# Solomusic's Scales Encyclopedia+ Science of Sound



## 009 (Jul 16, 2004)

This was copied from a website. ( too lazy to type...LOL  ). Here, it contains some general concepts as to other forms of important, but perhaps less known scales. There is a huge segment on the science and frequency of sound also, which leads to the topic why C sharp is not equalled to D flat and why u can get the same harmonics in other positions( layman way of saying ). Read it and it'll give u a clear picture of how the science/physics of harmonics, equal temperant, weel-tempered or the non-tempered work. This multi topic article is really long, but I've decided not to cut it, as it really gives u an in depth understanding of what I thought was essential. As it's quite tedious to read, so I've coloured and enlarged topics headings...hope it'll take the stress out. Take your time, read 1 paragraph everyday...and hopefully u'll complete it within a year! :lol:

SCALES IN MUSIC 
The musical term 'scale' (having nothing to do with fish or lime scale) derives from the Italian word scala, meaning ladder. There are many different types of scale, but the majority of western music uses only two forms, the major scale and the minor scale.

MAJOR AND MINOR SCALES
Major Scale
The major scale is the same as the Ionian mode (see below). After a while in mainstream western music (circa 1500) most of the modal scales dropped out of use, leaving the Ionian as our present day major scale, and the Aeolian as our minor scale. (See A Brief History of Western Music.)

Each note in a scale has a name, the equivalent note in C major is shown in brackets.

Tonic ©
Supertonic (D)
Mediant (E)
Subdominant (F)
Dominant (G)
Submediant (A)
Leading-note (B)
Tonic (C again)
The tonic is the most important in the scale. The second most important is the dominant. This gives rise to related keys; for a piece written in the key of C major, it is classically1 allowed to have a change of key to G major, since that is the dominant in the key of C. It would not be allowed to have a modulation (key change) to Db2 major, for example, because that is in no way related to the original key of C.

The pattern of tones and semitones3 in a major scale is this (T=tone, S=semitone):

T T S T T T S

There is a tone (in C major) from C to D, a tone from D to E, a semitone from E to F and so on.

Minor Scale
The minor scale is based around the Aeolian mode. However, in the Aeolian mode there is a tone between the leading note and the tonic, so the leading note doesn't lead to the tonic as it does in a major scale. To get around that there are two versions of the minor scale; the harmonic minor, which is modified so that it does have a semitone; and the melodic minor, which is different depending on whether the scale is ascending or descending.

This is the pattern of tones and semitones in a harmonic minor scale (T=tone, S=semitone 3/2=3 semitones):

T S T T S 3/2 S

In A minor, the notes are A B C D E F G# A.

The pattern of tones and semitones in a melodic minor scale is this (T=tone, S=semitone):

Ascending: T S T T T T S

Descending: T T S T T S T

In A minor, the notes are ascending: A B C D E F# G# A, and descending: A G F E D C B A.

CHROMATIC SCALES
The term chromatic comes from the Italian for colourful. There are twelve notes in a chromatic scale:

C
C#/Db (These are the same notes on a piano4.)
D
D#/Eb
E
F
F#/Gb
G
G#/Ab
A
A#/Bb
B
There is a semitone step between each note in this scale, making none of the notes more important than any of the others.

MODAL SCALES
The earliest types of scale to be used were the modal scales, used in renaissance music. These were derived from very early ancient Greek scales. There are eight modes:

Ionian - can be played from C to C on the white notes of a piano. This scale is the same as the Major scale. (see above)
Dorian - can be played from D to D on a piano.
Phrygian - can be played from E to E.
Lydian - can be played from F to F. This mode sounds like a major scale but with a wrong note in.
Mixolydian - can be played from G to G. This mode also sounds quite like a major scale.
Aeolian - can be played from A to A. This is the same as the Minor scale.
Locrian - can be played from B to B. This mode was (and is) very uncommon.
The names for these modes were supposed to be the same names that the Greeks gave them. However, they aren't the same.

Pentatonic Scale
Much traditional folk music is based around a pentatonic, or five note scale. This is also quite widely used in popular music and anything relating to blues. The E minor pentatonic scale is particularly easy to play on the guitar. For a pentanotic scale based on the key of C major, the five notes would be C, D, E, G and A.

Whole Tone Scale
The whole tone scale is perhaps the newest of these scales. It was used by composers such as Debussy. The whole tone scale is similar to the chromatic scale inasmuch as there is a constant gap between every note. In the case of the whole tone scale, this gap is a tone. The notes of a whole tone scale starting on C would be: C, D, E, F#, G# and A#.

The Science Bit
As music is sound, there is physics underlying the way the scale works. You pluck a string (or blow into a pipe) of length X, and you get a particular note (the fundamental). You also get small percentages of other notes (the harmonics).

