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如何設計一種『自然語言』？能夠聯繫『世界物種』！可以溝通『遙遠星際』！！一如『大霹靂』的『同源共起』之時？可稱之為『宇宙』語言的呢？？維基百科詞條講︰

宇宙語言（拉丁語：Lincos，源自lingua cosmica的縮寫），又稱為宇際語言，是一種在 1960 年由荷蘭數學家和天文學家弗勒登塞爾博士提出的一種人工語言。當年，他在其個人著作《Lincos：一種為宇宙間溝通而設計的語言 (第一部)》(Lincos: Design of a Language for Cosmic Intercourse, Part 1)提出這種語言的概念。他設計這語言的概念，是希望所有可能的外星生物物種都能夠明白，並且可以透過星際間的無線電信號傳達。弗勒登塞爾博士認為：這種語言必須能夠容易讓地球以外的物種明白，所以其設計不能夠與地球的任何生物有關聯。而且，Lincos的設計要能夠足以封裝(encapsulate)「我們知識的大部份」(“the whole bulk of our knowledge“)。

假設

雖然目前還無法證實除地球外宇宙中還有其他地方存在生命，但是許多科學家認為宇宙中存在其他生命的可能性應該是有的。基於這種假設，因此把宇宙中其他星球上的智能生物所說的所有語言統稱為宇宙語言。下面的敘述也是全部基於這種假設。

根據宇宙語言學的研究，宇宙中存在的語言有無數種。而人類由於對此還一無所知，因此也很難解讀這些宇宙語言。在這種情況下，人類的科學家試圖設計一種宇宙人能夠識別的人造語言，用來和他們進行聯繫。

研究和設計宇宙語言，是基於這樣一種依據：科學家認為宇宙人應該是比地球人更先進，更發達的智能生物，因此他們的語言也應該很發達。而標誌之一就是語言的高度數學化。因此，科學家試圖把語言用數學方法來表達，這就是宇宙語言的基本思路。因此，弗勒登塞爾博士提出過一套宇宙語言的設計方法，就是以數學符號為基礎，有一整套的規則和代碼，並且用數學的方法料表達句法。

他的基本信息運算非常簡單：例如

....>..
...<......
...=...
..+...=.....
......-....=..

雖然弗勒登塞爾博士對這種語言有一套整全的體系，但直到現在為止，他所設計的語言還未有機會進行廣播。因此，他聲稱能夠「與外太空智慧溝通」的理念上的方案。

可行性

義大利數學家、物理學家及天文學家伽利略早在17世紀初就指出，數學語言是解讀宇宙語言的鑰匙。當代美國天文學家薩根也指出，宇宙中的技術文明無論差異多大，都有一種共同的語言——數學語言。中國數學家和語言學家周海中教授在《宇宙語言：設計、發送與監聽》一文認為，數學語言具有明確性、單義性、緊湊性、普適性、可觀察性等優點，是宇宙交際的理想工具。另外還有專家認為,圖象語言和音樂語言也可作為宇宙交際的共同語言。

───

假使果真存在『宇宙語言』，若是說『人類理解』都得花費時間！不能『一讀即懂』。將之推於『乾坤創制』，怕那外星人也無法『一見就知』，恐還需要『考古』研究的吧！！

為什麼『老鼠』愛『大米』？為什麼『蜜蜂』採『花粉』？？人類尚且爭論『除了人以外』，地球上什麼『物種』能有『語言』的呢？！如果講『言語』司『溝通』，『文字』載『故事』，世界之民生活，奈何必要『數學』語言的耶！？或許『納西人』

納西族是中國西南少數民族，聚居地分布於雲南、四川和西藏的交界處的麗江及其毗鄰地區，雲南省為納西族主要分布省份。中國（中國大陸）有納西族總人口309,477人（2000年人口普查），列第27大民族，其中雲南295,464人，占納西族人口的95.5%，其餘省市區均有分布。

歷史

納西族是一個古老的民族，是古代羌人的一支，據《華陽國志》記載，東漢（公元25－220年）末年，該地區有「摩沙夷」活動。唐代，納西族先民分布記錄較詳細，據《蠻書》所記，西至金沙江河谷、巨甸以北「鐵橋上下及大婆、小婆、三探覽（均在今麗江地區）」東至雅礱江流域「昆池（今鹽源）等川」，都是「磨些蠻」所居之地。元明時代，由於雲南麗江木氏土司崛起，不斷向外擴張，納西族活動地域更為廣闊，東北部可能到了今四川省康定及以南貢嘎嶺一帶，北到今巴塘、理塘和昌都，西到怒江流域。清雍正元年（1723年）木氏土司改土歸流後，西北部藏族土司實力南移，納西族土司式微，加之大批彝族西遷入雅礱江流域，人數就逐漸減少。

語言

納西族有自己的語言和文字。納西族的語言是納西語，一般歸入漢藏語系藏緬語族彝語支。它有兩種文字：東巴文和哥巴文，其中東巴文創造於一千多年前，是今天世界上唯一的仍被使用的一種象形文字。

