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== MOS vs MODMOS vs inflected MOS ==
== MOS vs MODMOS vs MOS inflection ==


A [[MOS scale]] is constructed by iterating a [[generator]] g inside of a [[period]] R. That is, for non-negative integers n, we take n*g and reduce it modulo the period R to the range 0 ≤ i < R. We stop only at points when there are exactly two sizes of [[interval]]s in the scale, the large interval L and the small interval s. If the period R is 1/P [[octave]] for some integer P, we obtain an octave-repeating periodic scale by conjoining P copies of the MOS scale inside R so as to produce a MOS scale for the whole octave.
A [[MOS scale]] is constructed by iterating a [[generator]] g inside of a [[period]] R. That is, for non-negative integers n, we take n*g and reduce it modulo the period R to the range 0 ≤ i < R. We stop only at points when there are exactly two sizes of [[interval]]s in the scale, the large interval L and the small interval s. If the period R is 1/P [[octave]] for some integer P, we obtain an octave-repeating periodic scale by conjoining P copies of the MOS scale inside R so as to produce a MOS scale for the whole octave.
Line 7: Line 7:
If we take a MOS scale, and adjust one or more its notes up or down by a chroma consistently, so that octave periodicity is respected, and do not obtain another MOS, then we obtain a [[MODMOS]] scale.
If we take a MOS scale, and adjust one or more its notes up or down by a chroma consistently, so that octave periodicity is respected, and do not obtain another MOS, then we obtain a [[MODMOS]] scale.


If we take a MOS scale, and adjust one or more its notes up or down by some ''other'' interval besides a chroma, we obtain an '''inflected MOS scale'''.
If we take a MOS scale, and adjust one or more its notes up or down by some ''other'' consistent interval besides a chroma, we instead obtain an inflected MOS scale. This process is called '''MOS inflection'''.


An inflected MOS lacks many useful structural properties of a MODMOS, most importantly [[epimorphism]]. Compared to a MODMOS, it also preserves less of the character of the original MOS scale, which could be useful or detrimental depending on one’s goals in making the scale.
A scale being a MODMOS guarantees a number of useful characteristics, most importantly [[epimorphism]]. A MOS inflection ''might'' sometimes have these properties, but it often doesn't. Compared to a MODMOS, it also typically preserves less of the 'character' of the original MOS scale, which could be useful or detrimental depending on one’s goals in making the scale.


== Types of inflected MOS ==
== Types of MOS inflection ==
{{Idiosyncratic terms|The terms for the different kinds of MOS inflection, proposed by [[Budjarn Lambeth]]}}


It’s likely that many more types could be conceived of. Feel free to edit this page and add any which you find useful or interesting.
It’s likely that many more types could be conceived of. Feel free to edit this page and add any which you find useful or interesting.


=== Robust inflected MOS ===
=== Robust MOS inflection ===


A robust inflected MOS is an inflected MOS where the interval of adjustment is one of the intervals which was already present within the original MOS scale.
Robust MOS inflection is where the interval of adjustment is one of the intervals which was already present within the original MOS scale.


This will result in a scale which is a subset of some larger MOS scale with the same generator and period.
This will result in a scale which is a subset of some larger MOS scale with the same generator and period.


Compared to other inflected MOS scales, a robust inflected MOS more closely preserves the harmonic flavors of the original MOS, though it still doesn’t preserve them as well as a MODMOS would.
Compared to other types of MOS inflection, robust MOS inflection more closely preserves the harmonic flavors of the original MOS, though it still doesn’t preserve them as well as a MODMOS would.


For example, if you take this [[Negri]][8] MOS:
For example, if you take this [[Negri]][8] MOS:
Line 34: Line 35:
* 1200.000
* 1200.000


Then alter two of its steps by increments of 125.373 [[cents]], and then [[octave reduce]] the result, you get the following robust inflected MOS:
Then alter two of its steps by increments of 125.373 [[cents]], and then [[octave reduction|octave-reduce]] the result, you get the following robust inflected MOS:


* 125.373
* 125.373
Line 47: Line 48:
Which itself is a subset of the larger MOS, Negri[19].
Which itself is a subset of the larger MOS, Negri[19].


