Rhythm

State the modulation the way you would write it over the barline. Set the old tempo and what it counts (e.g. quarter = 120). Then state the equivalence: the old value on the left, the new value on the right — old dotted eighth = new quarter is the textbook case. Dots and tuplet ratios (entered as n:m, “n in the time of m”) can apply to either side. Choose what the new tempo should count and press Solve.

The result is exact. If the answer is not an integer the display shows the fraction (e.g. 1280/3), because writing ♩= 106.7 in a score is precisely the kind of rounding this tool exists to prevent. Use result as next start feeds the answer back in, so you can chain successive modulations and read the whole path in the log table.

The duration of every notated value is held as an exact fraction of a whole note. A dotted value multiplies by 3/2 (7/4 for double dots); a tuplet n:m multiplies by m/n. The solver computes new = old × (old beat / new beat) × (new value / old value) — note the direction: when a shorter old value takes over the new beat, the music gets faster. Check the canonical case: ♩=120, dotted eighth = new quarter, answer 160.

A full “rhythm workstation” with a notation view and MusicXML export was considered and rejected: a static site is the wrong home for an editing surface, and a modulation plan is a number you copy into your sketch. What gets used at the desk is the arithmetic — done exactly, including the tuplet cases that are error-prone late at night. The chain log covers the practice of long modulation paths where each local equivalence is simple but the cumulative ratio is not.

Cowell’s New Musical Resources (1930) had already proposed deriving tempo relationships from whole-number ratios; Carter turned the idea into a notational practice. Relevant repertoire: Elliott Carter, Cello Sonata (1948) and String Quartet No. 1 (1951), whose openings chain modulations much as the log table does; Nancarrow’s player-piano studies (Study No. 36 is a 17:18:19:20 canon); Kampela’s “micro-metric modulation” inside the beat; proportional tempo relations in Birtwistle and Adès.

把轉調照你會寫在小節線上方的樣子陳述出來。設定舊速度及其計拍單位 （例如四分音符 = 120）。然後陳述等價關係：左邊是舊值，右邊是新值—— 舊附點八分音符 = 新四分音符 即教科書式的範例。附點與連音比率（以 n:m 輸入，意為「在 m 的時間內奏 n 個」）可施於任一側。選定新速度的 計拍單位，按 Solve。

結果是精確的。若答案非整數，顯示會給出分數（例如 1280/3），因為在 樂譜上寫 ♩= 106.7 正是本工具存在以杜絕的那種取整。Use result as next start 會把答案再餵回去，於是你可以串接連續的轉調，並在記錄表中 讀出整條路徑。

每個記譜時值的長度都以全音符的精確分數保存。附點值乘以 3/2（複附點 為 7/4）；連音 n:m 乘以 m/n。求解器計算 新 = 舊 × (舊拍 / 新拍) × (新值 / 舊值)——注意方向：當較短的舊值 接管新拍時，音樂會變快。驗證標準範例：♩=120，附點八分音符 = 新四分 音符，答案 160。

一個附帶記譜檢視與 MusicXML 匯出的完整「節奏工作站」曾被考慮並否決： 靜態網站不是編輯介面該待的地方，而一份轉調方案不過是你抄進草稿的一個 數字。在書桌上真正會用到的是算術——精確地完成，包括那些深夜容易出錯的 連音情形。記錄表則涵蓋了長串轉調路徑的實務：其中每一步局部等價都很 簡單，累積比率卻不然。

Cowell 的 New Musical Resources（1930）早已提出由整數比率推導速度 關係的構想；Carter 則把這個構想化為一種記譜實踐。相關曲目：Elliott Carter，Cello Sonata（1948）與 String Quartet No. 1（1951），其 開頭如本記錄表一般串接轉調；Nancarrow 的自動鋼琴練習曲（第 36 號是 17:18:19:20 的卡農）；Kampela 在拍內的「微觀節拍轉調」；以及 Birtwistle 與 Adès 中的比例式速度關係。

The lab has three connected instruments: a coincidence calculator, a phase canon, and a multi-track metronome. All three share one premise — several strata of pulse heard against each other — at three levels of freedom: fixed integer ratios, a displacement free to sit anywhere on the continuum, and fully independent tempo streams.

