Neutral and interordinal intervals in MOS scales: Difference between revisions
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# Given 1 ≤ ''k'' ≤ a + b − 2, and assuming that the larger ''k''-step < the smaller (''k'' + 1)-step, the '''interordinal''' between ''k''-steps and (''k'' + 1)-steps, denoted ''k''x(''k'' + 1)s or ''k''X(''k'' + 1)s (read "''k'' cross (''k'' + 1) step" or "''k'' inter (''k'' + 1) step"), is the interval exactly halfway between the larger ''k''-step and the smaller (''k'' + 1)-step. The following always holds for a given interordinal interval ''k''-inter-(''k'' + 1)-step: | # Given 1 ≤ ''k'' ≤ a + b − 2, and assuming that the larger ''k''-step < the smaller (''k'' + 1)-step, the '''interordinal''' between ''k''-steps and (''k'' + 1)-steps, denoted ''k''x(''k'' + 1)s or ''k''X(''k'' + 1)s (read "''k'' cross (''k'' + 1) step" or "''k'' inter (''k'' + 1) step"), is the interval exactly halfway between the larger ''k''-step and the smaller (''k'' + 1)-step. The following always holds for a given interordinal interval ''k''-inter-(''k'' + 1)-step: | ||
#: ''k''-inter-(''k'' + 1)-step = larger ''k''-step + s/2 = smaller (''k'' + 1)-step − s/2. | #: ''k''-inter-(''k'' + 1)-step = larger ''k''-step + s/2 = smaller (''k'' + 1)-step − s/2. | ||
Though the term ''interordinal'' is intended to be JI-agnostic and generalizable to non-diatonic mosses, the term comes from the fact that ''k''-steps in the diatonic mos are conventionally called "(''k'' + 1)ths". | Neutral k-steps generalize diatonic neutral intervals, which are: | ||
* neutral 1-diastep = neutral 2nd (A-Bd) | |||
* neutral 2-diastep = neutral 3rd (A-Ct) | |||
* semiperfect 3-diastep = semiperfect 4th (A-Dt) | |||
* semiperfect 4-diastep = semiperfect 5th (A-Ed) | |||
* neutral 5-diastep = neutral 6th (A-Ft) | |||
* neutral 6-diastep = neutral 7th (A-Gt) | |||
Though the term ''interordinal'' is intended to be JI-agnostic and generalizable to non-diatonic mosses, the term comes from the fact that ''k''-steps in the diatonic mos are conventionally called "(''k'' + 1)ths". Interordinals in other mosses generalize diatonic interseptimals, which are: | |||
* 1-inter-2-diastep = semifourth = chthonic = ultramajor 2nd | |||
* 2-inter-3-diastep = semisixth = naiadic = ultramajor 3rd | |||
* 4-inter-5-diastep = semitenth = cocytic = ultraminor 6th | |||
* 5-inter-6-diastep = semitwelfth = ouranic = ultraminor 7th | |||
Given a primitive mos aLbs with a > b, it's easy to observe the following properties of the simplest equal tunings for the mos, due to the way they divide the small step (s) and the chroma (c = L − s). Note that s separates different ordinal categories while c separates larger and smaller intervals in the same ordinal category. | Given a primitive mos aLbs with a > b, it's easy to observe the following properties of the simplest equal tunings for the mos, due to the way they divide the small step (s) and the chroma (c = L − s). Note that s separates different ordinal categories while c separates larger and smaller intervals in the same ordinal category. | ||