3-thai music - ThaiScience

°“√«‘‡§√“–Àå√–¥—∫‡ ’¬ß¥πµ√’‰∑¬
ƒ¥’ √— µ πå ™‘ 𠇫™°‘ ® «“π‘ ™ ¬å 1*, √“«ÿ ≤‘ ÿ ®‘ µ ®√ 2, °‘ µ µ‘ Õ— µ ∂°‘ ® ¡ß§≈ 3 ·≈–
‡æ’¬√ ‚µ∑à“‚√ß4
Chinvejkitvanich, R.1*, Sujitjorn, S.2, Attakitmongcol, K.3, and Totarong, P.4 (2004). Thai Musical Signal
Analysis. Suranaree J. Sci. Technol. 11:179-192.
Recieved: Mar 4, 2004; Revised: Aug 10, 2004; Accepted: Aug 14, 2004
Abstract
Analysis of Thai principal notes was carried out for the octave referred to as ‘‘Thang Phiang Aw’’.
Four different techniques of digital processing, namely discrete Fourier transform (DFT), short-time
Fourier transform (STFT), autoregressive model (AR), and modal distribution (MD) werd used.
The results indicated that the modal distribution technique was superior to the others. The main Phiang
Aw octave covered the frequencies of 465-940 Hz. The pitches were not constant, which was in contrast
to the previous hypothesis of Morton’s.
Keywords: Musical signal analysis, traditional Thai music
∫∑§—¥¬àÕ
°“√«‘‡§√“–Àå√–¥—∫‡ ’¬ß‡ ’¬ß¥πµ√’‰∑¬ ¥”‡π‘πß“π°—∫∑∫‡ ’¬ß°≈“ß∑“ߥπµ√’‰∑¬ ∑’‡Ë √’¬°«à“ ç∑“߇撬ßÕÕé
‰¥âπ”‡∑§π‘§°“√ª√–¡«≈º≈ —≠≠“≥¥‘®‘µÕ≈ ’Ë·∫∫¡“ª√–¬ÿ°µå ‰¥â·°à ‡∑§π‘§°“√·ª≈ßøŸ√‘‡¬√凵Á¡Àπ૬
‡∑§π‘§°“√·ª≈ßøŸ√‘‡¬√å„π™à«ß‡«≈“ —Èπ ·∫∫®”≈Õ߇ÕÕ“√å ·≈–°“√°√–®“¬‡™‘ß‚¡¥ ‡¡◊ËÕ‡ª√’¬∫‡∑’¬∫º≈æ∫
«à“°“√°√–®“¬‡™‘ß‚¡¥„Àâº≈¥’∑ ’Ë ¥ÿ ‡ ’¬ß‡æ’¬ßÕÕ¢Õ߉∑¬„πÀπ÷ßË ∑∫‡ ’¬ß ¡’§«“¡∂’„Ë π¬à“π 465-940 ‡Œ‘√µ´å
¡’√–¬–摵™å‰¡à§ß∑’Ë¡‘‰¥â‡ªìπ‰ªµ“¡ ¡¡ÿµ‘∞“π¥—È߇¥‘¡¢Õß¡Õ√åµ—π
1
π—°»÷°…“∫—≥±‘µ»÷°…“ “¢“«‘™“«‘»«°√√¡‰øøÑ“ ”π—°«‘™“«‘»«°√√¡»“ µ√å ¡À“«‘∑¬“≈—¬‡∑§‚π‚≈¬’ ÿ√π“√’
Õ”‡¿Õ‡¡◊Õß ®—ßÀ«—¥π§√√“™ ’¡“ 30000 ‚∑√»—æ∑å 0-4422-4400 E-mail: [email protected]
2
Ph.D., √Õß»“ µ√“®“√¬å À—«Àπâ“ “¢“«‘™“«‘»«°√√¡‰øøÑ“ ¡À“«‘∑¬“≈—¬‡∑§‚π‚≈¬’ ÿ√π“√’
3
Ph.D., ºŸâ™à«¬»“ µ√“®“√¬å “¢“«‘™“«‘»«°√√¡‰øøÑ“ ¡À“«‘∑¬“≈—¬‡∑§‚π‚≈¬’ ÿ√π“√’
4
æ≈Õ“°“»µ√’ ºŸâÕ”π«¬°“√ ∂“∫—π¡“µ√«‘∑¬“·Ààß™“µ‘
*
ºŸâ‡¢’¬π∑’Ë„Àâ°“√µ‘¥µàÕ
«“√ “√‡∑§‚π‚≈¬’ ÿ√π“√’ 11:179-192
180
°“√«‘‡§√“–Àå√–¥—∫‡ ’¬ß¥πµ√’‰∑¬
∫∑π”
‡ ’¬ß‡√‘¡Ë ‡°‘¥¢÷πÈ ‡¡◊ÕË «—µ∂ÿÀ√◊Õ·À≈àß°”‡π‘¥‡ ’¬ß¡’°“√ —πË
àߺ≈µàÕ°“√‡§≈◊ÕË π∑’¢Ë Õß‚¡‡≈°ÿ≈Õ“°“»∑’ÕË ¬Ÿ‚à ¥¬√Õ∫
‡°‘¥°“√‡§≈◊ÕË π∑’‡Ë ¢â“™π°—π·≈–·¬°ÕÕ°®“°°—π ∑”„Àâ
§«“¡¥—π¢ÕßÕ“°“»‡æ‘Ë¡¢÷Èπ·≈–≈¥≈ß®“°ª°µ‘ ´÷Ëß
°“√‡§≈◊ËÕπ∑’Ëπ’ȇ°‘¥ ≈—∫°—π‰ª¡“ °àÕ°”‡π‘¥‡ªìπ
§≈◊Ëπ‡ ’¬ß¢÷Èπ (Alten, 1999) ¡πÿ…¬å‰¥â¬‘π‡ ’¬ß∑’Ë¡’
§«“¡∂’ËÕ¬Ÿà„π™à«ßª√–¡“≥ 20 ∂÷ß 20,000 ‡Œ‘√µ´å
·≈–√—∫√Ÿâ≈—°…≥–¢Õ߇ ’¬ß¥â«¬§«“¡∑ÿâ¡·À≈¡¢Õß
‡ ’ ¬ ßÀ√◊ Õ æ‘ µ ™å (pitch) ·≈–§«“¡¥— ß ¢Õ߇ ’ ¬ ß
(loudness) (∫√‘…∑— √’¥‡¥Õ√å ‰¥‡® ∑å (ª√–‡∑»‰∑¬)
®”°—¥, 2541) ‡ ’¬ß¥πµ√’‡ªìπ‡ ’¬ß™π‘¥Àπ÷Ë߇°‘¥®“°
°“√ —Ëπ‡™‘ß°≈¢Õßµ—« —Ëπ (oscillator) ´÷Ë߉¥â√—∫°“√