───

的『歷史』，正述說『咫尺天涯』的吧！！因此『純思辨』的議論，若失了天地『可經驗』之脈落，即使文字誠『真象形』，解釋成什麼還是『很難說』的哩！或可知 M♪o 的『算籌』程式語言，淵源根深久遠的耶？？

……

夏日晨起天光好，微風吹花氣味佳，忽而眼皮直跳，不知是何兆？欲法梅花心易，急讀《》觀水之法︰

…

『水』是生命三要素── 陽光、空氣、水 ──之一，中國古人認為『天地的大德曰生』，強調『生生不息』，或許是喜歡『論水』的重要原因。這三要素包括在『伏羲八卦』的『卦象』裡︰

乾 ☰ 天，兌 ☱ 澤，離 ☲ 火，震 ☳ 雷，巽 ☴ 風，坎 ☵ 水，艮 ☶ 山，坤 ☷ 地。

之後又把『日』比之『離之火』，『月』擬之『坎之水』；於是在時空中，自然和社會的舞台上，述說著『人事物』的故事。由於每個人的感知經驗不同，使得每一個人的世界大異其趣。對於這樣的錯綜複雜之人事物『萬象』，又要如何去理解呢？既然萬象坐落在時空裡，就很難『經久不變』；而又為人感知，哪能夠『沒有異同』。也就是說自然和社會構成的『現象世界』即使人人相同，每個人心中的『印象世界』卻是各各不同，習染著個人的風采與印記，選擇了自己的信仰和道路。試想『科學的實驗』雖然人人時時都可以『驗證』，尚且不免有理論對錯的爭議；那麼『感受的社會』由於刻刻彼此間的『差異』，或許只能求同存異，致力於中和之道了！！在制器尚象一文中，我們主要談的是『象』，這裡將說說『觀象』之觀。首先要如何看待卦象呢？【易繫辭‧第七章】孔子說︰

易之為書也不可遠，為道也屢遷，變動不居，周流六虛，上下無常，剛柔相易，不可為典要，唯變所適。其出入以度，外內使知懼，又明於憂患與故。无有師保，如臨父母。初率其辭而揆其方，既有典常。苟非其人，道不虛行。《易》之為書也，原始要終，以為質也。六爻相雜，唯其時物也。其初難知，其上易知，本末也。初辭擬之，卒成之終。若夫雜物撰德，辯是與非，則非其中爻不備。噫亦要存亡吉凶，則居可知矣。知者觀其彖辭，則思過半矣。

教育之旨重在『啟蒙』，那就藉著『山水蒙』來談談『與時變遷』的卦象── 時物 ──，的『卦變』之法︰

掛名	卦象上下	卦象左右	卦變
山水蒙	☶ ☵	☶ ☵	所觀之象
澤火革	☱ ☲	☱ ☲	錯卦
水雷屯	☵ ☳	☵ ☳	綜卦
火風鼎	☲ ☴	☲ ☴	綜卦之錯
地雷復	☷ ☳	☷ ☳	交互卦
天風姤	☰ ☴	☰ ☴	交互卦之錯
山地剝	☶ ☷	☶ ☷	交互卦之綜
澤天夬	☱ ☰	☱ ☰	交互卦之綜卦之錯
水山蹇	☵ ☶	☵ ☶	外內反卦
火澤睽	☲ ☱	☲ ☱	外內反卦之錯
雷水解	☳ ☵	☳ ☵	外內反卦之綜
風火家人	☴ ☲	☴ ☲	外內反卦之綜卦之錯

易經有六十四卦，一卦『六爻』，由下往上數，第一爻叫『初』，這是因為現象流變不停，『此時』之觀不能夠『窮究』象來之因緣，所以說其初難知。『卦』比之擬之是『象』的模型，可以用來『推敲』象的變化因果，所以把第六爻稱作『上』，以至於說其上易知。古有三易之說，連山、歸藏和周易，這個模型尚且『為道屢遷』，所以不可拘泥於『典要』，必須要『為變所適』。這個『時中』── 時空現象中獨立自主的人 ──就是孔子主張的『中觀』精神。

再將這六爻，兩兩一對從下至上分成『三才』，象徵著人足立於地頭頂著天，期待能如乾卦九三所說的『終日乾乾』，此處九說的是陽，三處於上下兩卦之際，意味著蛻變之義；而乾卦九四所講的『或躍在淵』，表述著人各有志，抉擇的行業也就不同。所以一個人如果上下不分又顛三倒四的『志‧行』混亂，勢成為俗語稱作的不三不四。而這三四兩爻位總是位於卦之中，或想說著人的『存亡吉凶』自來自『是與非之辯』。那要如何『審時度勢』呢？用『彖』── 當‧機立斷 ──；那又要如何『當‧機』呢？觀『錯綜複雜』又『上下無常』的『卦變』。

