The Riemann zeta function and tuning: Difference between revisions

As a result, although the zeta function is best known for its use in analytic number theory, the zeta function is present in the background of some tuning theory—the [[harmonic entropy]] model of [[concordance]] can be shown to be related to the Fourier transform of the zeta function, and several tuning-theoretic metrics, if extended to the infinite-limit, yield expressions that are related to the zeta function. Sometimes these are in terms of the "prime zeta function", which is closely related and can also be derived as a simple expression of the zeta function.

There are other metrics besides zeta for other definitions of "approximating well", which you can find in: [[:Category:Regular temperament tuning|optimised regular temperament tunings]].

; Riemann zeta function ("zeta") : A mathematical function which is tied to the harmonic series and to prime numbers, used in tuning theory as an "edo goodness" function to evaluate how close to JI an edo is.

; Zeta record edo : An equal tuning that sets some kind of record in regards to the zeta function compared to all smaller equal tunings.

; Record zeta peak : An equal tuning which is closer to JI than any previous tuning, and is usually an edo with compressed or stretched octaves, evaluated by the absolute "goodness" of the edo according to the zeta function.

; Record zeta peak integer : A zeta record edo by absolute "goodness", when compared only to other edos (i.e. ignoring stretched/compressed equivalences).

; Zero : A point where the Riemann zeta function is equal to zero, such as ~2.759edo, representing an equal tuning that does not represent JI much at all. Edos close to zeroes are called ''zeta valley edos''; all known zeroes are on the "critical line" used to obtain tuning information.  

; Record z gap : A zeta record edo by the size of the gap between its surrounding zeroes, adjusted for the fact that zeroes generally become more dense with larger inputs.

; Record zeta integral : A zeta record edo by the size of the area enclosed by the shape of the function between the edo's surrounding zeroes.

Zeta peak edos (tempered octaves): {{EDOs|1, 2, 3, 4, 5, 7, 10, 12, 19, 22, 27, 31, 41, 53, 72, 99, 118, 130, 152, 171, 217, 224, 270, … }}

 

Zeta peak integer edos (pure octaves): {{EDOs|1, 2, 3, 5, 7, 10, 12, 19, 22, 31, 41, 53, 87, 118, 130, 171, 224, 270, 311, … }}

 

A link to the graph of zeta can be found at [https://samuelj.li/complex-function-plotter/#abs(zeta(i*2*pi*real(z)%2Fln(2)%2Bimag(z))) Zeta in Samuelj Plotter]. (In the top left menu, make sure that "Enable Checkerboard" is unticked and "Invert Gradient" and "Continuous Gradient" are ticked.) The function has been reoriented to place edo size along the horizontal axis and weight along the vertical axis, and also scaled by {{sfrac|2π|ln(2)}} to ensure that the real number line aligns with edos. One can see that with higher weights, the function approaches a cyclic function with a period of 1; this corresponds to the prime 2 dominating more and more extremely as other harmonics are weighted less with higher weights. You can see this easier by raising the entire expression to an absurdly high power, such as 100. Note, however, that this visualization is inaccurate beyond a couple hundred: around 146.5, 324.5 and 473.5, and in many cases after, there appear to be zeroes that are not on the critical line; this is an artifact of the way the function is approximated and is the ultimate reason why the Riemann hypothesis remains unsolved. These actually correspond to zeroes that are very close together but on the critical line.   

You may also view the graph of zeta along the critical line on Desmos: [https://www.desmos.com/calculator/dstp7wnidf Zeta in Desmos]. This makes it easier to see peaks, but only works for {{nowrap| σ {{=}} {{sfrac|1|2}} }}.  

