审阅范围 · Review scope
公开来源文字与原创综合已审阅。请保留正文中的出处与限制；不代表全库最终验收、实际演奏或学习效果验证。
Public-source text and original synthesis reviewed. Keep the evidence and limitations stated in the text in view. This is not final corpus acceptance or validation of performance or learning outcomes.
公開來源第十四批：構造、動機與音律 / Public Batch 14: Patterns, Motifs, and Temperament
編製 / Prepared: 2026-10-22T15:14:47.754960+00:00
審閱六篇指定公開原作者文章的完整正文；保留 ID、網址與原標題。四張原聲鋼琴小卡、兩張紙上音律卡；另有一篇僅局部閱讀的技術補充。
Review the full bodies of six assigned public original-author articles, preserving IDs, URLs, and exact titles. Four acoustic-piano cards and two paper temperament cards; one technical supplement is read only in part.
覆蓋與界線 / Coverage and Boundaries
指定六篇、完整正文六篇；影音、圖像、工具、連結頁、實際演奏驗證皆零；另讀技術補充一篇的局部。
Six assigned sources, six full article bodies; zero audiovisual, image, tool, linked-page, or performance verifications; one technical supplement read in part.
原文是作者本人的教學，不是所有史學、風格或成效主張的一手研究。此批只確認文字；音訊、影片、譜圖、連結 PDF、工具及實際演奏都未驗證。
These are the author’s own lessons, not primary research establishing every historical, stylistic, or efficacy claim. This batch verifies text only; audio, video, score images, linked PDFs, tools, and actual performance are unverified.
英文標題是編輯翻譯；行號定位本次擷取，不是永久段落號。來源短述中英共用二百單位上限，含標題、用途、限制、證據與覆蓋；中文字元與英數詞皆計入，正文直接引用為零。
English titles are editorial translations; line numbers locate this extraction, not permanent paragraphs. Bilingual source capsules share a 200-unit limit including titles, role, caveats, evidence, and coverage; CJK characters and alphanumeric tokens count, with zero verbatim body quotations.
譜例、步驟、預測問題及紙上對照為獨立編輯設計，不轉抄原文旋律或完整樂譜，也不只是移調原作者例子。共同樂理概念不代表已取得教材重製授權。
Examples, procedures, prediction questions, and paper comparisons are independent editorial designs, not reproductions of article melodies or full scores, nor merely transposed author examples. Shared theory concepts do not imply permission to reproduce source teaching materials.
本批沒有重用第六批四篇補充來源的解說文字或新增支持主張；保留其既有預算，不重設額度。新增補充有自己的合併中英預算。
This batch reuses neither explanatory text nor new supported claims from the four batch-6 supplements; their prior budgets remain intact, not reset. The new supplement has its own combined bilingual budget.
零起點入口 / Zero-Start Entry
中央 C 寫 C4；C3 低一個八度，C5 高一個八度。C 是兩個黑鍵左邊的白鍵；白鍵依序 C D E F G A B。♯ 往右最近一鍵，♭ 往左最近一鍵，黑白都算；E–F、B–C 也只差一鍵。先找各卡列出的音，不假裝看譜即會彈。
Middle C is C4; C3 is one octave below and C5 one octave above. C is the white key just left of a two-black-key group; white keys proceed C D E F G A B. A sharp moves to the nearest key right, a flat left, counting black and white keys; E–F and B–C are also adjacent. Locate each card’s notes first rather than assuming reading equals playing.
鋼琴卡用 4/4，每小節四個四分音符拍。數「1 & 2 & 3 & 4 &」，& 在兩拍正中；1.5=1&，2.5=2&。表中拍長 0.5 是半拍、1 是一拍、2 是兩拍；明列休止時要放開，不是延續前音。
Piano cards use 4/4, four quarter-note beats per bar. Count “1 & 2 & 3 & 4 &”; & lies halfway between beats, so 1.5=1& and 2.5=2&. A table duration of 0.5 is half a beat, 1 one beat, and 2 two beats. An explicit rest means release, not sustain.
每手分開列事件；某手的發聲或休止從起拍延續指定拍長，到下一列才換。方括號內的音同時按，單音列只按一鍵。所有小節均列完整，沒有未提供的旋律或伴奏。
Events are listed separately for each hand. Each sound or rest lasts its stated duration from its onset until the next event. Notes in brackets sound together; a one-note event uses one key. Every bar is fully specified, with no missing melody or accompaniment.
先在紙上算拍、在鍵盤指音；第一至四卡每次只選一張。能無痛找到音才加入節拍；先分手，再試合手。第五、六卡可完全不碰琴，不需要認譜或開根號心算。
Count on paper and point to keys first; choose only one of cards 1–4 at a time. Add pulse only when notes can be found without pain; separate hands before combining. Cards 5–6 need no piano, notation reading, or mental root calculation.
初學詞彙 / Beginner Glossary
半音：此處指十二鍵排列的相鄰鍵；全音是兩個半音。卡二用鍵位模型，不量測真琴音分。
Semitone: here, neighboring positions in a twelve-key layout; a whole tone is two semitones. Card 2 uses a key-position model, not measured piano cents.
音高類別忽略八度，例如 C3、C4、C5 同屬 C 類。模 12 是把超過 11 的格號減 12，繞回 0–11；例如 10+3=13，繞回格 1。
Pitch class ignores octave: C3, C4, and C5 belong to C. Modulo 12 wraps slot numbers back into 0–11 by subtracting 12; for example, 10+3=13 wraps to slot 1.
和弦性質：例如大三和弦的根音以上 0、4、7 半音。功能是和弦在上下文中的作用，不能只由同一種構造決定。
Chord quality: for example, a major triad has offsets 0, 4, and 7 semitones above its root. Function concerns contextual harmonic behavior and is not determined by shared structure alone.
根音是和弦命名與間隔的參照，不一定是最低音。頻率是每秒振動次數，單位 Hz；紙上比值不需要指定一個實際 Hz 起音。
A root is the reference for chord naming and intervals, not necessarily the lowest note. Frequency counts vibrations per second in Hz; the paper ratios need no actual starting frequency in Hz.
p 輕聲，mp 中弱；兩者只是相對力度，沒有分貝要求。指定較突出不等於用力敲；保持舒適。
p means soft and mp moderately soft; these are relative dynamics, not decibel requirements. A more prominent part does not require striking forcefully; stay comfortable.
頻率比是相除，不是 Hz 差。音分用 c=1200 log₂(r)，八度比 2:1 是 1200 音分；本批紙算都假設理想、精確的八度。
A frequency ratio is a quotient, not a difference in Hz. Cents use c=1200 log₂(r); a 2:1 octave is 1200 cents. All paper calculations here assume ideal exact octaves.
編輯護欄 / Editorial Guardrails
固定的是構造，不是調性 / Structure, not tonality, is fixed
卡一的移動會保留根音相對間隔，但實際音集合改變；同樣都是大和弦，不代表同調、同功能、同一琴鍵形狀或不可用其他理論分析。本卡選四個連續同質和弦作比較，不制定唯一通用命名門檻。
Card 1 preserves root-relative intervals while changing actual pitch collections. All-major chords need not share key, function, keyboard shape, or immunity to other analysis. This card chooses four successive same-quality chords for comparison, without imposing a universal naming threshold.
音集合、拼寫與適配不同 / Collection, spelling, and fit differ
卡二的相同集合只指十二音高類別；同起音的半全與全半不同。等音異名可指同鍵，但拼寫要依聲部與和聲角色決定；本卡選一種易讀鍵名，不把它當所有樂句的標準。未指定和弦時不宣稱整個音階可任意套在任何屬和弦或減和弦。
Card 2’s identical collections concern twelve pitch classes only; half–whole and whole–half on the same starting note differ. Enharmonic names may share a key, but spelling depends on voice-leading and harmonic role. This card chooses readable key labels, not universal phrase spelling. Without a specified chord, no blanket fit over every dominant or diminished chord is asserted.