【錯】︰將一卦六爻，陰陽相反，陽變陰，陰變陽

乾天之錯成坤地；離火之錯為坎水。這就是一般人們常說的『立場不同』，雖然『所求之事』未必不同。也許『兩極的對話』當是『相反相成』！！

【綜】︰一卦的鏡象

物理的定律，即使『時間倒流』依然不變；雖然愛因斯坦的『相對論』打破了現象的『同時性』，把『觀察者』帶進了物理世界。綜卦將一個卦『翻面』，以至於原卦之『上』成了新卦之『初』，彷彿『逆果導因』，鏡裡看著現象。

【交互】︰一卦去掉初‧上，將所剩四爻，以【五‧四‧三】作上卦，以【四‧三‧二】為下卦形成的卦

觀『互』字，所象就是以人 ── 三‧四 ──為中心，立於地上 ── 二 ──，位於天之下── 五 ──，的象中看象。；不知能否清楚『來龍去脈』。

【外內反】︰一卦的上下兩體交換

天地之否轉眼成了地天之泰；水火既濟瞬間變作火水未濟。從三才中人的角度看，『行四』化為新卦之『初』，『志三』轉作新卦之『上』，到底何謂『起點』？又哪是『終點』？真不知如何說『輸贏』的『起跑點』或『終止線』；反而誤將教育的『啟蒙』變成學習的『蹇難』！！

細看卦變之法，何止眼觀四面耳聽八方，如果知道人人一太極、事事一太極以及物物一太極，觀『卦變』的『心法』就是觀『機』── 關鍵點、轉捩點、槓桿點、…… ──。

宋朝的邵康節號安樂先生，居地叫做『安樂窩』，著有『皇極經世』一書，將歷史紀年『卦象化』說能推知數百千年的現象；傳聞又有『梅花心易』一本，專講『見機』起卦之法，其中用著『京房八宮卦』，論斷之準真真神乎其技！！歷史上『科學』雖與『巫術』有千絲萬縷的關聯，當細思

子不語：怪、力、亂、神。

季路問事鬼神。子曰：未能事人，焉能事鬼？敢問死？曰：末知生，焉知死？

；然而【易繫辭】裡卻又有

易與天地準，故能彌綸天地之道。仰以觀於天文，俯於察於地理。是故知幽明之故。原始反終，故知死生之說。是故君子所居而安者，易之序也，所樂而玩者，爻之辭也。是故居則觀其象，而玩其辭，動則觀其變而玩其占，是以自天佑之，吉無不利。

這個『死生之說』怎麼說？這是從大自然一年四季的循環，草木的一歲一枯榮，『原始反終』推論而知，絲毫沒有『迷信』的色彩。

……

，眼為離 ䷝ 目，左通心，其跳吉。跳者眉居離眼上，或是喜上眉梢之兆。剛過夏至日，兆體取為乾 ䷀ ，目離變二爻，眉梢動上爻，樹欲靜，風不止，定前後。蓋指夬 ䷪ 之革 ䷰ ，當是『己日乃革』，『君子豹變』之象。

因是速往學堂。剛上迴廊，就見課堂外擺著桌子，三五同學看到我來，忙向前導引桌前，在那『未曾有』的『簽到簿』上畫卯，還得『工筆』寫句『勵志』語，只覺興筆寫下

，卻瞧見同學搔首笑說︰這可是句『畫』。趕緊添補上一筆

學而時習之不亦悅乎。

一進教室大吃一驚，桌子圍成了『圓』，拱著講桌而排，中央拼作『方』，上有派生碼訊，正播放著學習日子裡的點點滴滴，甚至還配著音樂呢！？一時如在夢中，以為今兒已經是『禮拜一』了，將要開『同樂會』？？正惶惑間，……學長走進了教室

─── 引自《 M♪o 之學習筆記本《巳》文章︰【䷪】有戎勿恤》

※ 若說總有人談

Raspberry Pi As Your Next Linux PC

，或許只在個人電腦何謂耶？！

樹莓派、樹莓派之學習、樹莓派之教育

勇闖新世界︰ W!o《卡夫卡村》變形祭︰品味科學‧教具教材‧ Hi-Jack‧Sonic π 之節拍體驗

2015-11-06 懸鉤子

在《【Sonic π】聲波之傳播原理︰共振篇《四下》》文本裡，我們談到了一根『半波長』之『估量的尺』，意在表明『欣賞自然』與『理解科學』的『靜觀自得』之境。此境將周遭萬象與人『關聯』起來，於是我們可以更深入的認識『自然』與『人造』物的世界︰