Here, the numerator represents the relative error on each prime, and the denominator represents a weight factor of the logarithm of each prime<ref group="note">Specifically, one reason we use the weighting 1/log₂(''p'') is because of certain desirable properties it has that singles it out as of unique interest: if the complexity of a prime ''p'' is {{nowrap| log₂(''p'') }}, then ''p''^''n'' is ''n'' times as complex as ''p''. Using {{nowrap| log₂(''p'') }} as the complexity also means that the complexity of a harmonic, according to its prime factorization, exactly matches where it's found in the harmonic series, so that e.g. {{nowrap| 25 {{=}} 5 × 5 }} is slightly less complex than {{nowrap| 26 {{=}} 2 × 13 }} is slightly less complex than {{nowrap| 27 {{=}} 3 × 3 × 3 }}. Therefore, the {{nowrap| 1/log₂(''p'') }} weighting is a kind of natural inverse-complexity weighting, that is, a simplicity weighting.</ref>, so the function represents a ''p''-limit badness metric. Also, for those unfamiliar, squaring the error is commonly done because it solves the flaws of two alternative ways of measuring error. Specifically, if you look only at the maximum error, you miss opportunities to make the tuning much better ''overall'' by allowing slightly more damage on the most damaged intervals, while if you look only at the average error, then it may be that you are unnecessarily damaging a few intervals a lot just to get intervals that are already in-tune slightly more in-tune, so both extremes have pathological behaviours, and using the squared error counters both of these behaviours so that it represents a more balanced approach to optimization that is used in a variety of disciplines.

This function has local minima, corresponding to associated generalized patent vals. The minima occur for values of ''x'' which are the [[Tenney–Euclidean tuning]]s of the octaves of the associated vals, while ξ_''p'' for these minima is the square of the [[Tenney–Euclidean relative error]] of the val—equal to the TE error times the TE complexity, and sometimes known as "TE simple badness."

=== Zeta yields relative error over all rationals ===

Above, Gene proves that the zeta function measures the [[Tenney–Euclidean relative error]], sometimes called ''Tenney–Euclidean simple badness'', of any edo, taken over all prime powers. The relative error is simply equal to the tuning error times the size of the edo, so we can easily get the raw absolute tuning error from this as well by simply dividing by the size of the edo.

=== Interpretation of results: cosine relative error ===

For every strictly positive rational ''n''/''d'', there is a cosine with period {{nowrap|log₂{{pars|{{sfrac|''n''|''d''}}|1.8}}}}. This cosine peaks at {{nowrap|''x'' {{=}} {{sfrac|''N''|log₂(''n''/''d'')}}}} for all integers ''N'', or in other words, the ''N''-th equal division of the rational number {{frac|''n''|''d''}}, and hits troughs midway between.

Our mysterious substitution above was chosen to set the units for this up nicely. The variable ''x'' now happens to be measured in divisions of the octave. (The original variable ''t'', which was the imaginary part of the zeta argument ''s'', can be thought of as the number of divisions of the interval {{nowrap|''e''^2π ≈ 535.49}}, or what [[Keenan Pepper]] has called the "[[zetave|natural interval]].")

As mentioned in Gene's original zeta derivation, these cosine functions can be thought of as good approximations to the terms in the TE error computation, which are all the squared errors for the different primes. Rather than taking the square of the error, we instead put the error through the function {{sfrac|1 − cos(''x'')|2}}, which is close enough for small values of ''x''. Since we are always rounding off to the best mapping, this error is never more 0.5 steps of the edo, so since we have {{nowrap| −0.5 < ''x'' < 0.5 }} we have a decent enough approximation.

There are three major differences between our cosine error functions, and the way we are incorporating them into the result, and what TE is doing:

# First, the function here is flipped upside down—that is, we are measuring accuracy rather than error—as well as shifted vertically down along the ''y''-axis. Since it is trivial to convert between the two, and since we only care about the relative rankings of edos, it is clear that we're measuring essentially the same thing.

# Instead of only looking at the primes, as we do in TE, we are now looking at ''all'' intervals, and in particular looking at the best mapping for each interval.

# If we only want the primes, rather than all intervals, we can use something called the ''prime zeta function'' to get those kinds of summations.

# If we do want the true TE squared error rather than our cosine error, then we would end up getting something called ''parabolic waves'' rather than cosine waves for each interval. A parabolic wave is the antiderivative of a sawtooth wave, and as it is a periodic signal, it has a Fourier series and can be expressed as a sum of sinusoids. We can use this to get a derivation of the squared error as an infinite sum of zeta functions.

