Acoustics

The converter takes whatever you have — a note name (F#3, Bb5), a frequency (440 or 440Hz), or a MIDI number — and returns the rest: frequency, fractional MIDI, the nearest 12-EDO pitch with its deviation in cents. The A4 reference is adjustable for period pitch (415, 430, 442…) and everything downstream respects it.

The just-intonation panel takes a ratio (3/2, 81/64, 7/4) above a reference note and reports its size in cents and its nearest tempered spelling; the dyad can be played. The EDO panel tabulates any equal division of the octave (2–96) against 12-EDO and plays the scale.

Ratios are kept as exact integer fractions; cents and Hz are floats — displayed rounded, computed at full precision. A numeric input under 130 without Hz is read as MIDI; anything else as frequency. Deviations use the convention that +14¢ means sharp of the named pitch.

Microtonal accidental suggestions (Johnston, Helmholtz–Ellis, Sagittal) from the original proposal were left out deliberately: accidental systems are notational policies, not facts about the pitch, and suggesting them without correct glyph rendering would produce confident-looking nonsense. The cents readout is system-neutral; any standard microtonal notation can be derived from it, and that choice should stay with the composer. What the tool does guarantee is the arithmetic: 3/2 = 701.955¢, every time.

Relevant repertoire and practice: Ben Johnston’s string quartets (JI ratios as primary material), Partch, Wyschnegradsky’s quarter-tone works, Haas (limited approximations, sixth-tone pianos), and the Helmholtz–Ellis school around Marc Sabat and Wolfgang von Schweinitz.

工具接受你手上任何一種輸入——音名（F#3、Bb5）、頻率（440 或 440Hz），或 MIDI 數字——並返回其餘各項：頻率、帶小數的 MIDI 值，以及最接近的十二平均律音高及其以音分計的偏差。A4 基準可調整以對應 歷史音高（415、430、442…），其後所有計算都依此基準。

純律面板接受相對於某參考音的一個比率（3/2、81/64、7/4），報告其 音分大小與最接近的平均律記法；該雙音可以播放。等分八度（EDO）面板可將 任意等分八度（2–96）與十二平均律對照列表，並播放該音階。

比率以精確的整數分數保存；音分與赫茲為浮點數——顯示時取整，計算時保持 完整精度。小於 130 且未標 Hz 的數字輸入視為 MIDI，其餘視為頻率。偏差 採用 +14¢ 表示高於所標音高的慣例。

原始提案中的微分音臨時記號建議（Johnston、Helmholtz–Ellis、Sagittal） 被刻意略去：記號系統是記譜上的約定，而非音高本身的事實；在沒有正確字形 渲染的情況下給出這些建議，只會產生看似可信的胡言。音分讀數與系統無關； 任何標準的微分音記法都能由它推導，而這個選擇應留給作曲者。本工具所保證 的是算術：3/2 = 701.955¢，每一次都如此。

相關曲目與實踐：Ben Johnston 的弦樂四重奏（以純律比率作為主要素材）、 Partch、Wyschnegradsky 的四分音作品、Haas（limited approximations， 六分音鋼琴），以及圍繞 Marc Sabat 與 Wolfgang von Schweinitz 的 Helmholtz–Ellis 學派。

Give a fundamental (note name or Hz) and a number of partials. The table lists each partial’s frequency, nearest tempered pitch, deviation in cents, and octave-reduced ratio. An inline meter on each row plots that deviation against the tempered pitch (centre line, ±20¢ ticks); bars turn red past 20¢ from 12-EDO — the 7th, 11th, 13th and their octaves — the ones that need microtonal notation or retuned instruments.

Check any subset of partials and Sustain selected to hold the stack as an additive drone: while it sounds, checking and unchecking partials adds and removes them live, and changing the fundamental or B retunes the sounding stack — it plays until you stop it. Hear series arpeggiates the whole table. Defaults select partials 4–7, the dominant-seventh-like cell from which much spectral harmony grows.

The inharmonicity coefficient B switches to a stiff-string model: fₙ = n·f₀·√(1+Bn²). Real piano strings have B roughly between 0.0002 (long bass strings) and 0.001+ (short treble strings); upper partials stretch sharp (Fletcher & Rossing, 1998). This is why pianos are stretch-tuned and why a “harmonic” spectrum on paper is not what a piano hands you.

Deviations are measured from the nearest 12-EDO pitch with A4 = 440. The standard checkpoints: 3rd partial +2¢, 5th −14¢, 7th −31¢, 11th −49¢. The proposal’s adjustable amplitude-decay model was fixed to a reasonable default instead, since it does not affect the table; spectral envelope matters for the roughness calculator (No. 05), where it is a parameter.

Relevant repertoire: Grisey, Partiels (1975) — the trombone E spectrum orchestrated; Murail, Gondwana; Radulescu; Saariaho’s harmony-timbre continuum; on the just-intonation side, La Monte Young’s The Well-Tuned Piano and Tenney’s Spectral CANON for CONLON Nancarrow.

