Filtered Point-Symmetry (FiPS) is a tool for modeling relationships between iterated maximally even sets. Common musical relationships can be studied by using FiPS to model chords contained within a specific scalar context (such as C major or the 01-octatonic collection), and by capturing those relationships in a FiPS configuration space. In this work, many FiPS configuration spaces are presented; some are isomorphic to commonly referenced voice-leading spaces like the neo-Riemannian Tonne , and others show tonal networks that have not previously been explored. A displacement operation is introduced to codify the traversal of a configuration space, and a short music analysis is provided to demonstrate some benefits of the approach. DOI: 10.30535/mto.25.2.3 Volume 25, Number 2, July 2019 Copyright © 2019 Society for Music Theory [1.1] Scale theory and transformational theory have converged in some interesting and exciting ways in music theory, notably in the presentation of neo-Riemannian theory and other transformational theories involving frugal pitch-class changes (where “frugal” here is meant to include, but not exclusively refer to, parsimonious voice leading). An important visualization of this convergence is the neo-Riemannian Tonne , which ushered in a wave of thought about how music moves in time through a constructed (or generated) space. In many of these constructions, there is a substantive reliance on the relative evenness of a collection of pitches or pitch classes. I want to draw your a ention to three particular approaches, the last of which will be the focus of this article. [1.2] First, Jason Yust’s approach to discrete Fourier transform (DFT) phase space (Yust 2015b): the DFT, as discussed by David Lewin (2001) and, later, Ian Quinn (2006, 2007), can give us a relative numeric representation of evenness in any pc set. Broadly, it allows us to parse evenness along a clear continuum relative to a totally even distribution. Additionally, while two different whole tone collections would have the same magnitude of evenness as measured by the DFT, the phases of those collections would be in opposition. Yust, in DFT phase space, explores how phases from the DFT tell a story about harmonic relationships. Second, the Callender, Quinn, and Tymoczko (CQT) (2008) approach to voice-leading space: As with DFT phase space, the voice-leading space constructed by Callender, Quinn, and Tymoczko maintains evenness as a core principal, and distances between elements involve considerations of near-evenness and note-to-note proximity. Lastly, the approach to modeling iterated maximally even sets through Filtered Point-Symmetry: Filtered Point-Symmetry, or “FiPS,” is a transformational system built upon the manipulation of iterated maximally even sets over time. It is a geometric visualization of John Clough and Jack Douthe ’s (1991) function. [1.3] This article deeply examines the utility of FiPS-based models. Although some models overlap with those that can be expressed in a CQT voice-leading space, there are different theoretical implications. Notably, only FiPS-based models are able to dissociate harmonic proximity from minimal harmonic change. Also, FiPS-based models allow us to rigorously compare harmonic relationships using different scales, such that a neo-Riemannian transformation within a diatonic collection can be distinguished from the same-sounding transformation when octatonic collections are involved. This paper sets out to codify and streamline FiPS-based models as distinct from these other approaches to musical evenness, and presents new theories based on what can be described using FiPS-based models.(1) Maximally Even Sets and FiPS A familiar distribution [2.1.1] Though I presume knowledge of set theory, as well as a familiarity with some existing transformational theory, it is not my expectation that the reader has specialized training in mathematics, nor is acquainted with the publication that lays out the basic premises of Jack Douthe ’s (2008) theory of Filtered Point-Symmetry. Even if the reader is well-versed in this literature, it is worthwhile having the topics presented in a way that is tightly integrated with the manner in which we are going to explore the theory. Let us begin by laying a preliminary conceptual and geometric foundation. [2.1.2] A maximally even (ME) set is simply a distribution of a group of things over a limited amount of space, where the things are spread as evenly apart as possible. To visualize this, picture the familiar clock face. Each hour is numbered, and each number could be