An octave is an exact 2:1 ratio in pitch, regardless of tuning system. In equal temperament, you get there by multiplying each frequency by the 12th root of two 12 times. In any just temperament an octave is one of the fundamental intervals and is defined by the ratio 2:1. In well-tempered scales, they screw around with the fourths and fifths, but they leave the octaves alone. It would be a mess otherwise.
As to not being able to find certain notes in the instructions, is your ocarina a chromatic one? Meaning, is it meant to play all 12 tones? Because most ocarinas aren’t…they play a certain set of notes but not the entire chromatic scale. Some instruments are just made that way, due to physical considerations and tradition. Harmonicas are generally the same way…you have to actually buy a chromatic harmonica to get one that plays all 12 tones.
This thread went very wrong very quickly.
In a standard Western chromatic scale (i.e. containing all notes) there are 12 notes in total. Starting on C, they are C - C#/Dflat - D - D#/Eflat - E - F - F#/Gflat - G - G#/Aflat - A - A#/Bflat - B - (C).
If you want to know why the scale looks like that, it’s the result of a long, long evolution going back (as noted) to ancient times and Pythagorus and mathematical ratios and the harmonic series, and thence onto medieval church modes and different types of musical notation systems, and that doesn’t even get into tuning systems which are a whole other ball of wax. You could easily do a semester or two in college on all this. So “that’s how it is” is not the worst explanation in the world for your purposes. But rest assured, it has nothing to do with the elements.
And I second the advice to forget the H unless you really want to do your head in. H in German is indeed B natural, vs B which is B flat. You may also see Es (E flat) or As (A flat). Those Germans are a wacky bunch.
Technical nitpick, but you can play a diatonic harp chromatically using bends, overblows, and overdraws (yes, a full chromatic scale from hole 1 up to hole 10.) But, yes, they’re designed so that simple blows and draws on the holes outline a certain scale (usually major).
There’s two inter-related questions here:
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Why 12 notes in the scale?
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Why the names?
The answer to the first revolves around the existence of “harmonics”. These are also called “overtones”. They are produced naturally when a string is plucked or a wind instrument is sounded.
Now, the overtones exist in a very distinct and defined pattern. It can be seen in stringed instruments. The first overtone has a “node” (a place where the string doesn’t vibrate) halfway along the length of the string. This produces a note which is one “octave” up from the first note. It sounds to our ears like the same note only higher. It is this first interval, the “octave”, which we are dividing into smaller notes.
The second overtone has a node at 1/3 and 2/3 of the way along the string. The third overtone has a node at 1/4, 1/2 (2/4), and 3/4 of the way along the string. And so on. It is an immutable mathematical sequence, with no steps in between. There is no “third and a half” overtone.
This produces a natural sequence of sounds which we find pleasing, probably because they are in all the sounds we hear all day.
Now, why 12 notes in the “even tempered” (equally spaced) scale? The answer is that a 12 note scale comes the closest to containing the exact notes of the overtones. If we have 11 notes, or 13, or 9, they don’t land as near to the overtones as does a 12 note scale. To our ears, they sound dissonant.
In the East, they use an even tempered 24 note scale. Why? Same reason. If we start looking at scales with more than 12 notes, the first one that does a better job (is closer to exactly reproducing the overtones) contains 24 notes. Seventeen or twenty-one notes won’t do it, they don’t hit the overtones nearly as well.
So that’s why even tempered scales have either 12 or 24 notes. Because they are best at reproducing the natural sounds of the overtones
Regarding the other question, why use “B#” or “F#” as a way to name one of the notes in the scale, that’s all tied in to history of musical notation. As someone mentioned upthread, that’s a one semester college course. They could just as easily be numbered one through twelve.
However, the use of alternate names (e.g. A sharp = B flat) has a simpler explanation.