波的傳播除了『頻率』之外，『波長』也是重要的因素，尤其在考慮波的『輻射』與『散射』現象的時候。常溫下聲波速度約為 343.2 米/秒，從一個『邊長』是 $d$ 的共振腔共振時『半波長』的『整數倍』來看，它是否可以用來度量『聲源』的『尺寸』大小的呢？假使以『音律』中『十二平均律』的鋼琴『中央 C』261.6 赫茲來作計算，聲波波長大約是 1.312 米。『紅嘴相思鳥』鳴叫聲，以基本音調為主，頻率範圍大約為 2.50 ~ 3.80 千赫，主峰在 1.82 千赫，波長是 18.86 公分。當一隻鳥在發現略食者在周遭時，會發出警告同伴的鳴叫聲，它的頻率大約是 7000 Hz，波長約為 4.90 公分。『蟋蟀』的蟲鳴聲頻率範圍很廣 3 ~ 50 千赫，通常是相當純的律音，主峰在四、五千赫，次峰在十四千赫。以主峰 4.5 千赫計算大約 7.63 公分。『人類』的發聲頻率範圍約為 85 ~ 1100 赫茲，假使說以低音 85 赫茲來講，波長為 4.04 米；『狗』的發聲頻率範圍是 452~ 1800 赫茲，波長是 75.93 公分；『貓』的發聲頻率範圍是 760 ~ 1500 赫茲，波長是 45.15 公分。如此看來『半波長』果然可以看成一根『估量的尺』，假使一個物體沒有明顯的『長寬高』，比方說像一個『球』，它的共振波長 $\lambda_H \propto \sqrt[3] V$ 正比於『體積立方根』也就是『想當然耳』的了！！

然而人的『聽覺範圍』是從 20 到 20k Hz，一個二十赫茲的聲音波長有十七點一六米那麼長，這樣一個『音響喇叭』又要多大才能夠『共振』的呢？因為即使對『中央 C』來講，半波長也有 65.6 公分那麼長。所以一般動力『揚聲器』Loudspeaker 是用『聲波輻射』原理來設計的。

聲學裡一個『脈動球』 Pulsataing Sphere 『聲源』

是一個『半徑』在 $r_0$ 附近 $dr << r_0$ 以『頻率』 $f$ 作『簡諧振動』的球，假使球的半徑遠小於聲波『波長』 $\lambda >> r_0$ ，多個波長距離之外的遠處『聲場強度』 $I \propto I_0^2 \cdot \frac {f^2}{{r_0}^4}$ ，此處 $I_0$ 是聲源振幅。其實假使 $r_0 \rightarrow 0$ 它可以看成『點聲源』，比方講這就可以計算典型『揚聲器』的圓形『振動膜』所產生的『聲場』。因此也就可以了解為了追求『高傳真』 HiFi 的『聲音品質』，揚聲器分開了『高音』、『中音』以及『低音』喇叭設計的原故。以及現今為了加強『影音』的『震撼力』與『臨場感』採用杜比 AC3 5.1 規格，它有五個『喇叭』加上一個『超重低音』音箱的因由。

當人們在戶外，有時『拗著手』作『大聲公』狀來『叫人』，它是可以用『共振』和『輻射』來作解釋的嗎？每當夜深人靜時，『蟲鳴鳥叫』聲又為什麼會『擾人清夢』的呢？？

也許越『理解』大自然，人們就越可能『欣賞』大自然的『美』吧！又或許果真是『處處靜觀皆自得』的啊！！

───

若是人們知道『十二平均律』發現的歷史︰

十二平均律

十二平均律，又稱十二等程律，是一種音樂的定律方法，將一個八度平均分成十二等份，每等分稱為半音，是最主要的調音法。

歷史

公元 400 年左右，中國南朝數學家何承天提出世界歷史上最早有記載的十二平均律數列 900 849 802 758 715 677 638 601 570 536 509.5 479 450（原文：……黃鐘長九寸，太簇長八寸二厘，林鐘長六寸一厘，應鐘長四寸七分九厘強）^[1] .

義大利的物理學家伽利略·伽利萊的父親伽利略·文森佐曾試圖解決十二平均率問題，但他用的倍率是 18:17 而不是 $\sqrt [12] {2}$ ，因此自乘12次後只得 1.98556，不是2，八度走了音，他的系統只可算近似十二音階平均律^[2]

1605年荷蘭數學家西蒙·斯特芬在一篇未完成的手稿「Van de Spiegheling der singconst」^[3]提出用 $\sqrt [12] {1/2}$ 計算十二平均律，但因計算精度不夠，他算出的弦長數字，有些偏離正確數字一至二單位之多^[4]

西蒙·斯特芬的弦長表^[5]

音	弦 10000	比率	正確的弦長
半音	9438	1.0595465	9438.7
全音	8909	1.0593781
1.5 音	8404	1.0600904	8409
2 倍全音	7936	1.0594758	7937
2.5 音	7491	1.0594046	7491.5
3 音	7071	1.0593975	7071.1
3.5 音	6674	1.0594845	6674.2
4 音	6298	1.0597014	6299
4.5 音	5944	1.0595558	5946
5 音	5611	1.0593477	5612.3
5.5 音	5296	1.0594788	5297.2
八度		1.0592000

