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It took me a while to get to this, partially from being busy, partially from being lazy, and just as much from not knowing whether to take a short and simple approach, or a deeper theoretical dive. I’ve decided to do both.
SHORT AND SIMPLE
Seconds are the smallest non-unison intervals in diatonic units. They are the building blocks of most scales and thus they sound very familiar to us. There are two basic seconds, major and minor.
A major second is also called a whole step, or a whole tone. It is the distance between the first two notes of a major scale. It is also the distance of two frets on a guitar. Ascending it’s simply the first two notes of a major scale (or major pentatonic), or Frere Jacques. Descending, think Three Blind Mice.
A minor second is also called a half step or a semitone. It is the distance between the 7th and 8th degrees of a major scaled “Ti Do” of DoReMiFaSolLaTiDo. (Also “MiFa”). It is one fret on the guitar. Ascending, think of the bass in Jaws, or the Pink Panther theme. Descending, think Fur Elise by Beethoven, or Joy to the World (Handel, not Three Dog Night), or the first notes of a Major Scale going down instead of up.
If you divide an octave up into nothing but whole notes, you get a whole tone scale, which sounds odd to us, but has been the basis for some beautiful music. There are only two whole tone scales: CDEF#G#A#C, or C#D#FGABC#. Here’s a short video showing the whole tone scale on a bass:
And here is Debussy making some beautiful, airy music using mainly whole tones in the melody:
If you divided an octave up into nothing but semitones, you have a chromatic scale. There is only one chromatic scale. If someone asks you what scales you play over a certain chord, you can be snarky and tell them that you only know one scale, the chromatic scale, and you choose which notes to play from that.
BB King was a big fan of the chromatic scale for warming up, and for exploration:
Here’s an amazing piece by Chopin which is based almost entirely on chromatic runs.
If you alternate whole steps and half steps, you get a diminished scale. For example C D Eb F Gb Ab A B C. There are three diminished scales, one starting on C, one on Db, and one on D. When you get to Eb, you simply repeat the same pattern as the C diminished scale, but up a minor third. I said this part would be short (I lied), and there’s too much to go into simply on diminished scales, so I will simply note that Robben Ford is a big fan of their use in the blues:
DEEPER DIVE
Where do the Major and Minor second come from? The first thing to note is that despite being called major and minor, they are both very dissonant intervals. For a long time, they were considered to be dissonant, but we have grown accustomed to them in melodies because of their use in scales. When sounded together harmonically, they still sound very dissonant to most people.
Remember that the main consonances from the harmonic series are the unison/octave, the fifth and the fourth. These have ratios of 2:1 for the octave. 3:2 for the fifth, and 4:3 for the fourth. The next consonant interval in the overtone series is the major third, which has a nice 5:4 ratio.
Now, lets say that we want to smooth out the octave and be able to sing notes in between these notes. One way to go about this is to derive the notes from what we already have. The first thing we might do is look at the interval between the fourth and the fifth.
The fourth occurs at 4/3 the root, and the fifth at 3/2. The ratio of 3/2:4/3 is 9/8ths. What happens if we take a fifth above a fifth (which would give us the ninth), and lower it an octave to make a second. The fifth is 3/2. The fifth above that is 3/2*3/2= 9/4. And lowering it an octave makes it 9/8ths. And behold, its the same interval as between the fourth and fifth. Thus, we are going to call this our second (for the fifth of a fifth), and it becomes our major second or whole tone.
We now have 1, 2, 3, 4, 5, and 8. What about the 6 and 7 degrees. With our consonances, we ended up making a very nice major chord with the 1, 3, and 5. And we like the 4 and 5 scale degrees a lot. So lets make a major chord over both of those as well.
The major third over the fourth gives us 4/3 * 5/4 = 5/3. And that is the just intonation interval for a major sixth. So far so good.
The major third over the fifth gives us 3/2 * 5/4 = 15/8ths, which we will take as our major 7th, or leading tone. So lets look at the distance between the major 7th and the octave. To complete the octave we need an interval of 16/15ths. We also have a small interval between the 3rd and 4th. Examining that we see that its the ratio of 4/3:5/4, which again is 16/15ths. Thus the ratio from the third step to the fourth step is the same as from the 7th to the Octave. And this is the semitone.
Everything looks great, right? Well, not really. When we divide the scale up exactly this way, it turns out we get two different sized major seconds. The distance from 4 to 5, and from 1 to 2 (and also from 6 to 7) are all 9/8ths. But what about from 2 to 3? Remember 2 is 9/8 and 3 is 5/4. That ratio is 10:9. The same is true of the distance from 5 to 6. This could create an intonation problem for just intonation, and it creates severe intonation problems for any justly tuned instrument in one key that tries to play in a different, and perhaps distant key.
One of the early tuning systems for keyboards was called “meantone” and it basically split the difference on the 9/8 major seconds and the 10/9 major seconds. The former was squeezed some, while the latter was stretched. Other tuning systems took different approaches to this same, inherent problem.
We ultimately “solved” that problem by creating equal temperament, where every semitone is defined as the ratio of the twelfth root of two to one. This makes everything but octaves (and tritones oddly enough) just a little out of key, but sort of approximates the just intervals where we got our scales from.
Only intervals I have left are the sevenths and compound intervals. Then I will move on to some stuff that has more direct application to guitar, like triads and then seventh chords.
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