°√–µÿâ π „π√Ÿ ª ·∫∫„¥√Ÿ ª ·∫∫Àπ÷Ë ß à ß º≈∑”„Àâ
à«πµà“ß Ê ¢Õ߇§√◊ËÕߥπµ√’ —Ëπ (Mitra, 2001)
”À√—∫‡§√◊ËÕߥπµ√’ª√–‡¿∑‡§√◊ËÕ߇ªÉ“ ‡™àπ ¢≈ÿà¬
∑—ßÈ ¢Õßµà“ߪ√–‡∑»·≈–¢Õ߉∑¬ ‡¡◊ÕË ‡ªÉ“≈¡‡¢â“‰ª„π
™àÕß«à“ß¿“¬„π°√–∫Õ° ‚¡‡≈°ÿ≈¢ÕßÕ“°“»∑’ËÕ¬Ÿà
¿“¬„π®–‡§≈◊ËÕπ∑’Ë°≈—∫‰ª°≈—∫¡“µ“¡§«“¡¬“«¢Õß
°√–∫Õ°‡°‘¥‡ªìπ‡ ’¬ß ´÷Ë߇ ’¬ß¥—ß°≈à“«π’ÈÀ“°«—¥√Ÿª
§≈◊πË ‰¥â ®–æ∫«à“¡’≈°— …≥–„°≈⇧’¬ß°—∫√Ÿª§≈◊πË ‰´πå
(Cannon, 1976) ‡ ’¬ß¥πµ√’®÷߇°‘¥®“°°“√ —Ëπ
∑—ÈßÀ¡¥∑’Ë√«¡°—πÕ¬Ÿà„π‡§√◊ËÕߥπµ√’™‘Èππ—Èπ Ê ∑”„Àâ
‡ ’¬ß¥πµ√’ª√–°Õ∫¥â«¬À≈“¬§«“¡∂’∑Ë À’Ë ‰Ÿ ¡à “¡“√∂
·¬°·¬–‰¥â«à“¡’§à“§«“¡∂’Ë„¥∫â“ß ·µà®–‰¥â¬‘π‡ ’¬ß
‚¥¬√«¡∑’ˇªìπº≈¢ÕßÀ≈“¬§«“¡∂’Ëæ√âÕ¡ Ê °—π
‡ ’¬ß¥πµ√’‡ªìπ‡ ’¬ß∑’Ë¡πÿ…¬å„À⧫“¡ π„®
»÷°…“√“¬≈–‡Õ’¬¥ ß“π«‘®—¬∑’Ë∑”°“√«‘‡§√“–Àå‡ ’¬ß
¥πµ√’ ∑’Ë ◊ ∫ §â π ‰¥â ‡ ªì π ¢Õߥπµ√’ µ –«— π µ°‡°◊ Õ ∫
∑—ÈßÀ¡¥ ¥πµ√’‰∑¬·≈–¥πµ√’ “°≈¡’§«“¡∂’Ë¢Õß
·µà≈–‡ ’¬ßµ—«‚πⵉ¡à‡∑à“°—π ‡π◊ËÕß®“°¥πµ√’ “°≈
·∫àßÀπ÷Ëß∑∫‡ ’¬ß (octave) ÕÕ°‡ªìπÀⓇ ’¬ß‡µÁ¡
(whole tone) ·≈– Õߧ√÷Ë߇ ’¬ß (semi tone)
à«π¥πµ√’‰∑¬π—Èπ ·∫àßÀπ÷Ëß∑∫‡ ’¬ßÕÕ°‡ªìπ‡®Á¥
‡ ’¬ß‡µÁ¡ ´÷Ëß¡’§«“¡‡™◊ËÕ«à“‚πâµ·µà≈–‡ ’¬ß¡’√–¬–
Àà“ß∑“ߧ«“¡∂’ˇ∑à“ Ê °—πÀ¡¥ ( —≥∑—¥ µ—≥±π—π∑å,
2542) ¥πµ√’‰∑¬¬—߉¡à¡°’ “√ª√–°“»√–¥—∫‡ ’¬ß∑’‡Ë ªìπ
¡“µ√∞“π„π‡™‘ß«‘∑¬“»“ µ√å∑”πÕ߇¥’¬«°—∫¢Õß
¥πµ√’ “°≈ ”À√—∫°“√‡∑’¬∫‡ ’¬ß ·µà¡’°“√°≈à“«∂÷ß
‚¥¬ Õÿ∑‘» 𓧠«— ¥‘Ï (2514) «à“ Õ—µ√“ à«π§«“¡∂’Ë
√–À«à“߇ ’¬ß‚πâµ∑’˵‘¥°—π (√“¬≈–‡Õ’¬¥ª√“°Ø„π
º≈°“√∑¥≈Õß·≈–«‘®“√≥å) ¡’§à“‡∑à“°—∫ 1.09745 ·µà
¡‘‰¥âÕ∏‘∫“¬∑’Ë¡“¢Õßµ—«‡≈¢¥—ß°≈à“« Õ¬à“߉√°Áµ“¡
À“°„™â°“√ª√–¡“≥Õ“®æ‘®“√≥“«à“§à“¥—ß°≈à“«‡∑à“°—∫
1.1 ‚¥¬ª√–¡“≥ (ºŸâ‡¢’¬π) πÕ°®“°π—Èπ„π‡Õ° “√
¥— ß °≈à “ «°Á ‰ ¥â 𔇠πÕ§à “ §«“¡∂’Ë „ π‡™‘ ß ∑ƒ…Æ’ ´÷Ë ß
Õâ“ßÕ‘ß®“°§«“¡∂’Ë¢Õ߇ ’¬ß‡ªï¬‚π (µ“√“ß∑’Ë 1) ‚¥¬
¡‘‰¥â· ¥ß∑’Ë¡“ µ“¡ª°µ‘·≈â«¥πµ√’‰∑¬‡∑’¬∫‡ ’¬ß
¥â«¬§«“¡§ÿâπ‡§¬·≈–§«“¡™”π“≠¢ÕߺŸâ‡ªìπÀ≈—°
¢Õ߫ߥπµ√’ °“√‡∑’¬∫‡ ’¬ß¢Õ߇§√◊ËÕߥπµ√’∑’Ë®–
º ¡‡ªìπ«ß‡¥’¬«°—ππ—Èπ ¬÷¥‡ ’¬ß¢Õ߇§√◊ËÕߥπµ√’„π
«ß∑’ˇ≈◊ËÕπ≈¥‡ ’¬ß‰¡à‰¥â‡ªìπÀ≈—°‡æ◊ËÕ‡∑’¬∫‡ ’¬ß ‡™àπ
¢≈ÿ¬à ªïò √–𓥇À≈Á° ‡ªìπµâπ (¡πµ√’ µ√“‚¡∑, 2540)
·≈–‰¥â¡’°“√„™â âÕ¡‡ ’¬ß (turning fork) ‡™àπ‡¥’¬«
°—∫¥πµ√’ “°≈
∫∑§«“¡π’πÈ ”‡ πÕ«‘∏°’ “√«‘‡§√“–Àå√–¥—∫‡ ’¬ß
¢Õß —≠≠“≥‡ ’¬ß¥πµ√’‰∑¬ ‚¥¬¡ÿà߉ª∑’ˇ ’¬ß‚πâµ
À≈—°¢Õߢ≈ÿଇ撬ßÕÕ·≈–√–π“¥‡Õ°‡À≈Á° ´÷Ë߇ªìπ
‡§√◊Ë Õ ß¥πµ√’ À ≈— ° ”À√— ∫ °“√‡∑’ ¬ ∫‡ ’ ¬ ߫߇§√◊Ë Õ ß
“¬‰∑¬·≈–¡‚À√’ ¥â«¬‡∑§π‘§°“√·ª≈ßøŸ√‘‡¬√凵Á¡
Àπ૬ (discrete Fourier transform À√◊Õ DFT)
‡∑§π‘§°“√·ª≈ßøŸ√‘‡¬√å„π™à«ß‡«≈“ —Èπ (short-time
Fourier transform À√◊ Õ STFT) ‡∑§π‘ § °“√
«‘‡§√“–Àå —≠≠“≥¥â«¬·∫∫®”≈Õ߇ÕÕ“√å (autoregressive model À√◊Õ AR) ·≈–‡∑§π‘§°“√°√–®“¬
‡™‘ß‚¡¥ (modal distribution À√◊Õ MD) ·≈â«π”
º≈∑’ˉ¥â®“°°“√§”π«≥¡“‡ª√’¬∫‡∑’¬∫°—π‡æ◊ËÕ§«“¡
∂Ÿ°µâÕß·¡à𬔠‡π◊ËÕß®“°‡ ’¬ß¥πµ√’‡ªìπ —≠≠“≥∑’Ë
º—π·ª√µ“¡‡«≈“·≈–‰¡à¡√’ “¬§“∫™—¥‡®π ‡∑§π‘§ DFT