風格不是比例測驗 / Style is not a ratio test
卡三只借一種伴奏分工作入門協調練習，不代表全部 Bossa Nova，更不代表所有巴西或拉丁美洲音樂。正反拍各半、延伸和弦多或省五音，都不是風格充分或必要條件。不能用未聽的示範背書「已道地」。
Card 3 uses one accompaniment division as a coordination scaffold, not all Bossa Nova and certainly not all Brazilian or Latin American music. Equal on/off-beat counts, many extensions, or omitted fifths are neither sufficient nor necessary style tests. Unheard demonstrations cannot certify authenticity.
變形規則要寫清 / State transformation rules
卡四把「精確音程反向」定義為每個有向半音數換正負；只上下反轉輪廓則不必保持大小。倒過來讀音符順序是另一操作，不能一概叫同一件事；本卡使用獨立編寫的四音素材。
Card 4 defines exact interval inversion by reversing every signed semitone displacement; contour reversal alone need not retain sizes. Reading note order backwards is a different operation, not universally the same thing. This card uses an independently written four-note seed.
紙上模型不是歷史或調音指示 / Paper models are not history or tuning instructions
原文人物對話、發明動機、優劣評語與年代保留為作者敘事，未獨立認證史學時間線或人人相同聽感。紙算只核對所列限制的數學結果；不調整、測量或診斷使用者鋼琴，也不把某一理想模型當真琴的精確現況。
The source’s dialogues, inventor motives, value judgments, and dates remain the author’s narrative; no historical timeline or universal listening preference is independently certified. Paper work checks mathematical consequences under stated constraints, without adjusting, measuring, or diagnosing the user’s piano or equating an ideal model with its actual tuning.
原創共用流程 / Original Shared Workflow
以下六卡是原創、尚未施測的設計。卡一至四可用一架原聲鋼琴；卡五六只用紙筆。沒有要求唱歌、麥克風、錄音、MIDI、付費軟體或伴奏音軌。
The six cards are original, untested designs. Cards 1–4 use one acoustic piano; cards 5–6 use paper and pencil only. No singing, microphone, recording, MIDI, paid software, or backing track is required.
主表同手同時最大跨度為完全五度；不固定指法、不強拉手。有痛就停；可由教師完整演奏，學習者只指音數拍，或改逐音核對並明記沒有測試同時和聲。
The main grids require no simultaneous same-hand span larger than a perfect fifth. No fixed fingering or forced stretch; stop pain. A teacher may play all parts while the learner points and counts, or use successive-note checking while recording that simultaneous harmony was not tested.
在目標首次發聲或揭開計算結果前先寫預測，可寫「不確定」。預備示範不要先彈目標。若已看答案或聽過就記「有提示」；看配方推算不是純聽覺測驗。全部印出的 A/B 及答案都已公開，不是保留遷移題。
Write a prediction, including “uncertain,” before the first target sound or numerical reveal. Preparatory demonstrations should not sound the target first. If an answer or sound was already encountered, mark prompted. Recipe-based reasoning is not a purely auditory test. All printed A/B variants and answers are disclosed practice, not held-out transfer.
以下速度與力度是本卡編輯標記，不是來源的原速或表情指示。四分音符=48 表示每分鐘 48 個四分音符拍（BPM）；下列加減 8 或 4 都指 BPM。每張鋼琴卡列出確切入門速度；可先用該速度再慢 8 的速度，但把實用數字記下，A/B 用同速。下一次先原速未預練重測；只有連續兩次無停頓、拍位可控而且舒適，才可加 4。這是進度規則，不是成效證據。
The tempos and dynamics below are editorial markings for these cards, not source performance directions. Quarter=48 means 48 quarter-note beats per minute (BPM); all changes by 8 or 4 below are BPM. Each piano card gives an exact starting tempo; start 8 slower if needed, recording the actual number and keeping A/B equal. Next session retest at that tempo before rehearsal. Increase by 4 only after two uninterrupted, pulse-controlled, comfortable trials. This is a progression rule, not efficacy evidence.
奏後先核對音名、起拍、休止及指定力度，再只修正一項。偏好另記，不能把「較喜歡」當作和聲、風格或內聽已證實。若要遷移，由教師另寫未公開新題並保留至預測時；本批未做。
After playing, check notes, onsets, rests, and dynamics, then correct just one item. Log preference separately; liking one version does not establish harmony, style, or inner hearing. Genuine transfer requires a new teacher-written item withheld until prediction; none was tested here.
記錄欄位 / Log Fields
卡號／版本；已見已聽；實際速度；分手、合手或紙上；舒適度。
Card/version; prior sight/sound; actual tempo; separate hands, together, or paper; comfort.
發聲或揭答前預測；觀察／算出結果；一次修正；偏好與不確定。
Pre-sound or pre-reveal prediction; observed/computed result; one correction; preference and uncertainty.
下次未預練重測；保留遷移未測；不填虛構得分或改善百分比。
Next-session unrehearsed retest; held-out transfer untested; no invented score or improvement percentage.
不調琴、不拆琴、不碰電線、不開大聲或做聽力／拍頻實驗。紙上音律卡沒有播放要求；數字差異不承諾人人可聽出。
No retuning, disassembly, electrical contact, loud playback, or hearing/beat experiment. The paper temperament cards require no playback; numeric differences do not promise audibility for every listener.
1. 一起來玩「恆定結構」（Constant Structure）！ / Playing with Constant Structure
來源 / Source: 一起來玩「恆定結構」（Constant Structure）！ · WEB-301a22a0da68
頁面日期 / Page date: 2025-09-08; 頁面字數標記 / length label: 1766; 存取 / accessed: 2026-10-22.
狀態 / Status: indexed_only → reviewed_public_article_text_only; full_public_article.
文字範圍 / Text coverage: 30–106; 108–119; 標題日期 / title/date: 25–28; 總行 / total lines: 120; web_open.
完整正文已讀；圖與影音未核。
Full body read; images and audiovisual media unreviewed.
媒體旗標 / Media flags: 完整公開正文 true；文件說明、聲音、影片、圖片、連結頁與互動 false。
Full public article text true; documentation, audio, video, images, linked pages, and interactive execution false.
來源重點 / Source Insight
作者用連續同質和弦，替代先定調的配和弦思路。
The author uses successive same-quality chords instead of choosing a key first.
知識庫用途 / Knowledge-Base Role
和聲構造比較。
Compare harmonic structures.
證據種類 / Evidence Type
原作者教學文字。
Author’s instructional text.
來源限制 / Source Caveat
同質不等於同調或同功能；聲響形容屬作者感受。
Shared quality is not shared key or function; sonic descriptions are the author’s impressions.
定位 / Anchor: 53–106 — 正文依據 / Body evidence
合併中英來源預算 / Combined bilingual source-derived budget: 128/200, 偏好上限 / preferred ceiling 170; 正文直接引用 / verbatim body quotes: 0.
計入 / Counted: title_zh, title_en_editorial, source_claim_summary, content_role, evidence_type, limits, coverage.note, evidence_anchors.label.
原創卡 / Original Card: 四張相同構造的和弦卡 / Four chords with one structure
先備 / Prerequisites
先學根音加 4、7 半音的大三和弦。先用不在目標內的 G3–B3–D4 指鍵示範，再逐個找 C、E、F、D 大三和弦；不先奏目標。
First learn a major triad as root plus 4 and 7 semitones. Point through G3–B3–D4, outside the target, then locate C, E, F, and D major; do not sound the target first.