=== Semirobust inflected MOS ===
=== Semirobust MOS inflection ===


A semirobust inflected MOS is an inflected MOS where the interval of adjustment is an interval which was not already present within the original MOS scale, but which is present within a larger MOS of the same generator and period.
Semirobust MOS inflection is where the interval of adjustment is an interval which was not already present within the original MOS scale, but which is present within a larger MOS of the same generator and period.


For example, if you take this Negri[8] MOS:
For example, if you take this Negri[8] MOS:
Line 77: Line 78:
Like robust scales, semirobust ones also turn out to be subsets of some larger MOS. However, said larger MOS are sometimes large enough that they lose the character of the original tuning: Especially so if the MOS belongs to a low [[complexity]] [[temperament]]. If preserving the harmonic character of the original MOS is desired, then it might be preferable to use only a small number of adjustments.
Like robust scales, semirobust ones also turn out to be subsets of some larger MOS. However, said larger MOS are sometimes large enough that they lose the character of the original tuning: Especially so if the MOS belongs to a low [[complexity]] [[temperament]]. If preserving the harmonic character of the original MOS is desired, then it might be preferable to use only a small number of adjustments.


=== Frankenstein inflected MOS ===
=== Frankenstein MOS inflection ===


A Frankenstein inflected MOS is an inflected MOS where the interval of adjustment is the chroma of another MOS. This approach lends itself to situations where the two MOS scales are tuned using the same [[equal temperament]] (though that isn’t compulsory).
Frankenstein MOS inflection is where the interval of adjustment is the chroma of another MOS. This approach lends itself to situations where the two MOS scales are tuned using the same [[equal temperament]] (though that isn’t compulsory).


For example, if you take this Negri[7] MOS in [[67edo]]:
For example, if you take this Negri[7] MOS in [[67edo]]:
Line 103: Line 104:
* 1200.000
* 1200.000


=== Justly inflected MOS ===
=== Aspirational MOS inflection ===


A justly inflected MOS is where one step of a MOS is adjusted by exactly the right amount to tune it to a pure [[JI]] ratio, relative to the [[tonic]]. Then any other step(s) may be adjusted by increments of the same amount that initial interval was modified by.  
Aspirational MOS inflection is where one step of a MOS is adjusted by exactly the right amount to tune it to a pure [[JI]] ratio, relative to the [[tonic]]. Then any other step(s) may be adjusted by increments of the same amount that initial interval was modified by.  


For example, if you take this Negri[7] MOS in 67edo:
For example, if you take this Negri[7] MOS in 67edo:
Line 129: Line 130:
* 1200.000
* 1200.000


Then alter, say, two of its other steps by increments of the same amount. Then you get the following justly inflected MOS:
Then alter, say, two of its other steps by increments of the same amount. Then you get the following aspirational inflected MOS:


* 125.373
* 125.373
Line 140: Line 141:
* 1200.000
* 1200.000


A justly inflected MOS may be constructed with similar goals to a [[well temperament]]: it may be designed to blunt the rough edges of an MOS by sanding over one or two of its [[wolf interval]]s.
Aspirational MOS inflection may be conducted with similar goals to [[well temperament]]: it may be used to blunt the rough edges of an MOS by sanding over one or two of its [[wolf interval]]s.


[[Category:MOS scale]][[Category:MODMOS]]
[[Category:MOS scale]]

Latest revision as of 23:24, 27 April 2026

MOS vs MODMOS vs MOS inflection

A MOS scale is constructed by iterating a generator g inside of a period R. That is, for non-negative integers n, we take n*g and reduce it modulo the period R to the range 0 ≤ i < R. We stop only at points when there are exactly two sizes of intervals in the scale, the large interval L and the small interval s. If the period R is 1/P octave for some integer P, we obtain an octave-repeating periodic scale by conjoining P copies of the MOS scale inside R so as to produce a MOS scale for the whole octave.

If we continue by adding another interval, it will either fall short of R, or exceed it, but either way we will now have an interval c = L-s, which has been called the "chroma".

If we take a MOS scale, and adjust one or more its notes up or down by a chroma consistently, so that octave periodicity is respected, and do not obtain another MOS, then we obtain a MODMOS scale.

If we take a MOS scale, and adjust one or more its notes up or down by some other consistent interval besides a chroma, we instead obtain an inflected MOS scale. This process is called MOS inflection.