This is that in different keys, the notators wanted to keep the “ABCDEFG” notation. To avoid having two notes with the same letter (e.g. “G”) in a scale that started at G flat, rather than the second note being called “G sharp” (which gives two G’s), they called the second note of that scale “A flat”.
Math and history …
Well said. The more succinct version I sometimes give to pupils who continue to ask “but why?” is “I could tell you the answer, but it’ll take a looooong time. Shall I do that, or do you want to play some music?”
Plus, in response to those explaining everything away as ‘because of ratios’, go and listen to a signal generator producing sine waves at an equal-tempered major third, and tell me it sounds nice.
Well, a sine wave generator won’t produce the harmonic overtones that a real instrument does. Those harmonics are also based on simple whole-number ratios, and their behavior is one of the things that affects the timbre of an instrument. (Why C on a piano sounds different than C on a trumpet.)
One of my favorite experiments when demonstrating harmonics can be done on any piano. Hold down a C key so the damper is off the string but not making any noise. Then quickly bang on the C an octave lower once. After the low C is damped, you’ll hear the high C vibrating softly. This is sympathetic vibration induced by the 2:1 harmonic on the lower C string.
And that’s not to mention double sharps and double flats. Yes, they do exist, but of course, in the common tempered scale, have enharmonic equivalents (G## and Bbb both sound like A, B## sounds like C#/Db).
So let’s not mention it.
…and a major third, on a violin or a piano, sounds rich and sonorous. The instruments and the music we have exploits the way physics causes overtones to clash against one another, rather than just finding ways to make things fit nicely. This doesn’t mean that the sounds created (or rather, the notes perceived by our western ears) seem to clash. It also explains how intervals of a fourth can sometimes be considered dissonant or sometimes consonant. Or how Stravinsky could take a similar approach to sevenths.
Edit: ‘helps to explain’ would be better than ‘explains’!
What is that a response to? You’re saying that scales have nothing to do with the ratios between frequencies??
Apart from 2:1, focussing in on ratios distracts from the evolution of scales - especially if suggesting that the semitones E-F and B-C are to make scales work. Melodic lines, and hexachord theory (with a semitone at the centre of six notes), created a system which then was adapted into a major/minor chromatic system. Even then, the assumption that all intervals should be equivalent was a long way off, the big difference between a ‘true’ major third and an equal-tempered one being just one obstacle.
One comment I want to fully endorse, even if it might have been meant as an aside, is this:
Gorillaman, not sure if you were saying that I was the one “explaining everything away because of ratios”. My points were simple:
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We divide the octave into 12 equally tempered notes (rather than say 9 or 13) because that gives the best fit to the natural overtones. As you point out, it’s not perfect. For example, the 11th overtone is way out … and I’ve read that piano hammers strike the strings 1/11th of the way along the length to minimize that overtone.
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If we want to divide the octave into more than 12 equally tempered notes, the next number that gives a better fit than 12 notes is 24 equally tempered notes. That’s why that scale is used in Eastern music.
If that’s what you call “explaining everything away because of ratios”, I’m happy to be proven wrong, bring on the math.
And if not … well, thanks for your input.
12 equally tempered notes is bad bad bad, if you want any overtones to coincide beyond octaves. Stating that ‘eastern music’ uses 24 tones…what about gamelans tuned to create interference between the various pentatonic pitches?
Gorillaman, 12 equally tempered notes is bad. I would not describe it as “bad, bad, bad” as you say, it’s reasonable, otherwise a piano would sound horrible. We use it because it is closer to the overtones than say 9 or 13 or any other number.
I know this how? Not because I read it in a book. Because being a musician all my life, in about 1974 I became very curious why there was music for 12 even-tempered notes, and 24 even tempered notes, but no other even tempered scales … so I did the calculations, and that’s what I found out.
12 notes is bad … but any other number is worse, far worse.
For example, the second overtone is the fifth above the octave. If we start at A=440 Hz, the first overtone is the octave at 880 Hz. The first non-octave overtone is the fifth, at 1320 Hz.