Layers & coincidences. Enter the attack counts of up to four layers — 3, 4 is the familiar hemiola; 5, 7, 11 is already a texture. The linear grid shows each layer’s attacks across one cycle; the circle shows the same cycle bent round. Attacks that coincide turn red, with a red diamond on the time axis — lines are reserved for the playhead. Playback loops until Stop, with a playhead sweeping the grid and a radius turning on the circle; drag the linear grid while playing to move the playhead and the loop continues from there. Coincidences only plays nothing but the meeting points — the skeleton of the polyrhythm rather than its surface. Attack positions are exact fractions, so coincidence is integer arithmetic, never floating-point near-equality; the summary reports the smallest common grid a notator would need (for 5:7 already 35 divisions — usually the moment one decides to notate in two voices).

Canon & phase. State a pattern — note names for a pitch canon (E4 F#4 B4 C#5 …; the Piano Phase pattern is the default and a preset), or x/. steps for pure rhythm (Clapping Music is a preset) — and a second voice states it displaced. The displacement is the point: in locked offset mode it is a slider over the whole continuum from 0 to 1 cycle, not just the step grid, and it moves while the canon plays. At exact step multiples you hear Clapping Music’s rotations or a strict canon at the distance of n steps; between them, the in-between worlds that notation does not reach. In gradual phasing mode voice 2 instead runs at a slightly different tempo (ratio slider, also live), and the readout reports the lap time — how long until the voices realign, which is the formal span of a Reich phase section: with a 12-step pattern at ♩=72 and ratio 1.01, one lap of the pattern takes about a minute and a half, which is why those pieces breathe at the pace they do. In stepped phase mode, voice 2 instead jumps ahead by one pattern step at the start of each cycle, so the rotations are discrete rather than gradual. A transposition control turns the pitch canon into a canon at the interval. Pitched patterns play as sustained notes (_ extends a note across steps), and the displacement can be viewed two ways: as two rows, voice 2 sliding against voice 1 — the layout familiar from the Clapping Music videos — or as two rings, the cycle bent round; a red playhead sweeps either view during playback.

Multi-track metronome. Up to four click tracks. Each has its own quarter-note tempo, clicks on its own note value, and accents its own bar length — so it covers both independent tempi (Nancarrow, Carter’s simultaneous speeds) and one tempo articulated in conflicting values (a dotted-eighth track against a quarter track sounds the 3:4 of a metric modulation before you commit it to paper, in tandem with Tool No. 01). Changes apply live on the next scheduled click. All scheduling in this lab runs on the WebAudio clock with a look-ahead scheduler; nothing is timed from the UI thread, so the clicks stay sample-accurate while you drag the sliders.

Relevant repertoire: Reich, Piano Phase (1967), Clapping Music (1972), Drumming (1971); Nancarrow’s tempo canons (Studies 24, 36, 37 — canon by ratio rather than displacement, i.e. this lab’s phasing mode held permanently); the mensuration canons of Ockeghem’s Missa prolationum, the historical ancestor of the metronome panel’s one-tempo-many-values case; Ligeti’s études and Birtwistle’s pulse labyrinths for the coincidence panel; Messiaen’s superimposed rhythmic canons, codified in his Technique, for the canon panel; and the ensemble repertoires analyzed by Arom, whose cycle diagrams the circular view reproduces.

本實驗室有三件相連的樂器：一個重合計算器、一個相位卡農，以及一個多軌 節拍器。三者共享同一前提——數層脈衝相互對置而聽——並提供三個層級的 自由度：固定整數比、可落在連續區間任一處的位移，以及完全獨立的速度流。

層與重合。 輸入至多四層的觸發次數——3, 4 即熟悉的二對三； 5, 7, 11 已然是一片織體。線性格點顯示每一層在一個循環內的觸發；圓形 則把同一循環彎成一圈。重合的觸發轉為紅色，並在時間軸上標出紅色菱形—— 直線則保留給播放頭。播放會循環到按 Stop 為止，播放頭掃過格點、半徑在 圓上轉動；播放時拖動線性格點 可移動播放頭，循環即由該處續行。 Coincidences only 只奏出相遇之點——複節奏的骨架，而非其表面。觸發 位置是精確分數，因此重合是整數運算，絕非浮點數的近似相等；摘要會報告 記譜者所需的最小公共格（5:7 已需 35 等分——通常正是決定分成兩個聲部 記譜的時刻）。