The first harmonic is one octave above the fundamental; the second is 1.5 octaves above. These are heard (usually subconsciously), and so the brain thinks they sound right. That's why all modes and scales have a tonic, a dominant (the second harmonic, but an octave low) and the next tonic (the first harmonic). From here, the various modes part company (depending on who cut the string or chopped the pipes), but they do have a note (the median) to go in the gap between tonic and dominant. The other notes just fill in the gaps (or not much, in pentatonic).

In any single key, you can generate all the notes in the scale by starting with a long enough string\pipe and continuing to cut it in half. That accounts for the ideal intervals between all the notes.

Unfortunately, if you start from a different fundamental (like from G instead of C, for example), the seconds, fourths and sevenths that you generate fall at slightly different frequencies from what should be 'the same' notes obtained from different keys. The difference, is called a 'comma', and it was what drove harpsichord tuners nuts. The 'equal temperament' tuning (the reason Bach wrote his famous Well Tempered Clavier, a collection of Preludes and Fugues in every key) was a compromise which equalled out all the 'commas' over the whole range of the keyboard. Bach wrote the 48, as these have become known, to prove that this method of tuning worked well for every single key.

There are many ways of tuning, some of which may put C# above Db, but by far the most put it below - other than Equal Temperament which equates them. The 'natural scale' is not a matter of universal agreement, but all do agree that the natural major third is smaller, not bigger, than the equal-tempered one; A-C# is less of an interval in natural harmonics (on a bugle for example) than in equal temperament. Two notes which sound the same but are written differently are called enharmonic notes. For more information on the mathematics of this, visit Sol-Fa - The Key to Temperament

Sol-Fa - The Key to Temperament  
This entry continues on from 'Sol-Fa - The Key to the Riddle of Staff Notation', and briefly discusses the subject of temperament in music, using Sol-Fa for reference.

Proportion
The six notes of the Guidonian hexachord (see the first entry in this series) fall naturally into the kinship of pure mathematical relations. The medieval and renaissance theorists didn't know about frequency, which is how we define pitch nowadays, but they knew the underlying maths. Indeed it had been known since Pythagoras, c500 BC, who worked out the mathematical relations for pipe lengths, string lengths, and bell weights. These all hold the same relationships as the inverse of the frequencies.

Frequency is measured in cycles (vibrations) per second, the units of measurement being named after scientist Heinrich Rudolf Hertz. A standard tuning fork defines the note A above middle C as 440 cycles per second, or 440Hz. Non-standard forks can also be bought.

If Ut has a frequency of 8 times x then Re is 9x, and Mi is 10x. The same relations bind Fa with Sol and La: if Fa is 8y, Sol is 9y and La is 10y. Furthermore, with Fa at 8y, Ut will be 6y; a relationship of 4 to 31. To put them all in a tidy line, we need to use bigger numbers. Taking x for granted from now on, we get these harmonious proportions:

La 40 
Sol 36 
Fa 32 
Mi 30 
Re 27 
Ut 24

Now comparing those numbers we see that:

Ut-Re and Fa-Sol are both 8:9
Re-Mi and Sol-La are both 9:10
Ut-Mi and Fa-La are both 4:5
Ut-Fa, Re-Sol and Mi-La are all 3:4
Ut-Sol is 2:3
Ut-La is 3:5
Furthermore, the relation of any note to its octave above (e.g. C-c, D-d) is always 1:2.

Harmony everywhere! Well, almost: the only pairs whose relationships don't reduce to smaller numbers are Re-Fa and Re-La.

The names given to the intervals are: 1:2 perfect octave 
2:3 perfect fifth 
3:4 perfect fourth 
4:5 major third 
5:6 minor third 
8:9 greater tone 
9:10 lesser tone 
15:16 diatonic semitone2

By tradition, the thirds, fourths, fifths and octaves are concordant3, and the tones and semitones are discords, when two notes are sounded together. The same consonances and dissonances apply to the inversion of each interval, that is, the intervals occurring when one of a pair of notes is flipped to its octave - making thirds become sixths and seconds become sevenths; sixths are concords, sevenths discords.

Prime numbers greater than five are not dealt with in classical European harmony at all, nor are their multiples - the Greeks counted the numbers up to four as concordant; five was added to the list in the 16th Century by the theorist Zarlino, having been the favourite consonance in practice for most of a century.