®÷ßÕ“®‰¡à‡À¡“–∑’Ë®–„™â°—∫ —≠≠“≥‡ ’¬ßª√–‡¿∑π’È
(Pielemeier et al., 1996) ·µà∂â“°“√‡≈àπ‡§√◊ËÕߥπµ√’
ºŸâ ‡ ≈à π ®ß„®∑”„Àâ ‡ °‘ ¥ ‡ ’ ¬ ß‚πâ µ ‡¥’Ë ¬ «Õ¬à “ ßµà Õ ‡π◊Ë Õ ß
¡Ë”‡ ¡Õ ‡ ’¬ßπ—ÈπÕ“®· ¥ß§«“¡‡ªìπ√“¬§“∫
‡∑§π‘§ DFT ®÷ßÕ“®π”¡“ª√–¬ÿ°µå‰¥â„π‡∫◊ÈÕßµâπ
‡∑§π‘§ STFT ‡ªìπ«‘∏’æ◊Èπ∞“π„π°“√«‘‡§√“–Àå
181
«“√ “√‡∑§‚π‚≈¬’ ÿ√π“√’ ªï∑’Ë 11 ©∫—∫∑’Ë 3, °√°Æ“§¡-°—𬓬π 2547
—≠≠“≥∑’Ë·ª√µ“¡‡«≈“·≈–§«“¡∂’Ë (Cohen, 1989)
‚¥¬®–π”øíß°å™—πÀπ⓵à“ß (window function) ¡“
§Ÿ≥°—∫ —≠≠“≥‡æ◊ËÕ·∫àß —≠≠“≥ÕÕ°‡ªìπ™à«ß °“√
‡≈◊Õ°øíß°å™π— Àπ⓵à“ß„Àâ‡À¡“– ¡¬—ߧ߇ªìπªí≠À“Õ¬Ÿà
‡æ√“–µâÕ߇≈◊Õ°√–À«à“ߧ«“¡ “¡“√∂·¬°‡«≈“ (time
resolution) °—∫§«“¡ “¡“√∂·¬°§«“¡∂’Ë (frequency
resolution) °≈à“«§◊Õ∂⓵âÕß°“√§«“¡ “¡“√∂∑’Ë¥’„π
°“√·¬°§«“¡∂’Ë ®”‡ªìπµâÕß„™âøíß°å™—πÀπ⓵à“ß∑’Ë¡’
§«“¡°«â“ß¡“° Ê ®–∑”„Àâ‡∑§π‘§ STFT ¡’§«“¡
“¡“√∂·¬°‡«≈“‰¥â‰¡à¥π’ °— (Pielemeier et al., 1996)
‡∑§π‘§ MD ‰¥â√—∫°“√æ—≤π“¢÷Èπ‡æ◊ËÕ≈¥¢âÕ®”°—¥
¢Õ߇∑§π‘§ STFT ·≈–‡∑§π‘§°“√°√–®“¬«‘°‡πÕ√å
(Wigner distribution À√◊Õ WD) ·≈– “¡“√∂
µ√«®®— ∫ Õß§å ª √–°Õ∫∑“ߧ«“¡∂’Ë ∑’Ë ‡ ª≈’Ë ¬ π·ª≈ß
µ“¡‡«≈“Õ¬à “ ß√«¥‡√Á « ¢Õß — ≠ ≠“≥‡ ’ ¬ ߥπµ√’
‰¥â¥’ (Pielemeier and Wakefield, 1996) à«π‡∑§π‘§
AR π—Èπ¡’§«“¡ “¡“√∂„π°“√µ√«®®—∫§«“¡∂’Ë∑’Ë
‡ª≈’¬Ë π·ª≈ßµ“¡‡«≈“Õ¬à“ß√«¥‡√Á«¢Õß —≠≠“≥‡ ’¬ß
查‰¥â¥’ (Totarong, 1983) ´÷ËßÕ“®„Àâº≈¥’µàÕ°“√
«‘‡§√“–Àå‡ ’¬ß¥πµ√’‰¥â Õ¬à“߉√°Áµ“¡ °“√‡≈◊Õ°
Õ—π¥—∫¢Õß·∫∫®”≈Õß AR „Àâ‡À¡“– ¡¬—߉¡à¡’
À≈—°‡°≥±å∑’Ë·πàπÕπ º≈∑’ˉ¥â®“°°“√«‘®—¬π’È®–™à«¬
„Àâ∑√“∫‰¥â«à“‡∑§π‘§°“√«‘‡§√“–Àå√Ÿª·∫∫„¥®–„Àâ
º≈¥’°«à“·∫∫Õ◊Ëπ Ê ´÷Ëß®–‡ªìπª√–‚¬™πåµàÕ°“√π”
‰ªª√–¬ÿ°µå‡æ◊ËÕ«‘‡§√“–Àå‡ ’¬ß¥πµ√’‰∑¬∑’Ë´—∫´âÕπ
¬‘Ëߢ÷È𠇙àπ ‡ ’¬ß°“√∫√√‡≈ߥ⫬‡∑§π‘§æ‘‡»…µà“ß Ê
‡ªìπµâπ πÕ°®“°π—Èπº≈«‘®—¬®–„Àâ¢âÕ¡Ÿ≈√“¬≈–‡Õ’¬¥
¢Õߧ«“¡∂’Ë „ πÀπ÷Ë ß ∑∫‡ ’ ¬ ß µ≈Õ¥®π≈— ° …≥–
‡©æ“–¢Õ߇ ’¬ß¥πµ√’‰∑¬∑’Ë∫√√‡≈ß‚πⵇ¥’ˬ«
«— ¥ÿ Õÿª°√≥å ·≈–«‘∏’°“√
Õÿª°√≥å∑’Ë„™â„π°“√‡°Á∫¢âÕ¡Ÿ≈¡’¥—ßπ’È
● ¢≈ÿଇ撬ßÕÕ 1 ‡≈“ (‰¡â¡–√‘¥)
● √–π“¥‡Õ°‡À≈Á° 1 √“ß (‡§√◊ËÕߪ√–®”«ß¥πµ√’
‰∑¬ ¡À“«‘∑¬“≈—¬‡∑§‚π‚≈¬’ ÿ√π“√’)
● ‰¡‚§√‚øπ AV-JFET √ÿàπ AVL110EM
● ÕÕ ´‘≈‚≈ ‚§ª Tektronic √ÿàπ TDS 420A
● ‡∑Õ√å‚¡¡‘‡µÕ√å (thermometer)
● ‰Œ‚°√¡‘‡µÕ√å (hygrometer)
● ¡‘‡µÕ√å«¥
— √–¥—∫‡ ’¬ß¥‘®µ‘ Õ≈ (digital sound level
meter) √ÿàπ TES-1352
¥”‡π‘π°“√‡°Á∫¢âÕ¡Ÿ≈∑’ËÀâÕߥπµ√’‰∑¬ Õ“§“√
à«π°‘®°“√π—°»÷°…“ ¡À“«‘∑¬“≈—¬‡∑§‚π‚≈¬’ √ÿ π“√’
´÷Ë߇ªìπÀâÕߪæ◊ËÕªÑÕß°—π‡ ’¬ß√∫°«π Õÿ≥À¿Ÿ¡‘
¿“¬„πÀâÕ߇©≈’ˬլŸà∑’Ë 27 Õß»“‡´≈‡´’¬ §«“¡™◊Èπ
—¡æ—∑∏åª√–¡“≥ 67 ‡ªÕ√凴Áπµå §«“¡¥—ߢÕ߇ ’¬ß
¢≥–∫√√‡≈ߢ≈ÿà ¬ ‡æ’ ¬ ßÕÕ·≈–√–π“¥‡Õ°‡À≈Á °
§«∫§ÿ¡‰«â„ÀâÕ¬Ÿà∑’˪√–¡“≥ 76 ‡¥´‘‡∫≈ ·≈– 80
‡¥´‘‡∫≈ µ“¡≈”¥—∫ °“√‡ªÉ“¢≈ÿ଄™â°“√√—°…“§«“¡
µà Õ ‡π◊Ë Õ ß¢Õß≈¡ ·≈–√— ° …“§«“¡ ¡Ë” ‡ ¡Õ¢Õß
§«“¡·√ß≈¡ ‚¥¬‰¡à®”‡ªìπµâÕß ç√–∫“¬≈¡é °“√µ’
Table 1. Comparison of Thai and Western notes in one octave (previous results).