完整配方 / Complete Recipe
A 四小節 C→E→F→D 大三和弦：左手單一根音，右手三音與五音的雙音，最高音就是完整旋律 G4–B4–C5–A4，沒有額外聲部。各小節第 1 拍同時按住三拍，第 4 拍兩手休止。B 只把第二小節 G♯4 換成 G4，成為 E 小三和弦，其餘相同。
A has four bars of C→E→F→D major triads: a single LH root and RH third/fifth dyad. The top notes G4–B4–C5–A4 are the complete melody, with no extra part. Both hands attack on beat 1, hold three beats, and rest on beat 4. B changes only bar 2’s G♯4 to G4, producing E minor; everything else stays fixed.
速度、節奏與力度 / Tempo, Rhythm, and Dynamics
4/4，四分音符=48；全程左手 p、右手 mp，不漸快、不加重最後和弦；無踏板。每手每版 16 拍，含四拍休止。右手同時最大大三度，音域 C3–C5。
4/4, quarter=48; LH p and RH mp throughout, without acceleration or a final-chord accent; no pedal. Each hand has 16 beats per version, including four resting beats. Maximum RH simultaneous span is a major third; total range C3–C5.
首次發聲或揭答前 / Before First Sound or Reveal
在首次 A、B 發聲前分別寫：哪版四個和弦都保持 0、4、7？哪一小節會改變構造？不要先判斷哪版比較好聽。
Before first sounding A and B, write which version retains 0,4,7 for all four chords and which bar changes structure. Do not predeclare the better-sounding version.
版本 / Version A — 4 小節 / bars
左手 / LH
| 小節 Bar | 起拍 Onset | 拍長 Beats | 音 Notes |
|---|---|---|---|
| 1 | 1 | 3 | [C3] |
| 1 | 4 | 1 | 休止 / Rest |
| 2 | 1 | 3 | [E3] |
| 2 | 4 | 1 | 休止 / Rest |
| 3 | 1 | 3 | [F3] |
| 3 | 4 | 1 | 休止 / Rest |
| 4 | 1 | 3 | [D3] |
| 4 | 4 | 1 | 休止 / Rest |
右手 / RH
| 小節 Bar | 起拍 Onset | 拍長 Beats | 音 Notes |
|---|---|---|---|
| 1 | 1 | 3 | [E4 G4] |
| 1 | 4 | 1 | 休止 / Rest |
| 2 | 1 | 3 | [G♯4 B4] |
| 2 | 4 | 1 | 休止 / Rest |
| 3 | 1 | 3 | [A4 C5] |
| 3 | 4 | 1 | 休止 / Rest |
| 4 | 1 | 3 | [F♯4 A4] |
| 4 | 4 | 1 | 休止 / Rest |
版本 / Version B — 4 小節 / bars
左手 / LH
| 小節 Bar | 起拍 Onset | 拍長 Beats | 音 Notes |
|---|---|---|---|
| 1 | 1 | 3 | [C3] |
| 1 | 4 | 1 | 休止 / Rest |
| 2 | 1 | 3 | [E3] |
| 2 | 4 | 1 | 休止 / Rest |
| 3 | 1 | 3 | [F3] |
| 3 | 4 | 1 | 休止 / Rest |
| 4 | 1 | 3 | [D3] |
| 4 | 4 | 1 | 休止 / Rest |
右手 / RH
| 小節 Bar | 起拍 Onset | 拍長 Beats | 音 Notes |
|---|---|---|---|
| 1 | 1 | 3 | [E4 G4] |
| 1 | 4 | 1 | 休止 / Rest |
| 2 | 1 | 3 | [G4 B4] |
| 2 | 4 | 1 | 休止 / Rest |
| 3 | 1 | 3 | [A4 C5] |
| 3 | 4 | 1 | 休止 / Rest |
| 4 | 1 | 3 | [F♯4 A4] |
| 4 | 4 | 1 | 休止 / Rest |
1. 分手指音後合手彈 A；每小節第 4 拍放鬆、準備下一位置，勿邊跳邊強拉。
After pointing and checking hands separately, play A together. Release and prepare the next position on beat 4 rather than stretching during the move.
2. 彈 B，核對只有第二小節的三音降低一鍵。A 保持性質，B 不再全程保持；兩版的根音路徑與最高音均不變。
Play B and verify that only bar 2’s third moves down one key. A preserves quality throughout; B does not. Root path and top voice remain unchanged in both.
3. 把四組根音相對間隔寫出，另列實際音名；不把同構造標成同一實際集合或同一調。
Write root-relative offsets separately from actual note names; do not label a shared structure as one actual collection or one key.
完成核對 / Completion Check
能指出固定構造與改變的鍵位、找到唯一變化，並按休止換位；喜好可相同或不確定。
Identify fixed structure versus changing keys, find the single alteration, and move during the rest; preference may be equal or uncertain.
替代方式 / Accessible Alternative
同時雙音不舒適就逐音核對，或教師演奏全表；改逐音後不能聲稱已比較原本同時和弦的聽感。
If dyads are uncomfortable, check notes successively or have a teacher perform the complete grid. Successive notes do not establish the original simultaneous-chord listening comparison.
驗證狀態 / Verification Status
未演奏、未施測；數據是配方或核算，不是學習結果。
Not performed or tested; data are recipes or calculations, not learner outcomes.
2. Diminished Scale 減音階！ / Diminished Scales
來源 / Source: Diminished Scale 減音階！ · WEB-b1a09f3d6175
頁面日期 / Page date: 2024-01-16; 頁面字數標記 / length label: 1572; 存取 / accessed: 2026-10-22.
狀態 / Status: indexed_only → reviewed_public_article_text_only; full_public_article.
文字範圍 / Text coverage: 30–81; 83–95; 標題日期 / title/date: 25–28; 總行 / total lines: 96; web_open.
完整正文已讀；圖與影音未核。
Full body read; images and audiovisual media unreviewed.
媒體旗標 / Media flags: 完整公開正文 true；文件說明、聲音、影片、圖片、連結頁與互動 false。
Full public article text true; documentation, audio, video, images, linked pages, and interactive execution false.
來源重點 / Source Insight
兩種八音階交替半全或全半；十二音類別下僅三組集合。
Two octatonic forms alternate half–whole or whole–half steps; twelve pitch classes yield three collections.
知識庫用途 / Knowledge-Base Role
音程與集合辨析。
Distinguish intervals and collections.
證據種類 / Evidence Type
原作者教學文字。
Author’s instructional text.
來源限制 / Source Caveat
第二圖說誤寫 C♯；正文是 D。等音拼寫須看情境。
The second caption incorrectly says C♯; the body says D. Enharmonic spelling is contextual.
定位 / Anchor: 33–81 — 正文依據 / Body evidence
合併中英來源預算 / Combined bilingual source-derived budget: 127/200, 偏好上限 / preferred ceiling 170; 正文直接引用 / verbatim body quotes: 0.
計入 / Counted: title_zh, title_en_editorial, source_claim_summary, content_role, evidence_type, limits, coverage.note, evidence_anchors.label.
原創卡 / Original Card: 十二格遮罩與兩句短旋律 / Twelve-slot masks and two short phrases
先備 / Prerequisites
先從 F3 數相鄰鍵：F3→G♭3 是半音，F3→G3 是全音；不彈目標。紙上畫 0–11 十二格，把 C 定為 0。
First count neighboring keys from F3: F3→G♭3 is a semitone and F3→G3 a whole tone, without playing the target. Draw twelve paper slots numbered 0–11, with C at 0.
完整配方 / Complete Recipe
A 遮罩 0,1,3,4,6,7,9,10；B 遮罩 0,2,3,5,6,8,9,11。先依交替 1,2 與 2,1 走滿一圈，每次記到達格，回到 12=C5 不新增音類別。表內只選遮罩中的六個事件形成兩小節，不是完整上行音階練習。左手全休止。
A mask: 0,1,3,4,6,7,9,10; B mask: 0,2,3,5,6,8,9,11. First walk alternating 1,2 and 2,1 around a full cycle, marking each arrival; 12=C5 returns without adding a pitch class. The grids select six events from each mask for two-bar phrases, not a complete ascending-scale drill. LH rests throughout.