Going back to the infinite summation above, we note that these cosine error (or really cosine accuracy) functions are being weighted by {{sfrac|1|(''nd'')^σ}}. Note that ''σ'', which is the real part of the zeta argument ''s'', serves as sort of a complexity weighting—it determines how quickly complex rational numbers become irrelevant. Framed another way, we can think of it as the degree of rolloff formed by the resultant (musical, not mathematical) harmonic series formed by those rationals with {{nowrap| ''d'' {{=}} 1 }}. Note that this rolloff is much stronger than the usual {{sfrac|1|log(''nd'')}} rolloff exhibited by TE error, which is one reason that zeta converges to something coherent for all rational numbers, whereas TE fails to converge as the limit increases. We will use the term ''rolloff'' to identify the variable σ below.

Putting this all together, we can take the approach to fix ''σ'', specifying a rolloff, and then let ''x'' (or ''t'') vary, specifying an edo. The resulting function gives us the measured accuracy of edos across all unreduced rational numbers with respect to the chosen rolloff. Taking it all together, we get a Tenney-weighted sum of cosine accuracy over all unreduced rationals. QED.

It is extremely noteworthy to mention how composite rationals are treated differently than with TE error. In addition to our usual error metric on the primes, we also go to each rational, look for the best [[direct approximation]] of that rational within the edo, and add ''that'' to the edo's score. In particular, we do this even when the best mapping for some rational does not match up with the mapping you would get from it just looking at the primes.

So, for instance, in 16edo, the best mapping for 3/2 is 9 steps out of 16, and using that mapping, we get that 9/8 is 2 steps, since {{nowrap| 9 × 2 − 16 {{=}} 2 }}. However, there is a better mapping for 9/8 at 3 steps—one which ignores the fact that it is no longer equal to two 3/2's. This can be particularly useful for playing chords: 16edo's direct approximation for 9 is useful when playing the chord 4:5:7:9, and the val mapping for 9 is useful when playing the major ninth chord 8:10:12:15:18. We can think of the zeta function as rewarding equal temperaments not just for having a good approximation of the primes, but also for having good extra approximations of rationals which can be used in this way. And although 16edo is pretty high error, similar phenomena can be found for any edo which becomes [[inconsistent]] for some chord of interest.

One way to frame this in the usual group-theoretic paradigm is to consider the group in which each strictly positive rational number is given its own linearly independent basis element. In other words, look at the {{w|free group}} over the strictly positive rationals, which we'll call ''meta-JI''. The zeta function can then be thought of as yielding an error for all meta-JI [[generalized patent val]]s. Whether this can be extended to all meta-JI vals, or modified to yield something nice like a norm on the group of meta-JI vals, is an open question. Regardless, this may be a useful conceptual bridge to understand how to relate the zeta function to ordinary regular temperament theory.

Now, one nitpick to notice above is that this expression technically involves all unreduced rationals, e.g. there will be a cosine error term not just for 3/2, but also for 6/4, 9/6, etc. However, we can easily show that the same expression also measures the cosine relative error for reduced rationals:

Let us go back to this expression here:

Note that since there is no restriction that ''n'' and ''d'' be coprime, the rationals we are using here do not have to be reduced. So this shows that zeta yields an error metric over all unreduced rationals, but leaves open the question of how reduced rationals are handled. It turns out that the same function also measures the error of reduced rationals, scaled only by a rolloff-dependent constant factor across all edos.

To see this, let us first note that every unreduced rational {{sfrac|''n''|''d''}} can be decomposed into the product of a reduced rational {{sfrac|''n''{{``}}|''d''{{-`}}}} and a common factor {{sfrac|''c''|''c''}}. Furthermore, note that for any reduced rational {{sfrac|''n''{{``}}|''d''{{-`}}}}, we can generate all unreduced rationals {{sfrac|''n''|''d''}} corresponding to it by multiplying it by all such common factors {{sfrac|''c''|''c''}}, where ''c'' is a strictly positive natural number.

This allows us to change our original summation so that it's over three variables, ''n''{{``}}, ''d''{{-`}}, and ''c''{{-`}}, where ''n''{{``}} and ''d''{{-`}} are coprime, and ''c'' is a strictly positive natural number:

Now, since we are still assuming that {{nowrap| ''σ'' > 1 }} and everything is absolutely convergent, we can decompose this into a product of series as follows

Now, since we are fixing ''σ'' and letting ''t'' vary, the left zeta term is constant for all edos. This demonstrates that the zeta function also measures cosine error over all the reduced rationals, up to a constant factor. QED.

