Pitch

Enter pitch classes as integers (0 1 4 6, with T/E for 10/11) or as note names without octaves (C E G). The analyzer returns normal order, prime form, Forte class, interval vector, the Z-partner where one exists, and the complement’s class. The clockface draws the set as a polygon, so inversional symmetry and interval saturation are visible at a glance. The set can be played as a chord or as an ascending succession from middle C.

Prime forms follow Rahn’s packing-from-the-right algorithm, the one most software implements (music21 included). For five set classes (5-20 among them) Forte’s own table differs by one rotation; both spellings name the same class. Forte numbers for cardinalities 7–9 follow the complement-ordinal convention (the complement of 4-Z15 is 8-Z15).

The catalogue is not typed in from a book: all 4,096 subsets of the aggregate are enumerated and classified at load time. The hand-entered table contributes only Forte’s labels (verified against the published lists), and the test suite checks the class counts (12/29/38/50), the idempotence of every prime form, and the interval-vector identity of every Z-pair. Microtonal input is rejected rather than silently rounded — set-class theory is a 12-EDO instrument.

Subset/superset browsing was cut: the complete lattice is large, rarely consulted in practice, and better served by Straus’s tables. What remains is what gets used at the piano: what is this collection, what is it related to, does it have a twin.

Relevant repertoire and literature: Forte on Webern and Stravinsky; Perle on the Second Viennese School; Carter’s Harmony Book practice; Babbitt and Wuorinen as a matter of course; and any composer auditing their own harmonic vocabulary.

以整數輸入音級（0 1 4 6，並以 T／E 表 10／11），或以不帶八度的 音名輸入（C E G）。分析器會返回標準序、原型、Forte 類、音程向量、 Z 夥伴（若存在），以及補集的類別。時鐘面把集合畫成多邊形，於是倒影 對稱與音程飽和度一目了然。集合可作為和弦、或自中央 C 起的上行音列 播放。

原型採用 Rahn 由右側緊縮的算法，即多數軟體（含 music21）所實作者。對 五個集合類（其中包括 5-20）而言，Forte 自己的表相差一次旋轉；兩種 寫法指的是同一個類。基數 7–9 的 Forte 編號沿用補集序號的約定（4-Z15 的補集是 8-Z15）。

這份目錄並非自書本鍵入：全部 4,096 個全集子集在載入時即被枚舉並分類。 手動鍵入的表只貢獻 Forte 的標籤（已對照已出版的清單核驗），測試套件 則檢查各類數目（12/29/38/50）、每個原型的冪等性，以及每一對 Z 的 音程向量相同性。微分音輸入會被拒絕，而非默默取整——集合類理論是一件 十二平均律的樂器。

子集／母集瀏覽被略去：完整的格結構龐大、實務上鮮少查閱，且由 Straus 的表處理更為妥當。留下的是在鋼琴前真正會用到的東西：這是什麼集合、它 與什麼相關、它有沒有一個雙生。

相關曲目與文獻：Forte 論 Webern 與 Stravinsky；Perle 論第二維也納 樂派；Carter 在 Harmony Book 中的實踐；Babbitt 與 Wuorinen 自不 待言；以及任何在審視自身和聲詞彙的作曲者。

Enter twelve distinct pitch classes (integers with T/E, or note names), or load one of the historical rows. The calculator validates the row (eleven entries or a duplicated pc is an error, not a warning), prints the ordered interval string, flags all-interval rows, and builds the standard matrix: rows are P forms read left to right, columns are I forms read top down, retrogrades read the other way; the labels on all four edges follow this convention.

Select any form (P/I/R/RI at any index) to spell it in pcs and note names, and play it from middle C. The combinatoriality panel lists every form whose first hexachord completes the aggregate against P’s first hexachord — Babbitt’s criterion for clean aggregate polyphony.

Form labels use the starting pitch class: P4 begins on pc 4, I9 begins on pc 9 (Straus’s convention; analysts trained on zero-centered labels should translate accordingly — the matrix itself is identical). The interval string lists ordered pc intervals 1–11.

The proposal’s “play the matrix” feature was dropped: a matrix is a reference table, not a piece. Audition is per-form against a neutral pulse, which is how one actually checks a row’s melodic profile. The historical rows double as test fixtures — the Berg must report all-interval, the Webern must show its derived combinatorial structure, and the test suite holds the calculator to both.

Relevant repertoire: Schoenberg op. 25; Berg, Lyric Suite (all-interval row, included); Webern, Concerto op. 24 (derived from [014] trichords — see its interval string and combinatorial forms); Dallapiccola, Quaderno musicale di Annalibera; Babbitt, Three Compositions for Piano (1947), where hexachordal combinatoriality became a system; late Stravinsky and Boulez.

