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Frequency

Let's imagine you have a long hollow tube. If you hit it, you get a fairly constant sound because hitting it produces a shock-wave which oscillates (travels up and down) the tube. This oscillation or vibration is what we hear as pitch.

The speed of oscillation or vibration is called "Frequency". Frequency is measured in Hertz (Hz), which is oscillations per second. If the hollow tube vibrates at 200 cycles per second, the frequency is 200 Hz.

When you hit a hollow tube, the shock-wave is actually travelling at a constant speed. What determines the frequency is the length of the hollow tube. The longer the tube, the further the shock-wave has to travel, hence, the lower the frequency... and vice versa.

Notes and Octaves

A "Note" is a given name to describe a musical frequency. It describes the pitch of a piano key or guitar string. By convention, notes are named as :-

A , A# , B , C , C# , D , D# , E , F , F# , G , G#.

The suffix "#" denotes sharp and "b" denotes flat.
Also note that A# = Bb, C# = Db, D# = Eb, F# = Gb and G# = Ab.
The names chosen are the de facto standard for nearly all music.

"Octaves" of a note are just multiples of the original frequency. Let's say that a length of hollow tube has a frequency of 264 Hz and we'll call it "C".

• If the length is half of the original length, the frequency will be double. This creates another "C" but one octave higher than the first (264 x 2 = 528 Hz).
• If the length is quarter of the original, the frequency will be quadruple. This creates yet another "C" but two octaves higher than the original (264 x 4 = 1,056 Hz).
• If the length is double, the frequency is halved. This creates "C" again but one octave lower than the original (264 / 2 = 132 Hz).

We can summarise the relationship between octaves and frequency as follows:

For most cultures, the "just intonation" tuning has been in use for thousands of years. This makes sense because we are using multiples of the original length (and then normalising them to the octave) to create notes.

The just-intonation tuning system works fine and sounds beautiful. However, it has only one drawback... you cannot transpose a song (ie you can only play songs in any key but "C"). When you play in another key (eg "D"), the tuning sounds wrong.

The "equal-tempered" tuning was developed to overcome this problem.

Equal-Tempered Tuning

How does it work? Well, if you think about it, tuning is not linear. You can double the frequency to get the next octave up but you have to quadruple it to get the next octave after that. Consequently, the notes within a scale are not equally distributed in frequency (nor in length).

This is how it's worked out... "A4" (the note "A" at the fourth octave) is deemed to be at 440 Hz and, therefore, "A5" will be at 880 Hz. We then take logarithms of A4 and A5 frequencies. Next, we mark in 11 equally spaced points between log(A4) and log(A5). On the logarithmic scale, this is the same as having 12 equally spaced notes per octave. We then apply arc-logarithms to those points and arrive the equal-tempered tuning.

For most of us, we will only probably need to know 2 scales: the Major scale; and, the Minor scale. Using a root of "C", the Major scale comprises C, D, E, F, G, A, B while the Minor scale comprises A, B, C, D, E ,F, G. Both of these scales have 7 notes per octave.

Note - The Melodic Minor is played differently when ascending (Asc) and descending (Desc).
* denotes Names conjured by myself to reflect the mood of the Scales.
For more examples of scales, see the Scales Reference section at the end of this document.

As you can see, there are many scales and there is nothing to stop you from creating your own. After all, scales are just a series of notes. Different cultures have developed different scales because they find some series of notes more pleasing than others.

Major and Minor Scales

The Major scale and Minor scale share many similarities. For example, the white notes on a piano concur for both "C Major" as well as "A minor". More precisely, "C Major" comprises C, D, E, F, G, A and B whilst "A Minor" comprises A, B, C, D, E, F and G. The difference is the starting point or root.

The Major scale will always have semitone jumps of 2 2 1 2 2 2 1 while a Minor scale has semitone jumps of 2 1 2 2 1 2 2. Semitone means the next note so one semitone up from "C" is "C#". In any major scale, the 6th note will be the equivalent minor scale. Similarly, in any minor scale, the 3rd note will be the equivalent major scale.

When we transpose, we are changing key (ie root). The scale is always maintained.

I have not included the Major scales for Db, Eb, F#, Ab and Bb but that should be easy for you to work out.

