View source for Neutral and interordinal intervals in MOS scales

: ''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 {{nowrap|''a''L ''b''s{{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 {{nowrap|''a''L ''b''s{{angbr|E}}}}:
# Given {{nowrap|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 {{nowrap|''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 ({{nowrap|''a'' + ''b'' − 1}})-steps, inclusive:
#: {{nowrap|neutral ''k''-step {{=}} smaller ''k''-step + ''c''/2}} {{nowrap|{{=}} larger ''k''-step − ''c''/2}}
# Given {{nowrap|0 ≤ ''k'' ≤ ''a'' + ''b'' − 1}}, and assuming that the larger ''k''-step < the smaller ({{nowrap|''k'' + 1}})-step, the '''interordinal''' between ''k''-steps and ({{nowrap|''k'' + 1}})-steps, denoted {{nowrap|''k'' × (''k'' + 1)(m)s}} (read "''k'' cross ({{nowrap|''k'' + 1}}) (mos)step" or "''k'' inter ({{nowrap|''k'' + 1}}) (mos)step"), is the interval exactly halfway between the larger ''k''-step and the smaller ({{nowrap|''k'' + 1}})-step.
#: If the smaller ({{nowrap|''k'' + 1}})-step is ''strictly larger'' than the larger ''k''-step in ''basic'' ''a''L&nbsp;''b''s, {{nowrap|''k'' × (''k'' + 1)}} is called a '''proper interordinal'''; otherwise, it is called an '''improper interordinal'''. If {{nowrap|''a'' > ''b''}}, then {{nowrap|''a''L ''b''s{{angbr|E}}}} has {{nowrap|''a'' + 1}} proper interordinals, including {{nowrap|0 × 1ms}} and {{nowrap|(''a'' + ''b'' − 1) × (''a'' + ''b'')ms}}.
#: We call s/2 (or {{nowrap|0 × 1ms}}) the '''interizer'''{{idiosyncratic}}. The interizer is of note since the following holds for any proper interordinal interval ''k''-inter-({{nowrap|''k'' + 1}})-step:
#: {{nowrap|''k''-inter-(''k'' + 1)-step {{=}} larger ''k''-step + s/2}} {{nowrap|{{=}} smaller (''k'' + 1)-step − s/2}}.

Neutral ''k''-steps generalize neutral interval categories based on the diatonic MOS, which are:
* neutral 1-diastep = neutral 2nd (A–B{{demiflat2}})
* neutral 2-diastep = neutral 3rd (A–C{{demisharp2}})
* semiperfect 3-diastep = semiperfect 4th (A–D{{demisharp2}})
* semiperfect 4-diastep = semiperfect 5th (A–E{{demiflat2}})
* neutral 5-diastep = neutral 6th (A–F{{demisharp2}})
* neutral 6-diastep = neutral 7th (A–G{{demisharp2}})

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 "({{nowrap|''k'' + 1}})ths". Proper interordinals in other mosses generalize [[interordinal]] categories that are novel with respect to diatonic (aka "interseptimals"), which are:
* 0-inter-1-diastep = "unison-inter-2nd" = s/2
* 1-inter-2-diastep = "2nd-inter-3rd" = semifourth = chthonic = ultramajor 2nd
* 2-inter-3-diastep = "3rd-inter-4th" = semisixth = naiadic = ultramajor 3rd
* 4-inter-5-diastep = "5th-inter-6th" = semitenth = cocytic = inframinor 6th
* 5-inter-6-diastep = "6th-inter-7th" = semitwelfth = ouranic = inframinor 7th
* 6-inter-7-diastep = "7th-inter-octave" = {{nowrap|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 ''a''L&nbsp;''b''s with {{nowrap|''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 ({{nowrap|''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.
* The basic equal tuning ({{nowrap|2''a'' + b}})-edE contains neither neutrals nor interordinals, since both ''s'' and ''c'' are one edo step. (For the diatonic MOS, this tuning is [[12edo]].)
* The monohard equal tuning ({{nowrap|3''a'' + ''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]].)
* The monosoft equal tuning ({{nowrap|3''a'' + 2''b''}})-edE contains interordinals but not neutrals, since ''c'' is one edo step and ''s'' is two edo steps. (For diatonic, this tuning is [[19edo]].)
* 2({{nowrap|2''a'' + ''b''}})-edE, twice the basic equal tuning, contains both types of intervals, since both ''c'' and ''s'' are two edo steps. (For diatonic, this tuning is [[24edo]].)

