Neutral and interordinal intervals in MOS scales: Difference between revisions
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*** > (A+B)-(C+D)-2 + (4a+2b)/(a+b) | *** > (A+B)-(C+D)-2 + (4a+2b)/(a+b) | ||
*** >= (A+B)-(C+D)-2 + 3 = (A+B)-(C+D)+1 | *** >= (A+B)-(C+D)-2 + 3 = (A+B)-(C+D)+1 | ||
*** Hence, (C+D)-(A | *** Hence, (C+D)-(A+B) > 2. | ||
* As s is the chroma of bL(a-b)s, it *would* be the difference between major and minor intervals in the parent mos, assuming these interval sizes (smaller k+1-step, larger k-step) occur in the parent; so kX(k+1) would become neutral or semiperfect. | * As s is the chroma of bL(a-b)s, it *would* be the difference between major and minor intervals in the parent mos, assuming these interval sizes (smaller k+1-step, larger k-step) occur in the parent; so kX(k+1) would become neutral or semiperfect. | ||
* To show that these actually occur in bL(a-b)s, consider smaller and larger j-steps ( 1 <= j <= a-1) in the parent mos. These intervals also occur in the mos aLbs separated by s, and the number of j’s (“junctures”) that correspond to these places in aLbs is exactly a-1. Note that we are considering “junctures” between k-steps and k+1-steps in aLbs, excluding k = 0 and k = a+b-1, so the total number of “junctures” to consider is a+b-2. This proves part (2). | * To show that these actually occur in bL(a-b)s, consider smaller and larger j-steps ( 1 <= j <= a-1) in the parent mos. These intervals also occur in the mos aLbs separated by s, and the number of j’s (“junctures”) that correspond to these places in aLbs is exactly a-1. Note that we are considering “junctures” between k-steps and k+1-steps in aLbs, excluding k = 0 and k = a+b-1, so the total number of “junctures” to consider is a+b-2. This proves part (2). | ||