Since the second is a simple monotonic transformation of the first, we can see that the same function basically measures both the relative error on just the prime powers, and also on all unreduced rationals, at least in the sense that edos will be ranked identically by both measures. The third function is really just the second function divided by a constant, since we only really care about letting ''t'' vary—we instead typically set ''σ'' to some value which represents the weighting rolloff on rationals. So, all three of these functions will rank edos identically.

It turns out that using the same principles of derivation above, we can also derive another expression, this time for the relative error on only the harmonics—i.e. those intervals of the form 1/1, 2/1, 3/1, …, ''n''/1, …. This was studied in a paper by Peter Buch called [[:File:Zetamusic5.pdf|"Favored cardinalities of scales"]]. The expression is:

Note that, although the last four expressions were all monotonic transformations of one another, this one is not—this is the ''real part'' of the zeta function, whereas the others were all some simple monotonic function of the ''absolute value'' of the zeta function. The results, however, are very similar—in particular, the peaks are approximately to one another, shifted by only a small amount (at least for reasonably-sized edos up to a few hundred).

is, up to a flip in sign, the Fourier transform of the unnormalized Harmonic Shannon Entropy for {{nowrap| ''N'' {{=}} &infin; }}, where φ(''t'') is the characteristic function (aka Fourier transform) of the spreading distribution and {{overline|φ(''t'')}} denotes complex conjugation.

More can be found at the page on [[Harmonic entropy #Extending_HE_to_.5Bmath.5DN.3D.5Cinfty.5B.2Fmath.5D:_zeta-HE|harmonic entropy]], including a generalization to Renyi entropy for arbitrary <math>a</math>.

So long as {{nowrap| ''s'' ≥ 1 }}, the absolute value of the zeta function can be seen as a relative error measurement. However, the rationale for that view of things departs when {{nowrap| ''s'' < 1 }}, particularly in the [http://mathworld.wolfram.com/CriticalStrip.html critical strip], when {{nowrap| 0 < ''s'' < 1 }}. As s approaches the value {{nowrap|''s'' {{=}} {{sfrac|1|2}}}} of the [http://mathworld.wolfram.com/CriticalLine.html critical line], the "information content" of the zeta function concerning higher primes increases and it behaves increasingly like a badness measure (or more correctly, since we have inverted it, like a goodness measure.) The quasi-symmetric [https://planetmath.org/encyclopedia/FunctionalEquationOfTheRiemannZetaFunction.html functional equation] of the zeta function tells us that past the critical line the information content starts to decrease again, with {{nowrap| 1 − ''s'' }} and ''s'' having the same information content; that is, for ''s'' > {{sfrac|1|2}}, {{nowrap|1 − ''s''}} essentially multiplies the zeta function at ''s'' by a fixed, monotonic increasing function. Hence it is the zeta function between {{nowrap|''s'' {{=}} {{sfrac|1|2}}}} and {{nowrap|''s'' {{=}} 1}}, and especially the zeta function along the critical line {{nowrap|''s'' {{=}} {{sfrac|1|2}}}}, which is of the most interest.

As {{nowrap| ''s'' > 1 }} gets larger, the Dirichlet series for the zeta function is increasingly dominated by the 2 term, getting ever closer to simply {{nowrap|1 + 2^−''z''}}, which approaches 1 as {{nowrap|''s'' {{=}} Re(''z'')}} becomes larger. When {{nowrap|''s'' ≫ 1}} and ''x'' is an integer, the real part of zeta is approximately {{nowrap|1 + 2^−''s''}}, and the imaginary part is approximately zero; that is, zeta is approximately real. Starting from {{nowrap|''s'' {{=}} +&infin;}} with ''x'' an integer, we can trace a line back towards the critical strip on which zeta is real. Since when {{nowrap|''s'' ≫ 1}} the derivative is approximately −{{sfrac|ln(2)|2^''s''}}, it is negative on this line of real values for zeta, meaning that the real value for zeta increases as ''s'' decreases. The zeta function approaches 1 uniformly as ''s'' increases to infinity, so as ''s'' decreases, the real-valued zeta function along this line of real values continues to increase though all real values from 1 to infinity monotonically. When it crosses the critical line where {{nowrap| ''s'' {{=}} {{sfrac|1|2}} }}, it produces a real value of zeta on the critical line. Points on the critical line where {{nowrap| ζ({{frac|1|2}} + ''ig'') }} are real are called "Gram points", after {{w|Jørgen Pedersen Gram}}. We thus have associated pure-octave edos, where ''x'' is an integer, to a value near to the pure octave, at the special sorts of Gram points which corresponds to edos.

