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2. The following three involve double contractions (coincidences of momenta) which imply [MATH] . This contribution is [MATH] 3. |
The last term gives no contribution, since it implies that [MATH] and [MATH] . This will always yield a factor of [MATH] Consequently, the four momentum sums of Eq. 75 have the value |
[EQUATION] This procedure is trivially generalized to any number of phase factors, although the structure becomes rather complicated for higher order terms. Fortunately, we will see that in certain limits, we may ignore the contributions coming from the coincidences of momenta, and only the pairwise contractions will c... |
Phase Averages in the Gaussian Disorder Case In the case of the gaussian correlated disorder, the normalization of the Fourier transform implies that [MATH] The momentum sums give a factor of [MATH] , which means that the two terms in Eq. 79 will be of order |
[EQUATION] This means that the second term is negligible in the thermodynamic limit. This argument can actually be carried through to any order, since any term of the form [MATH] goes to zero in the limit [MATH] , which renders all the contributions coming from the coincidence of indices irrelevant in this limit. |
Therefore, if we want to calculate a general [MATH] we may only consider the sum of all pairwise contractions of momenta. The total number of different contractions is [MATH] and each one contributes with a term [MATH] to the sum over momenta. Hence, we have |
[EQUATION] Phase Averages in the Power-Law Disorder Case For the case of Power-Law Correlated Disorder, the Eq. 79 is still valid, but one cannot generally ignore the [MATH] term. Let us consider only the cases where [MATH] , meaning that |
[EQUATION] with the normalization [EQUATION] Like before, we have [MATH] but the calculation of [MATH] is now, slightly different, i.e. |
[EQUATION] In the large [MATH] limit, the last sum converges if [MATH] and it gives [MATH] . Using Eq. 82 we finally obtain [MATH] |
which does not scale with the system size [MATH] . This interesting result suggests that the argument made for the gaussian case does not work here, and any calculation of the moments of [MATH] must account for the coincidences of momenta. In fact, this is easily seen to be true for any term of the form [MATH] , yieldi... |
[EQUATION] Nevertheless, a special case happens when [MATH] . In this limit, the denominator of Eq. 83 diverges as [MATH] , while the numerator remains finite near [MATH] . This means that, for [MATH] the corrections due to the coincidence of momenta become negligible, and we have [EQUATION] |
# Source: arxiv 1806.06703 # Title: A 5-Dimensional Tonnetz for Nearly Symmetric Hexachords # Sections: all # Downloaded: 2026-03-03T02:44:58.120939+00:00 |
A 5-Dimensional Tonnetz for Nearly Symmetric Hexachords Abstract The standard 2-dimensional Tonnetz describes parsimonious voice-leading connections between major and minor triads as the 3-dimensional Tonnetz does for dominant seventh and half-diminished seventh chords. In this paper, I present a geometric model for a ... |
Introduction In this paper, I constrct a 5-dimensional Tonnetze for nearly symmetric hexachords, known as the mystic and Wozzeck chords. Cohn (1996) describes that it should be possible for chords of the [MATH] set class of 6-34 to exhibit voice-leading parsimony in ways similar to the major and minor triads of 3-11. H... |
The Boretz spider region and Childs’ (1998) octatonic region similarly contain Neo-Riemannian transformations that appropriately describe parsimonious voice-leading between dominant seventh and half-diminished seventh chords, which are perturbations of the symmetric fully diminished chord. These chords are visually rep... |
The author has previously developed the dodecatonic and centipede regions for voice-leading between chords obtained from the perturbation of the symmetric whole-tone scale: the mystic and Wozzeck chords (Mohanty 2018). In this paper, I mathematically construct the 5D Tonnetz for voice-leading between mystic and Wozzeck... |
II The coordinate space In this section, I construct the geometric positions and identities of individual pitch classes in 5D space. Like the 2D and 3D Tonnetze , equally spaced points in the space represent pitch classes, and simplices bounded by [MATH] vertices in [MATH] correspond to nearly symmetric chords. |
II.1 The Tonnetz basis of [MATH] In both the 2D and 3D note spaces, the axes along which individual pitch classes lie are not mutually orthogonal. Writing the unit vectors pointing along these axes as [MATH] , where [MATH] is the cardinality of the chord, it is easy to see that the [MATH] -th and [MATH] -th unit vector... |
[EQUATION] I generalize this relation to the [MATH] case so that all 5 axes in the 5-dimensional note space will be oriented at [MATH] degrees with respect to one another. Imposing the conditions in eq. ( ) separately on the 5 axes, we find that the unit vectors [MATH] can be written in the Cartesian basis [MATH] as |
[EQUATION] Any vector [MATH] in the Tonnetz basis [MATH] can be represented in the Cartesian basis as [MATH] by the unitary transformation |
