1. Frequency
2. Notes and Octaves
3. Tuning Notes
4. Equal-Tempered Tuning
5. Scales
6. Major and Minor Scales
7. Major & Minor Transforms
8. Modes
9. Modal Transforms
10. Pentatonics
11. Modal-Pentatonic Transforms
12. Example Application of Transforms
13. Scales Reference
14. Conforming to Classical Notation

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.

"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).

For simplicity, let's call 132 Hz = "C2", 264 Hz = "C3", 528 Hz = "C4" and 1,056 Hz = "C5". By convention, the first note in a numbered octave is "A" (ie G#3 is followed by A4).

If we use fractions where the numerator and denominator are whole numbers, we are creating the "just intonation" sysem of tuning. The fractions are listed in the table below and are referenced to "C".

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.

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 (base 2) 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.

Since this tuning is mathematically derived, a song will sound "correct" when played in a different key.

Special note - The decision to use A4 = 440 Hz, 12 notes per octave and naming them A to G was due to historical circumstances. Any other combination would also be valid. However, the equal-tempered tuning is now the de facto system.

A Scale is usually referenced to a "root" note (eg C). Typically, we use notes from the "equal-tempered" tuning comprising 12 notes per octave; C, C#, D, D#, E, F, F#, G, G#, A, A# and B.

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.

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.

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.

By a process called "transposition", we can workout the major or minor scale for every key (ie root). Transposition is basically starting from another key but still maintaining the separation of notes by following the same sequence of semitone jumps. In other words, we are shifting the scale to a different starting note. We can calculate the "Db Major" scale as being Db, Eb, F, Gb, Ab, Bb and C. The concurring minor for the "Db major" scale will be "Bb minor".

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

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.

The table below highlights the one-note transforms for the major scale. These particular transforms only involve Key changes (not scale).

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.

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

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.

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.

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.

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.

If you have a sequencer, try this modal pentatonic experiment:
- Write a short pattern using only the black notes... name it "pFCG".
- Using "pFCG", convert every "A#" into "A"... name it "pD".
- Using "pD", convert every "D#" into "D"... name it "pA".
- Using "pA", convert every "G#" into "G"... name it "pE".
- Using "pE", convert every "C#" into "C"... name it "pB".
- Using "pFCG" again, convert every "F#" into "E"... name it "pD^+7".
- Using "pD^+7", convert every "C#" into "B"... name it "pA^+2".
- Using "pFCG" again, convert every "F#" into "G"... name it "pB^+1".
- Using "pB^+1", convert every "A#" into "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.