輸入十二個相異的音級（整數並用 T／E，或音名），或載入一條歷史 序列。計算器會驗證序列（只有十一個項、或音級重複是錯誤，而非警告）， 印出有序音程串，標示全音程序列，並建立標準矩陣：橫列由左至右讀為 P 形式，直行由上至下讀為 I 形式，逆行則反向讀；四邊的標籤皆依此約定。

選定任一形式（任意序號的 P/I/R/RI），即可用音級與音名拼出，並自中央 C 播放。組合性面板會列出所有「其首六音與 P 的首六音合起來補足全集」的 形式——即 Babbitt 對潔淨全集複音的判準。

形式標籤以起始音級命名：P4 始於音級 4，I9 始於音級 9（Straus 的約定； 習於以零為中心標記的分析者應據此換算——矩陣本身完全相同）。音程串 列出有序音級音程 1–11。

提案中的「播放矩陣」功能被捨棄：矩陣是參考表，不是樂曲。試聽以單一 形式對著中性脈衝進行，這才是真正檢查一條序列旋律輪廓的方式。歷史 序列兼作測試夾具——Berg 那條必須回報全音程，Webern 那條必須顯示其 衍生組合結構，測試套件對兩者都加以檢核。

相關曲目：Schoenberg 作品 25；Berg，Lyric Suite（全音程序列，已 收錄）；Webern，協奏曲作品 24（衍生自 [014] 三音組——見其音程串與 組合形式）；Dallapiccola，Quaderno musicale di Annalibera；Babbitt， Three Compositions for Piano（1947），其中六音組合性成為一套系統； 以及晚期 Stravinsky 與 Boulez。

Pick a mode and a transposition — the selector only offers the distinct ones (two for Mode 1, three for Mode 2, four for Mode 3, six for the rest; the counts are computed from the symmetry, not asserted). The clockface shows the mode as a polygon with its inversional axes dashed; the panel lists the notes and symmetry facts. The mode can be played ascending or as a sustained field.

The overlap checker takes your own pitches and reports what falls inside and outside the chosen transposition — the practical question (“can this passage live in Mode 3², and which notes would have to go?”) that one otherwise answers by squinting.

Modes are templates from C using Messiaen’s numbering (chapter XVI of the Technique); transposition Tₙ means the template shifted up n semitones. Mode 1 is the whole-tone scale (6-35), Mode 2 the octatonic (8-28) — the test suite verifies both identifications and all transposition counts. In pc space every one of these modes is inversionally symmetric: the modes are the pitch-side counterpart of nonretrogradable rhythm, which the axis display makes literal.

What is deliberately absent: the chord tables, added values, and nonretrogradable rhythm generators from the book. The rhythm tools (Nos. 01–02) and this one share a workbench, and a generator that stamps out palindromes adds nothing a pencil doesn’t. Software is genuinely better at the symmetry bookkeeping and the membership query; both are here.

Relevant repertoire: Messiaen throughout (Quatuor pour la fin du temps, Vingt regards, Turangalîla); Mode 2 back through Stravinsky to Rimsky-Korsakov and Scriabin; Takemitsu via Messiaen; Dutilleux at the edges; early Boulez (Notations); Murail and Grisey, who kept the symmetry thinking and changed the material.

選一種調式與一個移位——選擇器只提供相異者（調式 1 有兩個、調式 2 有 三個、調式 3 有四個、其餘各六個；這些數目由對稱性算出，而非斷言）。 時鐘面把調式畫成多邊形，其倒影軸以虛線表示；面板列出音與對稱性事實。 調式可上行播放，或作為持續音場播放。

重疊檢查器接受你自己的音，報告哪些落在所選移位之內、哪些在外——那個 原本要瞇著眼才答得出的實務問題（「這個樂段能否安身於調式 3²，又有 哪些音得割捨？」）。

調式是自 C 起、依梅湘編號（Technique 第十六章）的模板；移位 Tₙ 意為 模板上移 n 個半音。調式 1 是全音階（6-35），調式 2 是八聲音階 （8-28）——測試套件對這兩項辨識與所有移位數目都加以驗證。在音級空間 中，這些調式每一個都是倒影對稱的：它們是不可逆行節奏在音高一側的 對應物，軸線顯示把這點化為具體。

刻意缺席的是：書中的和弦表、附加時值，以及不可逆行節奏的產生器。節奏 工具（第 1–2 號）與本工具共用一張工作檯，而一個只會印出迴文的產生器 並不比鉛筆多給什麼。在對稱性簿記與成員查詢上，軟體確實更為擅長；兩者 都在此。

相關曲目：通篇的梅湘（Quatuor pour la fin du temps、Vingt regards、 Turangalîla）；調式 2 上溯經 Stravinsky 至 Rimsky-Korsakov 與 Scriabin；經由梅湘的 Takemitsu；邊緣處的 Dutilleux；早期 Boulez （Notations）；以及 Murail 與 Grisey，他們保留了對稱性的思路而更換了 素材。