速度、節奏與力度 / Tempo, Rhythm, and Dynamics
4/4，四分音符=52；兩版每小節 1+1+2 拍，右手全程 mp，第二小節最後長音 p；不加速、不用踏板。每手每版 8 拍，右手單音，音域 C4–A♭4。
4/4, quarter=52; each bar is 1+1+2 beats in both versions. RH mp except the final long note in bar 2, p; no acceleration or pedal. Each hand has eight beats per version; RH plays single notes, range C4–A♭4.
首次發聲或揭答前 / Before First Sound or Reveal
未發聲前，預測 A 與 B 的第一個移動各走幾鍵、第二小節前兩音哪版距離更大；再圈出兩遮罩共同的格。
Before sound, predict how many key-steps each first move takes, which version has the larger move between bar 2’s first two notes, and which slots the masks share.
版本 / Version A — 2 小節 / bars
左手 / LH
| 小節 Bar | 起拍 Onset | 拍長 Beats | 音 Notes |
|---|---|---|---|
| 1 | 1 | 4 | 休止 / Rest |
| 2 | 1 | 4 | 休止 / Rest |
右手 / RH
| 小節 Bar | 起拍 Onset | 拍長 Beats | 音 Notes |
|---|---|---|---|
| 1 | 1 | 1 | [C4] |
| 1 | 2 | 1 | [D♭4] |
| 1 | 3 | 2 | [E4] |
| 2 | 1 | 1 | [G4] |
| 2 | 2 | 1 | [F♯4] |
| 2 | 3 | 2 | [E4] |
版本 / Version B — 2 小節 / bars
左手 / LH
| 小節 Bar | 起拍 Onset | 拍長 Beats | 音 Notes |
|---|---|---|---|
| 1 | 1 | 4 | 休止 / Rest |
| 2 | 1 | 4 | 休止 / Rest |
右手 / RH
| 小節 Bar | 起拍 Onset | 拍長 Beats | 音 Notes |
|---|---|---|---|
| 1 | 1 | 1 | [C4] |
| 1 | 2 | 1 | [D4] |
| 1 | 3 | 2 | [F4] |
| 2 | 1 | 1 | [A♭4] |
| 2 | 2 | 1 | [G♭4] |
| 2 | 3 | 2 | [F4] |
1. 奏 A 再奏 B，核對 A 首步 1、B 首步 2；第二小節 G→F♯ 是 1，A♭→G♭ 是 2。只比較指定局部，不把兩句說成處處只差一步。
Play A then B: their first steps are 1 and 2; bar 2 starts G→F♯, one step, versus A♭→G♭, two. Compare these specified places, not a claim that the whole phrases differ by one step.
2. 紙上把 A 的起點改從格 1 開始，會先走 2；它是 D♭ 起始全半，集合仍是 A，不是 C 起始全半。再把 A 各格加 3 並取模 12，核對集合不變。
On paper, start A from slot 1: the first step is 2. This is D♭ whole–half with A’s collection, not C whole–half. Then add 3 to every A slot modulo 12 and verify that its collection is unchanged.
3. 一鍵可用不同音名，但這裡 F♯ 與 G♭ 的寫法只是各句標籤；尚未指定和聲功能，不能由鍵位等同推出功能等同。
One key may have different names; F♯ and G♭ here label the respective phrases. No harmonic function is specified, so shared keys do not establish shared function.
完成核對 / Completion Check
能依首步命名半全／全半、區分同起音兩集合，按兩小節時值完成。三集合結論見獨立枚舉，不等於所有拼寫或調性相同。
Name half–whole versus whole–half by the first step, distinguish same-start collections, and complete the two-bar durations. Independent enumeration below yields three collections, not identical spellings or tonalities.
替代方式 / Accessible Alternative
只在十二格紙圖上指移動及拍數也可完成結構辨析；未彈就記聲響與手部操作未測。
Pointing through the twelve-slot paper map and durations can complete the structural comparison; if unplayed, record sound and motor execution as untested.
驗證狀態 / Verification Status
未演奏、未施測；數據是配方或核算，不是學習結果。
Not performed or tested; data are recipes or calculations, not learner outcomes.
3. 如何用鋼琴彈 Bossa Nova 伴奏？ / Piano Bossa Nova Accompaniment
來源 / Source: 如何用鋼琴彈 Bossa Nova 伴奏？ · WEB-7998c1a3b80e
頁面日期 / Page date: 2024-05-28; 頁面字數標記 / length label: 1569; 存取 / accessed: 2026-10-22.
狀態 / Status: indexed_only → reviewed_public_article_text_only; full_public_article.
文字範圍 / Text coverage: 38–114; 115–138; 標題日期 / title/date: 24–27; 總行 / total lines: 139; web_open.
完整正文已讀；圖與影音未核。
Full body read; images and audiovisual media unreviewed.
媒體旗標 / Media flags: 完整公開正文 true；文件說明、聲音、影片、圖片、連結頁與互動 false。
Full public article text true; documentation, audio, video, images, linked pages, and interactive execution false.
來源重點 / Source Insight
作者分開低音、和弦與切分節奏教伴奏，並提供擴展選項。
The author separates bass, chords, and syncopation for accompaniment, offering richer options.
知識庫用途 / Knowledge-Base Role
雙手節奏分工。
Coordinate rhythmic layers.
證據種類 / Evidence Type
原作者教學文字。
Author’s instructional text.
來源限制 / Source Caveat
是入門配方，不是巴西音樂全貌；本文連結譜未讀。
An introductory recipe, not all Brazilian music; the linked score was unread.
定位 / Anchor: 42–114 — 正文依據 / Body evidence
合併中英來源預算 / Combined bilingual source-derived budget: 127/200, 偏好上限 / preferred ceiling 170; 正文直接引用 / verbatim body quotes: 0.
計入 / Counted: title_zh, title_en_editorial, source_claim_summary, content_role, evidence_type, limits, coverage.note, evidence_anchors.label.
原創卡 / Original Card: 低音不動，移動一次和弦落點 / Steady bass, one shifted chord onset
先備 / Prerequisites
先不按琴：左手在 1、3 輕點桌面，右手只在 2& 輕點一次；不必使用任何錄音。找 F3、C3、D3、A2，再找右手雙音。
Before the piano, lightly tap LH on 1 and 3 and RH once on 2&, without recording. Locate F3, C3, D3, and A2, then the RH dyads.
完整配方 / Complete Recipe
兩小節 Fmaj7→Dm7。左手根音與五音各兩拍；右手只按三音與七音：A3–E4、F3–C4。總和聲因此有根、三、五、七，不貼 maj9 或 m9 標籤。這是一段完整的純伴奏練習，沒有主旋律，也不要求唱出或補上旋律。A 右手六次起音為 1/2&/4 | 1&/3/4&，每次半拍；B 只把第一小節 2& 提前到 2。
Two bars: Fmaj7→Dm7. LH alternates root and fifth for two beats each; RH uses only thirds and sevenths, A3–E4 and F3–C4. The total harmony thus supplies root, third, fifth, and seventh, not maj9 or m9. This is a complete accompaniment-only exercise with no melody to sing or supply. A’s six RH onsets are 1/2&/4 | 1&/3/4&, each half a beat; B moves only bar 1’s 2& onset to 2.