[EQUATION] II.2 Pitches in the coordinate space Let [MATH] denote the set of tones in the Tonnetz coordinate space; in particular, [MATH] includes all linear combinations of the Tonnetz basis vectors [MATH] with integer coefficients. That is, |
[EQUATION] We define a map [MATH] such that [MATH] for [MATH] returns an integer [MATH] that corresponds to a particular pitch class [MATH] , and assignment is inherently arbitrary. However, throughout this paper, I use the standard convention of [MATH] [MATH] , etc. I will also use an ordered pair of integers [MATH] t... |
Now, I explicitly construct [MATH] by following the conventions of the 2D and 3D Tonnetze . We can succinctly state that [EQUATION] |
where [MATH] returns the remainder of the argument divided by 12. From this definition, one may see that, starting at the origin, the notes along the [MATH] in the positiive direction are [MATH] [MATH] [MATH] , etc. The notes along the [MATH] in the positive direction are [MATH] [MATH] [MATH] , etc. Similar logic can b... |
III Nearly symmetric hexachords Now that the coordinate space has been constructed precisely, I now introduce the geometric definitions of the mystic and Wozzeck chords. As described in section 1.3, the mystic and Wozzeck chords are inversionally related nearly symmetric hexachords, and I will show that the particular ... |
III.1 Wozzeck chords A Wozzeck chord is obtained from the downward perturbation of any tone in a whole-tone scale and will be denoted with a [MATH] symbol such that “ [MATH] Wozzeck” can be written as [MATH] . By the convention presented in an earlier work (Mohanty 2018), a Wozzeck chord will be labeled by the lower of... |
In the coordinate space defined in the previous section, a Wozzeck chord which has its root at the point [MATH] is given by the collection of tones [MATH] . This corresponds to the collection of vertices of a 5-simplex in [MATH] with orientation we will refer to as [MATH] |
III.2 Mystic chords A mystic chord is given by a upward perturbation of a tone within the whole-tone scale; these chords are denoted with the [MATH] symbol. Thus, [MATH] refers to the “ [MATH] mystic” chord and is comprised of the pitch classes [MATH] . A mystic chord is given by the collection of 6 tones [MATH] . Thes... |
III.3 Duality in the mystic-Wozzeck genus As major and minor triads—represented by triangles (or 2-simplices) in the 2D Tonnetz —have opposite graphical orientations, the dominant seventh and half-diminished seventh chords in Gollin’s (1998) 3D Tonnetz also are “upside down” images of each other. This notion of orienta... |
IV Neighbors in the 5D Tonnetz As described by Cohn (2012), the nearly symmetric hexachords exhibit parsimonious voice leading, and small-displacement chord transitions are fully described by a set of Neo-Riemannian transformations, which are defined in the author’s previous work (Mohanty 2018). The table of these Neo-... |
Starting with some arbitary chord on the 2D Tonnetz , it is easy to see that applying the standard triadic Neo-Riemannian transformations [MATH] [MATH] [MATH] [MATH] , and [MATH] to the starting chord result in a chord that shares either an edge or a corner with the starting chord. In the 2D and 3D Tonnetze , not all o... |
Since the full 5-dimensional space cannot be directly visualized in spatial coordinates, I have produced several reduced images in Figures 1 through . In each figure 1-7, the central chord, which appears as a hexagon with several diagonal lines, is the arbitrarily chosen [MATH] . This hexagon represents an orthographic... |
Tonnetz space. Despite varying lengths of the solid lines in this projection, each line represents the same [MATH] distance, which is precisely unit distance using the standard Euclidean metric. |
The permutation of the vertex labels of a given chord in different Figures 1 through allow for easy visualization of neighbors. A 5-simplex has 15 edges and 6 corners, so “true” picture of the 5D Tonnetz is a simultaneous superposition of all 7 panels shown in Figures 1 through . In the figure, the Neo-Riemannian trans... |
The rules for the chord neighbors shown in Figures 1 through generally hold for any Wozzeck chord, and the neighbors for any mystic chord can be quickly deduced by symmetry properties. A limitation of the 5D Tonnetz is the inevitable fact that the entire Tonnetz cannot be visualized with accurate representation of all ... |
Conclusion In this article, I have presented an explicit construction of the 5D Tonnetz for voice-leading between nearly symmetric hexachords. As discussed in previous work (Mohanty 2018; Cohn 2012), Mystic and Wozzeck chords obey voice-leading rules similar to those for the major-minor triadic complex as well as the T... |
References {hangparas} 1cm1 Childs, Adrian P. 1998. “Moving beyond Neo-Riemannian Triads: Exploring a Transformational Model for Seventh Chords.” Journal of Music Theory 42, no. 2: 181-193. |
Cohn, Richard. 1996. “Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progressions.” Music Analysis 15, no. 1: 9-40. |
Cohn, Richard. 2012. Audacious Euphony: Chromatic Harmony and the Triad’s Second Nature . 2nd Edition. New York: Oxford University Press. |