西蒙·斯特芬的頻率比，每音一率，且各不相同，這是不正確的^[6]

朱載堉發明十二平均律

中國明代音樂家朱載堉於萬曆十二年（1584年）首次提出「新法密率」（見《律呂精義》、《樂律全書》），推算出以比率 $\sqrt [12] {2}$ 將八度音等分為十二等分的算法，並製造出十二平均律律管及律準，是世界上最早的十二平均律樂器。他用九九八十一位算盤計算出來準確到25位數字新法密率為：

律名	比率
正黃鐘	1.000000000000000000000000
倍應鍾	1.059463094359295264561825
倍無射	1.122462048309372981433533
倍南呂	1.189207115002721066717500
倍夷則	1.259921049894873164767211
倍林鍾	1.334839854170034364830832
倍蕤賓	1.414213562373095048801689
倍仲呂	1.498307076876681498799281
倍姑洗	1.587401051968199474751706
倍夾鍾	1.681792830507429086062251
倍太蔟	1.781797436280678609480452
倍大呂	1.887748625363386993283826
倍黃鐘	2.000000000000000000000000

───

自將更能了解『 MIDI 碼』之制定與『十二平均律』的關係︰

MIDI 碼 = $69 + 12 \cdot log_2 ( \frac{f}{440} )$

MIDI音樂鍵盤

進而明白『鋼琴』琴鍵安排的規律。這一切的種種都離不開人類的『聽覺』，自然也沒有古今東西之人性差異。憑藉『感官知覺』之天賦，人們區分了『聲』與『樂』，『光』和『色』，居處於聲光樂色如斯美好的宇宙。這也許也說明了『感覺』終究是『物理』以及『心理』之『統覺』的吧！那根『豆芽』之理竟如是難啃的耶？所以我們需要『聽得見』又『看得到』的『學習環境』來『體驗』那既『具體』又『抽象』之『感覺』實在的乎！？

也許一個『拍頻』 beat note 『現象』之問題，足以表達那難言之隱的吧！！假使回答以

差頻（英文：beat note）一詞來源於聲學上兩個頻率相近但不同的聲波的干涉，所得到的干涉信號的頻率是原先兩個聲波的頻率之差，因此叫做差頻。

───

，怕是難明究理。即使再補之以

Beat (acoustics)

In acoustics, a beat is an interference between two sounds of slightly different frequencies, perceived as periodic variations in volume whose rate is the difference between the two frequencies.

With tuning instruments that can produce sustained tones, beats can readily be recognized. Tuning two tones to a unison will present a peculiar effect: when the two tones are close in pitch but not identical, the difference in frequency generates the beating. The volume varies like in a tremolo as the sounds alternately interfere constructively and destructively. As the two tones gradually approach unison, the beating slows down and may become so slow as to be imperceptible.

Diagram of beat frequency

Mathematics and physics of beat tones

This phenomenon manifests acoustically. If a graph is drawn to show the function corresponding to the total sound of two strings, it can be seen that maxima and minima are no longer constant as when a pure note is played, but change over time: when the two waves are nearly 180 degrees out of phase the maxima of each cancel the minima of the other, whereas when they are nearly in phase their maxima sum up, raising the perceived volume.

It can be proven (see List of trigonometric identities) that the successive values of maxima and minima form a wave whose frequency equals the difference between the frequencies of the two starting waves. Let’s demonstrate the simplest case, between two sine waves of unit amplitude:

{ \cos(2\pi f_1t)+\cos(2\pi f_2t) } = { 2\cos\left(2\pi\frac{f_1+f_2}{2}t\right)\cos\left(2\pi\frac{f_1-f_2}{2}t\right) }

^[1]

If the two starting frequencies are quite close (for example, a difference of approximately twelve hertz^[2]), the frequency of the cosine of the right side of the expression above, that is (f₁−f₂)/2, is often too slow(low) to be perceived as a pitch. Instead, it is perceived as a periodic variation of the first in the expression above (it can be said that the lower frequency cosine term, i.e. the second one, is an envelope for the faster wave, i.e. the first cosine term), whose frequency is (f₁ + f₂)/2, that is, the average of the two frequencies. However, because the human ear is not sensitive to the phase, only the amplitude or intensity of the sound, only the absolute value of the envelope is heard. Therefore, subjectively, the frequency of the envelope seems to have twice the frequency of the cosine, which means the audible beat frequency is:

f_{beat}=f_1-f_2\,

This can be seen on the diagram on the right.

The sum (blue) of two sine waves (red, green) is shown as one of the waves increases in frequency. The two waves are initially identical, then the frequency of the green wave is gradually increased by 25%. Constructive and destructive interference can be seen.

A physical interpretation is that when $\cos\left(2\pi\frac{f_1-f_2}{2}t\right)$ equals one, the two waves are in phase and they interfere constructively. When it is zero, they are out of phase and interfere destructively. Beats occur also in more complex sounds, or in sounds of different volumes, though calculating them mathematically is not so easy.

