We find that each overtone of 2X is also present in the overtone series of X. Thus. they will resonate together. For this reason, Xhz and 2Xhz will interact in similar ways with some other tone at Yhz as well. This, in part, is why we perceive tones where one is double (or quadruple, or multiplied by 2ⁿ for any integer n) the other. This, we call octave equivalence, and it'll be somewhat important later on.