Gollin, Edward. 1998. “Some Aspects of Three-Dimensional ‘ Tonnetze ’.” Journal of Music Theory 42, no. 2: 195-206. Mohanty, Vaibhav. 2018. “Dodecatonic Cycles and Parsimonious Voice-Leading in the Mystic-Wozzeck Genus.” Submitted for publication. Preprint: arXiv:1805.11087. Appendix: Figures |
# Source: arxiv 1806.06933 # Title: Delegated Search Approximates Efficient Search # Sections: all # Downloaded: 2026-03-02T09:23:02.983204+00:00 |
Delegated Search Approximates Efficient Search Abstract There are many settings in which a principal performs a task by delegating it to an agent, who searches over possible solutions and proposes one to the principal. This describes many aspects of the workflow within organizations, as well as many of the activities u... |
In this work, we investigate how much the principal loses – quantitatively, in terms of the objective they are trying to optimize – when they delegate to an agent. We develop a methodology for bounding this loss of efficiency, and show that in a very general model of delegation, there is a family of mechanisms achievin... |
Introduction There are many settings in which a decision-maker is faced with a difficult problem that they cannot solve on their own, and so they instead approach it in two steps: they first delegate the search for possible solutions to an agent who is able to invest more time in the process, and then they evaluate the... |
(Aghion and Tirole,, 1997 ; Armstrong and Vickers,, 2010 ; Li et al.,, 2017 A second example arises in regulation, where a governmental agency needs to decide whether there is a way to structure a proposed corporate merger in a way that is compatible with regulatory guidelines; the companies seeking to merge study poss... |
The interesting tension in all these situations is that the decision-maker who delegates the task (henceforth referred to as the “principal”) has a particular objective function that they are seeking to optimize; but the agent who actually performs the task might have interests that are not directly aligned with the pr... |
A rich literature has developed in economics around the formalization and analysis of delegation, focusing on this tension between the conflicting objectives of the principal and the agent; see Holmstrom, ( 1977 1984 |
for influential early research, and Alonso and Matouschek, ( 2008 ); Armstrong and Vickers, ( 2010 ); Amador and Bagwell, ( 2013 ); Ambrus and Egorov, ( 2017 |
for recent work. A dominant theme in this line of work is that the principal does not offer monetary compensation to the agent as a way of favoring certain proposed solutions over others (though see |
Krishna and Morgan, ( 2008 ); this is consistent with the motivating applications, in which for example regulators in many contexts can accept or reject proposals from companies, but cannot selectively offer varying amounts of compensation to these companies based on the content of the proposal. This lack of monetary t... |
(Alonso and Matouschek,, 2008 ; Armstrong and Vickers,, 2010 ; Melumad and Shibano,, 1991 The Present Work Given how broadly delegation is used across a range of contexts, it is interesting to consider how precarious a process it is — the principal is ceding control of their search problem to an agent whose interests m... |
In their most natural formulation, these are inherently comparative questions, since they seek to relate the solution quality obtained through delegation to the solution quality in an alternate, ideal setting where delegation was not necessary. As such, they address an issue fundamentally distinct from the primary focu... |
There is a natural benchmark to use for our comparison: we could measure the quality of the outcome under delegation versus the quality of the solution that the principal could obtain were they to perform the search task themself, investing the same level of effort in the search that the agent does. Now, there are many... |
In this paper we develop a methodology to bound the performance of delegated search, relative to the benchmark in which the principal searches for a solution on their own. Our methodology builds on a set of links that we identify between bounds on delegated search and the analysis of some fundamental models of decision... |
1.1 Overview of Models A Distributional Model We begin by describing the models in which we perform our analysis. Our main model, which is essentially the one considered in |
Armstrong and Vickers, ( 2010 , has the principal and the agent agree that the agent will consider [MATH] candidate solutions and propose one to the principal; |
[MATH] thus represents the level of effort that the agent commits to the problem. The principal will only see the solution that is proposed, not the other [MATH] that the agent also considers. |
What does it means for the agent to consider a candidate solution? We assume that the solutions belong to an abstract space [MATH] with a probability measure on it, and the agent’s search for a solution consists of performing [MATH] independent and identically distributed draws from [MATH] resulting in a set of candida... |