assigned to a chromatic pitch class. If you choose six notes from the chromatic scale, with the goal of maximal evenness, you would want to choose six notes each a whole step apart. On the clock face (here, abstracted into a collection of points around a circle), this is a selection of every other point (Example 1). [2.1.3] Such a selection is a totally even distribution, with each selected item separated from the next by an equal amount. The diatonic set is not totally even, but instead a maximally even distribution of seven notes over the chromatic scale (wri en 7→12). This distribution cannot be totally even, because seven shares no common factors with twelve. So it is as even as possible, with the two anomalous half steps placed as far apart from each other as they can be (Example 2a). A different visualization shows this selection of seven notes out of one chromatic segment on a piano keyboard (Example 2b). [2.1.4] Another familiar maximally even distribution is a triad within a diatonic key. Instead of a clock face with twelve parts, picture a clock face of only seven parts, where each of those parts represents one of the seven diatonic notes. Selecting three notes as far apart as possible (a maximally even distribution of 3→7), Example 3 shows the C+ triad as one possible result. The difference between 7→12 and 3→7 is subtle: In the former, the circle is divided into 12 equal parts, just like 12-note equal temperament equally divides the octave. Because of this, 7→12 gives us a ME distribution that is easily and accurately associated with chromatic space. In the la er, the circle is divided into 7 equal parts. A diatonic collection is not made of an equal division of the octave into 7 parts, though we can conceive of this evenly-spaced representation as a scale-step space. When the two independent diagrams are joined, the image of the scale-step space in 7→12 yields an explicit diatonic key, and the image of 3→7 yields an explicit triad within that key.(2) A triad, then, is a maximally even distribution over a diatonic subset, and to get this diatonic subset, we use the results of 7→12 to constrain the results of 3→7. This is more conventionally notated as 3→7→12, and is known as an iterated ME set (Example 4). J

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和弦接近度、简约性和滤波点对称性分析

滤波点对称 (FiPS) 是一种用于对最大偶集迭代集之间的关系进行建模的工具。可以通过使用 FiPS 对包含在特定标量上下文（例如 C 大调或 01 八度音集）中的和弦进行建模，并在 FiPS 配置空间中捕获这些关系来研究常见的音乐关系。在这项工作中，提出了许多 FiPS 配置空间；有些与新黎曼 Tonne 等通常引用的语音引导空间同构，而另一些则显示了以前未曾探索过的音调网络。引入了位移操作来编码配置空间的遍历，并提供了一个简短的音乐分析来展示该方法的一些好处。DOI: 10.30535/mto.25.2.3 Volume 25, Number 2, July 2019 版权所有 © 2019 Society for Music Theory [1. 1] 音阶理论和转换理论在音乐理论中以一些有趣和令人兴奋的方式融合，特别是在新黎曼理论和其他涉及节俭音阶变化的变革理论的介绍中（这里的“节俭”是指包括，但不完全是指，简约的声音领先）。这种融合的一个重要形象是新黎曼音调，它引发了关于音乐如何在构建（或生成）空间中随时间移动的思想浪潮。在许多这些构造中，实质上依赖于一组音高或音级的相对均匀性。我想请您关注三种特定的方法，最后一种将是本文的重点。[1.2] 首先，Jason Yust 的离散傅里叶变换 (DFT) 相空间方法 (Yust 2015b)：DFT，正如 David Lewin (2001) 和后来的 Ian Quinn (2006, 2007) 所讨论的，可以给我们一个相对数字表示任何电脑套装。从广义上讲，它允许我们沿着相对于完全均匀分布的清晰连续体来解析均匀度。此外，虽然两个不同的全音集合将具有由 DFT 测量的相同幅度的均匀度，但这些集合的相位将相反。Yust 在 DFT 相空间中探索了 DFT 中的相如何讲述有关谐波关系的故事。其次，Callender、Quinn 和 Tymoczko (CQT) (2008) 处理语音引导空间的方法：与 DFT 相位空间一样，Callender、Quinn 和 Tymoczko 构建的语音引导空间保持均匀性作为核心原则，元素之间的距离涉及到接近均匀性和音符间接近度的考虑。最后，通过滤波点对称对最大偶数集进行建模的方法：滤波点对称或“FiPS”是一种转换系统，它建立在随着时间的推移对最大偶数集进行操作的基础上。它是 John Clough 和 Jack Douthe (1991) 函数的几何可视化。[1.3] 本文深入探讨了基于 FiPS 的模型的实用性。尽管某些模型与可以在 CQT 语音引导空间中表达的模型重叠，但存在不同的理论含义。值得注意的是，只有基于 FiPS 的模型能够将谐波接近度与最小谐波变化分离。此外，基于 FiPS 的模型允许我们使用不同的尺度严格比较谐波关系，这样，当涉及八音集合时，可以将全音集合中的新黎曼变换与同音变换区分开来。本文着手对基于 FiPS 的模型进行编纂和简化，以区别于这些其他音乐均匀性方法，并基于可以使用基于 FiPS 的模型进行描述的内容提出新理论。(1) 最大偶数集和 FiPS 熟悉的分布 [ 2.1.1] 虽然我假设了解集合论，并且熟悉一些现有的转换理论，但我并不期望读者在数学方面有专门的训练，也不熟悉列出基本前提的出版物Jack Douthe (2008) 的滤波点对称理论。即使读者精通这些文献，以与我们将要探索理论的方式紧密结合的方式呈现主题是值得的。让我们首先奠定初步的概念和几何基础。[2.1.2] 最大偶数 (ME) 集只是一组事物在有限空间内的分布，其中事物尽可能均匀地分布。为了想象这一点，想象一下熟悉的钟面。每个小时都有编号，每个数字都可以分配给一个半音阶。如果您从半音阶中选择六个音符，以达到最大均匀度为目标，您会希望选择六个音符，每个音符相隔一整步。在钟面上（这里，抽象为围绕一个圆的一组点），这是每隔一个点的选择（示例 1）。[2.1.3] 这样的选择是完全均匀的分布，每个选定的项目与下一个等量分开。全音阶不是完全均匀的，而是在半音阶上最大程度地均匀分布七个音符（wri en 7→12）。这种分布不可能完全均匀，因为 7 与 12 没有公因子。所以它是尽可能均匀的，两个异常的半阶尽可能彼此远离（示例 2a）。不同的可视化显示了从钢琴键盘上的一个半音段中选择的七个音符（示例 2b）。[2.1.4] 另一个熟悉的最大均匀分布是全音调内的三和弦。不是有十二个部分的钟面，而是想象一个只有七个部分的钟面，其中每个部分代表七个全音音符之一。选择尽可能远的三个音符（3→7 的最大均匀分布），示例 3 将 C+ 三和弦显示为一种可能的结果。7→12 和 3→7 的区别很微妙：前者是把圆圈分成 12 个等份，就像 12 音符等律平分八度一样。正因为如此，7→12 为我们提供了一个 ME 分布，它可以轻松准确地与色空间相关联。在图层中，圆圈被分成 7 个相等的部分。全音阶集合不是由八度音程等分的 7 个部分组成，尽管我们可以将这种均匀间隔的表示视为尺度步长空间。当两个独立的图连接起来时，7→12 中的尺度步长空间的图像产生一个明确的全音调，而 3→7 的图像产生一个显式的三元组在那个键中。 (2) 一个三元组，那么，是一个全音子子集上的最大均匀分布，为了得到这个全音子子集，我们使用 7→12 的结果来约束 3→7 的结果。这更传统地表示为 3→7→12，并且被称为迭代 ME 集（示例 4）。J