The Syntonic and Pythagorean Commas
Temperament issues arise from the fact that some notes require retuning if they are to join with others in perfect harmony. This happens not only between different hexachords, but even within the same hexachord: as we saw above, Re-Fa and Re-La are not harmonic intervals within a hexachord. Here are some notes for comparison between the original medieval 'hard', 'natural' and 'soft' hexachords:

Hard Hexachord Natural Hexachord Soft Hexachord
d La ≠ d Sol 
c Sol = c Fa 
b Fa / b Mi 
a La = a Mi ≠ a Re 
G Sol = G Re = G Ut 
F Fa = F Ut 
E La = E Mi 
D Sol = D Re 
C Fa = C Ut 
B Mi 
A Re 
Γ Ut

From here on in this entry the medieval distinction between capital and small letter-names is abandoned, and all members of any pitch class4 are taken as equivalent.

The notes B Fa and B Mi are a semitone apart, as shown in the first entry in this series. This is not a question of temperament. There are however other differences, small but problematic, between notes that are treated as equal.

Two As, Two Ds

'A La Mi' is not as high as A Re; and D La is not as high as D Sol.

If F Fa is 64 and G Re is 72 then A Mi is 80; but with G Ut still at 72, A Re is 81. Similarly, if C Sol is 72 then D La is 80; but with C Fa at 72, D Sol is 81.

This difference, the difference in size between the greater and lesser tone, was known to the ancient Greeks as the syntonic comma. It is about a fifth of a semitone, which is clearly audible to untrained ears.

Flats versus Sharps: Never the Twain Shall Meet
Additional hexachords can be generated in both directions: denoting F as Sol will invoke a hexachord starting on Bb, introducing the note Eb, and so on towards the flat side; denoting G as Fa will produce one on D and introduce F#, and so on to the sharp side.

The sharps will never meet the flats; on the one side we reach Ab and on the other G#, but there is no reason why these two should be at the same pitch; and indeed they are not. The difference between Ab and G#, going round the cycle of fifths, is slightly larger than a syntonic comma5, and was called the Pythagorean comma by the ancient Greeks.

Practical Solutions
Some instruments such as concertinas have been traditionally made with separate buttons for Ab and G#. If one pitch is to serve for both then either:

One or more bad ('wolf') intervals must be tolerated between certain notes, or
Equal temperament or some other 'well-tempered' system must divide out the discrepancy among some or all of the intervals.

Some Temperaments
The tuning system most recommended in the medieval period was Pythagorean, which disregards the pleasantness or unpleasantness of the thirds altogether (after all, they were officially dissonances before Zarlino) and simply proceeds by pure fifths and fourths (2:3 and 3:4). Tuning all twelve notes of a keyboard we will end up with the 'wolf' interval between our first note and the last: too big by a Pythagorean comma, if that interval is a fourth, too small if it is a fifth. Most of the major thirds (Ut-Mi) that come out as a result are unpleasantly wide (by a syntonic comma, since we generate only greater tones, and no lesser tones). However, a strange coincidence happens towards the end of our tuning: a third that straddles the wolf fourth will become wonderfully pure, as the two wrongs in this case make a right, and the Pythagorean comma almost exactly cancels the syntonic comma. Since we can choose which will be our first and last notes to tune, we can place the wolf fourth wherever we like. Putting it between D and A (this means tuning A as Bbb) would result in beautifully-tuned chords of F major, C major and G major. D major would have a lovely third but a very nasty fifth.

A partial solution, widely used in the Renaissance, is mean-tone tuning, which yields a fair number of good chords. It is done by dividing Ut-Mi (4:5) into exactly equal steps - Ut:Re:Mi becomes 4:√20:5 - and it sounds very pleasant in keys with few sharps or flats. Its incompleteness, however, led to many different versions being tried: quarter-comma mean-tone, fifth-comma, sixth-comma and so on.

Tuning a guitar to four perfect fourths and a pure major third will leave the top E flatter than the bass E by exactly a syntonic comma. Widening all the intervals equally will approximate to fifth-comma mean-tone tuning. This is complicated by the placing of the frets, which are normally equal-tempered, though many guitar-makers (particularly in Spain) flatten6 the second fret; experience shows that an equal-tempered second fret always sounds too sharp. String physics, rather than mathematical considerations, may account for this. A solution that many guitarists use is to tune all the bottom strings fairly pure and tune the top two more widely; this will sweeten the chord of A, the guitar's home key, and also D and G; but the E chord will have a bit of a rasp with its flat fifth.

Temperament makes a great difference to an organ, since a triumphant long-drawn-out final organ chord is heard without any decay of the sound, and without any chance of adjustment. For this reason, organ tuners in England held on to mean-tone tuning, and only gave in to equal temperament around 1850.

Some viol players split their first fret, to give a sharp Bb on an A string but a flat C# on a C string. Others adjust the tuning as they go, as a large alteration can be made by pushing or pulling a string with the fretting finger.