(Õÿ∑‘» 𓧠«— ¥‘Ï, 2514)
One octave of
Thai musical notes
do
rae
mee
fa
sol
la
tee
do'
Frequency of piano
(Hz)
524
587
659
698
784
880
988
1,048
Frequency of Thai music
(Hz)
524
579
639
705
779
860
949
1,048
182
°“√«‘‡§√“–Àå√–¥—∫‡ ’¬ß¥πµ√’‰∑¬
√–𓥇À≈Á°„™â°“√ ç√—«é √—°…“§«“¡ ¡Ë”‡ ¡Õ
¢Õß·√ß‚¥¬√—«‰≈à‰ª∑’≈–≈Ÿ°√–π“¥®π§√∫∑∫‡ ’¬ß
„π¢≥–∑”°“√‡°Á ∫ ¢â Õ ¡Ÿ ≈ µ”·Àπà ß ¢Õß°“√«“ß
Õÿª°√≥å §ß∑’Ë (√Ÿª∑’Ë 1) ‰¡‚§√‚øπ∑’Ë„™â‡ªìπ
™π‘¥Õ‘‡≈Á°‡∑√¥ (electret) ¡’·∫∫√Ÿª°“√·ºàæ≈—ßß“π
(radiation pattern) √Õ∫∑‘»∑“ß (omnidirection)
¡’§à“§«“¡‰« -52 ‡¥´‘‡∫≈ ·≈–¡’º≈µÕ∫ πÕß∑“ß
§«“¡∂’Ë„π™à«ß 50 ‡Œ‘√µ´å ∂÷ß 18,000 ‡Œ‘√µ´å ´÷Ëß¡’
≈—°…≥– ¡∫—µ‘„°≈⇧’¬ß°—∫‰¡‚§√‚øπ∑’Ë ◊∫§âπ‰¥â
«à“‡À¡“– ”À√—∫„™â«¥— ‡ ’¬ß¥πµ√’ª√–‡¿∑‡§√◊ÕË ß≈¡‰¡â
(woodwind instrument) ( √“«ÿ≤‘ ÿ®‘µ®√, 2545)
°√≥’∫π— ∑÷°‡ ’¬ß¢≈ÿ¬à ‡æ’¬ßÕÕ ‰¡‚§√‚øπ®–«“ß∑”¡ÿ¡
50 Õß»“°—∫·∑àπ®—∫ Õ¬ŸàÀà“ß®“°¢≈ÿଠ3 π‘È« ·≈–
°√≥’∫π— ∑÷°‡ ’¬ß√–π“¥‡Õ°‡À≈Á° ‰¡‚§√‚øπ®–«“ß
∑”¡ÿ¡ 80 Õß»“°—∫·∑àπ®—∫·≈–Õ¬Ÿàµ√ßµ”·Àπàß
≈Ÿ°´Õ≈ Àà“ß®“°µ—«√–π“¥ 5 π‘È« °“√«“ßµ”·Àπàß
¢Õ߉¡‚§√‚øπ„π≈— ° …≥–¥— ß °≈à “ «°Á ‡ æ◊Ë Õ „Àâ
‰¡‚§√‚øπÕ¬Ÿà °÷Ë ß °≈“ߢÕß·À≈à ß °”‡π‘ ¥ ‡ ’ ¬ ß
à«πÕÕ ´‘≈‚≈ ‚§ª«“ßÀà“߇ªìπ√–¬– ÿ¥§«“¡¬“«
“¬¢Õ߉¡‚§√‚øπ‡æ◊ËÕªÑÕß°—π°“√‡°‘¥ —≠≠“≥
√∫°«π√–À«à“ß “¬ ·≈–ªÑÕß°—π‰¡à„À⧫“¡¥—ß
¢Õ߇ ’¬ß‡§√◊ËÕßÕÕ ´‘≈‚≈ ‚§ª¢≥–∑”ß“π·∑√°
Õ¥≈߉¡‚§√‚øπ‰¥â °“√∫—π∑÷°√Ÿª§≈◊Ëπ‡ ’¬ß‚πâµ
‡¥’ˬ«¢Õߢ≈ÿଇ撬ßÕÕ ·≈–√–π“¥‡Õ°‡À≈Á°∑—Èß·ª¥
' „π°√≥’π’È∂◊Õ
‡ ’¬ß (‚¥ ‡√ ¡’ ø“ ´Õ≈ ≈“ ∑’ ‚¥)
‰¥â«à“ ‡ ’¬ß∑’ˇ°‘¥¢÷Èπ¡’§«“¡ ¡Ë”‡ ¡Õ °“√∫—π∑÷°‰¥â
„™âÕÕ ´‘≈‚≈ ‚§ª∑”°“√‡°Á∫¢âÕ¡Ÿ≈®”π«π 2,500
®ÿ¥ ∑’ËÕ—µ√“°“√™—°µ—«Õ¬à“ß (sampling rate) 250
°‘‚≈·´¡‡ªîô≈µàÕ«‘π“∑’ (kS/sec) ¢âÕ¡Ÿ≈∑’Ë∫—π∑÷°‰¥âπ’È
𔉪ª√–¡«≈º≈‚¥¬Õ“»—¬‚ª√·°√¡ MATLABTM
√Ÿª∑’Ë 2 (a) · ¥ß¿“æ√Ÿª§≈◊Ëπ‡ ’¬ß´Õ≈¢Õß
¢≈ÿ¬à ‡æ’¬ßÕÕ∑’∫Ë π— ∑÷°‰¥â ®– —߇°µ‡ÀÁπ«à“√Ÿª§≈◊πË ‡Õ’¬ß
≈“¥≈ß ª√“°Ø°“√·∑√° Õ¥®“° —≠≠“≥√∫°«π
æ◊πÈ À≈—ß (background noise) §«“¡∂’µË Ë”„π™à«ß 0 ∂÷ß
Voltage (mV)
Voltage (mV)
Figure 1. Instruments setup.
Time (ms)
Time (ms)
(a)
(b)
Figure 2. Signal of “sol” note played by a middle pitch Thai flute recorded signal (a) and
filtered signal (b).
183
«“√ “√‡∑§‚π‚≈¬’ ÿ√π“√’ ªï∑’Ë 11 ©∫—∫∑’Ë 3, °√°Æ“§¡-°—𬓬π 2547
150 ‡Œ‘√µ´å ·≈–§«“¡∂’Ë Ÿß‡°‘π 20,000 ‡Œ‘√µ´å
(æ‘®“√≥“®“°º≈°“√·ª≈ßøŸ√‘‡¬√å) ¥—ßπ—Èπ°àÕπ∑’Ë®–
∑”°“√«‘ ‡ §√“–Àå · µà ≈ –‡ ’ ¬ ß ®÷ ß µâ Õ ßπ”¢â Õ ¡Ÿ ≈ ∑’Ë
∫—π∑÷°‰¥â‰ªºà“π°“√°√Õß —≠≠“≥·∫∫∫—µ‡µÕ√凫‘√∏å
ºà“π·∂∫ (bandpass Butterworth filtering) ¡’§à“
§«“¡∂’µË ¥— (cutoff frequency) Õ¬Ÿ∑à ’Ë 175 ·≈– 25,000
‡Œ‘ √ µ´å — ≠ ≠“≥‡ ’ ¬ ßÀ≈— ß ®“°∑’Ë ºà “ π°“√°√Õß
—≠≠“≥√∫°«π∑‘Èß·≈â« ¡’√Ÿª≈—°…≥–§ß∑’Ë (√Ÿª∑’Ë 2
(b))
‡∑§π‘§°“√«‘‡§√“–Àå —≠≠“≥
‡∑§π‘§°“√«‘‡§√“–Àå —≠≠“≥∑’Ë®–°≈à“«∂÷ßµàÕ‰ªπ’È
µâÕß°“√§à“æ“√“¡‘‡µÕ√åµà“ß Ê ∑’Ë∂Ÿ°°”Àπ¥¢÷ÈπÕ¬à“ß
‡À¡“– ¡ §«“¡‡À¡“– ¡π—Èπ¥Ÿ‰¥â®“°º≈«‘‡§√“–Àå