速度、節奏與力度 / Tempo, Rhythm, and Dynamics
4/4，四分音符=56，等分八分音符，不搖擺。左手全程 p；右手 mp，第二小節 4& 最後一下改 p，沒有突然重拍。無踏板，右手各半拍後確實放開。兩手每版各 8 拍；右手最寬完全五度，整體 A2–E4。
4/4, quarter=56, evenly divided eighth notes rather than swing. LH stays p; RH mp, except the final attack on bar 2’s 4&, p, without a sudden accent. No pedal; release RH after each half beat. Each hand totals eight beats per version; maximum RH span is a perfect fifth, total range A2–E4.
首次發聲或揭答前 / Before First Sound or Reveal
首次發聲前先標記：A 第一小節的第二次右手起音在左手第 3 拍之前還是之後？B 提早多少拍？左手有沒有改？
Before first sound, mark whether A’s second RH onset comes before or after LH beat 3, how far B moves it earlier, and whether LH changes.
版本 / Version A — 2 小節 / bars
左手 / LH
| 小節 Bar | 起拍 Onset | 拍長 Beats | 音 Notes |
|---|---|---|---|
| 1 | 1 | 2 | [F3] |
| 1 | 3 | 2 | [C3] |
| 2 | 1 | 2 | [D3] |
| 2 | 3 | 2 | [A2] |
右手 / RH
| 小節 Bar | 起拍 Onset | 拍長 Beats | 音 Notes |
|---|---|---|---|
| 1 | 1 | 0.5 | [A3 E4] |
| 1 | 1.5 | 1.0 | 休止 / Rest |
| 1 | 2.5 | 0.5 | [A3 E4] |
| 1 | 3.0 | 1.0 | 休止 / Rest |
| 1 | 4 | 0.5 | [A3 E4] |
| 1 | 4.5 | 0.5 | 休止 / Rest |
| 2 | 1 | 0.5 | 休止 / Rest |
| 2 | 1.5 | 0.5 | [F3 C4] |
| 2 | 2.0 | 1.0 | 休止 / Rest |
| 2 | 3 | 0.5 | [F3 C4] |
| 2 | 3.5 | 1.0 | 休止 / Rest |
| 2 | 4.5 | 0.5 | [F3 C4] |
版本 / Version B — 2 小節 / bars
左手 / LH
| 小節 Bar | 起拍 Onset | 拍長 Beats | 音 Notes |
|---|---|---|---|
| 1 | 1 | 2 | [F3] |
| 1 | 3 | 2 | [C3] |
| 2 | 1 | 2 | [D3] |
| 2 | 3 | 2 | [A2] |
右手 / RH
| 小節 Bar | 起拍 Onset | 拍長 Beats | 音 Notes |
|---|---|---|---|
| 1 | 1 | 0.5 | [A3 E4] |
| 1 | 1.5 | 0.5 | 休止 / Rest |
| 1 | 2 | 0.5 | [A3 E4] |
| 1 | 2.5 | 1.5 | 休止 / Rest |
| 1 | 4 | 0.5 | [A3 E4] |
| 1 | 4.5 | 0.5 | 休止 / Rest |
| 2 | 1 | 0.5 | 休止 / Rest |
| 2 | 1.5 | 0.5 | [F3 C4] |
| 2 | 2.0 | 1.0 | 休止 / Rest |
| 2 | 3 | 0.5 | [F3 C4] |
| 2 | 3.5 | 1.0 | 休止 / Rest |
| 2 | 4.5 | 0.5 | [F3 C4] |
1. 先按 A 左手彈兩小節，接著右手照休止練；合手前保留書面預測。第二小節第 1 拍只有左手，不要偷偷補右手。
Play A’s LH for two bars, then RH with its rests; retain the written prediction before combining. Only LH plays on bar 2 beat 1; do not add an RH attack.
2. 合手 A 後奏 B；核對第二次右手起音由 2.5 到 2，早半拍，左手固定。其餘起音、音長與力度相同。
After A together, play B: the second RH onset moves from 2.5 to 2, half a beat earlier, while LH stays fixed. All other onsets, durations, and dynamics match.
3. 記下哪版較容易守住脈拍，以及是否出現搶拍或漏休止；不以六次起音正反拍比例評定道地性。
Record which version makes pulse easier to maintain and any rushing or missed releases, without scoring authenticity from the on/off-beat ratio of six attacks.
完成核對 / Completion Check
能完整彈兩小節，保留右手休止，指出唯一半拍位移；不要求判定哪版比較 Bossa Nova。
Complete two bars with RH rests and identify the single half-beat displacement; no judgment of which is more Bossa Nova is required.
替代方式 / Accessible Alternative
可只做雙手桌面節奏或在紙上圈起音；若雙音不舒適由教師照表奏。桌面版僅測拍位，不算已奏和聲。
Use two-hand tabletop rhythm or circle onsets on paper; if dyads are uncomfortable, a teacher can perform the grid. The tabletop version checks timing only, not harmony execution.
驗證狀態 / Verification Status
未演奏、未施測；數據是配方或核算，不是學習結果。
Not performed or tested; data are recipes or calculations, not learner outcomes.
4. 動機發展：寫旋律時，沒有靈感怎麼辦？ / Motivic Development: Writing without Inspiration
來源 / Source: 動機發展：寫旋律時，沒有靈感怎麼辦？ · WEB-e6431bbce4ca
頁面日期 / Page date: 2025-10-24; 頁面字數標記 / length label: 1527; 存取 / accessed: 2026-10-22.
狀態 / Status: indexed_only → reviewed_public_article_text_only; full_public_article.
文字範圍 / Text coverage: 38–109; 111–128; 標題日期 / title/date: 24–27; 總行 / total lines: 129; web_open.
完整正文已讀；圖與影音未核。
Full body read; images and audiovisual media unreviewed.
媒體旗標 / Media flags: 完整公開正文 true；文件說明、聲音、影片、圖片、連結頁與互動 false。
Full public article text true; documentation, audio, video, images, linked pages, and interactive execution false.
來源重點 / Source Insight
重複小片段，再改音高、方向及時值，可建立有關聯的段落。
Repeating a fragment and changing pitch, direction, and duration can build related passages.
知識庫用途 / Knowledge-Base Role
受限素材創作。
Compose with constrained material.
證據種類 / Evidence Type
原作者教學文字。
Author’s instructional text.
來源限制 / Source Caveat
示範不證明無需靈感；輪廓反向不必是精確半音反射。
A demonstration does not prove inspiration unnecessary; contour reversal need not be exact semitone reflection.
定位 / Anchor: 53–109 — 正文依據 / Body evidence
合併中英來源預算 / Combined bilingual source-derived budget: 140/200, 偏好上限 / preferred ceiling 170; 正文直接引用 / verbatim body quotes: 0.
計入 / Counted: title_zh, title_en_editorial, source_claim_summary, content_role, evidence_type, limits, coverage.note, evidence_anchors.label.
原創卡 / Original Card: 四音問句與反向答句 / A four-note question and inverted answer
先備 / Prerequisites
先學把上行寫正、下行寫負，以鍵位半音數計大小；用別的二音 F4→G4 示範 +2，再看目標。全曲只需右手單音，左手休止。
First label ascent positive and descent negative, counting semitone key-steps; demonstrate +2 with the separate pair F4→G4 before the target. The whole piece uses RH single notes with LH resting.
完整配方 / Complete Recipe
原始四音 C4–E4–D4–G4，時值半拍、半拍、一拍、兩拍，輪廓 +4、−2、+5。A 第二小節原樣重複；B 第二小節改 E4–C4–D4–A3，輪廓 −4、+2、−5，時值不變。這是把原段起音 C4 對到 E4 的有向音程反向，不說它以 E4 為固定鏡軸。C 只改 A 第二小節時值為一拍、半拍、半拍、兩拍。
The seed C4–E4–D4–G4 lasts half, half, one, and two beats, with contour +4,−2,+5. A repeats it unchanged in bar 2. B uses E4–C4–D4–A3 in bar 2, contour −4,+2,−5, retaining durations. This reverses directed intervals while mapping the seed’s opening C4 to E4; it is not called a reflection fixed about E4. C changes only A’s second-bar durations to one, half, half, and two beats.
