Motif

Load a series of notes with octaves (C4 D4 F4 E4 G4). Then apply operations in any order: transposition by semitones, inversion around an explicit axis note, retrograde, rotation. Every step is appended to the chain readout (T+2 → I(C4) → R), and Undo pops one step — the chain is the document. The graph shows the original contour dashed and the current result in red; both can be played at a neutral pulse.

Operations act on absolute pitch (MIDI), not pitch classes: inversion around C4 means registral mirroring through middle C, the strict dodecaphonic sense. For pc-space reflection — register preserved, only chroma flipped — use Tool No. 12, which exists precisely because conflating the two domains is the classic error. Double inversion around the same axis restores the original; the test suite holds the operations to their group properties.

Grander operations from the proposal — contour-preserving remaps, spacing normalization, anchor-note preservation — are each a composite of the four primitives plus judgment, and judgment should stay with the composer; a tool that “normalizes spacing” makes compositional decisions silently. The provenance chain, by contrast, is pure tooling value: undo, reproducibility, and a sketch diary at once. This is also the module the other tools build on — the reflection and serial operations are its special cases.

Relevant repertoire: the contrapuntal canon from Bach (Musical Offering) through Webern’s mirror forms; Hindemith, Ludus Tonalis (postlude = prelude inverted and retrograded); Dallapiccola and Berio; anyone writing canons by inversion at the fifth still does exactly this arithmetic.

載入一串帶八度的音符（C4 D4 F4 E4 G4）。然後以任意順序施加操作：以 半音移位、繞某明確軸音倒影、逆行、旋轉。每一步都會附加到操作鏈讀數上 （T+2 → I(C4) → R），Undo 則彈出一步——操作鏈即是文件。圖以虛線 顯示原型輪廓、以紅線顯示當前結果；兩者都能以中性脈衝播放。

操作作用於絕對音高（MIDI），而非音級：繞 C4 倒影意指以中央 C 為軸的 音區鏡射，即嚴格十二音的意義。若要在音級空間反射——保留音區、只翻轉 音名——請用第 12 號工具，它的存在正因為混淆這兩個範疇是典型的錯誤。 繞同一軸兩次倒影會還原原型；測試套件對這些操作的群性質加以檢核。

提案中更宏大的操作——保輪廓的重映射、音距正規化、錨音保留——每一個都是 那四種基本操作加上判斷的合成，而判斷應留給作曲者；一個會「正規化音距」 的工具是在默默地替你做作曲決定。相對地，這條來源鏈是純粹的工具價值： 復原、可重現性，與一本草稿日記，三者兼具。這也是其他工具所倚賴的模組 ——反射與序列操作都是它的特例。

相關曲目：自 Bach（Musical Offering）到 Webern 鏡像形式的對位卡農； Hindemith，Ludus Tonalis（終曲 = 前奏的倒影兼逆行）；Dallapiccola 與 Berio；任何以五度倒影寫卡農的人，做的仍正是這套算術。

Choose a starting triad and type a chain — letters P, L, R, S, N in any order (PLR, LPLPLP…). The table shows every intermediate triad, its pitch classes, and the common tones retained from the previous step, which is the point of the theory: these operations move triads while holding the maximum in place. Hexatonic cycle runs the alternating P–L cycle from your triad (six chords, back to the start). Playback uses nearest-voice-leading realization — common tones stay put in the audio, semitone moves sound as semitones.

Conventions: P exchanges C↔c (parallel); R sends C→a, c→E♭ (relative); L sends C→e, c→A♭ (leittonwechsel); S is the slide C→c♯ (shared third); N the nebenverwandt C→f. The domain is exactly the 24 consonant triads, and the tool rejects anything else — that restriction is the theory, not a limitation: the parsimonious behavior of L, P, and R is a special algebraic fact about how the major/minor triad sits inside the chromatic 12, and extending the buttons to seventh chords would require a different theory. Each of P, L, R, S is an involution — applied twice it returns home — and the tests enforce this.

A Tonnetz/graph visualization was considered and set aside: the step table with common-tone accounting contains more analytic information than a lattice rendering of the same six chords.

Relevant repertoire and literature: Cohn on Schubert; Wagner (the Tarnhelm progression), Franck, and late-Romantic chromaticism generally; in contemporary pan-triadic writing, Adès (Asyla, slow movement), John Adams’s harmonic drift, and much post-tonal film scoring — the hexatonic cycle is the standard map of triadic motion without functional syntax.