∑’Ë®–„Àâ§à“øÕ√å·¡π∑å¢Õ߇ ’¬ß‚¥·≈–‚¥' ¡’ —¥ à«π
‡ªìπ Õ߇∑à“·°à°—πµ“¡§«“¡À¡“¬¢ÕßÀπ÷Ëß∑∫‡ ’¬ß
‡∑§π‘§ DFT (Oppenheim and Schafer, 1989)
—≠≠“≥‡µÁ¡Àπ૬∑’ˇªìπ√“¬§“∫ ¡’≈—°…≥–
¢ÕߢâÕ¡Ÿ≈‡ªìπ≈”¥—∫·≈–¡’§“∫‡∑à“°—∫ N xƒ(n))
Õ“®‡¢’¬π·∑π‰¥â¥«â ¬º≈√«¡¢Õß≈”¥—∫‡Õ°´å‚æ‡ππ‡™’¬≈‡™‘ß´âÕπ ek (n) ´÷Ëß e k (n) = e j(2π / N)kn ‡¡◊ËÕ
k ‡ªìπ®”π«π‡µÁ¡ ¥—ß ¡°“√ (1) º≈√«¡π’ȇ√’¬°«à“
Õπÿ°√¡øŸ√‡‘ ¬√凵Á¡Àπ૬ (discrete Fourier series À√◊Õ
DFS) Xƒ ( k ) §◊Õ§à“ —¡ª√– ‘∑∏åÕπÿ°√¡øŸ√‘‡¬√凵Á¡
Àπ૬ ‚¥¬§”π«≥‰¥â®“° xƒ(n) „π√Ÿª·∫∫§«“¡
—¡æ—π∏嵓¡ ¡°“√ (2)
2π
N −1
ƒ
x(n)
=
j( )kn
1
ƒ e N
X(k)
N k =0
∑
N −1
ƒ (k) =
X
∑
xƒ( n ) e
− j(
2π
) kn
N
(1)
(2)
n=0
‡¡◊ËÕæ‘®“√≥“ —≠≠“≥‡µÁ¡Àπ૬∑’Ë¡’≈—°…≥–
¢ÕߢâÕ¡Ÿ≈‡ªìπ≈”¥—∫ x(n) ·≈–¡’®”π«π N ¢âÕ¡Ÿ≈
´÷Ëß∂Ÿ°°”Àπ¥„À⇪ìπÀπ÷Ëߧ“∫¢Õß xƒ(n) ·≈– x(n)
¡’§à“Õ¬Ÿà„π™à«ß 0 < n < N-1 §à“ —¡ª√– ‘∑∏‘Ï¢Õß
x(n) ®÷ßÀ“‰¥â®“°°“√ª√–¬ÿ°µå DFS ¢Õß xƒ(n) ‚¥¬
§”π«≥ ‡æ’¬ßÀπ÷Ëߧ“∫ · ¥ß¥—ß ¡°“√ (3)
N-1
X(k) =
∑
-j(
x(n)e
2π
) kn
N
,0
≤ k ≤ N −1
(3)
n=0
‡√’¬°«à“°“√·ª≈ßøŸ√‘‡¬√凵Á¡Àπ૬ (discrete Fourier
transform À√◊Õ DFT) X(k) ‡ªìπª√‘¡“≥‡™‘ß´âÕπ
‡ª°µ√—¡¢π“¥¢Õß X(k) „Àâ¢âÕ¡Ÿ≈§à“ x(n) ´÷Ëß
· ¥ß∂÷ߧ«“¡∂’Ë‚πâµÀ≈—°¢Õ߇ ’¬ß¥πµ√’ ´÷Ë߇√’¬°
°—π«à“ øÕ√å·¡π∑å (formant) ( √“«ÿ≤‘ ÿ®µ‘ ®√, 2545)
°√“ø¢Õß ‡ª°µ√— ¡ ¢π“¥¢Õß X(k) ®÷ ß µ’ · ºà
øÕ√å·¡π∑å¢Õß‚πâµ·µà≈–‡ ’¬ß °“√§”π«≥¥â«¬
‚ª√·°√¡ MATLABTM °”Àπ¥®”π«π®ÿ¥„π°“√
§”π«≥ DFT ‡∑à“°—∫ 17,500 ®ÿ¥ ´÷Ë߇ªìπ°“√‡æ‘Ë¡
®”π«π®ÿ¥„π°“√§”π«≥ DFT „Àâ¡“°°«à“®”π«π
¢ÕߢâÕ¡Ÿ≈‚¥¬®”π«π®ÿ¥∑’ˇæ‘Ë¡¢÷Èπ§◊Õ§à“»Ÿπ¬å ∑”„Àâ
µ”·ÀπàߢÕ߇ â𠇪°µ√—¡¡’§«“¡∂Ÿ°µâÕß·¡àπ¬”
¡“°¢÷Èπ ®”π«π 17,500 ®ÿ¥π’È¡“®“°°“√ ÿࡇ≈◊Õ°
§à“µà“ß Ê ‡™àπ 5,000 7,500 8,192 10,000 ·≈–
16,384 ‡ªìπµâπ π”¡“∑¥≈Õߧ”π«≥ DFT ¢Õß
‚πâ µ ∑—È ß ·ª¥‡ ’ ¬ ß ·≈–‡≈◊ Õ °®”π«π®ÿ ¥ ∑’Ë „ Àâ §à “
§«“¡∂’Ë¢Õ߇ ’¬ß‚¥' „°≈⇧’¬ßÀ√◊Õ‡ªìπ Õ߇∑à“¢Õß
‡ ’¬ß‚¥
‡∑§π‘§ STFT (Mitra, 2001)
‡ ¡◊Ë Õ ∑” ° “ √ «‘ ‡ § √ “ – Àå — ≠ ≠ “ ≥ ∑’Ë ‰ ¡à π‘Ë ß
(stationary signal) °“√·ª≈ß∑’ËÕ“®π”¡“„™â‡æ◊ËÕ
«‘‡§√“–À凪ìπ°“√·ª≈ßøŸ√‡‘ ¬√剡àÕ ‘ √–∑“߇«≈“ (timedependent Fourier transform) À√◊Õ∑’ËÕ“®‡√’¬°«à“
°“√·ª≈ßøŸ√‘‡¬√å„π™à«ß‡«≈“ —Èπ (short-time Fourier
transform À√◊Õ STFT) °“√·ª≈ߥ—ß°≈à“«®–·∫àß
—≠≠“≥ x(n) ÕÕ°‡ªìπ à«π —Èπ Ê ´÷Ëß·µà≈– à«π¡’
™à«ß‡«≈“∑’ˇ∑à“°—π ‚¥¬π”øíß°å™—πÀπ⓵à“ß w(n) ¡“
§Ÿ≥°—∫ x(n) ·µà≈– à«π∑’Ë·¬°ÕÕ°¡“π’È®–æ‘®“√≥“
«à“‡ªìπ —≠≠“≥∑’ˉ¡à·ª√µ“¡‡«≈“ ®“°π—Èπ§”π«≥
DFT ¢Õß —≠≠“≥·µà≈– à«π µ”·ÀπàߢÕß w(n)
„π™à«ß‡«≈“µà“ß Ê —¡æ—π∏å°∫— x(n) ·≈–§«“¡∂’·Ë ¥ß
¥—ß ¡°“√ (4)
∞
X STFT (e jω , n ) = ∑ x( n − m )w( m )e − jωm (4)
m =−∞
184
°“√«‘‡§√“–Àå√–¥—∫‡ ’¬ß¥πµ√’‰∑¬
n §◊Õµ—«·ª√‡«≈“‡ªìπ®”π«π‡µÁ¡ ·≈– ω §◊ Õ
µ— « ·ª√§«“¡∂’Ë µà Õ ‡π◊Ë Õ ß ∑”°“√™— ° µ— « Õ¬à “ ß
X STFT (e jω , n ) ¥â«¬§«“¡∂’Ë∑’Ë¡’√–¬–Àà“߇∑à“ Ê °—π
N ®”π«π ®–‰¥â ωk = 2πk / N ·≈– w(n) ¡’
§«“¡¬“« R ®”π«π ´÷Ëß N > R ‰¥â‡ªìπ XSTFT(k, n)
¥—ß ¡°“√ (5)
R −1
X STFT ( k, n ) =