速度、節奏與力度 / Tempo, Rhythm, and Dynamics
4/4，四分音符=60；第一小節 mp，第二小節 p，所有版本相同。無踏板、不在長音後搶拍；每手每版 8 拍。右手單音，音域 A3–G4，沒有需同時伸手的音程。
4/4, quarter=60; bar 1 mp and bar 2 p in every version. No pedal or rushing after the long note. Each hand has eight beats per version; RH single-note range A3–G4, with no simultaneous stretch.
首次發聲或揭答前 / Before First Sound or Reveal
首次彈 B 前寫第二小節前三次移動的方向；首次彈 C 前標出第二小節第二音的新起拍。已看公開答案就記有提示。
Before first playing B, write the directions of its three bar-2 moves. Before first playing C, mark the new onset of bar 2’s second note. If the disclosed answers were seen, mark prompted.
版本 / Version A — 2 小節 / bars
左手 / LH
| 小節 Bar | 起拍 Onset | 拍長 Beats | 音 Notes |
|---|---|---|---|
| 1 | 1 | 4 | 休止 / Rest |
| 2 | 1 | 4 | 休止 / Rest |
右手 / RH
| 小節 Bar | 起拍 Onset | 拍長 Beats | 音 Notes |
|---|---|---|---|
| 1 | 1 | 0.5 | [C4] |
| 1 | 1.5 | 0.5 | [E4] |
| 1 | 2.0 | 1 | [D4] |
| 1 | 3.0 | 2 | [G4] |
| 2 | 1 | 0.5 | [C4] |
| 2 | 1.5 | 0.5 | [E4] |
| 2 | 2.0 | 1 | [D4] |
| 2 | 3.0 | 2 | [G4] |
版本 / Version B — 2 小節 / bars
左手 / LH
| 小節 Bar | 起拍 Onset | 拍長 Beats | 音 Notes |
|---|---|---|---|
| 1 | 1 | 4 | 休止 / Rest |
| 2 | 1 | 4 | 休止 / Rest |
右手 / RH
| 小節 Bar | 起拍 Onset | 拍長 Beats | 音 Notes |
|---|---|---|---|
| 1 | 1 | 0.5 | [C4] |
| 1 | 1.5 | 0.5 | [E4] |
| 1 | 2.0 | 1 | [D4] |
| 1 | 3.0 | 2 | [G4] |
| 2 | 1 | 0.5 | [E4] |
| 2 | 1.5 | 0.5 | [C4] |
| 2 | 2.0 | 1 | [D4] |
| 2 | 3.0 | 2 | [A3] |
版本 / Version C — 2 小節 / bars
左手 / LH
| 小節 Bar | 起拍 Onset | 拍長 Beats | 音 Notes |
|---|---|---|---|
| 1 | 1 | 4 | 休止 / Rest |
| 2 | 1 | 4 | 休止 / Rest |
右手 / RH
| 小節 Bar | 起拍 Onset | 拍長 Beats | 音 Notes |
|---|---|---|---|
| 1 | 1 | 0.5 | [C4] |
| 1 | 1.5 | 0.5 | [E4] |
| 1 | 2.0 | 1 | [D4] |
| 1 | 3.0 | 2 | [G4] |
| 2 | 1 | 1 | [C4] |
| 2 | 2 | 0.5 | [E4] |
| 2 | 2.5 | 0.5 | [D4] |
| 2 | 3.0 | 2 | [G4] |
1. 先彈 A，圈出相同的音高、時值與動機；力度變輕並未改寫音符。再彈 B，只比較第二小節的有向音程，記下下、上、下。
Play A and mark repeated pitches, durations, and motif; the softer dynamic does not rewrite the notes. Then play B and compare only bar 2’s directed intervals: down, up, down.
2. 彈 C，核對第二小節起拍從 1/1.5/2/3 改成 1/2/2.5/3，仍滿四拍。比較的是已公開節奏變體，不是新題能力測驗。
Play C: bar 2’s onsets change from 1/1.5/2/3 to 1/2/2.5/3, still filling four beats. This is a disclosed rhythmic variant, not a novel-item ability test.
3. 可在紙上另選 A 或 B 當第二小節，先寫選擇理由再演奏兩小節；不得因寫完就宣稱已發展整首作品或證實作曲教學成效。
On paper, choose A or B for the second bar, write a reason, then play both bars. Completing it does not establish a whole composition or efficacy of composition instruction.
完成核對 / Completion Check
能分開音高反向、節奏改寫與力度變化，數滿每小節四拍；反向版不需要與原版同調或同情緒。
Distinguish pitch inversion, rhythmic change, and dynamics while counting four beats per bar. The inverted version need not share key or mood with the original.
替代方式 / Accessible Alternative
可畫三個方向箭頭並用手指走時值；教師可完整示範。未彈時只記紙上規則核對完成。
Draw three direction arrows and trace durations with a finger; a teacher may demonstrate completely. Without playing, record only completion of the paper-rule check.
驗證狀態 / Verification Status
未演奏、未施測；數據是配方或核算，不是學習結果。
Not performed or tested; data are recipes or calculations, not learner outcomes.
5. 中庸全音律，以及 Do-Sol 和 Do-Mi 的故事 / Meantone: The Story of Do–Sol and Do–Mi
來源 / Source: 中庸全音律，以及 Do-Sol 和 Do-Mi 的故事 · WEB-d164fcbe8cfe
頁面日期 / Page date: 2023-12-26; 頁面字數標記 / length label: 2314; 存取 / accessed: 2026-10-22.
狀態 / Status: indexed_only → reviewed_public_article_text_only; full_public_article.
文字範圍 / Text coverage: 30–108; 109–117; 標題日期 / title/date: 25–28; 總行 / total lines: 118; web_open.
完整正文已讀；圖與影音未核。
Full body read; images and audiovisual media unreviewed.
媒體旗標 / Media flags: 完整公開正文 true；文件說明、聲音、影片、圖片、連結頁與互動 false。
Full public article text true; documentation, audio, video, images, linked pages, and interactive execution false.
來源重點 / Source Insight
本文縮小五度以取得純大三度，仍留下閉合問題。
The article narrows fifths to obtain just major thirds while retaining a closure problem.
知識庫用途 / Knowledge-Base Role
比較音律取捨。
Compare tuning trade-offs.
證據種類 / Evidence Type
原作者教學文字。
Author’s instructional text.
來源限制 / Source Caveat
此文是取捨解說，不是歷史研究或聽感實測。
This explains trade-offs; it is not historical research or an empirical listening study.
定位 / Anchor: 51–108 — 正文依據 / Body evidence
合併中英來源預算 / Combined bilingual source-derived budget: 137/200, 偏好上限 / preferred ceiling 170; 正文直接引用 / verbatim body quotes: 0.
計入 / Counted: title_zh, title_en_editorial, source_claim_summary, content_role, evidence_type, limits, coverage.note, evidence_anchors.label.
原創卡 / Original Card: 先猜取捨，再算四條箭頭 / Predict the trade-off, then compute four arrows
先備 / Prerequisites
紙筆即可；不需動琴。先用不在表內的簡單乘法 2×2=4 理解「連乘」；再用 4÷4=1 理解移回兩個八度。會按計算機四次乘法即可，不要求心算根號。
Paper and pencil suffice; do not alter the piano. First use the separate example 2×2=4 for repeated multiplication and 4÷4=1 for folding down two octaves. Four calculator multiplications are enough; mental root calculation is unnecessary.
