Neutral and interordinal intervals in MOS scales: Difference between revisions

: ''This page assumes that the reader is familiar with [[TAMNAMS]] MOS interval and step ratio names.''

Given a tuning of a primitive (i.e. single-period) [[MOS]] pattern aLbs{{angbr|E}} with arbitrary [[equave]] E in a specific tuning (i.e. with a specific [[hardness]] value for L/s), we may define two types of notes "in the cracks of" interval categories defined by aLbs{{angbr|E}}:

# Given 1 ≤ ''k'' ≤ a + b − 1, the '''neutral''' ''k''-mosstep or ''k''-step (abbrev. n''k''ms, n''k''s) is the interval exactly halfway between the smaller ''k''-step and the larger ''k''-step of the MOS. When the MOS is generated by a (perfect) ''k''-step, this may instead be called the '''semiperfect''' ''k''-step (abbrev. sP''k''s), since it is halfway between the perfect and imperfect (either diminished or augmented, depending on whether the generator is bright or dark) ''k''-step. Depending on whether the imperfect generator is augmented or diminished, the corresponding semiperfect generator may be called '''semiaugmented''' or '''semidiminished'''.  

#: We use '''semichroma''' for the quantity c/2 where c = L − s is the [[chroma]] of the MOS. The semichroma represents the difference between ordinary MOS intervals and their neutralized counterparts, in the sense that the following holds for a given interval class of ''k''-steps between 1-steps and (a + b − 1)-steps, inclusive:

#: neutral ''k''-step = smaller ''k''-step + c/2 = larger ''k''-step − c/2

# Given 0 ≤ ''k'' ≤ a + b &minus; 1, and assuming that the larger ''k''-step < the smaller (''k'' + 1)-step, the '''interordinal''' between ''k''-steps and (''k'' + 1)-steps, denoted ''k''×(''k'' + 1)(m)s (read "''k'' cross (''k'' + 1) (mos)step" or "''k'' inter (''k'' + 1) (mos)step"), is the interval exactly halfway between the larger ''k''-step and the smaller (''k'' + 1)-step.

#: If the smaller (''k'' + 1)-step is ''strictly larger'' than the larger ''k''-step in ''basic'' aLbs, ''k''×(''k'' + 1) is called a '''proper interordinal'''; otherwise, it is called an '''improper interordinal'''. If a > b, then aLbs{{angbr|E}} has a + 1 proper interordinals, including 0×1ms and (a+b&minus;1)×(a+b)ms.

#: We call s/2 (or 0×1ms) the '''interizer'''{{idiosyncratic}}. The interizer is of note since the following holds for any proper interordinal interval ''k''-inter-(''k'' + 1)-step:

Neutral ''k''-steps generalize neutral interval categories based on the diatonic MOS, which are:

* neutral 1-diastep = neutral 2nd (A-(Bd)

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:

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 &minus; 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.

* The basic equal tuning (2a + b)-edE contains neither neutrals nor interordinals, since both s and c are one edo step. (For the diatonic MOS 5L2s, this tuning is [[12edo]].)

* The monohard equal tuning (3a + b)-edE contains neutrals of that MOS but not interordinals, since c is two edo steps but s is one edo step. (For diatonic, this tuning is [[17edo]].)

These charts show where neutrals and interordinals corresponding to example 2/1-equivalent mosses are located in 2(2a + b)-edo (twice the basic edo tuning). Intervals larger than 600c have been omitted, as the structure is symmetrical about 600c.

  Parent MOS: soft 2L3s (pentic, pt-)

  1\24 0×1dias  (1st×2nd)

Parent MOS: soft 5L2s (diatonic, dia-)

  1\38 0×1s

  Parent MOS: soft 1L3s (antetric, att-)

  1\18 0×1mans

  Parent MOS: soft 3L2s (anpentic, apt-)

  1\26 0×1oneis

== Interordinal-Neutral Theorem ==

The Interordinal-Neutral Theorem relates the neutral (resp. interordinal) intervals of aLbs (with a > b) with the interordinal (resp. neutral) intervals of its parent MOS, bL(a &minus; b)s, generalizing an observation by [[User:Godtone]] relating neutrals and interordinals of [[5L 2s]] to those of the parent MOS [[2L 3s]].

# Every proper interordinal of basic aLbs{{angbr|E}} save for 0×1ms and (a+b&minus;1)×(a+b)ms is a neutral or semiperfect interval of the parent MOS bL(a &minus; b)s{{angbr|E}}. The interizer of aLbs{{angbr|E}}, 0x1ms = s/2, is the semichroma of the parent MOS.  

# Every interordinal interval of the parent MOS bL(a &minus; b)s{{angbr|E}} of basic aLbs{{angbr|E}} excluding 0×1ms and (a+b&minus;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 &minus; 1)-step, every neutral or semiperfect interval of basic aLbs{{angbr|E}} is a proper interordinal of bL(a &minus; b)s{{angbr|E}}. The number (b &minus; 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.

# aLbs{{angbr|E}} has a + 1 proper interordinals and b &minus; 1 improper interordinals.

=== Proof ===

Consider a primitive MOS aLbs. Recall that (b &minus; 1) satisfies:

= #{k : 0 < k < a + b and larger k-step of basic aLbs = smaller (k + 1)-step of basic aLbs} = # of potential improprieties,

 

Also recall that the following are equivalent for a MOS aLbs:

* The parent MOS, which is bL(a&minus;b)s, has steps L + s and L. In this context, since aLbs is assumed to have hardness 2/1, bL(a&minus;b)s has hardness 3/2 thus is strictly proper.

Finally, recall that:

* A concrete MOS tuning is improper if and only if its hardness is > 2/1 and the number of s steps it has is > 1.

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 &minus;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 &minus;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 &minus;a generators, corresponding to s.

* As s is the chroma of bL(a &minus; 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 &minus; b)s, consider smaller and larger j-steps (1 ≤ j ≤ a &minus; 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 &minus; 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 &minus; 1, so the total number of “junctures” to consider is finite, namely a + b &minus; 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 &minus; smaller k-step = 2(L &minus; s) = 2s = L. The step L is 4 steps in 2n-edo. {{qed}}