Telicity

Telicity is a property of both EDOs and commas and how they relate to each other. Specifically, for EDOs, it is the quality or state of being able to successfully stack a number of instances of a given prime's patent interval to connect with an interval belonging to a chain created by a lower prime's patent interval – designated as the telos – without any of the tempered intervals on either prime's patent interval chain, or even the tempered comma connecting the two prime chains in question, failing to satisfy the following equation where N is the number of steps in a given EDO, r is the ratio of an interval in one of the two prime chains in question, and val (N) denotes the patent val of N-EDO:

[math]\displaystyle{ \text{val} (N) \cdot \text{monzo} (r) = \text{round} (N \log_2 (r)) }[/math]

Commas and EDOs that demonstrate this property are referred to as as being telic. When a given EDO is telic in a given multiprime relationship by more than one means, it can be said to be multitelic.

Telicity Range and Usable EDO Sizes

Each telic comma has a numerical range in which it can possibly be telic, that is, a telicity range. The size of a comma's telicity range is inversely correlated to the size of the comma itself – that is, the smaller the comma in question, the larger the size of that comma's telicity range. The size of a comma's telicity range is evaluated by comparing the size of a given EDO's step with the size of the comma in question – if the comma is more than half the size of the EDO's step, said EDO is outside the comma's telicity range. For example, 81/80 has a telicity range of 27, as 27 is the largest EDO with a step size that is more than twice the size of the comma, while Mercator's comma has a telicity range of 165, as 165 is the largest EDO with a step size that is more that twice the size of that comma. In addition, the concept of telicity can be used to evaluate the usefulness of EDOs relative to the JND of human pitch perception, and, since an interval smaller than 3.5 cents is unlikely to be noticed by even the most trained listeners, it can be said that the JND has a telicity range of 171, meaning that EDOs that are higher than 171 are not all that suitable for use as musical systems outside of pitch bends.

It should also be noted that while the tempering of a given comma that is larger than half a step in a given EDO can sometimes be accomplished to join primes without either prime exceeding the 50% relative error threshold in said EDO, the fact remains that the EDO in question lies outside the comma's telicity range. Thus, such a phenomenon in relation to telicity is analyzed as being a result of the EDO in question tempering out two or more commas that actually do meet the strict criteria requirements, and thus, having two or more forms of the same type of telicity. As an example, 31edo tempers out 81/80, which is larger than half of this EDO's step size, but this can be attributed to 31edo tempering out both the Würschmidt comma (393216/390625) and the Semicomma (2109375/2097152), with the former equating an octave-reduced chain of eight 5/4 intervals with 3/2, and the latter equating an octave-reduced chain of seven 5/4 intervals with 32/27, while both commas additionally satisfy the equation for telicity on account of being less than half an EDO step in size.

Telicity Type Designations

Different EDOs have different relationships among the various primes that are used to define their patent vals, and thus, there is a need to designate these relationships. Given that EDOs specifically are defined as equal divisions of the octave, the 2-prime – which defines the octave – is always listed last, with the largest prime being listed first, and other primes being listed in the middle from largest to smallest thus, for example a telic connection between the 3-prime and the 2-prime is denoted as 3-2 telicity. It should be noted that the only type of telicity available to the 3-prime is 3-2 telicity, as 2 is the only positive prime lower than 3, and since octave equivalency renders the unison as the only available target, that means that the 3-prime requires a complete circle of fifths without accumulating 50% relative error or more in order to achieve telicity. However, higher primes have more options for achieving a form of telicity as there are multiple lower primes to chose from to potentially connect with. For instance, the 5-prime has both 5-3 and 5-2 telicity available to it. Not only that, but in cases where multiple overlapping telic relationships exist for a given EDO without the largest tempered comma failing to satisfy the telicity equation, one can express all of these telic relationships within a single designation. For example, 12edo, which simultaneously demonstrates 3-2 telicity, 5-3 telicity, and 5-2 telicity, can be said to demonstrate 5-3-2 telicity.

K-Strong Telicity

While the telicity of EDOs with, say, 3-2 telicity and only a single circle of fifths, is independent, properly accounting for the same type of telicity in EDOs with multiple circles of fifths is another story, and for that, we need to work with K-Strong Telicity. K-Strong Telicity is k times as strict as normal telicity, which is to say that for any two generating intervals A and B, A^n * B^m for nonzero integers n,m should by patent val consistently be mapped to the right interval in both N EDO and kN EDO so that the error is less than 50%/k of a step in N EDO. Note that this also requires that the mapping for intervals A and B in kN EDO should be the same as the mapping for them in N EDO, and that it requires all the other things needed for telicity by default. Using this, we can see that 12edo is a 2-strong 3-2 telic system and 53edo is a 3-strong 3-2 telic system.

Applications

Given that different EDOs can temper out different commas to achieve the same type of telicity – for example, 12edo tempers out the Pythagorean comma to achieve 3-2 telicity, while 53edo tempers out Mercator's comma to achieve 3-2 telicity – it can thus be argued that sequences of different EDOs demonstrating one or more types of telicity can be compiled. For instance, the first nine EDOs to demonstrate 3-2 telicity specifically form the sequence of 1, 2, 5, 12, 24, 53, 106, 159, 306. In addition, one can compare multiple such telicity sequences, and see how frequently the various prime chains connect to one another across various EDOs, revealing which portions of the harmonic lattice are best utilized by any given EDO. Furthermore, this also enables one to examine the properties of the various prime chains themselves and provides cause to look for unexpectedly useful commas that, as of yet, are still unknown. As if all this weren't enough, telicity also useful in notation systems for establishing good positions for the "resets" in JI harmonic lattice representation that inevitably come about due to EDOs being closed systems in terms of their own harmonic lattices. All this makes telicity a viable endgame for the application of consistency to distance d, with which the concept of telicity is closely related.