If we have 11 even tempered notes in an octave, the nearest one to that fifth misses by 36 Hz. If we have 13 notes in an octave, the nearest misses by 28 Hz. With 12 even tempered notes, the nearest one misses by 1 Hz … yes, it’s not dead on, but it’s not “bad, bad, bad”. It’s very close.
And the twelve tone scale hits the next fifth up within 3 Hz, compared to 71 Hz for 11 notes and 84 Hz for 13 notes.
I encourage you to actually do the math. 12 has the least error in fitting the overtones. That’s why we use it.
Finally, pardon my lack of clarity, I should have said some eastern music uses a 24 note even tempered scale. I did not mean to imply that there were not other scales.
All of that is based on a big assumption, that of an overriding need for equal temperament at all. Most musics across the world, and most of the history of European classical music, have used other tunings.
Only because equal temperament hadn’t been invented yet.
There’s a reason ET became the dominant musical tuning so quickly and every historical tuning is now mostly a curiosity: because it makes things easy. Instruments tuned in equal temperament can play in any key without being retuned, which makes them infinitely more versatile for composers and performers.
Except in “Do-re-mi”, we learn that it’s “Sew, a needle pulling thread”. 
Quickly? You call many hundreds of years quick? There was massive resistance against ET. There’s also an element of cultural supremacy about the ‘because it hadn’t been invented yet’, suggesting that other musics, which use different tunings, are necessarily somehow lacking or backward.
I am by no means a musician but I am a mathematician and this is what I remember.
Our goal is the octave where the vibrating medium cycles twice as fast as the given note. But in addition, we want certain overtones (called harmonics) which can be expressed as a rational number. For example vibrating 4/3 as fast or 5/2 as fast. Certain harmonics are very pleasing such as the perfect fifth (a 3/2) ratio. If we just wanted one base note and the pleasing harmonics, life would be easy, but what if you want the base note to be B instead of C?
It’s the transposition that makes life difficult. To divide an octave into equal parts, each note is a root of 2. So for example a decimal scale (10 notes) would have each note vibrate 2^(1/10) = 1.07177346… thimes faster than the previous one. The problem is that roots are irrational numbers and as such, can only approximate rational numbers (the exception being 2 since we are using a root of 2)
As intention points out, using the twelfth root of 2 gives us the closest approximations of the rational numbers we want. I think music theory would be a lot easier if we just had twelve notes (A-L) but as others have pointed out, the twelve notes are divided into the chromatic and pentonic scale for historical reasons.
Yes, I call it quick, compared to the history of music and musical temperaments as a whole. The opposition to ET in the west, while indeed significant, came almost entirely from the Catholic clergy due to superstitious reasons, and not lay musicians or composers. Given that the clergy were, until the end of the Renaissance period, the primary sponsors of composition and musical performance, it’s not surprising that it took a little while to catch on.
There was also a significant shift in style that led to wider acceptance of ET. In particular, beginning with the baroque period, compositions focused more on complex melody, contrapuntal phrasing, and motif rather than the relatively simple harmonic phrasings of earlier Church music. The other side of the coin is Renaissance and baroque era folk music, which typically made use of only a small number of instruments and voices, and could therefore be played in a variety of pure tunings.
But putting together a versatile orchestra which can voice chromatic, diatonic melodies in any available timbre, and provide less perfect but adequate harmonies for them, without using equal temperament would be essentially impossible. And as this type of music became popular, ET became a necessity.
And it’s a good thing, too, because nearly all Western music we now enjoy is rooted in the music theory that was developed in the baroque and subsequent classical period.
Gee, thanks. I don’t know how the hell you get from my making a factual statement about the current primacy of equal-tempered tunings to me being an ignorant Amerikkkan cultural imperialist bastard, but I’m sure in your mind it makes perfect sense.
I would recommend learning a little more about the subject matter at hand instead of accusing others of cultural supremacy because they disagree with you.