Beating can also be heard between notes that are near to, but not exactly, a harmonic interval, due to some harmonic of the first note beating with a harmonic of the second note. For example, in the case of perfect fifth, the third harmonic (i.e. second overtone) of the bass note beats with the second harmonic (first overtone) of the other note. As well as with out-of tune notes, this can also happen with some correctly tuned equal temperament intervals, because of the differences between them and the corresponding just intonation intervals: see Harmonic series (music)#Harmonics and tuning.

A 110 Hz A sine wave (magenta; first 2 seconds), a 104 Hz G# sine wave (cyan; following 2 seconds), their sum (blue; final 2 seconds) and the corresponding envelope (red)

───

恐仍莫名其妙，或將知『學習環境』之重要的耶？？

就讓我們試著用 Sonic π 建造一個『節拍體驗』情境，看看能否自學『樂理』的哩？！

現代『聽覺』的研究表明，雖然不同的人『音感』不同，然而卻都有下圖的聽覺『共性現象』：用一參考音高 F1 、另一變化音高 F2 ，讓 F2 遠比 F1 小而漸增加，聽覺逐漸從『平穩』變成『不平穩』── 『臨界頻帶』，在頻率差小於 15 Hz 時轉成了『拍音區域』、一般人當頻率差小於 12.5 Hz 時會產生『拍音』之感覺，最後 F2 = F1 時感覺是同音高；同樣的如果 F2 由比 F1 大而減小，情況相同。

所以我們可以借

【 MIDI 碼‧音高頻率表】

及一點點『 Sonic π 』之編程，在 JACK 的環境中，

探究、體驗、玩索『節拍』現象的真實矣！！

樹莓派、樹莓派之學習、樹莓派之教育

勇闖新世界︰ W!o《卡夫卡村》變形祭︰品味科學‧教具教材‧ Hi-Jack‧應用隨筆

2015-11-05 懸鉤子

事實上『 JACK 』相關的『 Applications 』應用軟體族類眾多︰

‧ Audio File Editors
‧ Control Applications
‧ DJ Software
‧ Effects Processors
‧ Graphics Applications
‧ Live Loopers
‧ Media Players
‧ Metering and Analysis
‧ Mixers
‧ Multi-track sequencers and HDR systems
‧ Music Notation Editors
‧ Programming Libraries
‧ Radio
‧ Simple Recorders
‧ Sound Generators
‧ Streamers
‧ Tuners
‧ Utilities
‧ Video Applications
‧ VOIP
‧ VST

因為文本系列之目的，我們選擇了其中的一小部份。建議讀者自行瀏覽網頁，或許可以找到很多極好的『音樂』軟件。此處我們僅僅介紹兩個已經用過的軟體，導引讀者進入『插頭』的世界而已。

假使你順著『插頭』應用軟件的鏈結走，想要了解那個軟體的用法，也許你會訝異所找到的內容。舉例來說︰

【 jaaa 】

Welcome to FA’s LinuxAudio page

Jaaa

Jaaa (JACK and ALSA Audio Analyser, is an audio signal generator and spectrum analyser designed to make accurate measurements.

───

【 meterbridge 】

JACK Meterbridge

This is the webpage for a software meterbridge for the UNIX based JACK audio system.

It supports a number of different types of meter, rendered using the SDL library and user-editable pixmaps.

───

這裡所擷取的內容，就是那兩個軟體主要的說明了。就像

《概念拼圖︰Linux 手冊！》文本所言︰

The Linux Document Project 這個文件計畫始於 1992 年，二十二年過去了，到了今天也有了 WiKi ，雖然已經有了相當多的文件和資料，但是對大多數人來說，想要用的資料卻總是『找不到』。也許對許多軟體工程師來講，寫『使用手冊』是非常令人討厭的事，尤其針對快速發展，經常改版的軟體；當然也有很多開源軟體文件寫得很好，例如說功能強大的自由辦公室套件 Libre Office，它的文件就像真的書本一樣。那麼對一些已有『手冊』manual 的命令呢？很多使用過 man 來讀命令手冊的人 ── 比如打 man dmesg ︰dmesg 的命令手冊 ──，都覺得十分難讀！為什麼呢？試著再打一下『 man man 』，你就了解了。然而，為什麼不是呢？最早的『程式設計師』手冊，寫於 1971 年，已經四十多年前了，主撰者包括了Dennis Ritchie 先生，今天知名的『 C 語言』的創始者。只是那個時代最好的的電腦，功能都比不上今天的一個小微處裡器，更不會用『叫什麼 troff 』的文書處理程式下指令來寫文件？？世界上，或許不只空間會有距離，時間也會產生隔閡；也許今天更需要『閱讀』過去的辦法了！以前愛因斯坦特別推重『想像力』，他說他想著『騎在光上』而發現了相對論。那麼如果能想像的『設身處地 』，是否能了解不同的『時代』── 無論是『過去』或『未來』──打開時空之門！

……

此時最好的機會，也許就是『手冊』的了︰

【 man jaaa 】

JAAA(1) General Commands Manual JAAA(1)

NAME
jaaa — JACK and ALSA Audio Analyser

SYNOPSIS
jaaa [-h] [-C nchan] [-J] [-A [-ddevice] [-rrate] [-pperiod] [-nnfraqs] ]

DESCRIPTION
jaaa is an audio signal generator and spectrum analyser designed to make accurate measurements.