∑
x ( n − m ) w( m ) e
− j(
2π
) km
N
∫√‘‡«≥∑’Ë ‡ª°‚µ√·°√¡¡’¢π“¥ Ÿß ÿ¥
‡∑§π‘§ AR
°“√®”≈ÕߢâÕ¡Ÿ≈ x(n) ´÷ßË ¡’®”π«π¢âÕ¡Ÿ≈‡∑à“°—∫
N ¥â«¬·∫∫®”≈Õß AR (À√◊Õ all pole model)
®–æ‘®“√≥“®“°°“√ª√–¡“≥§à“ x(n) °≈à“«§◊Õ°“√
ª√–¡“≥§à“ªí®®ÿ∫—π¢Õß x(n) Õ“»—¬º≈√«¡‡™‘߇ âπ
¢ÕߢâÕ¡Ÿ≈ M ®”π«π∑’˪√“°Ø¢÷Èπ°àÕπÀπâ“ ‡¢’¬π
(5) · ¥ß‰¥â¥—ß ¡°“√ (6) (Totarong, 1983)
m=0
XSTFT(k, n) °Á§◊Õ DFT ¢Õß x(n-m)w(m) ·≈– k ¡’
§à“Õ¬Ÿà„π™à«ß 0 < k < N-1 °“√‡≈◊Õ°„™â w(n)
„Àâ‡À¡“– ¡¡’§«“¡ ”§—≠°≈à“«§◊Õ ∂â“ —≠≠“≥¡’
æ“√“¡‘‡µÕ√å¢Õß ‡ª°µ√—¡∑’Ë·ª√º—π„π™à«ß°«â“ß Ê
§«√≈¥®”π«π§«“¡¬“«¢Õß w(n) ≈߇æ◊ËÕ‡æ‘Ë¡§«“¡
“¡“√∂·¬°‡«≈“ ·µà„π¢≥–‡¥’¬«°—π°“√‡æ‘Ë¡§«“¡
¬“«¢Õß w(n) ®–∑”„À⧫“¡ “¡“√∂·¬°§«“¡∂’Ë
‡æ‘Ë¡¢÷Èπ °“√§”π«≥¥â«¬‚ª√·°√¡ MATLABTM
°”Àπ¥®”π«π¢âÕ¡Ÿ≈„π°“√§”π«≥ DFT ‡∑à“°—∫
17,500 ®ÿ¥ w(n) ∑’ˇ≈◊Õ°„™â‡ªìπ·∫∫ ’ˇÀ≈’ˬ¡º◊πºâ“
®”π«π 2,000 ®ÿ¥ π—Ëπ§◊Õ x(n) ®–‰¡à∂Ÿ°¢¬“¬·≈–
≈¥∑Õπ „π°“√§”π«≥¥—ß°≈à“«π’È w(n) ®–‡≈◊ËÕπ
‰ª®π ÿ¥§«“¡¬“«¢ÕߢâÕ¡Ÿ≈ ´÷Ëß°”Àπ¥„Àâ¡’°“√
‡À≈◊ËÕ¡ (overlap) °—π 1,950 ®ÿ¥ §à“æ“√“¡‘‡µÕ√å
¥—ß°≈à“«¡“®“°°“√∑¥≈Õߧ”π«≥ STFT ¢Õß‚πâµ
∑—ßÈ ·ª¥‡ ’¬ß‚¥¬‡ª≈’¬Ë π§à“®”π«π§«“¡¬“«¢Õß w(n)
·≈–®”π«π°“√‡À≈◊ËÕ¡ ‡™àπ 500 °—∫ 25 500 °—∫
450 1,000 °—∫ 500 ·≈– 1,500 °—∫ 1,000 ‡ªìπµâπ
´÷ßË ®”π«π°“√‡À≈◊ÕË ¡∑’¡Ë §’ “à ¡“°®– àߺ≈„Àâ¢π“¥¢Õß
XSTFT(k, n) ¡’§à“¡“°‰ª¥â«¬ ®“°π—Èπ®–æ‘®“√≥“
' ‡§’¬ß
‡≈◊Õ°§à“æ“√“¡‘‡µÕ√å∑„’Ë À⧫“¡∂’¢Ë Õ߇ ’¬ß‚¥„°≈â
À√◊Õ‡ªìπ Õ߇∑à“¢Õ߇ ’¬ß‚¥' XSTFT(k, n) ‡ªìπ
øíß°å™—π Õßµ—«·ª√ ¡’§à“∑—Èß„π‚¥‡¡π‡«≈“ (n) ·≈–
‚¥‡¡π§«“¡∂’Ë (k) °“√· ¥ß¢π“¥¢Õß XSTFT(k, n)
‡√’¬°«à“ ‡ª°‚µ√·°√¡ (spectrogram) ¡’≈—°…≥–
‡ªì π ¿“æ “¡¡‘ µ‘ · ¥ß„Àâ ‡ ÀÁ π Õß§å ª √–°Õ∫
∑“ߧ«“¡∂’Ë¢Õß x(n) ∑’Ë·ª√µ“¡‡«≈“ ´÷Ëß¢π“¥¢Õß
X STFT(k, n) Õ¬Ÿà „ π∑‘ » ∑“ß z ∫π√–π“∫ x-y
øÕ√å · ¡π∑å ¢ Õ߇ ’ ¬ ߥπµ√’ ®÷ ß ª√“°Ø‡¥à π ™— ¥ µ√ß
)
x( n ) = −
M
∑ a x( n − i )
(6)
i
i =1
)
x( n )
§◊Õ§à“ª√–¡“≥¢Õß x(n) M §◊ÕÕ—π¥—∫¢Õß
·∫∫®”≈Õß ·≈– ai §◊Õ§à“ —¡ª√– ‘∑∏‘¢Ï Õß·∫∫®”≈Õß
º≈µà“ߢÕß x(n) ·≈– x) (n) §◊Õ§à“º‘¥æ≈“¥®“°
°“√ª√–¡“≥‡™‘߇ âπ (linear prediction error, en)
· ¥ß¥—ß ¡°“√ (7) (Totarong, 1983)
)
x( n ) − x( n ) =
M
∑ a x( n − i )
i
= en
(7)
i=0
º≈°“√·ª≈ß z ¢Õß ¡°“√ (7) ‚¥¬„Àâ a0 = 1
®–‰¥â∑√“π ‡øÕ√åøíß°å™—π¢Õß·∫∫®”≈Õß AR ¡’
§à“¥—ß ¡°“√ (8) (Totarong, 1983)
Z[e n ]
X( z ) =
(8)
M
1+
∑a z
−i
i
i =1
Z[en] À¡“¬∂÷ߺ≈°“√·ª≈ß z ¢Õß en §à“§«“¡∂’Ë
¢Õß x(n) ª√–¡“≥‰¥â®“° ‡ª°µ√—¡¢π“¥¢Õß X(z)
´÷Ëß z = ejω ·≈–‡æ◊ËÕÀ≈’°‡≈’ˬ߰√≥’∑’Ë Z[en] = 0
®÷ߧ”π«≥ ‡ª°µ√—¡¢π“¥¢Õß X(z) µ“¡ ¡°“√ (9)
(Griffiths, 1975)
1
X( z ) =
(9)
M
1+
∑a z
−i
i
i =1
„π°“√ª√–¡“≥§à“ —¡ª√– ‘∑∏‘Ï¢Õß·∫∫®”≈Õß ®–
‡≈◊Õ°„™âÕ—≈°Õ√‘∑÷¡¢Õ߇∫‘√å° (Burg Algorithm) ´÷Ëß
¡’°“√§”π«≥‡ªìπ°√–∫«π°“√∑”´È”‚¥¬≈¥º≈√«¡
¢Õß§à “ º‘ ¥ æ≈“¥®“°°“√ª√–¡“≥‡™‘ ß ‡ â π (e m)
185
«“√ “√‡∑§‚π‚≈¬’ ÿ√π“√’ ªï∑’Ë 11 ©∫—∫∑’Ë 3, °√°Æ“§¡-°—𬓬π 2547
∑—Èß·∫∫¬âÕπ°≈—∫ (backward) ·≈–·∫∫‰ªÀπâ“
