Talk:Music Theory/Scales and Intervals

"Pentatonic scales all have five notes, although this is really a coincidence. They are called pentatonic scales because each note in the scale is a fifth (seven semitones) relative to another note."
 * I don't think this is true; it only holds true for the major pentatonic. -- Jimregan 08:49, 18 May 2004 (UTC)

I'm not 100% sure about it myself, but I think other "pentatonic" scales are just modes of the major pentatonic...I should perhaps look into this further. Speaking of modes, they already have their own chapter. They don't need to be discussed here, just mentioned in brief with a cross-reference.

I think the parts showing the scales really belonged on their own lines. On my video resolution the scales are currently word-wrapped, which is jarring and inhibits clarity. The only way to ensure they won't wrap is to put them on their own lines.

--Furrykef 11:22, 18 May 2004 (UTC)

Well, as it is it renders for me like this:


 * Here are two versions of the chromatic scale, one with sharps, the other with flats, each on its own line: C C# D D# E F F# G G# A A# B C Db D Eb E F Gb G Ab A Bb B

Maybe what you want is this

C C# D D# E F F# G G# A A# B C Db D Eb E F Gb G Ab A Bb B
 * Here are two versions of the chromatic scale, one with sharps, the other with flats, each on its own line:

-- Jimregan 18:49, 18 May 2004 (UTC)

Ah, yeah, sorry. I should check my previews more carefully...

--Furrykef 01:39, 19 May 2004 (UTC)

Tables
Do we really need those tone/semitone tables? I don't really think it's a good use of table markup. --Furrykef 15:02, 9 Jun 2004 (UTC)

Can the derivation of the pentatonic scale be discussed further? After reading the docs on pentatonic intervals, I'm a bit confused over the statement that all tones are in relation by a fifth. If one bases that statement on the circle of fifths, then the statement is somewhat true with the exception of E does not repeat to C again. I found the above link very interesting in regards to the derivation of a pentatonic scale.

In summary, to generate a pentonic scale, using the Pythagorean method. Use 5/4f to calcuate a fifth and 4/3f to calculate a forth (note however that the forth is not actually part of the pentatonic scale). So using the Pythagorian method, it looks something like:


 * C = 1f
 * D = 3/2f - 4/3f
 * E = ((3/2f - 4/3f) + 3/2f) - 4/3f
 * G = 3/2f
 * A = (3/2f - 4/3f) + 3/2f
 * c = 2f

That looks confusing, but really it's just a cycle of C up a fifth to G, down a forth to D, up a fifth to A, and back down a forth to E (C -> G -> D -> A -> E). The high c is in there, but really it can't be derived using this natural cycle. It's just there to show an octave.

As you play with the numbers, you'll find an exact even interval of 9/8f between C&D,D&E, and G&A, and two larger intervals of 32/27f between E&G and A&c.

A pentatonic scale can also be derived using straight harmonic calculations of:


 * C = 1f
 * D = 9/8f
 * E = 5/4f
 * G = 3/2f
 * A = 5/3f
 * c = 2f

If all these ratios look confusing, just think of 5/4f as f + 1/4f or the base/tonic freq plus a quarter of it's value. 2f of course being double the freq to achieve an octave.

So there you have it, two pentatonic scale derived naturally. The first is called Pythagorean temperament, the second is called Just temperament. Both are interesting in their own respects. The Just temperament uses natural harmonics for all notes. However the interval between each note varies. The Pythagorean temperament is interesting in that the intervals between each tone is exactly 9/8f!

Using the Pythagorean model, a daitonic scale can be produced by splitting the two larger intervals of 32/27f into 9/8f + 256/243f. That 256/243f interval is the definition of a Pythagorean semitone or half tone. When done, you have five 9/8 intervals and two 256/243 intervals in a WWHWWWH pattern and hence the definition of what a diatonic scale is.

I should point out that as cool as the Pythagorean method is, it's not exact. That's because the semitones between E & F and B & c are not exactly a half step (ie: half of 9/8f), it's something like 256/243f. For that reason, the Pythagorean scale is great for music written in one key like so many folk tunes. If one was to switch musical keys with instruments tuned to Pythagorean temperament, one would have to switch instruments as well (ie: one instrument for each musical key). This lead to the adoption of Even termperament. Even temperament essentially devides an octave into 12 equally tuned semitones. This has the advantage in that an Even tuned instrument can be played in any key. The disadvantage of Even temperament is that none of the actual tones in the scale are exact to natual harmonics (other than tonic and octave). The difference is very slight, but it does exist.

The calculations and comparisons between Just, Pythagorean, and Even respectively look something like:

... --releppes 16:18, 13 Jun 2005 (UTC)
 * C: J=1f, P=1f E=1f
 * D: J=9/8, P=9/8f, E=2^(2/12)f
 * E: J=5/4f, P=81/64f, E=2^(4/12)f

What about another table for the simple interval section with # intervals as 1 axis and # semitones as the other axis to show how the notation of music deals with the actual number of semitones between notes? i.e. a 13 (including 0)x 8 grid. I have found no such grid when trying to help my daughter with her grade iv theory and wanted to explain the whole concept. C Knox