Inversion of Intervals - page 1 Page   1    2         Inversion of intervals To invert the tones of an interval (or the tones of a chord) means to switch them around. That is, to switch their octave positions. This is accomplished by either of two means: by moving the higher pitched tone one octave lower, or by moving the lower pitched tone one octave higher. When all’s said and done, understanding and playing inverted chord voicings on the guitar fretboard (with knowledge and confidence, and without having to learn to read music or play the  piano) is what The Cipher and the upcoming book are all about. Inverting intervals and creating inverted chord voicings on the guitar fretboard is ultimately dependent on our knowledge of the Pattern of Unisons and Octaves. [The Pattern of Unisons and Octaves occupies an unprecedented 30 page section in the book and is reproduced on this web site in it’s entirely]. Before we can talk about inverted chords though we must understand the inversion of intervals generally, both on paper and on the fretboard. The concept of inversion is concerned with the naming of intervals. Specifically; the direction of measurement (order) — lower pitched to higher, higher pitched to lower, left to right, right to left —, and how direction or starting point effects the entity and it's name. The issues arose (centuries ago) when we recognized some inescapable occurrences within music, and for want of names of for those occurrences. The overriding concern and impetus was this; before settling on names (for our intervals, and by extension our chords), it would be prudent to know exactly how many distinct things there are to name, hence, what exactly defines distinctness in/of intervals and then chords. This is the heart of the matter of inversions. The importance of this issue becomes apparent later when analyzing chords. That this; we must eventually ask how many distinct chords are there? What classification scheme should be used to name our chords? And finally, names must be selected to reflect those relationships. The underlying issues can easily be lost or side-stepped if rule-format definitions are used. For that reason the issues are discussed here now. Simple intervals, that is; non-compound first-octave intervals, have so far been discussed solely in terms of measuring and naming the distance from a lower pitched tone up to a higher pitched tone, with both tones restricted to the original (and the same) octave. We haven't yet considered nor accounted for a very common type of occurrence. When working with multiple octaves of tones (the usual case) a given named-tone will appear over and over again, once within each octave. Indeed, all tones recognized in any octave are present in all octaves. So here's the problem; if tone C and tone D (played together) have been named a Major-second interval, can we still use the name Major-second regardless of which C note [that is, from which octave] and which D note we use? And, does their order (C to D, D to C) matter? The answer(s) must take into account many scenarios. Both the octave(s) in which the tones reside and their order do make a difference. And the single name, Major-second, is not adequate. Some definitions From Webster's Seventh Collegiate Dictionary: Interval — A space between objects, units, or states Invert — To reverse in position, order, or relationship (Alas, two terms in music theory that mean exactly what they say!) Inversion of intervals — definitions The members of an interval can be rearranged, called inversion, where the lower note can be placed one octave higher than it originally occurred, or the higher note can be placed one octave lower. The act of inversion will change the actual distance between the two tones involved and the name of the interval will change accordingly — calculated from the new lower note up. Inverted intervals (i.e. an interval plus it’s inversion) span a total of twelve half-steps (semitones), or one octave. [Again, these definitions are true enough, but they don’t acknowledge the real underlying issues — the need to identify just how many distinct things (intervals and chords) we’re dealing with and then how to name and classify those materials appropriately. The above definitions imply that inversions are merely a function of technique, a tool of the craft, a way to manipulate tones. Inversions are indeed powerful tools, but you understand my point.] Our topic involves distance, direction, and naming of intervals. Allowing for the exception of enharmonic names, there is only one interval name for any given distance. The name will apply regardless of the direction of measurement, low to high or high to low. If the distance between two tones is five half-steps (chromatic-steps, semitones, or frets) the interval is called a perfect-fourth. For example, C up to F is a perfect-fourth, and F down to C is still a perfect-fourth (Figure 1).     Figure 1       If the overall distance is the same (regardless of direction) the name is the same. On to inversions. In Figure 1 we used the letter names C and F, with C being the lower pitched tone. Now measure the distance from any F note up to the nearest C ( Figure 2).     Figure 2       Point of focus in Figure 2: F up to C is a P-5th (0°-7°) but F down to C is a P-4th (0°-5°). F up to C is seven half-steps. The interval is named a Perfect-Fifth. [Again, both “F up to C” and “C down to F” are Perfect-Fifth intervals. The distance is constant (See Figure 3).] The distance is different, the interval names are therefore different, and the musical effect may well be different (particularly in melody lines, but in general depending on circumstances). This distinction or difference applies to both melodic and harmonic intervals — the later being the simultaneous sounding of tones, as in chords. The difference here refers to distance. That is, the identity of the intervals, one wider, one narrower. Perfect-Fourth and Perfect-Fifth intervals do (however) exhibit an equality (or a kind of sameness) that should also be stressed. You probably noticed that the net result of moving either a P-5th up in pitch or a P-4th down is essentially the same. For example; if you start at F you’ll end up at C in both cases. This is the characteristic that defines inversions; A given interval and it’s inversion span a total distance of one octave (twelve half-steps). In our example (highlighted in Figure 4) two C notes are the boundaries of the octave. F can be found either five half-steps above the lower C or seven half-steps below the higher C. [Add up the half-steps; five + seven equals twelve or one octave.]     Figure 3             Figure 4       Part of the definition of inversions reads; The members of an interval can be rearranged by either placing the lower note one octave higher or placing the higher note one octave lower. Figures 5 and 6 illustrate this. See the text explanations numbered 1, 2, 3, below each of the two graphics.     Figure 5       Explanation of Figure 5: 1. C up to D is a Major-second — two half-steps. 2. First, move the higher note D an octave lower. The distance between D and C is now ten half-steps, called a minor-seventh interval. Double check by adding up the half-steps: (Maj.-2nd = 2°) and (min.-7th=10°) so 2+10 = 12. Recall, an interval and it’s inversion span one octave (twelve half-steps) so this is correct.) 3. This time, go back to the starting point (1) and move the lower note C an octave higher. Again, D and C are now ten half-steps apart. We can summarize those results with either of the following (alternately worded) statements: • “Major-second intervals invert to minor-sevenths”. . .  or . . • “minor-seventh is the inversion of Major-second”. Lastly, Figure 6 shows the reverse scenario proving that the reverse statements are also true, namely, that “minor-sevenths invert to Major-seconds”. See the explanatory text below the graphic.     Figure 6       Explanation of Figure 6: 1. Begin with a minor-seventh interval — this time we’ll use C up to A#. 2. Invert the C by placing it one octave higher. The distance between A# and C is then two half-steps — a Major-second interval. 3. Or, invert the A# by moving it an octave lower. Again, A# up to C is a Major-second interval. And again, a minor-seventh plus a Major-second equal one octave distance. These summary statements (the reverse of Figure 5) then, are also true: • “minor-seventh intervals invert to Major-seconds”. or • “Major-second is the inversion of minor-seventh”. Grand summary for the previous two figures: • Major-second inverts to minor-seventh. • minor-seventh inverts to Major-second. • any (simple) interval plus it’s inversion spans one octave. We can extrapolate from these example proofs. For example, the combined distance of a P-4th and a P-5th equals one octave of pitch (12 half-steps), so P-4th and P-5th must be inversions of each other, which is true. Table of inverted intervals (next page) Page   1    2         Up to Top of page