給定一個基音（音名或赫茲）與分音數目。表格列出每個分音的頻率、最接近的 平均律音高、以音分計的偏差，以及約化至一個八度內的比率。每一列都有一個 行內刻度尺，將該偏差對照其平均律音高繪出（中線，±20¢ 刻度）；偏離十二 平均律超過 20¢ 的長條轉為紅色——第 7、11、13 分音及其各八度——正是 需要微分音記譜或重新調音樂器的那些。

勾選任意分音子集，再按 Sustain selected，把這個音堆作為加法式持續音 保持：發聲期間，勾選與取消分音會即時加入或移除它們，改變基音或 B 值會 重新調整正在發聲的音堆——它會一直播放到你停止為止。Hear series 將 整張表琶音奏出。預設勾選第 4–7 分音，即許多頻譜和聲由之生長的那個 「屬七般」的音團。

非諧性係數 B 會切換為剛性弦模型：fₙ = n·f₀·√(1+Bn²)。真實鋼琴弦的 B 值大致介於 0.0002（低音長弦）與 0.001 以上（高音短弦）之間；高次分音會 被拉高（Fletcher & Rossing, 1998）。這正是鋼琴採用「擴展調音」的原因， 也是何以紙面上的「諧」頻譜並非鋼琴實際給你的東西。

偏差以 A4 = 440 之下最接近的十二平均律音高為基準量測。標準的對照點： 第 3 分音 +2¢，第 5 分音 −14¢，第 7 分音 −31¢，第 11 分音 −49¢。提案 中可調的振幅衰減模型在此固定為一個合理的預設值，因為它不影響表格；頻譜 包絡對粗糙度計算器（第 5 號工具）才重要，在那裡它是一個參數。

相關曲目：Grisey，Partiels（1975）——將長號 E 的頻譜配器化；Murail， Gondwana；Radulescu；Saariaho 的和聲—音色連續體；在純律一側， La Monte Young 的 The Well-Tuned Piano 與 Tenney 的 Spectral CANON for CONLON Nancarrow。

The dissonance curve fixes a lower tone and sweeps a second tone through one octave, plotting modeled roughness. With six harmonic partials the valleys fall at 5/4, 4/3, 3/2, 5/3, 2/1. Reduce the partial count to 1 and the valleys vanish: pure sine pairs have almost no interval-specific consonance. That is the central point — sensory consonance is a property of timbre × interval, not of the interval alone (Sethares’s claim, and the reason this tool has spectrum controls at all).

The voicing comparison scores two chords (note names with octaves) under the same spectrum model and shows the pairwise contribution matrix for voicing A, locating which pair of notes produces the bite. Try C3 E3 G3 against C3 E4 G5: same set class, very different roughness — the low-register third is responsible, as orchestration manuals have always said.

The model is Plomp & Levelt’s (1965) critical-band roughness as parameterized by Sethares (1993; b₁ = 3.5, b₂ = 5.75, s = 0.24/(0.0207·fmin + 18.96)), summed over all partial pairs. Values are unitless; only rankings within one spectrum model are meaningful. This is a model of sensory roughness, not of musical dissonance — context, voice leading, and style are outside its competence.

The proposal also asked for a second (Vassilakis) model and register-sweep heatmaps. The Vassilakis refinement changes amplitude weighting, not rankings, at the precision relevant for compositional choice; and instead of a heatmap, changing the register in the voicing input answers the same question directly.

Relevant repertoire: Tenney, Critical Band; Saariaho’s sound/noise axis; Grisey’s and Murail’s orchestral voicings; Haas (in vain), where tempered and spectral tunings collide; Sethares’s own music, which composes scales to match synthetic timbres.

不協和曲線固定一個低音，讓第二個音掃過一個八度，繪出模型化的粗糙度。在 六個諧分音的情況下，谷底落在 5/4、4/3、3/2、5/3、2/1。把分音數減到 1， 谷底便消失：純正弦音對幾乎沒有任何與特定音程相關的協和性。這正是核心 所在——感官協和是音色 × 音程的屬性，而非單由音程決定（Sethares 的論點， 也是本工具之所以設有頻譜控制項的緣由）。

聲部配置比較會在同一頻譜模型下為兩個和弦（帶八度的音名）評分，並顯示 配置 A 的兩兩貢獻矩陣，找出是哪一對音造成了刺耳之處。試試以 C3 E3 G3 對比 C3 E4 G5：同一集合類，粗糙度卻大不相同——元兇是低音區的那個 三度，正如配器法教本一向所言。

此模型為 Plomp 與 Levelt（1965）的臨界頻帶粗糙度，依 Sethares（1993； b₁ = 3.5、b₂ = 5.75、s = 0.24/(0.0207·fmin + 18.96)）參數化，對所有 分音對求和。數值無單位；只有同一頻譜模型內的排序才有意義。這是感官 粗糙度的模型，而非音樂不協和的模型——脈絡、聲部進行與風格都不在它的 能力範圍內。

提案還要求第二種（Vassilakis）模型與音區掃描熱圖。就作曲抉擇相關的精度 而言，Vassilakis 的修正改變的是振幅權重，而非排序；而與其用熱圖，不如 直接改變聲部配置輸入中的音區，便能直接回答同一問題。

相關曲目：Tenney，Critical Band；Saariaho 的聲音／噪音軸；Grisey 與 Murail 的管弦聲部配置；Haas（in vain），其中平均律與頻譜調音相互 碰撞；以及 Sethares 自己的音樂，他為配合合成音色而譜寫音階。