Octaves and scale formation

Some of the most fundamental concepts in music are the octave and how we form scales from them. It's the basis of how all music is written, and the relation of ocarinas in different keys.

I think that the easiest place to begin understanding this topic is with the nature of sound itself. If you take a bowl of water and tap the surface, you'll notice ripples moving on the surface. Sound is a similar ripple that travels through the air.

When you play an ocarina, the air in the chamber oscillates, and these oscillations push on the air in your environment, setting waves in motion that travel to your ears, and you can hear them as a sound.

Just like the ripples on the surface of water, we can represent sound as a graph. The pitch of the sound (how high or low it is) corresponds to the frequency of the sound wave, the number of peaks or troughs that pass a point in space in 1 second.

When one note is exactly double the frequency of another, the human mind perceives both with the same tone, which is called an octave. Incidentally, the phenomenon is not limited to a single doubling. If the high wave is again doubled, now four times the original frequency, all three notes are perceived equivalently.

Because of this, the notes used in music repeat. This repetition exists on all instruments, but is most obvious on the piano keyboard, where a clear pattern can be seen in the white and black keys:

If you pick any key within one octave and play a tune, then play the same sequence from that key in a different octave, it will sound like the same tune.

You can also play a sequence in two or more octaves at the same time. The highest note is heard as the melody, and the lower ones create a richer tone.

Timbre

As the frequency of a sound is determined only by its peaks, the shape of the wave in between can vary. These differences are perceived as tone colour, or timbre.

Timbre is what makes an ocarina sound like an ocarina, a violin sound like a violin, and a piano sound like a piano. Sounds can have the same pitch, but have a different timbre.

You can hear the same pitch played on a collection of instruments with different timbres in the following audio sample:

How to form a major scale

Western music divides the octave into 12 notes, 12 sequential white and black keys. After note 12, it returns to note 1 an octave higher or double the frequency.

Using the interactive keyboard below, have a play around using all of the keys, both the white and black ones. Try to make a melody and observe how it sounds.

What you may notice is that music formed from all of these notes sounds very 'ghostly' and ungrounded. The reason is that the distance between any two of these adjacent notes is called a 'half step' or 'semitone', and all semitones sound the same.

Note 1 to note 2 is a half step, Note 2 to note 3 is also a half step, and so on.

This 12 note scale is called the 'chromatic scale', and it is rarely used as a whole. Rather, we use a selection of pitches from it that follow an irregular pattern.

First, we need to introduce another interval called the whole step, simply two half steps in sequence:

And we can then form a major scale from the chromatic scale by choosing any note as a starting point, and following this pattern of intervals:

Whole, Whole, Half, Whole, Whole, Whole, Half

Thus, if you start from the note we are calling '1', you end up with the note sequence 1,3,5,6,8,10,12,1. The asymmetry gives each of the notes a unique sound and gives the ear a hook so that it can tell where it is.

In fact, we can follow the same pattern beginning from any of the 12 chromatic notes per octave, producing 12 major scales. These all sound remarkably similar, but are higher or lower in pitch.

As a side note, you may also find it interesting that the physical layout of many instruments is based on this pattern, including the ocarina. It is visible in the sizes of the finger holes:

Naming the notes

As most music uses only seven notes of the chromatic scale at a time, it would be cumbersome to think about the additional five notes which you are not using. For example, having to remember that the next note after 1 is 3, but that 6 follows 5 directly.

The standard note names are designed to hide these irregularities. One of the major scales was chosen in history to serve as a basis, and its notes are named using the first seven letters of the alphabet.

Taking the pattern 1,3,5,6,8,10,12,1, note 1 is called C, followed by D, E, F, G, A, B and finally C an octave higher. These are called the 'natural' notes.

The other notes are named in relation to these, and this can be done in two ways.

Sharps name these notes in relation to the semitone below. 'C♯' is the note one semitone above C:

And flats name these notes in relation to the semitone above. If you begin from the note B and add a flat, you descend to the chromatic note between A and B:

The reason we have both sharps and flats becomes apparent once you start considering how the notes would be named for scales beginning from different notes.

As it is the base of the system, the scale of C Major needs no accidentals, but if you form a scale from any other note they will be included. For example, if you start on D you have two sharps, F♯ and C♯.

But if you try forming scales using sharps beginning from other notes, you may notice a problem. Have a play with the following tool and see if you can spot it.

You may notice that some scales when notated in this way are not very elegant, for instance F major:

F, G, A, A♯, D, E, F

The name 'A' shows up twice, while 'B' is missing entirely. If we instead notate the scale using flats, that problem goes away:

F, G, A, B♭, C, D, E, F

The ability to give the same note two different names is called an 'enharmonic'.