(forward) ·≈–¬—ߧ߄Àâ§à“ —¡ª√– ‘∑∏‘Ï¢Õß·∫∫
®”≈Õ߇ªìπ‰ªµ“¡‡≈ø«‘π —π √’‡§Õ√å™—π (Levinson
recursion) µ—Èß·µàÕ—π¥—∫ 1 ∂÷ß M ‡æ◊ËÕ∑”„Àâµ—«°√Õß
§«“¡∂’Ë AR ¡’‡ ∂’¬√¿“æ (Marple, 1980) ¡°“√
(10) · ¥ß§«“¡ — ¡ æ— π ∏å ”À√— ∫ §”π«≥ a M,M
À√◊Õ‡√’¬°«à“ —¡ª√– ‘∑∏‘Ï°“√ –∑âÕπ (reflection
coefficient) ‚¥¬∑’Ë bM,k ·≈– fM,k §◊Õ§à“§«“¡
º‘¥æ≈“¥∑’ˇ°‘¥®“°°“√ª√–¡“≥‡™‘߇ âπ·∫∫¬âÕπ
°≈—∫·≈–·∫∫‰ªÀπ⓵“¡≈”¥—∫ ·≈– —≠≈—°…≥å *
À¡“¬∂÷ß§à“ —߬ÿ§‡™‘ß´âÕπ „π·µà≈–√Õ∫∑’Ë M ¡’§à“
‡æ‘Ë¡¢÷Èπ eM ´÷Ëߧ”π«≥µ“¡ ¡°“√ (11) ®–¡’¢π“¥
≈¥≈ß·≈–§à“ —¡ª√– ‘∑∏‘Ï¢Õß·∫∫®”≈Õߧ”π«≥
‰¥â®“° ¡°“√ (12) (Marple, 1980)
°√–®“¬«‘°‡πÕ√å‡∑’¬¡‡µÁ¡Àπ૬ (discrete pseudoWigner distribution À√◊Õ DPWD) ‚¥¬‡√‘Ë¡®“°
°“√§”π«≥øíß°å™—πÕ—µ À —¡æ—π∏å¢≥–Àπ÷Ëß·∫∫
‡µÁ¡Àπ૬ (discrete instantaneous autocorrelation
function, R s (n, l)) ¢ÕߢâÕ¡Ÿ≈ s(n) · ¥ß¥—ß
¡°“√ (13)
R s ( n, l) = s( n + l)s * ( n − l)
(13)
—≠≈—°…≥å * À¡“¬∂÷ß§à“ —߬ÿ§‡™‘ß´âÕππ” R s (n, l)
ºà “ π°“√°√Õß — ≠ ≠“≥¥â « ¬ h LP(n) ®–‰¥â ‡ ªì π
R s, t ( n, l) ¡’§à“µ“¡ ¡°“√ (14)
P
R s, t ( n , l ) =
∑ R (n − p, l)h
s
LP ( p)
(14)
p =− P
‚¥¬ —≠≈—°…≥å t À¡“¬∂÷ß°“√°√Õß —≠≠“≥∑“ß
‡«≈“·≈– P ¡’§à“‡∑à“°—∫ 2N µ—«·ª√ l „π ¡°“√
N−M
(13) ·≈– (14) ¡’§à“Õ¬Ÿà„π™à«ß 0 < l < L ‡π◊ËÕß®“°
2 ∑ b *M −1, k fM −1, k +1
a M,M = − N − M k =1
(10) R s (n, l) ·≈– R s,t (n, l) ¡’≈—°…≥– ¡¡“µ√ ®÷ß
2
2
§”π«≥·µà‡æ’¬ß¥â“π∫«°‡∑à“π—πÈ l §◊Õ§à“≈â“À≈—ß (lag)
b
+ fM −1, k +1 
∑
 M −1, k

·≈–¡’§à“¡“°∑’Ë ÿ¥‡∑à“°—∫ L ®“°π—Èππ” R s,t (n, l)
k =1
ºà“π°“√°√Õß —≠≠“≥∑“ߧ«“¡∂’˥⫬ gLP(n) ·≈–
2
(11)
e M = e M −1[1 − a M,M ]
∑”°“√·ª≈ßøŸ√‡‘ ¬√凵Á¡Àπ૬ ‰¥â‡ªìπ¥—ß ¡°“√ (15)
a M, k = a M −1, k + a M,M a *M −1,M − k
(12)
− j2 πkl
L
(
)
2L
(15)
M
(
n
,
k
)
=
R
(
n
,
l
)
g
(
l
)
e
s
s, t
LP
∑
TM
‡¡◊ËÕ„™â‚ª√·°√¡ MATLAB x(n) ®–‡ªìπ‡ ¡◊Õπ
l =− L
‡Õ“µåæÿµ¢Õß√–∫∫ AR ∑’Ë¡’Õ‘πæÿµ‡ªìπ —≠≠“≥ ´÷Ëß· ¥ß√Ÿª·∫∫‡µÁ¡Àπ૬¢Õß MD (Ms(n,k)) ¢Õß
√∫°«π¢“« (white noise) ·≈–°”Àπ¥„ÀâÕ—π¥—∫ ¢âÕ¡Ÿ≈ s(n) k ¡’§à“„π™à«ß 0 < k < L/2 µ—«·ª√ n
¢Õß·∫∫®”≈Õ߇∑à“°—∫ 18 Õ—π¥—∫¢Õß·∫∫®”≈Õß „π ¡°“√ (13) §◊Õ®”π«π¢âÕ¡Ÿ≈∑—ÈßÀ¡¥ à«π„π
π’‰È ¥â¡“®“°°“√∑¥≈Õß«‘‡§√“–Àå‡ ’¬ß‚πâµ∑—ßÈ ·ª¥‡ ’¬ß ¡°“√ (14) ·≈– (15) ®–§”π«≥‡©æ“– n ‡æ’¬ß
‚¥¬§”π«≥µ—Èß·µàÕ—π¥—∫ 1, 2, 3, ..., 50 ·≈– ∫“ߧà“∑’µË √ßµ”·Àπàß°“√™—°µ—«Õ¬à“ߢâÕ¡Ÿ≈ s(n) ¥â«¬
∑¥≈Õß ÿࡇ≈◊Õ°§à“Õ—π¥—∫ ‡™àπ 100, 500, 1,000 §«“¡∂’Ë ° “√™— ° µ— « Õ¬à “ ߇ªì π Õ߇∑à “ ¢Õß ∆ωmin
·≈– 1,500 ‡ªì π µâ π ´÷Ë ß æ∫«à “ Õ— π ¥— ∫ ¢Õß·∫∫ (§◊ Õ º≈µà “ ß∑“ߧ«“¡∂’Ë ∑’Ë πâ Õ ¬∑’Ë ÿ ¥ √–À«à “ ß Õß
®”≈Õß∑’Ë¡“°¢÷Èπ·≈–¡“°‡°‘π‰ª‰¡à‰¥â„Àâº≈∑’Ë¥’¢÷Èπ Õߧåª√–°Õ∫„¥ Ê „π —≠≠“≥ Àπ૬‡Œ‘√µ´å)
°“√æ‘ ® “√≥“‡≈◊ Õ °Õ— π ¥— ∫ 18 ‡æ√“–„Àâ º ≈°“√ hLP(n) ·≈– gLP(n) §◊Õøíß°å™—πÀπ⓵à“ß·∫∫·Œ¡¡‘Ëß
«‘‡§√“–Àå∑’Ë· ¥ß§«“¡‡ªìπ∑∫‡ ’¬ß¢Õß‚¥·≈–‚¥' (Hamming) ´÷Ëß¡’§«“¡¬“«„π™à«ß -N ∂÷ß N ·≈–