Major & Minor Transforms

"Transform" is a general term meaning to convert something into another. Here, transform is just a way to convert from one scale to another. It is not the same as transpose. Transpose changes the key but always maintains the scale. A transform can change the key and/or the scale. Transforms are a convenient way to convert a musical sequence into a different scale and/or key.

This document will concentrate on one-note transforms. If you have a song in C Major, then converting every occurrance of F to F# will transform it into G Major. Similarly, converting every B to A#/Bb will give you F Major.

A scale can be transformed into the one above it or below it simply by comparing the difference between them. The difference should only be one note.

When would you use a transform? Let's say you have a nice sequenced pattern running throughout a song. You have to accommodate a big key change but transposing it doesn't sound right. Then try transforming it instead. Transforming only a few notes will not detract too much from the original pattern and can sound more natural.

Modes

Modes are variant-scales developed from the Major scale simply by starting from a different note. Consider the C Major scale [C, D, E, F, G, A, B, C] which has 7 notes: If you start from D with the same 7 notes, you get a new scale [D, E, F, G, A, B, C, D]. Basically, starting the series from any of 7 notes would give you a different scale and these are called "Modes". Each mode also has a name taken from ancient Greece.

The table below shows the modal scales for the white notes on a piano.

Note - this looks is a bit like transposition but is actually completely different.
In transposition, the series of semitone jumps is the same. In other words, the separation of notes of the scale is maintained (ie the relative differences in frequencies between notes remains).

In modes, the series of semitone jumps changes. In other words, the separation of notes of the scale is different (ie the relative differences in frequencies between notes is not maintained).

The table below shows the same modal scales with a "C" root:-

What do they sound like? Well, Ionian mode is the same as the Major scale and Aeolian mode is the same as Minor scale. The rest sound strangely familiar but not quite right. For example, Dorian mode sounds like the band is playing in "D" but you're doing the melody in "C" instead.

Mode Transforms

We can look as the modes in terms of one-note transforms. The table below highlights the one-note transforms for modes. These particular transforms involve scale changes but not key changes.

If you have a song in C Major (ie Ionian), then converting every occurrance of B to A# (Bb) will give you Mixolydian. Similarly, converting every F to F# will give you Lydian.

The prefix "m" denotes mode. Therefore "mF" is Lydian. "Carets" (^) with plus or minus signs denote transposition up and down respectively and is enumerated in semitones.

Therefore "mF^-1" is Lydian transposed down one semitone.
A scale can be transformed into the one above it or below it simply by comparing the difference between them. The difference should only be one note.

Pentatonics

A pentatonic is simply a scale of five notes. A series of any five notes per octave will qualify as a pentatonic scale.

A Major pentatonic in "C" comprises C, D, E, G and A... which is a common scale used by most cultures in the world. This is achieved by removing the 4th and 7th notes.

What is interesting is that if we remove the 4th and 7th notes from the modal scales, we get quite remarkable results. The table below illustrates the modal pentatonics. This time I'm using the "black" notes on the piano.

What do they sound like (my interpretation)?

"pF, pC & pG" are exactly the same and as they are all the Major pentatonic. The major pentatonic is the mainstay of most Folk music.
"pA" is used mainly in Japanese and Balinese music.
"pE" is a popular scale in music from India (also used in Bali).
"pB" sounds like a mix of arab and indian music (or somewhere from Asia minor). You'll have to judge this one yourself.
"pD" sounds very serious indeed. You'll have to judge this one yourself too.

If we arrange the pentatonics in the same order as the previous one-note transforms, we get the following modal scale transforms:-

A scale can be transformed into the one above it or below it simply by comparing the difference between them. The difference should only be one note.

In addition to the above transforms, there are a further set of transforms for the modal-pentatonics. The table below is slightly different as it groups the possible transforms by each pentatonic. These particular transforms involve scale changes ans well as key changes.

Well, there you have it... all the posible one-note transforms for the pentatonics. If wish to transform from one pentatonic to another but no direct one-note transform is available, then you will have to do it in two or more steps.

Example Application of Transforms

Transforms are useful for converting from one scale and/or key to another. Of all the transforms described in this document, the modal pentatonic transforms are the most interesting to apply because the results are quite remarkable.