完整配方 / Complete Recipe
畫五個圓 C→G→D→A→E，每條箭頭乘同一 g；不逐步移回八度，最後一次把 g⁴ 除以 4，得到與 C 同一八度中的 E 比值。以表中三個 g 分別算，不把紙上箭頭當琴鍵實際音準。
Draw five circles C→G→D→A→E, multiplying by one g on each arrow. Do not fold after each step; divide g⁴ by 4 only at the end to obtain E’s ratio in C’s octave. Calculate separately for the three supplied g values, without equating paper arrows with actual piano tuning.
速度、節奏與力度 / Tempo, Rhythm, and Dynamics
這是自定步調的紙上題，速度、拍號、力度、左右手與音域不適用；無任何播放或聽辨要求。為每條箭頭核對一次乘法即可。
This is self-paced paper work; tempo, meter, dynamics, hands, and register are not applicable. No playback or listening discrimination is required; check one multiplication per arrow.
首次發聲或揭答前 / Before First Sound or Reveal
先遮住結果欄：把 g 從 1.5 稍微縮小後，五度與四次累積後的大三度都會縮小嗎？哪一模型要求最後正好 5/4？先寫預測，再揭表。
Cover result columns first: if g is reduced slightly from 1.5, do both the fifth and four-step major third shrink? Which model requires exactly 5/4 at the end? Write the prediction before revealing the table.
紙上圖與揭答表 / Paper Diagram and Reveal Table
C —×g→ G —×g→ D —×g→ A —×g→ E；終點 ÷4。
C —×g→ G —×g→ D —×g→ A —×g→ E; divide the endpoint by 4.
P: 純五度生成模型
Pure-fifth generator model
Q: 四分之一音差模型
Quarter-comma model
T: 十二平均律模型
12-TET model
| 模型 Model | g | 五度比 Fifth ratio | 五度音分 Fifth cents | g⁴/4 三度比 Third ratio | 三度音分 Third cents |
|---|---|---|---|---|---|
| P | 3/2 | 1.5 | 701.955001 | 1.265625 | 407.820003 |
| Q | 5^(1/4) | 1.4953487812 | 696.578428 | 1.25 | 386.313714 |
| T | 2^(7/12) | 1.4983070769 | 700.0 | 1.2599210499 | 400.0 |
1. 用列出的 g 算 g⁴÷4。Q 的 g⁴=5，所以結果 5/4；P 的結果 81/64；T 的結果 2^(1/3)。相減頻率與相減音分不可混用。
Compute g⁴÷4. Q has g⁴=5, hence 5/4; P gives 81/64 and T gives 2^(1/3). Do not confuse subtracting frequency values with subtracting cents.
2. Q 的術語與範圍見補充 SUP-BAEZ-QCOMMA。自行代入 81/80 核對等式，不把頻率比直接除以 4；計算只核對這張卡的參數。
See SUP-BAEZ-QCOMMA for Q’s term and scope. Substitute 81/80 to check the identity yourself rather than dividing the frequency ratio by 4; this checks only this card’s parameter.
3. 已公開比較 T：五度音程正好 700 音分、該大三度 400；與 Q 分欄記數值，不選「人人最好聽」。不以原聲鋼琴假裝切換三種音律。
Disclosed comparison T has a 700-cent fifth and a 400-cent major third. Record values separately from Q, without selecting a universally best sound or pretending an acoustic piano switches among these models.
完成核對 / Completion Check
能說明四條箭頭為何除以 4，並用計算指出五度縮小與純大三度的取捨；不宣稱已聽出差別。
Explain why four arrows require division by 4 and calculate the narrower-fifth/just-third trade-off, without claiming audible discrimination.
替代方式 / Accessible Alternative
若分數太難，使用已印小數把 P、Q、T 排大小；記為有提示的數值比較，不當作獨立推導。
If fractions are difficult, order P, Q, and T using the printed decimals. Record a prompted numerical comparison rather than independent derivation.
驗證狀態 / Verification Status
未演奏、未施測；數據是配方或核算，不是學習結果。
Not performed or tested; data are recipes or calculations, not learner outcomes.
6. 「狼五度」（Wolf Fifth）和不負責任的音樂學家 / The Wolf Fifth and Irresponsible Musicologists
來源 / Source: 「狼五度」（Wolf Fifth）和不負責任的音樂學家 · WEB-797cf24f69a4
頁面日期 / Page date: 2023-12-26; 頁面字數標記 / length label: 2456; 存取 / accessed: 2026-10-22.
狀態 / Status: indexed_only → reviewed_public_article_text_only; full_public_article.
文字範圍 / Text coverage: 33–117; 119–127; 標題日期 / title/date: 25–28; 總行 / total lines: 128; web_open.
完整正文已讀；圖與影音未核。
Full body read; images and audiovisual media unreviewed.
媒體旗標 / Media flags: 完整公開正文 true；文件說明、聲音、影片、圖片、連結頁與互動 false。
Full public article text true; documentation, audio, video, images, linked pages, and interactive execution false.
來源重點 / Source Insight
作者用五度鏈解說十二鍵閉合的狼音程。
The author uses a fifth chain to explain a wolf interval in twelve-key closure.
知識庫用途 / Knowledge-Base Role
檢查模型限制。
Check model constraints.
證據種類 / Evidence Type
原作者教學文字。
Author’s instructional text.
來源限制 / Source Caveat
差 0.027 的頻率倍數不是 0.027 八度；對話不是史證。
A frequency-factor difference of 0.027 is not 0.027 octave; dialogue is not historical evidence.
定位 / Anchor: 55–117 — 正文依據 / Body evidence
合併中英來源預算 / Combined bilingual source-derived budget: 136/200, 偏好上限 / preferred ceiling 170; 正文直接引用 / verbatim body quotes: 0.
計入 / Counted: title_zh, title_en_editorial, source_claim_summary, content_role, evidence_type, limits, coverage.note, evidence_anchors.label.
原創卡 / Original Card: 十一條相同邊，最後一邊誰負責 / Eleven equal edges: what must the last do?
先備 / Prerequisites
先完成卡五或使用印好的 g 值。紙上畫十二個節點 N0 至 N11，只需理解「乘完要等於 128」；不要求音階指法或調音。
Complete card 5 or use the printed g values. Draw twelve nodes N0 through N11; understand only that the full product must equal 128. No scale fingering or tuning is needed.
完整配方 / Complete Recipe
固定理想八度與十二鍵閉合：N0→N1→…→N11 的十一邊都乘 g，最後 N11→N0 的閉合邊乘 w，走完上行五度方向對應七個八度，故 g¹¹w=2⁷=128。無限延長而不閉合的鏈不受這條閉合要求。P、Q、T 是明示比較版本。
Assume ideal octaves and twelve-key closure: eleven edges N0→N1→…→N11 each multiply by g; the closing N11→N0 edge multiplies by w. A complete traversal in the upward-fifth direction spans seven octaves, so g¹¹w=2⁷=128. An indefinitely extended, unclosed chain does not face this closure requirement. P, Q, and T are disclosed comparison versions.
速度、節奏與力度 / Tempo, Rhythm, and Dynamics
紙上自定步調；速度、拍號、力度、手部與音域不適用。這不是在琴上試奏十二個純五度；不用發聲、調弦或測聽力。
Self-paced paper work; tempo, meter, dynamics, hands, and register do not apply. This is not a piano trial of twelve pure fifths; no sounding, retuning, or hearing test is involved.
首次發聲或揭答前 / Before First Sound or Reveal
先遮住結果，猜 P 的十二個 g 會超過或不足 128，因此最後 w 要比 g 大還是小？再對 Q 與 T 各寫預測。
Cover results and predict whether twelve P generators exceed or fall short of 128, and therefore whether the last w must be larger or smaller than g. Repeat for Q and T before revealing values.
紙上圖與揭答表 / Paper Diagram and Reveal Table
N0 —g→ N1 —g→ … —g→ N11 —w→ N0；十一個 g，加一個 w。
N0 —g→ N1 —g→ … —g→ N11 —w→ N0; eleven g edges plus one w edge.