卡農與相位。 陳述一個模式——音高卡農用音名（E4 F#4 B4 C#5 …； 預設並內建 Piano Phase 的模式），純節奏則用 x／. 的步格（內建 Clapping Music）——再由第二聲部位移地陳述它。位移正是重點：在 locked offset 模式下，它是一個遍及整個連續區間（0 到 1 個循環， 而不只是步格）的滑桿，且 在卡農播放時 移動。在精確的步數倍數處，你 會聽到 Clapping Music 的輪轉，或相距 n 步的嚴格卡農；在它們之間，則是 記譜觸及不到的中介世界。在 gradual phasing 模式下，第二聲部改以 略為不同的速度運行（比率滑桿，亦為即時），讀數會報告繞行一圈所需的 時間——即兩聲部重新對齊前的時長，也就是 Reich 相位段落的形式跨度： 以 ♩=72、比率 1.01 的 12 步模式而言，跑完一圈約需一分半，這正是那些 作品何以以那樣的步調呼吸。在 stepped phase 模式下，第二聲部則在 每個循環開始時往前跳一個模式步，因此輪轉是離散而非漸進的。一個移調 控制項可把音高卡農變為相距某音程的卡農。有音高的模式以持續音播放 （_ 把一個音延展跨越多步），位移可以兩種方式檢視：兩條 rows， 第二聲部對著第一聲部滑動——即 Clapping Music 影片中熟悉的版面——或 兩個 rings，把循環彎成一圈；播放時紅色播放頭掃過任一檢視。

多軌節拍器。 至多四條點擊軌。每條都有自己的四分音符速度、以自己的 時值點擊、並以自己的小節長度作重音——因此它既涵蓋獨立速度（Nancarrow、 Carter 的同時多速），也涵蓋同一速度以相衝突時值表達的情形（一條附點 八分軌對著一條四分軌，會在你把它落於紙面之前先奏出某個節拍轉調的 3:4，與第 1 號工具搭配使用）。更動會在下一個排定的點擊即時生效。本 實驗室所有排程都跑在 WebAudio 時鐘上，採用前瞻式排程器；沒有任何計時 來自 UI 執行緒，因此你拖動滑桿時點擊仍保持取樣精度。

相關曲目：Reich，Piano Phase（1967）、Clapping Music（1972）、 Drumming（1971）；Nancarrow 的速度卡農（第 24、36、37 號——以比率 而非位移構成卡農，亦即本實驗室相位模式的永久保持態）；Ockeghem Missa prolationum 的比例卡農，是節拍器面板「一速多值」情形的歷史 先祖；Ligeti 的練習曲與 Birtwistle 的脈衝迷宮對應重合面板；Messiaen 疊置的節奏卡農（其 Technique 已加以系統化）對應卡農面板；以及 Arom 所分析的合奏曲目，其循環圖正是圓形檢視所重現的。

Enter a subject with its rhythm: NOTE:duration tokens, where the durations are w h q e s (whole to sixteenth) with an optional dot, R is a rest, and a token without a duration repeats the previous one — so D4:e A4 F4 D4 is four eighths. The builder then lays out an exposition for two to four voices: each entry states the subject or the answer, transposed per the entry table, at the chosen octave. The preset menu loads a few worked expositions — the Art of Fugue principal subject and two from the Wohltemperiertes Klavier — including the C minor fugue’s tonal answer and regular countersubject, so the answer and countersubject fields fill in alongside the subject.

The answer defaults to the subject up a perfect fifth — the real answer, which is why “by fifths” is the default behavior. If your subject needs a tonal answer (most subjects that open on or leap to the dominant do), write the answer yourself in the same notation and the builder uses it verbatim; deciding where the adjustment falls is the composer’s first real decision in a fugue, and no tool should make it silently. A countersubject (entered as it should sound against entry 2) follows each voice’s own statement, shifted in parallel with the entry it accompanies.

Each entry also carries a rhythm transformation: state the material in augmentation (note values doubled, ×2 or ×4) or diminution (halved, ×½ or ×¼), the devices that stack a subject against itself at different speeds — the augmented entry runs underneath while quicker entries come and go above it (as in Art of Fugue Contrapunctus VII). Pitches are untouched; only the note values stretch or compress, so the statement lengthens or shortens while still entering on the grid.