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

=== Diatonic (5L&nbsp;2s) ===
<pre>
 Basic 5L&nbsp;2s (diatonic, dia-): 12edo
 Parent MOS: soft 2L&nbsp;3s (pentic, pt-)
 1\24 0×1dias   (1st×2nd)
 2\24 m1dias    (m2nd)
 3\24 n1dias    (n2nd)
 4\24 M1dias    (M2nd)      == m1pts
 5\24 1×2dias   (2nd×3rd)   == n1pts
 6\24 m2dias    (m3rd)      == M1pts
 7\24 n2dias    (n3rd)      == 1×2pts
 8\24 M2dias    (M3rd)      == d2pts
 9\24 2×3dias   (3rd×4th)   == sP2pts
10\24 P3dias    (P4th)      == P2pts
11\24 sP3dias   (sP4th)
12\24 A3/d4dias (A4th/d5th) == 2×3pts
</pre>

=== m-chromatic (7L&nbsp;5s) ===
<pre>
Basic 7L&nbsp;5s (m-chromatic, mchr-): 19edo
Parent MOS: soft 5L&nbsp;2s (diatonic, dia-)
 1\38 0×1s
 2\38 m1s
 3\38 n1s
 4\38 M1s      == m1dias (m2nd)
 5\38 1×2s     == n1dias (n2nd)
 6\38 m2s      == M1dias (M2nd)
 7\38 n2s
 8\38 M2s/m3s  == 1×2dias (2nd×3rd)
 9\38 n3s
10\38 M3s      == m2dias (m3rd)
11\38 3×4s     == n2dias (n3rd)
12\38 m4s      == M2dias (M3rd)
13\38 n4s
14\38 M4s/d5s  == 2×3dias (3rd×4th)
15\38 sPs
16\38 P5s      == P3dias (P4th)
17\38 5×6s     == sP3dias (sP4th)
18\38 m6s      == A3dias (A4th)
19\38 n6s      == 3×4dias (4th×5th)
</pre>

=== Manual (4L&nbsp;1s) ===
<pre>
 Basic 2/1-equivalent 4L&nbsp;1s (manual, man-): 9edo
 Parent MOS: soft 1L&nbsp;3s (antetric, att-)
 1\18 0×1mans
 2\18 d1mans
 3\18 sP1mans
 4\18 P1mans  == P1atts
 5\18 1×2mans == sP1atts
 6\18 m2mans  == A1atts
 7\18 n2mans  == 1×2atts
 8\18 M2mans  == m2atts
 9\18 2×3mans == n2atts
</pre>

=== Oneirotonic (5L&nbsp;3s) ===
<pre>
 Basic 5L&nbsp;3s (oneirotonic, onei-): 13edo
 Parent MOS: soft 3L&nbsp;2s (anpentic, apt-)
 1\26 0×1oneis
 2\26 m1oneis
 3\26 n1oneis
 4\26 M1oneis     == m1apts
 5\26 1×2oneis    == n1apts
 6\26 m2oneis     == M1apts
 7\26 n2oneis
 8\26 M2/d3oneis  == 1×2apts
 9\26 sP3oneis
10\26 P3oneis     == P2apts
11\26 3×4oneis    == sP2apts
12\26 m4oneis     == A2apts
13\26 n4oneis     == 2×3apts
</pre>

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

=== Statement ===
Suppose {{nowrap|''a'' > ''b''}} and {{nowrap|gcd(''a'', ''b'') {{=}} 1}}.
# Every proper interordinal of basic {{nowrap|''a''L ''b''s{{angbr|E}}}} save for {{nowrap|0 × 1ms}} and {{nowrap|(''a'' + ''b'' − 1) × (''a'' + ''b'')ms}} is a neutral or semiperfect interval of the parent MOS {{nowrap|''b''L (''a'' − ''b'')s{{angbr|E}}}}. The interizer of {{nowrap|''a''L ''b''s{{angbr|E}}}}, {{nowrap|0 x 1ms {{=}} s/2}}, is the semichroma of the parent MOS. 
# Every interordinal interval of the parent MOS {{nowrap|''b''L (''a'' − ''b'')s{{angbr|E}}}} of basic {{nowrap|''a''L ''b''s{{angbr|E}}}} excluding {{nowrap|0 × 1ms}} and {{nowrap|(''a'' + ''b'' − 1) × (''a'' + ''b'')ms}} is a neutral or semiperfect interval of basic {{nowrap|''a''L ''b''s{{angbr|E}}}}. 
# Except the neutral/semiperfect 1-step and the neutral/semiperfect ({{nowrap|''a'' + ''b'' − 1}})-step, every neutral or semiperfect interval of basic {{nowrap|''a''L ''b''s{{angbr|E}}}} is a proper interordinal of {{nowrap|''b''L (''a'' − ''b'')s{{angbr|E}}}}. The number ({{nowrap|''b'' − 1}}) counts the places in {{nowrap|2(2''a'' + ''b'')edE}} (twice the basic MOS tuning for {{nowrap|''a''L ''b''s{{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.
# {{nowrap|''a''L ''b''s{{angbr|E}}}} has {{nowrap|''a'' + 1}} proper interordinals and {{nowrap|''b'' − 1}} improper interordinals.

