Neutral and interordinal intervals in MOS scales: Difference between revisions
m →Proof |
m Added notes about improper interordinals. |
||
| Line 17: | Line 17: | ||
* neutral 6-diastep = neutral 7th (A-Gt) | * 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". Proper interordinals in other mosses generalize | 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". Proper interordinals in other mosses generalize [[interordinal]] categories that are novel with respect to diatonic (aka "interseptimals"), which are: | ||
* 0-inter-2-diastep = "unison-inter-2nd" = s/2 | * 0-inter-2-diastep = "unison-inter-2nd" = s/2 | ||
* 1-inter-2-diastep = "2nd-inter-3rd" = semifourth = chthonic = ultramajor 2nd | * 1-inter-2-diastep = "2nd-inter-3rd" = semifourth = chthonic = ultramajor 2nd | ||
| Line 24: | Line 24: | ||
* 5-inter-6-diastep = "6th-inter-7th" = semitwelfth = ouranic = inframinor 7th | * 5-inter-6-diastep = "6th-inter-7th" = semitwelfth = ouranic = inframinor 7th | ||
* 6-inter-7-diastep = "7th-ubter-octave" = octave − s/2 | * 6-inter-7-diastep = "7th-ubter-octave" = octave − s/2 | ||
Improper interordinals, in contrast, represent intervals that are technically between ordinal categories but occur within the MOS scale unlike proper interordinals which are wholly outside the interval categories defined by the MOS. The diatonic example of this is the tritone, which is interordinal but falls within diatonic interval categories as the [[12edo|basic tuning]] of diatonic tunes both the augmented 3-diastep and the diminished 4-diastep to 600 cents. | |||
Given a primitive MOS aLbs with a > b, one can 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 adjacent ordinal categories (i.e. [[interval class]]es) while c separates larger and smaller intervals in the same ordinal category. | Given a primitive MOS aLbs with a > b, one can 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 adjacent ordinal categories (i.e. [[interval class]]es) while c separates larger and smaller intervals in the same ordinal category. | ||
| Line 116: | Line 117: | ||
# Every interordinal interval of the parent MOS bL(a − b)s{{angbr|E}} of basic aLbs{{angbr|E}} excluding 0×1ms and (a+b−1)×(a+b)ms is a neutral or semiperfect interval of basic aLbs{{angbr|E}}. | # Every interordinal interval of the parent MOS bL(a − b)s{{angbr|E}} of basic aLbs{{angbr|E}} excluding 0×1ms and (a+b−1)×(a+b)ms is a neutral or semiperfect interval of basic aLbs{{angbr|E}}. | ||
# Except the neutral/semiperfect 1-step and the neutral/semiperfect (a + b − 1)-step, every neutral or semiperfect interval of basic aLbs{{angbr|E}} is a proper interordinal of bL(a − b)s{{angbr|E}}. The number (b − 1) counts the places in 2(2a + b)edE (twice the basic MOS tuning for aLbs{{angbr|E}}) where the parent's interordinal is improper, being two steps away, instead of one step away, from each of the adjacent ordinal categories. | # Except the neutral/semiperfect 1-step and the neutral/semiperfect (a + b − 1)-step, every neutral or semiperfect interval of basic aLbs{{angbr|E}} is a proper interordinal of bL(a − b)s{{angbr|E}}. The number (b − 1) counts the places in 2(2a + b)edE (twice the basic MOS tuning for aLbs{{angbr|E}}) where the parent's interordinal is improper, being two steps away, instead of one step away, from each of the adjacent ordinal categories. | ||
# The number of improper interordinals of aLbs{{angbr|E}} is b & minus; 1. | |||
=== Proof === | === Proof === | ||
| Line 134: | Line 136: | ||
Let n = 2a + b (the basic edo tuning of aLbs) and suppose that m\(2n) is an interordinal (where m must be odd) between k-steps and (k + 1)-steps, denoted k×(k + 1)ms. For parts (1) and (2): | Let n = 2a + b (the basic edo tuning of aLbs) and suppose that m\(2n) is an interordinal (where m must be odd) between k-steps and (k + 1)-steps, denoted k×(k + 1)ms. For parts (1) and (2): | ||
* | * (Note that we are assuming the basic tuning of the MOS) smaller (k + 1)-step of aLbs minus larger k-step of aLbs ≥ 0, with equality at improprieties (essentially by definition). At the values of k and k+1 that are proper, this equals s. To see why, observe that the number of generators represented by the difference must be b mod (a+b), since L is obtained by stacking b bright generators. The difference is at most −1 generators, since the larger interval of the difference between the larger k-step and the larger k+1 step, a chroma sharper, does occur (in the brightest mode of the MOS), and cannot be less than −a generators, lest the gap be nonpositive in the basic MOS, a contradiction since kx(k+1) is a proper interordinal. Hence the difference must be −a generators, corresponding to s. | ||
* 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 k×(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 k×(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. These j's correspond to values of k such that larger k-step < smaller (k + 1)-step. 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 finite, namely a + b − 2. This proves parts (1) and (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. These j's correspond to values of k such that larger k-step < smaller (k + 1)-step. 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 finite, namely a + b − 2. This proves parts (1) and (2). | ||
Part (3) is also immediate now: when larger k-step = smaller (k + 1)-step, larger (k + 1)-step − smaller k-step = 2(L − s) = 2s = L. The step L is 4 steps in 2n-edo. {{qed}} | Part (3) is also immediate now: when larger k-step = smaller (k + 1)-step, larger (k + 1)-step − smaller k-step = 2(L − s) = 2s = L. The step L is 4 steps in 2n-edo. Part (4) is definitionally true, as aforementioned. {{qed}} | ||
=== Corollary === | === Corollary === | ||