NAME	C	-	D	-	E	F	-	G	-	A	-	B	C	ALTERNATIVE
Chromatic	1	2	3	4	5	6	7	8	9	10	11	12	1	-
Spanish 8 Tone	1	2	-	3	4	5	6	-	7	-	8	-	1	-
Flamenco	1	2	-	3	4	5	-	6	7	-	8	-	1	-
Symmetrical	1	2	-	3	4	-	5	6	-	7	8	-	1	Inverted Diminished
Diminished	1	-	2	3	-	4	5	-	6	7	-	8	1	-
Whole Tone	1	-	2	-	3	-	4	-	5	-	6	-	1	-
Augmented	1	-	-	2	3	-	-	4	5	-	-	6	1	-
3 semitone	1	-	-	2	-	-	3	-	-	4	-	-	1	-
4 semitone	1	-	-	-	2	-	-	-	3	-	-	-	1	-
NAME	C	-	D	-	E	F	-	G	-	A	-	B	C	ALTERNATIVE
Ultra Locrian	1	2	-	3	4	-	5	-	6	7	-	-	1	-
Super Locrian	1	2	-	3	4	-	5	-	6	-	7	-	1	Ravel
Indian-ish*	1	2	-	3	4	-	-	5	6	-	7	-	1	-
Locrian	1	2	-	3	-	4	5	-	6	-	7	-	1	-
Phrygian	1	2	-	3	-	4	-	5	6	-	7	-	1	-
Neapolitan Minor	1	2	-	3	-	4	-	5	6	-	-	7	1	-
Javanese	1	2	-	3	-	4	-	5	-	6	7	-	1	-
Neapolitan Major	1	2	-	3	-	4	-	5	-	6	-	7	1	-
Todi (Indian)	1	2	-	3	-	-	4	5	6	-	-	7	1	-
Persian	1	2	-	-	3	4	5	-	6	-	-	7	1	-
Oriental	1	2	-	-	3	4	5	-	-	6	7	-	1	-
Maj.Phrygian (Dom)	1	2	-	-	3	4	-	5	6	-	7	-	1	Spanish/ Jewish
Double Harmonic	1	2	-	-	3	4	-	5	6	-	-	7	1	Gypsy/ Byzantine/ Charhargah
Marva (Indian)	1	2	-	-	3	-	4	5	-	6	-	7	1	-
Enigmatic	1	2	-	-	3	-	4	-	5	-	6	7	1	-
NAME	C	-	D	-	E	F	-	G	-	A	-	B	C	ALTERNATIVE
Locrian Natural 2nd	1	-	2	3	-	4	5	-	6	-	7	-	1	-
Minor (natural)	1	-	2	3	-	4	-	5	6	-	7	-	1	Aeolian/ Algerian (oct2)
Harmonic Minor	1	-	2	3	-	4	-	5	6	-	-	7	1	Mohammedan
Dorian	1	-	2	3	-	4	-	5	-	6	7	-	1	-
Melodic Minor (Asc)	1	-	2	3	-	4	-	5	-	6	-	7	1	Hawaiian
Hungarian Gypsy	1	-	2	3	-	-	4	5	6	-	7	-	1	-
Hungarian Minor	1	-	2	3	-	-	4	5	6	-	-	7	1	Algerian (oct1)
Romanian	1	-	2	3	-	-	4	5	-	6	7	-	1	-
NAME	C	-	D	-	E	F	-	G	-	A	-	B	C	ALTERNATIVE
Maj. Locrian	1	-	2	-	3	4	5	-	6	-	7	-	1	Arabian
Hindu	1	-	2	-	3	4	-	5	6	-	7	-	1	-
Ethiopian 1	1	-	2	-	3	4	-	5	6	-	-	7	1	-
Mixolydian	1	-	2	-	3	4	-	5	-	6	7	-	1	-
Major	1	-	2	-	3	4	-	5	-	6	-	7	1	Ionian
Mixolydian Aug.	1	-	2	-	3	4	-	-	5	6	7	-	1	-
Harmonic Major	1	-	2	-	3	4	-	-	5	6	-	7	1	-
Lydian Min.	1	-	2	-	3	-	4	5	6	-	7	-	1	-
Lydian Dominant	1	-	2	-	3	-	4	5	-	6	7	-	1	Overtone
Lydian	1	-	2	-	3	-	4	5	-	6	-	7	1	-
Lydian Aug.	1	-	2	-	3	-	4	-	5	6	7	-	1	-
Leading Whole Tone	1	-	2	-	3	-	4	-	5	-	6	7	1	-
Bluesy R&R*	1	-	-	2	3	4	-	5	-	6	7	-	1	-
Hungarian Major	1	-	-	2	3	-	4	5	-	6	7	-	1	Lydian sharp2nd
NAME	C	-	D	-	E	F	-	G	-	A	-	B	C	ALTERNATIVE
"pB"	1	2	-	3	-	-	4	-	5	-	-	-	1	-
Balinese 1	1	2	-	3	-	-	-	4	5	-	-	-	1	"pE"
Pelog (Balinese)	1	2	-	3	-	-	-	4	-	-	5	-	1	-
Iwato (Japanese)	1	2	-	-	-	3	4	-	-	-	5	-	1	-
Japanese 1	1	2	-	-	-	3	-	4	5	-	-	-	1	Kumoi
Hirajoshi (Japanese)	1	-	2	3	-	-	-	4	5	-	-	-	1	"pA"
"pD"	1	-	2	3	-	-	-	4	-	5	-	-	1	-
Pentatonic Major	1	-	2	-	3	-	-	4	-	5	-	-	1	Chinese 1/ Mongolian/ "pFCG"
Egyptian	1	-	2	-	-	3	-	4	-	-	5	-	1	-
Pentatonic Minor	1	-	-	2	-	3	-	4	-	-	5	-	1	-
Chinese 2	1	-	-	-	2	-	3	4	-	-	-	5	1	-

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.