Equal temperament was proposed by Aristoxenus around 350 BC, rejected for almost two thousand years, then championed by Rameau in the 18h Century, and only more-or-less-universally applied since around 1850. By this time the early-music movement had begun, which has reinstated unequal temperaments for performing period music. Equal temperament is arrived at by dividing the octave into twelve equal semitones, using the twelfth root of two. Aristoxenus seems to have done it by ear: he was a practical musician. Newton may have been the first to figure out the twelfth root of two mathematically. He didn't publish it, but the figures for a chromatic scale (which uses all twelve pitches in an octave), to many decimal places, have been found among his notes.

In equal temperament every interval except the octave is compromised - the harmonious mathematical relationships are hinted at rather than sounded. The biggest difference equal temperament makes to the sound of a chord is that the major thirds are too wide, by about one seventh of a semitone. Musicians other than keyboard players can and do bend the tuning of these notes to add a glow to significant chords. Of course they also bend notes for expressive reasons, as well as harmonious ones.

Plucked instruments work well in equal temperament, perhaps because their sound dies out and seems to adjust itself miraculously in the process. Pianos are in the worst position; because of the high tension on their frames, they suffer from a condition called 'inharmonicity', which is offset by tuning their octaves wide. String players accordingly need to adjust their intonation when playing with piano accompaniment.

Many varieties of mean-tone temperament were used in the Renaissance, and the 17th and 18th Centuries saw even more inventive systems come into vogue. Theorists searched for a system that would give as many good chords as possible in as many keys as possible. JS Bach wrote two sets of preludes and fugues in all twelve major and twelve minor keys, known as the Well Tempered Clavier7; it appears that when he wrote the first book of his '48', Bach did not use equal temperament but a well-tempered system close to those of Werckmeister and Neidhardt. However by the time he wrote the second book (the second set of twenty-four preludes and fugues) he had come round to equal or near-equal temperament. These systems sought to keep a pleasing variety between keys; others purposely made distant keys more discordant than familiar keys, so that a composer could increase the drama of his compositions by straying into those outlandish keys before returning to a comfortable conclusion.

Vallotti's 1754 system, now favoured by many viol players, makes several of the commonly used thirds harmonious by dividing the Pythagorean comma between all the 'natural' notes (six fifths: F-C-G-D-A-E-B), while the six fifths incorporating 'black' notes8 (B-F#-C#-G#-Eb-Bb-F) remain pure.

1 It is vital to distinguish proportion from arithmetical sequence. These relations between notes are all proportional, so the step from 8 to 9 is not the same size as the step from 9 to 10. Rather than 'an increase of one' it is 'an increase of one-eighth'. Therefore 24-to-27 is the same 'interval' as 32-to-36. It can be written as 8:9, or as a fraction.
2 The chromatic semitone, Bb-B, is smaller than the diatonic semitone. The proportion is 128:135, which is close to 18:19.
3 The fourth is an exception: a fourth is treated as discordant, unless there is another concordant note below it in a chord. This has to do with the language of harmony, rather than the science of acoustics.
4 A pitch class is defined as all the octaves of a note: all the As make up the pitch class A.
5 The proportion is 219:312, or 524288:531441, which is between 73:74 and 74:75 (for the purpose of comparison with the syntonic comma, 80:81), and almost a quarter of a semitone.
6 'Flatten' in this context means 'place closer to the nut'.
7 Also known as Bach's '48'. Clavier simply means keyboard.
8 Black keys on the piano keyboard.


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## max (Jul 17, 2004)

Wow, this is all GREAT info... Thanks DW!

Btw. if anybody needs to memorize their modes...

I
Don't
Play
Lousy
Music
After
Lunch

Ionian
Dorian
Phrygian
Lydian
Mixolydian
Aeolian
Locrian

Dorian starts on D. Just go up from each (and one Back for "I")
Play - E
Lousy - F
Music - G
After - A
Lunch - B
I - C

Also remember that Mixolydian is in the Middle (M for middle) so Lydian comes right before it... That way you can tell your L's apart. Locrian is spelled closer to Lunch (if you look at it)...

Don't know why I felt like telling you all this, but this has helped me on Music History and Theory exams... It tells you all the modes, in order of their notes. Write out the mode and then you can find the half steps/whole steps and then of course, you can transpose to any key.

Hope this helps somebody!


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## baroque flute (Jul 21, 2004)

I hadn't seen this before today. This is great stuff--thanks, DW!!!


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## Quaverion (Jul 20, 2004)

Yeah, thanks. What encyclopedia was it from?


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## Harvey (Jul 26, 2004)

Good stuff. If you continue to edit/refine/add to it it could be pinned.


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## 009 (Jul 16, 2004)

Max's memory method is really interesting, and so down to earth. I'm gonna print it and hand it to my students!


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## MoonlightSonata (Mar 29, 2014)

Very useful, I shall remember this next time I use modes.


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