Just like with zeta peak edos which have progressively higher {{nowrap|{{abs|Z(x)}}}} scores, we can also look at edos with progressively ''lower'' {{nowrap|{{abs|Z(x)}}}} for integer values of ''x''. This gives us {{EDOs| 1, 8, 18, 39, 55, 64, 79, 5941, 8294,}}… which correspond to ''zeta valley edos''. Zeta valley edos can be thought of as pure-octave tunings that tend to deviate from ''p''-limit JI as much as possible while still preserving octaves, and can serve as "more xenharmonic" tunings. Zeta valley edos are only measured with pure octaves, since "tempered-octave zeta valley edos" would simply be any zero of Z(x). Keep in mind, however, that the ''most'' xenharmonic tunings (essentially, tuning systems that avoid ''all'' ''p''-limit JI as much as possible) would not contain octaves at all.

=== Other lists ===

{{Idiosyncratic terms|"Absolute zeta peak edos" was coined by {{u|Godtone}}, "''k''-ary-peak edos" was coined by {{u|Akselai}}, the types of "local zeta" edos and "indecisive edos" were coined by [[Budjarn Lambeth]].}}

If we consider that zeta is a measure of relative error (that is, error measured relative to the step size), we realize that plenty of equal temperaments<ref group="note">Note importantly that we speak of ''equal temperaments'' rather than ''edos'' because generally a record peak ''does not'' correspond to an edo, which can have tangible consequences (a significant example is discussed in the next section).</ref> are excluded simply because, though practically speaking they have great tuning properties, they are not as "efficient" with their number of tones as the last record peak. Arguably what we are interested in is a sequence of edos that generally do increasingly better at tuning JI in terms of lowering the average cent error. Therefore, it suffices to multiply the score by the size of the equal temperament. Surprisingly, the list for ''s'' = 1/2 — which is supposedly where high-limit information is maximized — is ''almost identical'' to the one for ''s'' = 1 — which is the smallest value of ''s'' that we can assume to be meaningful without assuming that the analytic continuation preserves the tuning properties we are interested in — so that we have reassurance from the ''s'' = 1 list that the ''s'' = 1/2 list is meaningful wherever they agree. This is important because surprisingly, the two lists of equal temperament are ''identical up to [[311edo|311et]]'', with only one edo, [[8edo]], omitted from the list for {{nowrap| ''s'' {{=}} 1 }}. This list is {{EDOs| 1, 2, 3, 4, 5, 7, 9, 10, 12, 14, 15, 17, 19, 22, 24, 26, 27, 31, 34, 41, 46, 53, 58, 65, 68, 72, 84, 87, 94, 99, 111, 118, 130, 140, 152, 171, 183, 198, 212, 217, 224, 243, 270, 311, … }}.

==== Extended list of absolute zeta peak edos ====

If you look at the graph of zeta (for any zeta graph of interest), another issue quickly becomes evident: many equal temperaments of interest fail to have peaks of record height by only small amounts, so that we intuitively want to include them in a more comprehensive list. However, trying to "fix" this issue quickly leads into another issue: how many "nearly record" edos should we include, and why? The smallest alteration we can make is to allow an equal temperament that does better than the second-best-scoring equal temperament so far. But sometimes we have two very strong equal temperaments appear in quick succession, and given the motivation is to find a more comprehensive list anyways, here we'll include any equal temperament that does better than the third-best-scoring equal temperament so far. The motivation for this cutoff is that you intuitively might expect that the three best equal temperaments found so far represent roughly how good we can do in a given range of step sizes, so that they define what is "normal" for that range, that is, it's the heuristic of the "rule of three". Again, the list for ''s'' = 1/2 is almost identical to ''s'' = 1 for equal temperaments up to 311et, though this time the differences are less trivial: [[176edo|176et]] and [[202edo|202et]] only appear for ''s'' = 1/2, so are put in brackets. The list is {{EDOs| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 22, 24, 26, 27, 29, 31, 34, 36, 38, 39*, 41, 43, 45, 46, 48, 50, 53, 56, 58, 60, 63, 65, 68, 72, 77, 80, 84, 87, 89, 94, 96, 99, 103, 106, 111, 113, 118, 121, 125, 130, 137, 140, 145, 149, 152, 159, 161, 166, 171, (176,) 183, 190, 193, 198, (202,) 212, 217, 224, 229, 239, 243, 248, 255, 270, 277, 282, 289, 301, 311, … }}.