選一個起始三和弦，鍵入一條鏈——P、L、R、S、N 各字母任意順序（PLR、 LPLPLP…）。表格顯示每一個中間三和弦、其音級，以及自前一步保留的 共同音，這正是該理論的要點：這些操作在移動三和弦的同時，把最多的音 保持原位。Hexatonic cycle 自你的三和弦運行 P–L 交替循環（六個 和弦，回到起點）。播放採用最近聲部進行的實現——共同音在音響中留在 原位，半音的移動就聽成半音。

約定：P 交換 C↔c（同主音）；R 把 C→a、c→E♭（關係調）；L 把 C→e、 c→A♭（導音交換，leittonwechsel）；S 是滑移 C→c♯（共享三音）；N 是 nebenverwandt，C→f。其論域恰為 24 個協和三和弦，工具會拒絕其餘 一切——這個限制就是理論本身，而非侷限：L、P、R 的簡約行為，是大／小 三和弦如何嵌入半音十二之中的一項特殊代數事實，把這些按鈕擴展到七和弦 需要一套不同的理論。P、L、R、S 各自都是對合——施加兩次便回到原處—— 測試對此加以強制。

一個 Tonnetz／圖視覺化曾被考慮而擱置：帶共同音計帳的步驟表，比同樣 那六個和弦的格狀繪製包含更多分析資訊。

相關曲目與文獻：Cohn 論 Schubert；Wagner（隱身盔進行）、Franck，以及 晚期浪漫派的半音化整體；在當代泛三和弦寫作中，有 Adès（Asyla，慢 樂章）、John Adams 的和聲漂移，以及大量後調性電影配樂——六音循環是 無功能語法之三和弦運動的標準地圖。

Enter material as note names with octaves. Choose the domain: pitch classes reflects chroma and keeps each note near its original register; absolute pitch mirrors the staff upside-down around an explicit axis note. In pc mode, either pick the “negative harmony in key of X” preset — for C this is the axis between E♭ and E (sum 7), sending C→G, D→F, E→E♭, the mapping popularized from Levy’s book — or set any axis sum 0–11 manually. The chromatic mapping table shows the whole involution at once; fixed points (even sums only) are highlighted.

Reflection in pc space is the standard inversion operator I_s: pc ↦ s − pc (mod 12). Even s fixes two pitch classes (s/2 and s/2+6); odd s fixes none — the table makes this visible. Double reflection is always identity; the tests enforce it in both domains.

On the skepticism the source proposal already voiced: “negative harmony” as an internet phenomenon is a single inversion operator under a new name. The substantive content is Levy’s harmonic dualism — which concerns functional meaning under reflection, not just pitch mapping — and the general I_s operator, which post-tonal theory has owned for a century. So the engine is general (twelve axes, two domains) and the famous mapping is a labeled preset, no more privileged than I₀. For the dualist theory itself, read Levy and Harrison; this tool supplies the arithmetic and the audition.

Relevant repertoire and literature: Riemann’s dualism and its rehabilitation in Harrison (1994); Bartók’s axis thinking in Lendvai’s account; strict-inversion practice from Bach’s Musical Offering table canon to Webern’s mirror canons; recently, Steve Coleman and Jacob Collier put the C-axis preset into wide circulation.

以帶八度的音名輸入素材。選擇論域：音級會反射音名並使每個音留在接近 原本的音區；絕對音高則繞某明確軸音把譜表上下鏡射。在音級模式下，可選 「X 調的負和聲」預設——對 C 而言，軸在 E♭ 與 E 之間（和為 7），把 C→G、D→F、E→E♭，即由 Levy 一書推廣開來的映射——或手動設定任意 和為 0–11 的軸。半音映射表一次顯示整個對合；固定點（僅偶數和才有） 會被標亮。

音級空間的反射即標準倒影算子 I_s：pc ↦ s − pc（mod 12）。偶數 s 固定 兩個音級（s/2 與 s/2+6）；奇數 s 一個也不固定——表格把這點顯示出來。 兩次反射恆為恆等；測試在兩個論域都加以強制。

關於原始提案早已表達的懷疑：作為網路現象的「負和聲」，不過是同一個 倒影算子換了個新名字。有實質內容的是 Levy 的和聲二元論——它關乎反射 之下的功能意義，而不只是音高映射——以及一般的 I_s 算子，後者後調性 理論已掌握逾一世紀。因此引擎是通用的（十二個軸、兩個論域），而那個 著名的映射只是一個貼了標籤的預設，並不比 I₀ 更受偏待。至於二元論本身， 請讀 Levy 與 Harrison；本工具提供的是算術與試聽。

相關曲目與文獻：Riemann 的二元論及其在 Harrison（1994）中的復權； Lendvai 對 Bartók 軸心思維的論述；自 Bach Musical Offering 表式 卡農到 Webern 鏡像卡農的嚴格倒影實踐；晚近則有 Steve Coleman 與 Jacob Collier 把 C 軸預設帶入廣泛流傳。