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).

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").

MAJOR SCALE   R   -   2   -   3   4   -   5   -   6   -   7 
   C  maj.:   C   -   D   -   E   F   -   G   -   A   -   B 
   Db maj.:   Db  -   Eb  -   F   Gb  -   Ab  -   Bb  -   C
   D  maj.:   D   -   E   -   F#  G   -   A   -   B   -   C#
   Eb maj.:   Eb  -   F   -   G   Ab  -   Bb  -   C   -   D 
   E  maj.:   E   -   F#  -   G#  A   -   B   -   C#  -   D#
   F  maj.:   F   -   G   -   A   Bb  -   C   -   D   -   E 
   F# maj.:   F#  -   G#  -   A#  B   -   C#  -   D#  -  (E#)
   G  maj.:   G   -   A   -   B   C   -   D   -   E   -   F#
   Ab maj.:   Ab  -   Bb  -   C   Db  -   Eb  -   F   -   G 
   A  maj.:   A   -   B   -   C#  D   -   E   -   F#  -   G#
   Bb maj.:   Bb  -   C   -   D   Eb  -   F   -   G   -   A 
   B  maj.:   B   -   C#  -   D#  E   -   F#  -   G#  -   A#
MINOR SCALE   R   -   2   b3  -   4   -   5   b6  -   b7  -
   A  min.:   A   -   B   C   -   D   -   E   F   -   G   -
   Bb min.:   Bb  -   Cb  Db  -   Eb  -   F   Gb  -   Ab  -
   B  min.:   B   -   C#  D   -   E   -   F#  G   -   A   -
   C  min.:   C   -   D   Eb  -   F   -   G   Ab  -   Bb  -
   C# min.:   C#  -   D#  E   -   F#  -   G#  A   -   B   -
   D  min.:   D   -   E   F   -   G   -   A   Bb  -   C   -
   Eb min.:   Eb  -   F   Gb  -   Ab  -   Bb (Cb) -   Db  -
   E  min.:   E   -   F#  G   -   A   -   B   C   -   D   -
   F  min.:   F   -   G   Ab  -   Bb  -   C   Db  -   Eb  -
   F# min.:   F#  -   G#  A   -   B   -   C#  D   -   E   -
   G  min.:   G   -   A   Bb  -   C   -   D   Eb  -   F   -
   G# min.:   G#  -   A#  B   -   C#  -   D#  E   -   F#  -

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.

C# maj.:   C#    D#   (E#)   F#    G#    A#   (B#)
D# maj.:   D#   (E#)  [F##]  G#    A#   (B#)  [C##]
Gb maj.:   Gb    Ab    Bb   (Cb)   Db    Eb    F  
G# maj.:   G#    A#   (B#)   C#    D#   (E#)  [F##]
A# maj.:   A#   (B#)  [C##]  D#   (E#)  [F##] [G##]
Ab min.:   Ab    Bb   (Cb)   Db    Eb   (Fb)   Gb 
A# min.:   A#   (B#)   C#    D#   (E#)   F#    G# 
Db min.:   Db    Eb   (Fb)   Gb    Ab   [Bbb] (Cb)
D# min.:   D#   (E#)   F#    G#    A#    B     C# 
Gb min.:   Gb    Ab   [Bbb] (Cb)   Db   [Ebb] (Fb)

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.

Major only              Db                       Ab      
both Major & Minor   C      D  Eb  E   F  F#  G      A  Bb  B   C
Minor only              C#                       G#