[MATH] Each solution [MATH] drawn by the agent has a quality for the principal, denoted [MATH] , and a possibly different quality for the agent, denoted [MATH] The agent selects one of its candidate solutions, say [MATH] to present to the principal. (Below, we will discuss the contrast between the model in which the pr... |
Now, if the principal imposed no constraint on the agent’s behavior, then the agent would simply choose the solution [MATH] that maximizes [MATH] , and the principal would receive whatever corresponding |
[MATH] value resulted from this choice. To improve on this, the principal could specify at the outset that they will only accept [MATH] values that satisfy some predicate on |
[MATH] and (in the case that they can determine it) [MATH] we will refer to the set of all [MATH] satisfying the principal’s predicate as the eligible set of solutions. It is thus in the agent’s interest to propose a solution belonging to the eligible set; we ask whether one can design eligible sets that provide provab... |
with maximum [MATH] A Binary Model In our first, distributional model, the agent draws a set of candidate solutions [MATH] that the principal cannot observe, and then must choose one to present to the principal. This models a setting in which the agent explores a design space and cannot fully anticipate which [MATH] it... |
We model this version with publicly known binary options as follows. There is a set of options [MATH] and for each [MATH] , there is a known probability [MATH] such that option [MATH] is feasible with probability [MATH] , and |
infeasible otherwise. If option [MATH] turns out to be feasible, then it produces a known payoff of [MATH] for the principal and a known payoff [MATH] for the agent; if it turns out to be infeasible, then it produces a payoff of [MATH] for both. The only way to evaluate the feasibility of option [MATH] is to pay a cost... |
The principal delegates to the agent the task of proposing a feasible option, which the principal can either accept or reject. The principal will not be able to see which options the agent decides to pay to evaluate as part of this task, but again the principal can specify a predicate defining the eligible subset of [M... |
that they will accept. Subject to this constraint, the agent then must decide how to evaluate options in a way that maximizes its own benefit [MATH] |
from the option [MATH] it proposes, minus the evaluation cost. Here too we evaluate the principal’s payoff relative to the scenario in which they performed the evaluation of options themself. We also consider an extension of this model in which there is a budget of [MATH] on the number of options that the agent can eva... |
1.2 Overview of Results We begin by showing that for an arbitrary instance of the distributional model, there is a mechanism the principal can specify to the agent so that the principal’s expected payoff from delegation is within a factor of [MATH] of the expected payoff they’d receive were they able to search for the ... |
[MATH] candidate solutions and choose the one that was best for them.) This mechanism only requires knowledge of the principal’s |
[MATH] values, not the agent’s [MATH] values, and it has a very simple structure: depending on the distribution of values, it can be written as a threshold rule with either a weak threshold, in which the principal only accepts proposals [MATH] for which [MATH] for some [MATH] or a strict threshold, in which the princip... |
[MATH] in this bound can be improved to [MATH] There are several things worth remarking on about this result. First, the fact that arbitrary instances of the problem have mechanisms providing provable guarantees of this form suggests a qualitative argument for the robustness of delegation: no matter how misaligned the ... |
A connection to prophet inequalities These results on threshold mechanisms and their guarantees follow from a general result at the heart of our analysis — a close connection between bounds for delegated search and prophet inequalities Prophet inequalities are guarantees for the following type of decision under uncerta... |
Prophet inequalities tend to be established by designing carefully constructed threshold rules , in which the decision-maker accepts [MATH] if and only if |
[MATH] (weakly or strictly) exceeds a specified threshold [MATH] that can depend on the position [MATH] The key component of our analysis is to establish a close, though subtle, technical connection between delegated search and prophet inequalities: roughly speaking, the sequence of values |
[MATH] sampled by the agent from the set of possible solutions [MATH] plays the role of the process generating [MATH] and the principal and the agent jointly — through the principal’s specification of the threshold and the agent’s incentive to obey it — play the role of the decision-maker who uses a threshold rule for ... |
[MATH] that the agent generates; this is the sense in which the stopping rule is jointly constructed by the behavior of the principal and the agent together. |
Stronger bounds for independent values Using this connection to threshold rules for prophet inequalities, we can design a much more powerful policy for the principal in the case when the values of |
