Neutral and interordinal intervals in MOS scales: Difference between revisions
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#: neutral ''k''-step = smaller ''k''-step + c/2 = larger ''k''-step − c/2 | #: neutral ''k''-step = smaller ''k''-step + c/2 = larger ''k''-step − c/2 | ||
# Given 0 ≤ ''k'' ≤ a + b − 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. | # Given 0 ≤ ''k'' ≤ a + b − 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'''. If a > b, then aLbs{{angbr|E}} has a + 1 proper interordinals, including 0×1ms and (a+b−1)×(a+b)ms. | #: 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−1)×(a+b)ms. | ||
#: We call s/2 (or 0×1ms) '''interizer'''{{idiosyncratic}}. The interizer is of note since the following holds for any proper interordinal interval ''k''-inter-(''k'' + 1)-step: | #: 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: | ||
#: ''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. | ||
Neutral ''k''-steps generalize neutral interval categories based on the diatonic MOS, which are: | Neutral ''k''-steps generalize neutral interval categories based on the diatonic MOS, which are: | ||
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* 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- | * 0-inter-1-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 | ||
* 2-inter-3-diastep = "3rd-inter-4th" = semisixth = naiadic = ultramajor 3rd | * 2-inter-3-diastep = "3rd-inter-4th" = semisixth = naiadic = ultramajor 3rd | ||
* 4-inter-5-diastep = "5th-inter-6th" = semitenth = cocytic = inframinor 6th | * 4-inter-5-diastep = "5th-inter-6th" = semitenth = cocytic = inframinor 6th | ||
* 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- | * 6-inter-7-diastep = "7th-inter-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. | ||
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# 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. | ||
# aLbs{{angbr|E}} has a + 1 proper interordinals and b − 1 improper interordinals. | |||
=== Proof === | === Proof === | ||
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(b − 1) = |(brightest mode of basic aLbs, ignoring equaves) ∩ (darkest mode of basic aLbs, ignoring equaves)| | (b − 1) = |(brightest mode of basic aLbs, ignoring equaves) ∩ (darkest mode of basic aLbs, ignoring equaves)| | ||
= #{k : 0 < k < a + b and larger k-step of basic aLbs = smaller (k + 1)-step of basic aLbs} = # of | = #{k : 0 < k < a + b and larger k-step of basic aLbs = smaller (k + 1)-step of basic aLbs} = # of potential improprieties, | ||
where ''potential improprieties'' are pairs of adjacent interval classes that witness the impropriety of a hard-of-basic tuning of the MOS. Part (4) immediately follows. | |||
Also recall that the following are equivalent for a MOS aLbs: | Also recall that the following are equivalent for a MOS aLbs: | ||
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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). | ||
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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. {{qed}} | ||
[[Category:MOS scale]] | [[Category:MOS scale]] | ||
[[Category:Pages with proofs]] | [[Category:Pages with proofs]] | ||