OPTIONS
-h Show summary of options.

-C Number of channels (1..8) [4

-J Use JACK

-A Use ALSA (with following options)

-d device Alsa device [hw:0.0]

-r rate Sample frequency [48000]

-p period Period size [1024]

-n nfrag Number of fragments [2]

Either -J or -A must be given.

INTERACTIVE CONTROL
Input :
Select on of the four inputs.

Frequency and Amplitude :
These two sets of buttons set the display view. One of these six buttons, or 'Bandw', 'Peak', or 'Noise' discussed below, has an orange LED at its left side. The LED indicates the 'currenty
selected parameter' that usually can be modified in three ways :

· by typing a new value into the text widget, followed by ENTER

· by using the '<' or '>' buttons to decrement or increment,

· by mouse gestures

Frequency :
Buttons 'Min' and 'Max' set the min and max displayed frequencies. If either of these is selected then

· a horizontal Drag Left changes 'Min'

· a horizontal Drag Right changes 'Max'

Button 'Cent' is the frequency at the middle of the x-axis. Button 'Span' is 'Max' - 'Min', changing this value preserves 'Cent'. If either of these is selected then

· a horizontal Drag Left changes 'Cent'

· a horizontal Drag Right changes 'Span'

Button 'Cent' can also be set by Clicking in the frequency axis scale.

Amplitude :
Button 'Max' is the maximum value on the y-axis. Button 'Range' is the range of the y-axis. If either of these is selected then

· a vertical Drag Left changes 'Max'

· a vertical Drag Right changes 'Range'

So for the last four mouse gestures, a Drag Left will scroll the display, while a Drag Right will zoom in or out. Maybe I will add and automatic selection of the axis based on the direction of the
mouse gesture.

Analyser :
The analyser is based on a windowed FFT. Actually the windowing is performed by convolution after the FFT, and combined with interpolation. The windowing and interpolation ensure that displayed
peaks will be accurate to 0.25 dB even if the peak falls between the FFT bins. More accurate measurements can be made using the markers (see below).

Button 'Bandw' sets the FFT length, and hence the bandwidth of the analyser.

Depending on this value, the size of the display and the frequency range, you may sometimes see two traces.

This happens when the resolution of the analyser is better than the display, so that one pixel contains more than one analyser value.

In that case, the blue trace is the peak value over the frequency range represented by each pixel, and the gray one is the average value.

The first one is correct for discrete frequencies, and the latter should be used to read noise densities.

There is no mouse gesture to change the bandwidth.

Button 'VidAv' or video average, when switched on, averages the measured energy over time. This is mainly used to measure noise. The averaging length increases over time, to a maxumum of 1000
iterations. Changing the input or bandwidth resets and restarts the averaging.

Button 'Freeze' freezes the analyser, but not the display, so you can still scroll and zoom or use the markers discussed below.

Markers :
Markers are used in order to accurately read off values in the display. There can be up to two markers, set by clicking at the desired frequency inside the display. When there are two markers, the
second one will move with each click, while the first remains fixed. Measured values for the two markers, and their difference in frequency and level are displayed in the upper left corner of the
display.

Button 'Clear' clears the markers.

When 'Peak' is selected, clicking inside the display will set a marker at the nearest peak.

The exact frequency and level of the peak are found by interpolation, so the frequency can be much more accurate than the FFT step, and the level corresponds to the true peak value regardless of
display or analyser resolution.

When 'Noise' is selected, clicking inside the display will set a noise marker.

The noise density (energy per Hz) is calculated and displayed.

勇闖新世界︰ W!o《卡夫卡村》變形祭︰品味科學‧教具教材‧ Hi-Jack‧簡介

2015-11-04 懸鉤子

『插頭』 jack 概念之淵源久遠，早在那個『接線生』 Operator 尚為專業的『電話』時代，就操作著『通話配線板』

A telephone switchboard is a telecommunications system used in the public switched telephone network or in enterprises to interconnect circuits of telephones to establish telephone calls between the subscribers or users, or between other exchanges. The switchboard was an essential component of a manual telephone exchange, and was operated by one or more persons, called operators who either used electrical cords or switches to establish the connections.

The electromechanical automatic telephone exchange, invented by Almon Strowger in 1888, gradually replaced manual switchboards in central telephone exchanges starting in 1919 when the Bell System adopted automatic switching, but many manual branch exchanges remained operational during the last half of the 20th century in offices, hotels, or other enterprises. Later electronic devices and computer technology gave the operator access to an abundance of features. In modern businesses, a private branch exchange (PBX) often has an attendant console for the operator, or an auto-attendant, which bypasses the operator entirely.