[MATH] and [MATH] on a draw [MATH] from [MATH] are distributed independently, and when the principal can see both [MATH] and [MATH] (rather than only [MATH] in the solution [MATH] proposed by the agent. |
To do this, we begin with a stopping rule from the prophet inequality literature achieving an expected payoff that is at least [MATH] times the optimum when the distributions of the [MATH] values are independent and identically distributed (Abolhassani et al.,, 2017 ; Correa et al.,, 2017 ; Hill and Kertz,, 1982 ; Kert... |
that decrease with [MATH] , making the decision-maker naturally more prone to stop and accept a value as the end of the sequence nears — effectively following the idea that one should only accept a value early if it’s very good. |
In the context of delegated search when the principal can observe both [MATH] and [MATH] for a proposed solution [MATH] , a related concept is useful for designing mechanisms: the principal should only accept a solution [MATH] with [MATH] very large if [MATH] is very large as well. The analogy between requiring strong ... |
values (over the set of candidate solutions considered by the agent) can be used as a kind of “continuous time” parameter for deriving a threshold: if we think of the candidate solution [MATH] |
as arriving at continuous time [MATH] , then we can derive a threshold function [MATH] in which the principal only accepts [MATH] if |
[MATH] (weakly or strictly) exceeds [MATH] In this sense, the principal and agent again jointly construct the stopping rule, with the agent’s payoff providing a type of synthetic temporal ordering that is useful in formulating a threshold policy. |
Bounds for the binary model We also use the connection to prophet inequalities to derive bounds for the binary model, where the agent pays to evaluate the feasibility of pairs from a known list of options |
[MATH] Here too the principal can designate a predefined eligible set of proposals so that the mechanism that accepts any eligible proposed [MATH] |
yields an expected payoff that is within a factor of two of the benchmark in which the principal performs the search on their own. However, the eligibility criterion in this case is subtler: it depends not only on the principal’s assessment of the proposal’s quality, [MATH] but also on the cost [MATH] and the a priori |
probability of feasibility, [MATH] To establish this bound, we draw on both prophet inequality bounds and on work of Kleinberg et al., ( 2016 |
for the box problem (Weitzman,, 1979 by considering an ordering of the options by the notion of reservation price (or, equivalently, |
strike price ) defined in those works, we can establish a provable guarantee that correctly handles not only the payoff arising from the [MATH] and [MATH] values but also the cost incurred by the agent in evaluating the feasibility of options. |
Finally, we derive similar bounds in the more general case where the agent also has a budget of [MATH] on the number of options they can evaluate. The approach using reservation prices does not directly extend to this case, but we show that by combining the approach of Kleinberg et al., ( 2016 with bounds for stochasti... |
Asadpour and Nazerzadeh, ( 2016 , we can obtain more general bounds for a budgeted variant of the box problem that contains the case we need for our delegated search guarantee. |
We note that it would be a natural open question to consider a variant of the problem combining characteristics of the two main versions we consider here: as in the distributional model, the agent performs independent draws from a space [MATH] but as in the binary model, the agent does not have a fixed bound [MATH] |
on the number of allowed draws, instead incurring a cost to perform each draw that must be traded off against the eventual payoff from the sample selected. |
1.3 Further Discussion of Related Work on Delegation The theory of delegation in the economics literature is often viewed as beginning with Bengt Holmstrom’s Ph.D. thesis |
(Holmstrom,, 1977 1984 this work articulates the basic tension that we see in these models, between allowing an agent to optimize in a large space and restricting the agent’s freedom of action to prevent them from pursuing their own objectives too aggressively. Holmstrom’s model considered delegating an optimization pr... |
(Alonso and Matouschek,, 2008 ; Melumad and Shibano,, 1991 Armstrong and Vickers, ( 2010 propose a model that is very close to what we consider here, where the optimization takes place over a discrete set that the agent samples from an underlying distribution. By way of comparison between our work and that of |
Armstrong and Vickers, ( 2010 , we noted the key contrast earlier in this section: their paper is largely devoted to identifying cases of the delegated search problem for which the structure of the optimal mechanism can be identified, whereas we focus on bounding the inefficiency of delegated search relative to a bench... |
A distinct line of work in delegation relaxes the constraint that the principal may only allow or forbid each proposed solution, and instead allows the principal to add arbitrary amounts of cost to certain subsets of proposed solutions |
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