P: 純五度生成模型
Pure-fifth generator model
Q: 四分之一音差模型
Quarter-comma model
T: 十二平均律模型
12-TET model
| 模型 Model | g | g¹²/128 | 偏差音分 Error cents | w=128/g¹¹ | w 音分 Cents | c(w)−c(g) 音分 Cents |
|---|---|---|---|---|---|---|
| P | 3/2 | 1.013643264771 | 23.46001 | 1.479810552818 | 678.49499 | -23.46001 |
| Q | 5^(1/4) | 0.9765625 | -41.058858 | 1.531237151971 | 737.637287 | 41.058858 |
| T | 2^(7/12) | 1.0 | 0.0 | 1.498307076877 | 700.0 | -0.0 |
1. 算 w=128÷g¹¹，代回 g¹¹w 應等於 128。P 的 w=262144/177147，比 3/2 窄；Q 的 w 比自己的 g 寬；T 的 w=g。核對結果不是聽感投票。
Compute w=128÷g¹¹ and substitute back to obtain 128. P has w=262144/177147, narrower than 3/2; Q’s w is wider than its generator; T has w=g. This checks numbers, not a listening vote.
2. P 十二個純五度與七個八度的比為 531441/524288；不是把 2.02728653−2 當八度數。正確差是 log₂(531441/524288) 個八度，乘 1200 約 23.46 音分。
P’s twelve pure fifths versus seven octaves give 531441/524288, not an octave count of 2.02728653−2. The correct difference is log₂(531441/524288) octaves, multiplied by 1200 to give about 23.46 cents.
3. 選做標音名：N0–N11 可標 E♭–B♭–F–C–G–D–A–E–B–F♯–C♯–G♯。閉合 G♯→E♭ 按字母是減六度，十二鍵配置中占五度式關係；「狼五度」是此閉合位置的慣稱，不把拼寫硬說成完全五度。
Optional note labels for N0–N11 are E♭–B♭–F–C–G–D–A–E–B–F♯–C♯–G♯. Closing G♯→E♭ is a diminished sixth by spelling, occupying a fifth-like relation in the twelve-key layout. “Wolf fifth” is the conventional label for this closure, not a claim that the spelling is a perfect fifth.
4. 最後寫限制：一個特別閉合邊來自「十一邊保持 g、十二鍵、精確八度」這組選擇。可以改分配誤差；T 已展示不必把差全塞一邊。不能推論每種調律必有同一狼音或所有調都不可用。
Write the constraints: a distinct closing edge follows from choosing eleven unchanged g edges, twelve keys, and exact octaves. The discrepancy can be distributed differently; T already shows it need not be concentrated on one edge. Do not infer the same wolf in every tuning or that every key is unusable.
完成核對 / Completion Check
能驗回三模型閉合、解釋 P/Q 的補償方向相反，並分清頻率倍數、八度與音分。沒有聲音或歷史結論。
Verify closure in all three models, explain opposite correction directions for P and Q, and distinguish frequency factors, octaves, and cents. No sonic or historical conclusion follows.
替代方式 / Accessible Alternative
教師可提供乘積與表格，學習者只比較大於、等於、小於 1；記為輔助理解而非獨立算術。
A teacher may supply products and the table while the learner compares greater than, equal to, and less than 1. Record supported understanding rather than independent arithmetic.
驗證狀態 / Verification Status
未演奏、未施測；數據是配方或核算，不是學習結果。
Not performed or tested; data are recipes or calculations, not learner outcomes.
局部技術補充 / Partial Technical Supplement
音律系統的數學 / The Mathematics of Tuning Systems
The Mathematics of Tuning Systems · John C. Baez · SUP-BAEZ-QCOMMA · 2026-10-22
覆蓋 / Coverage
草稿書〈四分之一音差中庸律〉第 3 部分局部，行 812–824；非全書或圖像審閱。
Draft-book “Quarter-Comma Meantone, Part 3” excerpt, lines 812–824; not a whole-book or image review.
支持主張 / Supported Assertion
四分之一音差中庸律的五度為 ∜5=(3/2)/(81/80)^(1/4)；另有其他中庸方案。
The quarter-comma meantone fifth is ∜5=(3/2)/(81/80)^(1/4); other meantone variants exist.
用途 / Role
核對術語。
Check terminology.
限制 / Limits
僅用數學命名，不採歷史評述。
Used only for mathematical naming, not historical narrative.
媒體未核 / Media unreviewed. 合併中英預算 / Combined bilingual budget: 155/200; 正文直接引用 / body quotes: 0.
計入 / Counted: title, title_zh_editorial, authors, coverage, supported_assertion, content_role, limits.
跨文件計入 / Cross-document counted: sources[4].original_application.paper_grid.rows[1].label, sources[5].original_application.paper_grid.rows[1].label.
局部欄位 / Local fields: 133; 跨文件欄位 / cross-document fields: 22.
先前補充來源預算帳 / Prior Supplement Budget Ledger
| ID | 原批 Prior batch | 先前 Prior | 新增 Added | 累計 Aggregate | 重用 Reused |
|---|---|---|---|---|---|
| SUP-STIFF | public-batch6-acoustics | 130 | 0 | 130 | 否 / No |
| SUP-BEATS | public-batch6-acoustics | 96 | 0 | 96 | 否 / No |
| SUP-BINAURAL | public-batch6-acoustics | 98 | 0 | 98 | 否 / No |
| SUP-MAINS | public-batch6-acoustics | 60 | 0 | 60 | 否 / No |
獨立計算驗證 / Independent Calculation Verification
本地 Python 重新計算；只用標準庫，沒有聲音合成、外部模型或安裝。已核對音高差、各手逐小節拍數及閉合公式；這不是演奏驗證。
Recomputed locally with Python’s standard library only, without synthesis, external models, or installation. Pitch differences, each hand’s bar totals, and closure formulas were checked; this is not performance verification.
音分 c=1200 log₂(r)；四五度折回的大三度比 r₃=g⁴/4；閉合 w=2⁷/g¹¹。
Cents c=1200 log₂(r); folded four-fifth third ratio r₃=g⁴/4; closure w=2⁷/g¹¹.
表格音分四捨五入到小數六位；最後一位浮點誤差不解讀成音律差。
Table cents are rounded to six decimals; terminal floating-point error is not interpreted as a tuning difference.
三種八音集合；A 首步半音、B 首步全音；動機 +4,−2,+5 反向為 −4,+2,−5；P、Q、T 全部 g¹¹w=128。
Three octatonic collections; A starts with a semitone and B a whole tone; motif +4,−2,+5 inverts to −4,+2,−5; P, Q, and T all satisfy g¹¹w=128.
十二音類別八音集合 / Octatonic twelve-pitch-class collections:
[0, 1, 3, 4, 6, 7, 9, 10]
[0, 2, 3, 5, 6, 8, 9, 11]
[1, 2, 4, 5, 7, 8, 10, 11]
普通音差 / Syntonic comma: 81/80 = 21.50629 cents / 音分; 四分之一 / quarter: 5.376572.
畢氏音差 / Pythagorean comma: 531441/524288 = 23.46001 cents / 音分.
算術相對容差 / Arithmetic relative tolerance: 1e-12.
未核媒體與阻礙 / Unreviewed Media and Blockers
音訊、影片、譜圖、連結 PDF 及工具仍未審閱；不得把本批文字審閱升格為影音或演奏驗證。若日後需要媒體證據，須另外在授權範圍內逐項檢查。
Audio, video, score images, linked PDFs, and tools remain unreviewed. Do not promote text review into audiovisual or performance verification; any later media evidence needs separate item-by-item inspection within authorization.
無存取阻礙；未使用付費回退、安裝或本地模型推理。
No access blockers; no paid fallback, installation, or local model inference.