，完成『收發』兩方之『通訊』。其後或許因為時代進步，也許是教育發達，那個專業也變成了『直覺』而且『簡易』的了！於是乎應用範圍日增漸廣，其中最主要的應用，還是用在『音響設備』之『連接』上︰

Phone connector (audio)

In electronics, a phone connector, also known as phone jack, audio jack or jack plug, is a common family of connector typically used for analog signals, primarily audio. It is cylindrical in shape, typically with two, three or four contacts. Three-contact versions are known as TRS connectors, where T stands for “tip”, R stands for “ring” and S stands for “sleeve”. Similarly, two- and four-contact versions are called TS and TRRS connectors respectively.

The phone connector was invented for use in telephone switchboards in the 19th century and is still widely used. In its original configuration, the outside diameter of the “sleeve” conductor is ¹⁄₄ inch (exactly 6.35 mm). The “mini” connector has a diameter of 3.5 mm (approx. ¹⁄₈ inch) and the “sub-mini” connector has a diameter of 2.5 mm (approx. ³⁄₃₂ inch).

A pair of phone connectors: A phone plug (right) is inserted in a phone socket (left). For terms, see section Other connectors, other terms.

A 6.35 mm (¹⁄₄ inch) two-contact phone connector used for various signals including electric guitar, loudspeaker, microphone and line-level audio.

如此當可理解 JACK 這個專業級的『聲音伺服器』，之所以用著『反覆』之詞自名的哩！

JACK Audio Connection Kit

Introduction

JACK is a low-latency audio server, written for any operating system that is reasonably POSIX compliant. It currently exists for Linux, OS X, Solaris, FreeBSD and Windows. It can connect several client applications to an audio device, and allow them to share audio with each other. Clients can run as separate processes like normal applications, or within the JACK server as “plugins”.

JACK was designed from the ground up for professional audio work, and its design focuses on two key areas: synchronous execution of all clients, and low latency operation.

See also:: <http://jackaudio.org>

JACK Overview

Traditionally it has been hard if not impossible to write audio applications that can share data with each other. In addition, configuring and managing audio interface hardware has often been one of the most complex aspect of writing audio software.

JACK changes all this by providing an API that does several things:

1. provides a high level abstraction for programmers that removes the audio interface hardware from the picture and allows them to concentrate on the core functionality of their software.

2. allows applications to send and receive audio data to/from each other as well as the audio interface. There is no difference in how an application sends or receives data regardless of whether it comes from/goes to another application or an audio interface.

For programmers with experience of several other audio APIs such as PortAudio, Apple’s CoreAudio, Steinberg’s VST and ASIO as well as many others, JACK presents a familiar model: your program provides a “callback” function that will be executed at the right time. Your callback can send and receive data as well as do other signal processing tasks. You are not responsible for managing audio interfaces or threading, and there is no “format negotiation”: all audio data within JACK is represented as 32 bit floating point values.

For those with experiences rooted in the Unix world, JACK presents a somewhat unfamiliar API. Most Unix APIs are based on the read/write model spawned by the “everything is a file” abstraction that Unix is rightly famous for. The problem with this design is that it fails to take the realtime nature of audio interfaces into account, or more precisely, it fails to force application developers to pay sufficient attention to this aspect of their task. In addition, it becomes rather difficult to facilitate inter-application audio routing when different programs are not all running synchronously.

Using JACK within your program is very simple, and typically consists of just:

calling jack_client_open() to connect to the JACK server.
registering “ports” to enable data to be moved to and from your application.
registering a “process callback” which will be called at the right time by the JACK server.
telling JACK that your application is ready to start processing data.

There is a lot more that you can do with JACK’s interfaces, but for many applications, this is all that is needed. The simple_client.c example demonstrates a complete (simple!) JACK application that just copies the signal arriving at its input port to its output port. Similarly, inprocess.c shows how to write an internal client “plugin” that runs within the JACK server process.

───

。同時曉得 JACK 的操作既『直覺』又『簡易』，根本無須再說明的乎！！『出』連『入』，『入』接『出』，『入』『出』間流著『聲音數據』，如是而已矣？？！！

樹莓派、樹莓派之學習、樹莓派之教育

勇闖新世界︰ W!o《卡夫卡村》變形祭︰品味科學‧教具教材‧ Hi-Jack‧Sonic π 之設定

2015-11-03 懸鉤子

固然『插頭』 Jack 並非是為 Sonic π 而設計，由於 Sonic π 相關教育資源越來越豐富，因著此一利益，特別以此為中心來談 Jack 的設定。同時希望更快的進入主題，這裡的說法也不多作解釋，可算是個『快速開始』 quick start 的吧，隨著接續篇章之開展，再於適當處隨處補講的了。

【使用選單啟動 Sonic π 】