The equal temperaments added relative to the non-extended list of only things that are records proper are: {{EDOs| 6, 8, 11, 13, 16, 21, 29, 36, 38, 39, 43, 45, 48, 50, 56, 60, 63, 77, 80, 89, 96, 103, 106, 113, 121, 125, 137, 145, 149, 159, 161, 166, (176,) 190, 193, (202,) 229, 239, 248, 255, 277, 282, 289, 301 }}<ref group="note">39et is a notable example because 39edo corresponds to a zeta valley, so it is surprising that it would be included here; the reason that it is included is because this is ''not'' 39edo, but 39 ''equal temperament'', corresponding to a 3.8{{cent}} flat-tempered octave so that it is actually ~39.124edo, that is, it corresponds to the 173rd zeta peak, known by the shorthand 173zpi (where i stands for index). Therefore, this may prove a good testcase for investigating the effects of zeta-based octave-tempering, though given the size of the stretch, the difference is likely to be subtle, but the fact that it "changes zeta's mind" this much is itself interesting. You can also interpret this result differently, which is as evidence that you should not include equal temperaments worse than the third-best-scoring equal temperament so far, given the somewhat dubious inclusion of 39et, however it should be noted that this is more to do with that at the very beginning of the list there are not many equal temperaments to "beat" so that beating the third-best-scoring equal temperament so far is easy, though arguably this is not a flaw because people are often more likely to try a smaller equal temperament. It is also perhaps worth noting that 37et almost makes this extended list, but the omission of 37et is much better addressed by no-3's zeta.</ref>.

If we want to find the second-best edos ranked by zeta peaks, then given a full list of zeta peaks, we can remove the successively higher peaks to get another sequence of succesively higher peaks, which correspond to edos called '''2-ary peak edos''': defined as non-zeta-peak edos with a higher zeta peak than any smaller non-zeta-peak edo.  

This list can be used finding an alternative to any given zeta peak edo of similar size and still-okay accuracy, but with different regular temperament properties (e.g. 9 as alternative to 10, 17 as alternative to 19).

{{EDOs| 6, 8, 9, 14, 15, 17, 24, 34, 46, 58, 65, 77, 87, 111, 140, 183, 243, 301, 311, 460, 472, 525, 571, 581, 814, 836, 882, 1205,}} …

We can then remove those secondary peaks again to get '''3-ary peak edos''': non-zeta-peak edos with a higher zeta peak than any smaller edo that is neither a zeta peak nor a 2-ary peak.

We can repeat this process as many times as we want, resulting in '''''k''-ary-peak edos'''. The ordinary peak edos are 1-ary peak edos, then there are 2-ary peak edos, 3-ary peak edos, and so on. However keep in mind that the higher ''k'' gets, the less meaningful the peaks will get, especially for smaller edos (less than about 100).

We may define ''local zeta peak'' edos as a generalization of the ''zeta peak'' edos as those that do not necessarily have successively higher zeta peaks but simply have a higher zeta peak than the edos on either side of them. This lists a wide variety of edos that approximate primes well for their size, even when they aren't better than every smaller edo. It could be helpful for:

* finding edos with plenty of consonances in size ranges that lack any record-holding zeta edos (e.g. between 60 and 70 tones)

==== Local zeta valley edos ====

We may define ''local zeta valley'' edos as those with a ''lower'' best nearby zeta peak than the edos on either side of them. This is helpful for finding edos that force the use of methods other than traditional concordant harmony, or for composers seeking a challenge/limitation to inspire creativity.

Finally, ''indecisive'' edos can be defined as edos which are ''neither'' local zeta peak, nor local zeta valley. For some, these tunings might narrow down the range of compositional choices available so as to be not so many to promote indecision, but not so few as to promote frustration.

* {{subpage|appendix|u|s=Black magic formulas|text=How it can be shown what ETs have a sharp vs. flat tendency?}}