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.
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#.
"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:
Tube Length Note Octave Frequency
Original C Original 264 Hz = 264 Hz
Half C Up 1 264 x 2 = 528 Hz
Quarter C Up 2 264 x 4 = 1,056 Hz
Double C Down 1 264 / 2 = 132 Hz
For simplicity, let's call 132 Hz = "C1", 264 Hz = "C2", 528 Hz = "C3" and 1,056 Hz = "C4". By convention, the first note in a numbered octave is "A" (ie G#3 is followed by A4).
Let's look at the hollow tube length again. Halving it gives us an octave higher. What happens for lengths in between? Well, for lengths in between, we get the notes in between.
If we use fractions where the numerator and denominator are whole numbers, we are creating the "just intonation" system of tuning. The fractions are listed in the table below and are referenced to "C".
Tube Length Frequency Note
Original 264 x 1 = 264 Hz C3
3 / 4 264 x 4 / 3 = 352 Hz F3
2 / 3 264 x 3 / 2 = 396 Hz G3
3 / 5 264 x 5 / 3 = 440 Hz A4
4 / 5 264 x 5 / 4 = 330 Hz E3
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.
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 (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.
Calculation for Equal-Tempered tuning [A4 = 440Hz]Hertz Octave=1 Octave=2 Octave=3 Octave=4 Octave=5 Octave=6 0 A 55.000 110.000 220.000 440.000 880.000 1,760.000 1 A#/Bb 58.270 116.541 233.082 466.164 932.328 1,864.655 2 B 61.735 123.471 246.942 493.883 987.767 1,975.533 3 C 65.406 130.813 261.626 523.251 1,046.502 2,093.005 4 C#/Db 69.296 138.591 277.183 554.365 1,108.731 2,217.461 5 D 73.416 146.832 293.665 587.330 1,174.659 2,349.318 6 D#/Eb 77.782 155.563 311.127 622.254 1,244.508 2,489.016 7 E 82.407 164.814 329.628 659.255 1,318.510 2,637.020 8 F 87.307 174.614 349.228 698.456 1,396.913 2,793.826 9 F#/Gb 92.499 184.997 369.994 739.989 1,479.978 2,959.95510 G 97.999 195.998 391.995 783.991 1,567.982 3,135.96311 G#/Ab 103.826 207.652 415.305 830.609 1,661.219 3,322.43812 A 110.000 220.000 440.000 880.000 1,760.000 3,520.000Since 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.
Musicians compose and play songs. In order to ensure that the song is played correctly, we have to determine which notes are valid. A Scale is a series of notes which we define as "correct" or appropriate for a song. Normally, we only need to define the series within an octave and the same series will be used for all octaves.
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# & 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.
Examples of various Scales (Root = "C")Name C Db D Eb E F Gb G Ab A Bb B CMajor 1 2 3 4 5 6 7 1Minor (natural) 1 2 3 4 5 6 7 1Harmonic Minor 1 2 3 4 5 6 7 1Melodic Minor Asc 1 2 3 4 5 6 7 1Melodic Minor Desc 1 2 3 4 5 6 7 1Enigmatic 1 2 3 4 5 6 7 1Chromatic 1 2 3 4 5 6 7 8 9 10 11 12 1Diminished 1 2 3 4 5 6 7 8 1Whole Tone 1 2 3 4 5 6 1Augmented 1 2 3 4 5 6 1Pentatonic Major 1 2 3 4 5 1Pentatonic Minor 1 2 3 4 5 13 semitone 1 2 3 4 14 semitone 1 2 3 1Bluesy R&R* 1 2 3 4 5 6 7 1Indian-ish* 1 2 3 4 5 6 7 1As 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 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.
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".
The Major Scale Key C C# D D# E F F# G G# A A# B C C 1 2 3 4 5 6 7 1 D 7 1 2 3 4 5 6 E 6 7 1 2 3 4 5 F 5 6 7 1 2 3 4 5 G 4 5 6 7 1 2 3 4 A 3 4 5 6 7 1 2 B 2 3 4 5 6 7 1When 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.
"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 occurrence 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).
One Note Transforms - Comparisons for the Major Scale Key C C# D D# E F F# G G# A A# B C F 5 6 7 1 2 3 4 5 C 1 2 3 4 5 6 7 1 G 4 5 6 7 1 2 3 4 D 7 1 2 3 4 5 6 A 3 4 5 6 7 1 2 E 6 7 1 2 3 4 5 B 2 3 4 5 6 7 1A 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 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.
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 occurrence 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.
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 & A ; which is a common scale used by most cultures in the world. This is achieved by removing the 4th and 7th notes. The minor pentatonic refers to the same scale starting at "A" comprising A, C, D, E & G.
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.
Try playing them to get an idea of how they sound. Some are happy, some are sad, some are exotic sounding.
"pC, pF & pG" are all the Major pentatonic. They are exactly the same with semitone jumps as 2 2 3 2 3. The major pentatonic is basis of Chinese music and is also the mainstay of most Folk music including the Blues and Malayan gamelan. It has a familiar "happy" character.
"pA" and "pE" sound very similar with a "melancholic" character. The difference between "pA" and "pE" is only the second note of the scale.
"pA" is popularly used in Japanese music.
"pE" is the main scale in Balinese gamelan music as well as used in some Javan & Indian music.
"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.
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:
You now have 7 pentatonic patterns: 2 AsiaMins, 2 Japans, 2 Balis and 1 Serious.
Arrange the patterns in any order you like... you've now made one seriously ethnic-sounding new tune.
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.
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.