If you have a sequencer, try this modal pentatonic experiment:

- Write a short pattern using only the bblack notes... name it "pFCG".
- Using "pFCG", convert every "A#" into "A"... name it "pD".
- Using "pD", convert every "D#" into "DD"... name it "pA".
- Using "pA", convert every "G#" into "GG"... name it "pE".
- Using "pE", convert every "C#" into "CC"... name it "pB".
- Using "pFCG" again, convert every "F#"" into "E"... name it "pD^+7".
- Using "pD^+7", convert every "C#" intoo "B"... name it "pA^+2".
- Using "pFCG" again, convert every "F#"" into "G"... name it "pB^+1".
- Using "pB^+1", convert every "A#" intoo "C"... name it "pE^+6".
- Then delete "pFCG".

You now have 7 pentatonic patterns: 2 AsiaMins, 2 JapBalis, 2 Indians and 1 Serious. Arrange the patterns in any order you like... you've now made one seriously ethnic-sounding new tune.

Scales Reference

Below is a table of Scales. They are arranged into 3 sections: (a) Non 7 or 5 note scales, (b) 7 note scales, and (c) 5 note scales. They are sorted in order of distance from the root-key.

In general, the non-european scales have not been well documented and many of the names selected may not be representative of their music. For example, indian and indonesian music use a huge range of different scales. Arabic music also use quarter-tone tuning (there are notes in between the semitones). Algerian music can use one scale for the first octave and another for the next. Ethopian music can also use the minor, dorian and mixolydian scales. And this is only the tip of the iceberg. Also remember that the above table is only a guide to scales used and the actual tunings used can vary immensely. In the end, the best source for examining scales it to hear it for yourself and translate it.

Conforming to Classical Notation

You do not have to know how to read classical notation in order to use the information in this section. This information is provided as an additional guide to scales because of the limitations imposed by classical notation. For example, the scale of "A# major" and "Bb major" are exactly the same but classical notation only allows for "Bb major".

The classical notation system is well suited for instruments which are "pre-fingered" for the major scale (eg keyboards) but, for "linear" instruments (eg guitar, violin), it requires more familiarisation.

Classical Notation system:-

• The Range of Notes - is represented by a pair of "Staffs" (a Staff has 5 horizontal lines) and these are marked by "Clefs" (ie either treble or bass).
• Note Identification - Every line or space has been preassigned a note "letter" (ie C, D, E, F, G, A, B). The large space between the Staffs has an imaginary line which represents "middle C".
• The Duration of Notes - are represented by a set of Note-Symbols (usually containing some form of circular dot).
• The Notes to be played - are the placement of Symbols either on the lines or in the spaces.
• Sharp and Flat Notes - "#" and "b" can also be placed next to the Note-Symbols on the Staff.
• Specific Scales - are declared by a "Key-Signature" (a set of sharps or flats on the relevant line or space) at the beginning of the Staff (eg The scale of G major or E minor is declared by marking "#" on "F" locations at the start).

If you are using the scale of C major or A minor (the white notes on a piano), you will not have to pre-mark any sharps or flats as Key Signature. With any other scale, you will need to assign sharps or flats.

With classical notation, problems arises because the Staff represents notes by their "letter". This means that every note in the scale should have a different letter. For example, the scale of F major is F, G, A, Bb, C, D, E. You should not use A# instead of Bb, otherwise the "A#" will have to share the same line or space as "A" (and the "B" line or space will not be used at all). This will cause problems with the Key-Signature.

The table below gives Major and Minor Scales which conform to classical notation. Note - as you count the notes in the scale, you are also counting "letters" (ie In E major, the 6th note is "C#"... so counting 1, 2, 3, 4, 5, 6 is counting E, F, G, A, B, C... and "C" is letter no.6 from "E").

Note - F# major contains "E#" (which is "F") and that Eb minor contains "Cb" (which is "B"). This is a small discrepancy in the system.

The table below illustrates the "letter" problems of using the non-conforming keys. Notes in brackets () indicate small discrepancies. Notes in square brackets [] indicate serious problems.

These problems do not exist physically, scientifically or mathematically. The problems arise from the system itself. However, the classical notation system is the de facto "language" of music. Plus the system is fairly compact and concise. So perhaps this extra "learning" is not too bad.

The table below shows the Scales of the Major and Minor Keys which conform to classical notation. This may be easier to visualise and remember.

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