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Inversion of Intervals - page 2





Table of inverted intervals

Pairs of inverted (simple) intervals are catalogued in Figure 7. Intervals paired in the center column (highlighted in gray) are expressed using standard interval number-names. The two intervals in each line are inversions of each other. Corresponding half-step numbers (Cipher numbers) appear at the far left and right. Those numbers, if added together, always equal twelve degrees, i.e. one octave. Note, an octave interval (P-8), when inverted, becomes Unison — no distance between tones, i.e. two identically pitched tones. This interval is sometimes called Perfect-Prime (abbr.. PP) or P-1. Prime means first or one.



Figure 7




Figure 8 shows the interval inversion pairs rendered in chromatic numbers only. At the top are the Major scale’s Major and Perfect  intervals (top line are Major scale intervals, the lower line gives they’re inversions. Note only the Tonic, Octave, Fourth and Fifth are present in both lines, meaning 0°, 12°, 5°, and 7° respectively.) The bottum set of inversion pairs in Figure 8 are for the full 12 tone chromatic scale.



Figure 8






If you didn’t know the Cipher’s half-step values of intervals, where summing intervals to make to 12° (the octave) quickly identifies the inversion of any interval, standard interval number-names alone could be used to identify the inversion of any interval along with it’s complete number-name including qualifier-prefix. It’s a two step process treating the number and prefix parts separately:

  • First, the number portion of the interval number-names. Using standard interval number-names, the sum of an interval and it’s inversion (regardless of qualifying-prefixes Maj, min, aug, dim) always equals nine. In other words if you begin with some kind of third interval it’s inversion will be some kind of sixth — because 3 plus 6 equals 9.
  • Then consider the prefixes. Figure 9 shows how prefixes change when intervals are inverted:



Figure 9




Example: You start with a minor-third interval. You want to know what it’s inversion is. It’s inversion will carry the number six (because 3+6=9) and it’s prefix will be Major — because the original interval is minor. So the inversion of minor-third is Major-sixth.

Inverting compound intervals

Compound intervals, those spanning more than one octave (e.g. 9ths, 11ths, and 13ths) are special cases. The inversions of compound intervals do not follow the same rules nor exhibit the same patterns (when inverted) as simple intervals do. Specifically, the combined distance of a compound interval and it’s inversion does not equal one octave (nor two for that matter). Compound intervals invert to their first octave counterparts. That is, subtract one octave from the original interval. [e.g. Using half-step numbers; Major-ninth is 14°, subtract 12 degrees (14°-12°=2°). 2° is a Major-second. Using standard numbers; subtract 7 from the intervals original number-name (9-7=2) The original prefix (Major) is retained. So Maj.9th becomes Maj.2nd] The so-called or would-be inversion of Major-ninth is Major-second. But/and, the reciprocal in not true — that a Major-second inverts to Major-ninth. Major-second, we saw moments ago, inverts to minor-seventh — not Major-ninth. Compound intervals really don’t invert, as we know it. They just close the gap by one octave. They also retain their original letter order (see Figure 10, the letter order remains C up to D in all cases, i.e. before and after inversion).



Figure 10




Explanation of Figure 10 — inverting compound intervals:

1. Begin with a Major-ninth interval — C up to D in the next octave.
2. Move the lower note (C) one octave higher. Gives you C up to D again — but an octave narrower, i.e. a Major-second interval.
3. Or (go back to the beginning and), move the higher note (D) an octave lower. You still end up with an interval that reads C up to D but an octave narrower than the original interval. It’s another (C) Major-second interval.

Any difficulties or conceptual problems you have when dealing with compound intervals, inverted or not, can be side-stepped to a large degree if, as a rule, you always think of compound intervals as their first-octave counterparts. That is, any time you encounter a compound interval or need to perform some operation on it, always do a quick mental conversion first, i.e. convert it to it’s first-octave counterpart, then continue the operation. e.g. If what you really want is to invert the tones of a chord voicing (as we know [or will know] inversions with simple intervals) and you’re faced with a Major-ninth interval for example, think; Major-ninth is Major-second, Major-second inverts to minor-seventh, then employ minor-seventh as the inversion of Major-ninth. Given the limitations of the fretboard and the too few fingers on our chording hand Guitarists in particular must constantly convert compound intervals to simple intervals or visa versa. It’s a basic survival tool if you will.

Fretboard examples — Inversion of intervals

Creating or identifying inversions on the guitar fretboard is largely a matter of exercising our knowledge of The Pattern of Unisons and Octaves. The only problems we’ll ever have (somewhere down the road) are due to the limitations of the fretboard — sometimes we’ll run out of range, out of strings, and have to settle for the best compromise voicings available.

Figure 11 and 12 show the inversion of Perfect-fifth and Perfect-fourth intervals on the guitar fretboard.



Figure 11







Figure 12






Creating an inversion by intention, is far easier than taking two unidentified tones out of context and then trying to identify a relationship between them. For example, in the previous two illustrations, the lettered tones C and G were used throughout. But the root tones were clearly marked (as diamonds). If plain uniform dots had been used instead, giving no indication of the root, the relationship between those or any two tones would be more difficult to tell. There’d be more than one possible interpretation. If we look again at tones C and G as an interval pattern on the fretboard, see Figure 13, we can be sure that it’s a Perfect-fifth interval and that tones C and G are involved. But apart from any context we really can’t say which tone is the root, so we can’t completely identify the relationship between tones. If taken as a small chord (a harmonic interval), either tone could be the root. Also, depending on which tone you call the root, the other tone will be marked as either a 4 or a 5 — even though both intervals are actually Perfect-fifths! See Figure 14. This will probably confuse you at first but it’ll make complete sense after you’ve had a little more experience analyzing and inverting chord voicings — which we’ll do next. Just remember that ultimately the root defines the context and therefore the relationships between tones. [Note, this has nothing to do with the Cipher System per say. This is the standard stuff. This is how chord analysis and inverting chord tones works.]






Figure 13









Figure 14







Some points to remember:
1. The distance between tones is the most important consideration in determining the identity of an interval. Meaning, the order of any two lettered tones, which is higher in pitch and which is lower, does matter when identifying intervals.
2. A context, a root or key, is essential to even begin analysis.
3. The apparent inconsistencies of naming and numbering interval tones verses chord tones is at the heart of the matter of inversions mentioned at the outset. i.e. determining just how many distinct things there really are to name, and then how to name and classify them. This becomes most evident when analyzing chords and inverted chords in particular.

The relationship of inverted intervals to the octave interval is examined in Figure 15.



Figure 15




Having covered intervals and their inversion we can move to scales, triads, and then some simple progressions.




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© Copyright 2002   Roger Edward Blumberg


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