A scale being a MODMOS guarantees a number of useful characteristics, most importantly epimorphism. A MOS inflection might sometimes have these properties, but it often doesn't. Compared to a MODMOS, it also typically preserves less of the 'character' of the original MOS scale, which could be useful or detrimental depending on one’s goals in making the scale.

Types of MOS inflection

This article or section contains multiple idiosyncratic terms. Such terms are used by only a few people and are not regularly used within the community.

Terms: The terms for the different kinds of MOS inflection, proposed by Budjarn Lambeth

It’s likely that many more types could be conceived of. Feel free to edit this page and add any which you find useful or interesting.

Robust MOS inflection

Robust MOS inflection is where the interval of adjustment is one of the intervals which was already present within the original MOS scale.

This will result in a scale which is a subset of some larger MOS scale with the same generator and period.

Compared to other types of MOS inflection, robust MOS inflection more closely preserves the harmonic flavors of the original MOS, though it still doesn’t preserve them as well as a MODMOS would.

For example, if you take this Negri[8] MOS:

  • 125.373
  • 250.746
  • 376.119
  • 698.507
  • 823.881
  • 949.254
  • 1074.627
  • 1200.000

Then alter two of its steps by increments of 125.373 cents, and then octave-reduce the result, you get the following robust inflected MOS:

  • 125.373
  • 376.120
  • 501.493
  • 698.507
  • 823.881
  • 1020.896
  • 1074.627
  • 1200.000

Which itself is a subset of the larger MOS, Negri[19].

Semirobust MOS inflection

Semirobust MOS inflection is where the interval of adjustment is an interval which was not already present within the original MOS scale, but which is present within a larger MOS of the same generator and period.

For example, if you take this Negri[8] MOS:

  • 125.373
  • 250.746
  • 376.119
  • 698.507
  • 823.881
  • 949.254
  • 1074.627
  • 1200.000

Then alter one of its steps by increments of 1020.896 cents, and then octave reduce the result, you get the following semirobust inflected MOS:

  • 250.746
  • 376.119
  • 698.507
  • 823.881
  • 949.254
  • 967.111
  • 1074.627
  • 1200.000

Which itself is a subset of the larger MOS, Negri[28].

Like robust scales, semirobust ones also turn out to be subsets of some larger MOS. However, said larger MOS are sometimes large enough that they lose the character of the original tuning: Especially so if the MOS belongs to a low complexity temperament. If preserving the harmonic character of the original MOS is desired, then it might be preferable to use only a small number of adjustments.

Frankenstein MOS inflection

Frankenstein MOS inflection is where the interval of adjustment is the chroma of another MOS. This approach lends itself to situations where the two MOS scales are tuned using the same equal temperament (though that isn’t compulsory).

For example, if you take this Negri[7] MOS in 67edo:

  • 125.373
  • 250.746
  • 376.119
  • 698.507
  • 823.881
  • 949.254
  • 1074.627
  • 1200.000

Then alter one of its steps by increments of the chroma of the Meantone[7] MOS in 67edo (89.552), you get the following Frankenstein inflected MOS:

  • 125.373
  • 376.119
  • 519.402
  • 698.507
  • 823.881
  • 949.254
  • 1074.627
  • 1200.000

Aspirational MOS inflection

Aspirational MOS inflection is where one step of a MOS is adjusted by exactly the right amount to tune it to a pure JI ratio, relative to the tonic. Then any other step(s) may be adjusted by increments of the same amount that initial interval was modified by.

For example, if you take this Negri[7] MOS in 67edo:

  • 125.373
  • 250.746
  • 376.119
  • 698.507
  • 823.881
  • 949.254
  • 1074.627
  • 1200.000

Then alter one of its steps by 3.448 to line up perfectly with the just ratio 3/2 (701.955):

  • 125.373
  • 250.746
  • 376.119
  • 701.955
  • 823.881
  • 949.254
  • 1074.627
  • 1200.000

Then alter, say, two of its other steps by increments of the same amount. Then you get the following aspirational inflected MOS:

  • 125.373
  • 250.746
  • 379.607
  • 701.955
  • 823.881
  • 952.702
  • 1074.627
  • 1200.000

Aspirational MOS inflection may be conducted with similar goals to well temperament: it may be used to blunt the rough edges of an MOS by sanding over one or two of its wolf intervals.

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