Telicity

Telicity is a property of both equal temperaments 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 direct approximation to connect with an interval belonging to a chain created by a lower prime's direct approximation – designated as the telos – so that all of the tempered intervals on either prime's approximation chain, and the tempered comma connecting the two prime chains in question, 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 or simple map of N-EDO:

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

Commas and equal temperaments 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 and Mappings

In order to understand how and why telicity is useful, one must first realize that there are two ways of constructing a mapping for a given EDO.

The first is the construct a regular temperament, in which you first establish a set of approximations to prime intervals, putting the values into a temperament mapping matrix and, from there, just take the prime factorization and multiply to get other desired intervals. For example, in creating a simple map for 12edo, mapping matrix M of [12 19 28], from which we can calculate the following:

• M(5/4) = 4
• M(81/80) = 0
• M(81/80)^3 = 0

Now because M is linear, it satisfies the conditions:

• M(a) + M(b) = M(a * b) (for two JI intervals a, b)
• n*M(a) = M (a^n) (for some integer n)

The second way to construct a mapping is to just round, using the equation F(r) = round(N⋅log2(r)) in which N is the number of EDO-steps and r is the ratio in question. For 12edo, this is F(r) = round(12⋅log2(r)), from which we can calculate the following:

• F(5/4) = 4
• F(81/80) = 0
• F((81/80)^3) = 1 (!)

Because F is not a linear function, it does not satisfy any of the conditions above. So even though rounding sometimes gives us better approximations, it doesn't conserve interval arithmetic, and conserving interval arithmetic is why we care about RTT in the first place.

However, people who care about both good approximations and conserving interval arithmetic are seemingly presented with a dilemma- either go with a linear map and risk bad approximations, or go with a non-linear map and risk inconsistent interval arithmetic. What makes telicity so useful is that when one is able to work with the section of the EDO's harmonic lattice in which both mapping methods lead to the same result and limit the usable portion of the harmonic lattice to this section through means of harmonic lattice resets – which, as per the name, are places in which the harmonic lattice of an EDO is considered to "reset" either to the unison or to something that can easily access the unison – one has the best of both worlds. In order to do this mathematically, however, one has to use an equation that sets the results of both mapping methods against each other, hence why telicity is defined by equation val(N)⋅monzo(r) = round(N⋅log2(r)).

Integer and Rational Telicity

Harmonics, and especially primes, are fairly simple as both equaves and generators when it comes to telicity, and since all of these interval ratios are integers, this type of telicity can be referred to specifically as integer telicity. However, when the equave and or the generator are a combination of primes, things are more complicated, leading to the broader term rational telicity as a descriptor for this second type of telicity.

Telicity Type Designations

Different equal temperaments have different relationships among the various primes that are used to define their simple maps, 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 telicity that is k-Strong. 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, which is to say the error is less than 1\(2kN). 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.

Telicity on Subgroups

Telicity is often most useful in the accurate modelling of subgroups of interest; it therefore makes sense to define senses of telicity for subgroups. This builds on the idea that a telic connection between two generators in a rank one temperament can be k-strong. Consider a set of generators. As we are generalising the notion, the generators need not necessarily be prime, but ideally all generators are either harmonics (positive integers > 1) or at least (ideally low-complexity) intervals of significant musical interest. Then a subgroup (a set of rationals > 1 AKA a "set of generators") is k-strong pairwise telic if there is a k-strong telic connection between every pair of generators. If a subgroup is "almost" k-strong pairwise telic except for exactly n pairs of generators lacking a k-strong telic connection, then it is instead n-deficient k-strong pairwise-telic. (If a subgroup is "1-strong pairwise telic" it is simply pairwise telic.) This means that a subgroup can be both "pairwise telic" and "n-deficient k-strong pairwise-telic" simultaneously. This can be abbreviated as being n-weak k-strong pairwise-telic.

There is a yet weaker - but not unuseful - notion of telicity on subgroups, where every generator can be considered as a node in a graph, and every telic connection can be considered as an edge in that graph. Then the model of the subgroup that the rank one temperament provides is said to have connective telicity if the graph is connected, meaning every generator in that subgroup can be related to every other generator directly or indirectly through a path of telic connections to other generators. Then, if every one of those telic connections is k-strong, it is said to have k-strong connective telicity. (Having connective telicity is defined as having "1-strong connective telicity".) Analogously, a model of a subgroup may have "k-strong connective telicity" except for exactly n pairs of generators that do not have a k-strong telic connection, but which demonstrate 1-strong connective telicity with respect to the subgroup nonetheless. Then the subgroup demonstrates n-weak k-strong connective telicity.

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 equal temperament's step with the size of the comma in question – if the comma is more than half the size of the equal temperament's step, said equal temperament is outside the comma's telicity range. For example, in EDO systems, 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, if you take JND-sized commas as commas to be tempered. (If you instead want to have the melodic JND mapped to a single edostep, ~311 might be a reasonable upper bound, due to being barely above the JND and due to its theoretical significance.) Note that the melodic JND varies widely from person to person and depending on context.

It should also be noted that while the tempering of a given comma that is larger than half a step in a given equal temperament can sometimes be accomplished to join primes without either prime exceeding the 50% relative error threshold in said equal temperament, the fact remains that the equal temperament 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 equal temperament 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.

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.