This question relates to projecting pitches in the context of music theory. However, I'm looking for the appropriate mathematical term for this particular concept.
Here's the musical context:
Imagine you have a set setA = [0, 1, 2, 3] that represent scale degrees in a musical scale. What we want to to is project that set into different musical scales. If I were to project this set setA into a major scale (setM = [0, 2, 4, 5, 7, 9, 11]), I would get result = [0, 2, 4, 5]. If I were to project set setA into an octatonic scale (setO = [0, 2, 3, 5, 6, 8, 9, 11]) I would get result = [0, 2, 3, 5].
To accomplish this, we need to break it down into two types of sets. setA is an "index set" and setM, setO are each examples of a "projection set". Note that in this case, the projection sets that we have used so far ultimately map to the scale degrees of the chromatic scale.
Extending projections into octaves:
When have have an index set that includes numbers outside the cardinality of a projection set, we want to project the resultant pitches up or down by octaves. For example, imagine an index set setB such that setB = [0, 3, 5, 7, 10] that we want to project into the major scale (setM). Done properly, that result would be result = [0, 5, 11, 12, 17], since, once we go up to values at and beyond index 7 from setB the resultant projection is now up an octave, and we +12 to that projected pitch.
Extending projections to non-+12 "octaves":
In the examples above, we are ultimately projecting our pitches into the chromatic scale. However, not all projection will +12 at the octave. If we want to project the index set setN = [0, 2, 4, 6, 7] into a projection set representing scale degrees of a diatonic scale setD = [0, 1, 2, 3, 4, 5, 6] we will need to add a property to a projection set which will indicate what value will be added or subtracted when extending above or beyond the cardinality, in this case +7.
Intuitively, the resultant projection of setN onto setD would be [0, 2, 4, 6, 7] (and not [0, 2, 4, 6, 12]).
Update our Projection sets to include extensions:
To accommodate this, we need to update our definition of a "projection set" to include this value. Here are some projection sets with extensions included:
chromatic_scale_in_chromatic_space = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | 12]
diatonic_scale_in_chromatic_space = [0, 2, 4, 5, 7, 9, 11 | 12]
diatonic_scale_in_diatonic_space = [0, 1, 2, 3, 4, 5, 6 | 7]
triad_in_diatonic_space = [0, 2, 4 | 7]
major_triad_in_chromatic_space = [0, 4, 7 | 12]
Does mathematics have an analog or proper term for this extension value?
And finally, the question. What is the name for the property that +12 or +7 represents in our updated projection set? For lack of a more proper term, I'm using the noun "extension" to describe the property, and the verb "extends" to describe the action, but I'm hoping there is a more proper mathematical description for what I'm trying to describe.