Enharmonics are mostly seen in the naming of the accidentals, but in some cases, the natural notes can also be enharmonic. For example, if you write C♯ Major using the names of natural notes:

C♯, D♯, F, F♯, G♯, A♯, C, C♯

The names F and C appear twice, while E and B are not used at all. In this case, it is better to call F, 'E♯' and C, 'B♯', which allows each letter to appear only once:

C♯, D♯, E♯, F♯, G♯, A♯, B♯, C♯

We name the scales according to the note that we started from, followed by the type of the scale, for instance 'C Major'. The letter 'C' used alone would imply the major scale.

Here are the 12 Major scales with their common note names. There is no need to memorise these right now, as it's easy to look them up:

• C - C, D, E, F, G, A, B, C
• C♯ - C♯, D♯, E♯, F♯, G♯, A♯, B♯, C♯
• D - D, E, F♯, G, A, B, C♯, D
• E♭ - E♭, F, G, A♭, B♭, C, D, E♭
• E - E, F♯, G♯, A, B, C♯, D♯, E
• F - F, G, A, B♭, C, D, E, F
• F♯ - F♯, G♯, A♯, B, C♯, D♯, E♯, F♯
• G - G, A, B, C, D, E, F♯, G
• A♭ - A♭, B♭, C, D♭, E♭, F, G, A♭
• A - A, B, C♯, D, E, F♯, G♯, A
• B♭ - B♭, C, D, E♭, F, G, A, B♭
• B - B, C♯, D♯, E, F♯, G♯, A♯, B

Octave registers

At this point, you may be wondering how you differentiate some note like 'C' in one octave from the same note in a different octave. We do this by numbering the octaves.

Each note from C, to B that is 12 semitones higher is considered a single octave register, and we number them sequentially:

• The note C1 is C in octave register 1 and the highest note in this register is B1.
• After this, the next note is C2, and so on.

In total, seven full octaves are commonly used in music. In the following audio example, you may hear the note C played in all eight octaves.

Any single chambered ocarina can play a slice of this range. An alto C, for example, plays from C5 to F6. You can always work out how the pitch of one ocarina relates to another using octave register numbers, as these names are standardised.

Other naming systems, such as 'Alto' vs. 'Bass' can vary between makers. See 'Ocarina keys and pitch ranges' for more details.

Transposition

Because all of the major scales are based on the same pattern, they are equivalent. You can take a melody in one scale and 'transpose' it into a different scale, and the result will still sound like the same melody.

You can see and hear this for yourself using the tool below. The 'transposition' slider moves the notes. Press 'play' to hear how it sounds.

Another way of thinking about this is to realise that if two scales are based on the same pattern, they are equivalent. You can take the notes of both scales and number them starting from the first note. C and D major, for example:

C major scale degrees

D major scale degrees

If you substitute the note of one scale with the note of the other scale with the same number, you've transposed the melody. The notes 'E, C, D' in C major have the numbers '3, 1, 2'. So you substitute them for 'F♯, D, E'.

The ability to transpose music is very useful on the ocarina, as it means that you can alter melodies that would be impossible to play, and make them fit within the range of your instrument. Transposing also allows you to create variation in a performance, as music that remains in the same key for a long time can sound monotonous.

This topic is discussed further in 'Transposing sheet music to fit the ocarina'.

Closing notes, and other scales

These concepts will start to make sense pretty quickly once you start putting them into practice, looking out for the patterns in the music you play.

One thing worth noting is the difference between the terms 'scale' and 'key'. As discussed, the 'scale' names the selection of notes formed following a pattern, and the term 'key' basically means the same thing.

One difference between a scale and a key is that a scale has an implied order. One can 'play a scale', starting from its lowest note to the highest, but you don't 'play a key'. A key is an unordered selection of notes that a song makes use of.

After a while, you may start to wonder what would happen if you formed a scale by following a different pattern. The answer is that you get a different type of scale.

Quite a few different scales exist, including the major, minor, and blues scales which you can hear below:

The minor scale is one interesting case because it reveals why we name the notes of the major scale starting from C instead of A. It is built from the pattern:

Whole, Half, Whole, Whole, Half, Whole, Whole

The pattern that the minor scale is formed from is actually the same as the major scale, just rotated by a few steps. If you take 'Whole, Half' from the end of the major scale formula and put it at the start, you get the minor scale formula.

Thus, A minor and C major share the same notes, and differ only in the tonic note. The naming is convenient as it creates a clear relationship between the two scales:

• A minor - A, B, C, D, E, F, G, A
• C Major - C, D, E, F, G, A, B, C

Most music is tonal, meaning that one note is chosen as the tonal centre, and the remainder of the music is structured as a journey away from and returning to this note.

How we perceive a given sequence of notes depends on the note we use as the tonal centre. If you want to take this same concept further, it would be worth looking up the 'modes' of the major scale. Essentially, you can form a unique scale from all seven of the notes.