The entry distance defaults to the subject’s full length — each voice waits its turn, the textbook exposition. Shorten it and the entries overlap: stretto, on a continuum down to half a beat. The piano roll shows each voice as a horizontal line of pitches (red = voice 1, then fading grays; dashed = countersubject; sparse dashes = free counterpoint not yet written), so overlap, register spacing, and the answer’s tessitura are visible before a note is engraved. Playback states the scaffold in sustained tones at the chosen tempo.

Below the roll, the builder reports parallel perfect fifths and octaves at attack onsets between the stated materials — the first mechanical check any counterpoint teacher applies, and deliberately the only one: voice-leading judgment between the onsets, dissonance treatment, and everything in the dashed regions remain yours. What the tool guarantees is the bookkeeping: where each voice is, in what transposition, against what.

Relevant repertoire: Bach’s Wohltemperiertes Klavier for every exposition layout this tool can produce, and Die Kunst der Fuge for strettos at shrinking distances (Contrapunctus V–VII); Beethoven’s Große Fuge; Hindemith’s Ludus Tonalis and Shostakovich’s 24 Preludes and Fugues as the modern systematic cycles; Bartók’s Music for Strings, Percussion and Celesta, whose first movement enters by fifths around the entire circle — the default transposition scheme of this builder taken to its limit.

輸入一個 帶節奏的主題：以 音符:時值 為記號，時值為 w h q e s （全音符到十六分音符）並可加附點，R 為休止，不帶時值的記號則沿用 前一個——因此 D4:e A4 F4 D4 是四個八分音符。建構器隨即鋪排出兩到 四聲部的呈示部：每次進入陳述主題或答題，依進入表移調，置於所選的 八度。preset 選單載入幾個已寫好的呈示部——Art of Fugue 的主要 主題，以及兩個取自 Wohltemperiertes Klavier 者——包含 C 小調賦格的 守調答題與固定對題，因此答題與對題欄會連同主題一併填入。

答題 預設為主題上行純五度——即 實 答題，這也是「依五度」為預設 行為的原因。若你的主題需要 守調 答題（多數開頭落在屬音、或躍向屬音 的主題都需要），請以相同記法自行寫出答題，建構器會原樣採用；判斷調整 落在何處，是作曲者在一首賦格中第一個真正的決定，任何工具都不該擅自 代勞。對題（以它應對著第二次進入發聲的樣子輸入）跟在每一聲部自己 的陳述之後，並與它所伴隨的進入平行移位。

每次進入還可帶一種 節奏 變形：以 增值（時值加倍，×2 或 ×4）或 減值（減半，×½ 或 ×¼）陳述素材，這些手法使主題以不同速度疊置於 自身之上——增值的進入在下方運行，較快的進入在其上來去（一如 Art of Fugue 第七對位）。音高不變；只有時值伸展或壓縮，因此陳述 在仍對齊格點進入的同時變長或變短。

進入間距 預設為主題的全長——每一聲部各候其時，即教科書式的呈示部。 縮短它，進入便會重疊：緊接段，可連續縮短至半拍。鋼琴捲簾把每一聲部 顯示為一條水平的音高線（紅 = 第一聲部，其後為漸淡的灰；虛線 = 對題； 稀疏虛線 = 尚未寫出的自由對位），於是重疊、音區間距與答題的音域，在 一個音符付梓之前便已可見。播放會以所選速度、用持續音陳述這副骨架。

捲簾下方，建構器會報告所陳述素材之間 在觸發起點上的平行純五度與 八度——任何對位老師最先施行的機械檢查，也是刻意保留的唯一一項： 起點之間的聲部進行判斷、不協和處理，以及虛線區域中的一切，都仍歸你。 本工具所保證的是簿記：每一聲部在何處、作何移調、對著什麼。

相關曲目：Bach 的 Wohltemperiertes Klavier 涵蓋本工具所能產生的 每一種呈示部佈局，Die Kunst der Fuge 則示範間距漸縮的緊接段（第 五至七對位）；Beethoven 的 Große Fuge；Hindemith 的 Ludus Tonalis 與 Shostakovich 的二十四首前奏與賦格，作為現代的系統性套曲；以及 Bartók 的 Music for Strings, Percussion and Celesta，其第一樂章 繞行整個五度圈依次進入——正是本建構器預設移調方案推至極致的樣子。