=== Proof ===
Below we assume that the equave is 2/1, but the proof generalizes to any equave.

Consider a primitive MOS ''a''L&nbsp;''b''s. Recall that ({{nowrap|''b'' − 1}}) satisfies:

: {{nowrap|''b'' − 1}} = |(brightest mode of basic ''a''L&nbsp;''b''s, ignoring equaves) ∩ (darkest mode of basic ''a''L&nbsp;''b''s, ignoring equaves)| {{nowrap|{{=}} #{''k'' : 0 < ''k'' < ''a'' + ''b''}} and larger ''k''-step of basic ''a''L&nbsp;''b''s = smaller ({{nowrap|''k'' + 1}})-step of basic ''a''L&nbsp;''b''s} =&nbsp;#&nbsp;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 ''a''L&nbsp;''b''s:
* {{nowrap|''a'' > ''b''}}.
* The parent MOS, which is {{nowrap|''b''L (''a'' − ''b'')s}}, has steps {{nowrap|L + s}} and L. In this context, since ''a''L&nbsp;''b''s is assumed to have hardness 2/1, {{nowrap|''b''L (''a'' − ''b'')s}} has hardness 3/2 thus is strictly proper.

Finally, recall that:
* In basic ''a''L&nbsp;''b''s, {{nowrap|s {{=}} 1\''n'' {{=}} 2\(2''n'')}}. 
* A concrete MOS tuning is improper if and only if its hardness is greater than 2/1 and the number of ''s'' steps it has is greater than 1.

Let {{nowrap|''n'' {{=}} 2''a'' + ''b''}} (the basic edo tuning of ''a''L&nbsp;''b''s) and suppose that ''m''\(2''n'') is an interordinal (where ''m'' must be odd) between ''k''-steps and ({{nowrap|''k'' + 1}})-steps, denoted {{nowrap|''k'' × (''k'' + 1)ms}}. For parts (1) and (2):
* (Note that we are assuming the basic tuning of the MOS) smaller ({{nowrap|''k'' + 1}})-step of ''a''L&nbsp;''b''s minus the larger ''k''-step of ''a''L&nbsp;''b''s is nonnegative, with equality at improprieties (essentially by definition). At the values of ''k'' and {{nowrap|''k'' + 1}} that are proper, this equals ''s''. To see why, observe that the number of generators represented by the difference must be {{nowrap|''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 {{nowrap|''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 {{nowrap|''k'' x (''k'' + 1)}} is a proper interordinal. Hence the difference must be −''a'' generators, corresponding to ''s''.
* As ''s'' is the chroma of {{nowrap|''b''L (''a'' − ''b'')s}}, it ''would'' be the difference between major and minor intervals in the parent MOS, assuming these interval sizes (smaller ({{nowrap|''k'' + 1}})-step, larger ''k''-step) occur in the parent; so {{nowrap|''k'' × (''k'' + 1)}} would become neutral or semiperfect.
* To show that these actually occur in {{nowrap|''b'' L(''a'' − ''b'')s}}, consider smaller and larger ''j''-steps ({{nowrap|1 ≤ ''j'' ≤ ''a'' − 1}}) in the parent MOS. These intervals also occur in the MOS ''a''L&nbsp;''b''s separated by ''s'', and the number of ''j''{{`s}} ("junctures") that correspond to these places in ''a''L&nbsp;''b''s is exactly {{nowrap|''a'' − 1}}. These ''j''{{`s}} correspond to values of ''k'' such that {{nowrap|larger ''k''-step < smaller (''k'' + 1)-step}}. Note that we are considering "junctures" between ''k''-steps and ({{nowrap|''k'' + 1}})-steps in ''a''L&nbsp;''b''s, excluding {{nowrap|''k'' {{=}} 0}} and {{nowrap|''k'' {{=}} ''a'' + ''b'' − 1}}, so the total number of "junctures" to consider is finite, namely {{nowrap|''a'' + ''b'' − 2}}. This proves parts (1) and (2).

Part (3) is also immediate now: when {{nowrap|larger ''k''-step {{=}} smaller (''k'' + 1)-step}}, {{nowrap|larger (''k'' + 1)-step − smaller k-step}} {{nowrap|{{=}} 2(L − s)}} {{nowrap|{{=}} 2s {{=}} L}}. The step L is 4 steps in 2''n''-edo.{{qed}}

[[Category:MOS scale]]
[[Category:Pages with proofs]]