The "interval of equivalence" = "equivalence interval" = "formal octave" is that interval (much larger than a unison) which, when it occurs between two pitches, we consider them to be, in some sense, (formally if not perceptibly) the same note.
For most scales this is the octave 1:2, and when it is not the octave it is usually some other highly consonant interval such as the "tritave" 1:3. But the essential feature of the interval of equivalence in relation to definitions of scales and types of scales is that when we describe a scale we describe only the pitches that fall within a single interval of equivalence, and we leave it up to the instrument builder to decide the range of the instrument and therefore how many times (including fractions) the interval of equivalence should be repeated.
Another name for this is "identity-interval".
It should be noted that the set of nominals for a notation is typically defined for the pitches within one equivalence-interval and then copied, either unchanged or with some slight change, in other registers.
For example, the modern Western letter notation, first used around 1000 in the dialogus of pseudo-Odo, specifies the seven letters A B C D E F G in ascending order of pitch within one 2:1 ratio, then repeats those letters for all pitches at intervals with a frequency ratio of any power of 2 away from those seven. This was expressed formally in 1960 by Milton Babbitt in his definition of pitch-class.
As another example, the ancient Greeks used two different types of musical notation: one which they called "vocal" and another called "instrumental". The instrumental notation is based on the rotation of ancient letters for the three neighboring pitches of the pyknon, while the vocal notation simply runs thru the Greek alphabet in order as it labels descending pitches from the entire modal-system gamut. The instrumental notation is clearly based on the 4:3 ratio "perfect-4th" as an equivalence-interval. The full verbal Greek names also defer to equivalence at the 4:3, but originating not at the top or bottom but rather in the middle ("mese"): this is an artefact of the origin of the verbal names as descriptions of the 7 and 8 strings of the lyre -- its expansion into the PIS used 4:3 similarity from the center to copy the names into higher and lower registers.
Several composers and theorists have utilized the phi ratio as an equivalence-interval. Two examples from the work of Brink McGoogy are phinocchio tuning and the Pyramid of Mars symmetry.
Another intriguing example of a non-2:1 interval as identity is the stretched-octave scale of Charles Ives. Ives proposed two scales of apparently diatonic L and s composition, but with all intervals stretched larger, so that the identity-intervals were the minor-10th in one scale and the major-10th in the other.
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