‰¥â„°≈⇧’¬ß∑’Ë ÿ¥ Õߧåª√–°Õ∫∑“ߧ«“¡∂’Ë¢Õß x(n) -R ∂÷ß R µ“¡≈”¥—∫ ‚¥¬∑’Ë R ‡∑à“°—∫ L/2 §à“§ß∑’Ë
· ¥ß¥â«¬°√“ø¢Õß ‡ª°µ√—¡¢π“¥¢Õß X(z)
µà “ ß Ê „π°“√§”π«≥‰¥â · °à L = 8,192,
‡∑§π‘§ MD (Pielemeier and Wakefield, 1996) ∆ωmin = 183.11 rad/sec, N = 1,365 ·≈– n ‡æ‘Ë¡
°“√§”π«≥ MD ¡’ æ◊È π ∞“π¡“®“°°“√ ¢÷Èπ¢—Èπ≈– 4 ®”π«π °“√‡≈◊Õ°§à“æ“√“¡‘‡µÕ√åµà“ß Ê
186
°“√«‘‡§√“–Àå√–¥—∫‡ ’¬ß¥πµ√’‰∑¬
¥—ß°≈à“« æ‘®“√≥“¥—ßπ’È °≈à“«§◊Õ ‡≈◊Õ° L ‡ªìπ
§à“°”≈—ߢÕß Õß ∑”„Àâ„™âÕ≈— °Õ√‘∑¡÷ °“√·ª≈ßøŸ√‡‘ ¬√å
Õ¬à “ ß√«¥‡√Á « (FFT) §”π«≥ ¡°“√ (15) ‰¥â
∆ω min ¡“®“°º≈ ‡ª°‚µ√·°√¡¢Õß s(n) N
‡°‘¥®“°º≈À“√√–À«à“ßÕ—µ√“°“√™—°µ—«Õ¬à“ߢÕß s(n)
°—∫ ∆ωmin ·≈â«ªí¥„À⇪ìπ®”π«π‡µÁ¡ ·≈–‡≈◊Õ° n
µ√ßµ”·Àπàß ´÷Ëß¡’§«“¡∂’Ë°“√™—°µ—«Õ¬à“ß s(n) ¡“°
°«à “ Õ߇∑à “ ¢Õß ∆ω min ‡æ◊Ë Õ ®–‰¥â √ “¬≈–‡Õ’ ¬ ¥
¢Õߺ≈°“√«‘‡§√“–Àå∑“߇«≈“¡“°¢÷Èπ Ms(n,k) „π
¡°“√ (15) ‡ªìπøíß°å™—π Õßµ—«·ª√ ¡’§à“∑—Èß„π
‚¥‡¡π‡«≈“ n ·≈–‚¥‡¡π§«“¡∂’Ë k Õߧåª√–°Õ∫
∑“ߧ«“¡∂’Ë¢Õß s(n) ∑’Ë·ª√µ“¡‡«≈“· ¥ß¥â«¬¿“æ
“¡¡‘µ‘ ´÷Ëß¢π“¥¢Õß Ms(n,k) ®–Õ¬Ÿà„π∑‘»∑“ß z
∫π√–π“∫ x-y ·≈–∫√‘‡«≥∑’Ë ‡ª°µ√—¡¢π“¥¢Õß
Ms(n,k) ¡’§à“ Ÿß ÿ¥®–· ¥ß∂÷ßøÕ√å·¡π∑å¢Õ߇ ’¬ß
¥πµ√’
º≈°“√∑¥≈Õß·≈–«‘®“√≥å
‡¡◊ËÕ∑”°“√ª√–¡«≈º≈ —≠≠“≥‡ ’¬ß¢≈ÿଇ撬ßÕÕ
¥â«¬‡∑§π‘§µà“ß Ê ∑’ˉ¥â𔇠πÕ‰ª·≈â« º≈°“√
¥”‡π‘πß“π∫“ß à«πÕ“®π”¡“· ¥ß‰¥â¥—ß√Ÿª∑’Ë 3, 4,
5 ·≈– 6 µ“¡≈”¥—∫ ∑—Èß ’Ë√Ÿª· ¥ßøÕ√å·¡π∑å¢Õß
‡ ’¬ß ‚¥ ¡’ ´Õ≈ ·≈– ‚¥' Õ¬à“ß™—¥‡®π øÕ√å·¡π∑å
¢Õ߇ ’ ¬ ߥπµ√’ ‰ ∑¬„π™à « ßÀπ÷Ë ß ∑∫‡ ’ ¬ ߢÕß
¢≈ÿ¬à ‡æ’¬ßÕÕ ‰¥â√∫— °“√𔉪 √ÿª√«¡‰«â„πµ“√“ß∑’Ë 2
¢âÕ¡Ÿ≈∑’‰Ë ¥â®“°°“√«‘‡§√“–Àå‡ ’¬ß®√‘ߥ—ßµ“√“ß∑’Ë 2 π’È
‡ªìπ‡§√◊ËÕ߬◊π¬—π«à“§à“§«“¡∂’Ë¢Õ߇ ’¬ß¥πµ√’‰∑¬¥—ß
µ“√“ß∑’Ë 1 §≈“¥‡§≈◊ÕË π‰ª®“°§«“¡‡ªìπ®√‘ß¡“° ·≈–
¬—ßÕ“® —߇°µ‰¥â«à“‡ ’¬ß¥πµ√’‰∑¬π—ÈπµË”°«à“‡ ’¬ß
¥πµ√’ “°≈ «‘∏’ DFT ·≈– STFT „Àâ§à“§«“¡∂’Ë
øÕ√å·¡π∑å·µ°µà“ß°—π‡æ’¬ß∫“߇ ’¬ß‡∑à“π—Èπ´÷Ë߉¡à
¡“°π—°·≈–¡’§«“¡∂’ˇæ‘Ë¡¢÷Èπ®“°‡ ’¬ßµË”‰ª¬—߇ ’¬ß
Ÿß‡ªìπ≈”¥—∫ ‡æ’¬ß·µà«à“º≈°“√«‘‡§√“–À宓° Õß«‘∏’
π’È„À⧫“¡∂’Ë∫“ß‚πⵇªìπ‡≈¢≈ßµ—« ‰¥â·°à 500, 600
·≈– 900 ‡Œ‘√µ´å π—∫«à“‡ªìπ ‘ßË º‘¥ª°µ‘∑“ß∏√√¡™“µ‘
¢Õߥπµ√’ ‰ ∑¬‡ªì π Õ¬à “ ߬‘Ë ß ‡æ√“–°“√µ—È ß ‡ ’ ¬ ß
‡§√◊ËÕߥπµ√’‰∑¬‡¡◊ËÕº≈‘µ ‰¡à¡’°“√„™â‡∑§‚π‚≈¬’
‡§√◊ËÕß¡◊Õ«—¥≈–‡Õ’¬¥Õ¬à“߇™àπ°“√º≈‘µ‡§√◊ËÕߥπµ√’
¢Õßµ–«—πµ° À“°·µà¥πµ√’‰∑¬„™â∏√√¡™“µ‘¢Õß
°“√øí߇∑’¬∫‡§√◊ËÕߥπµ√’ Õß™‘Èπ¥â«¬°—π·≈–ª√—∫
µ—È߇ ’¬ß º≈®“°«‘∏’ AR π—Èπ §à“§«“¡∂’Ë¢Õ߇ ’¬ß¡’
(657.14 ‡Œ‘√µ´å) Ÿß°«à“‡ ’¬ßø“ (650 ‡Œ‘√µ´å) ´÷Ëß
‰¡à‡ªìπ‰ªµ“¡°“√‰≈à√–¥—∫‡ ’¬ß «‘∏’ AR ·≈–
Õ—≈°Õ√‘∑÷¡µ≈Õ¥®πæ“√“¡‘‡µÕ√å∑’ˇ≈◊Õ°„™â ®÷߬—ß
‰¡à‡À¡“– ¡ ”À√—∫°“√«‘‡§√“–Àå‡ ’¬ß¥πµ√’‰∑¬
«‘∏’ MD ®÷ß„Àâº≈°“√«‘‡§√“–Àå∑’Ëπà“‡™◊ËÕ∂◊Õ¡“°°«à“
‡æ√“–øÕ√å·¡π∑å∑’ˉ¥â¡’°“√‰≈à‡√’¬ß‰ªµ“¡≈”¥—∫
µË” Ÿß¢Õ߇ ’¬ß ·≈–Àπ÷Ëß∑∫‡ ’¬ß°Áª√“°Ø§«“¡∂’Ë
„°≈⇧’¬ß Õ߇∑à“µàÕ°—π
®“°°“√∑’Ë«‘∏’ MD „Àâº≈≈—æ∏å∑’Ë¥’∑’Ë ÿ¥ ®÷߉¥â
§”π«≥Õ—µ√“ à«π§«“¡∂’Ë (frequency ratio) ·≈–
√–¬–摵™å (pitch interval) ¢Õߺ≈¥—ß°≈à“« · ¥ß
¢âÕ¡Ÿ≈∑’˧”π«≥‰¥â‡ª√’¬∫‡∑’¬∫‰«â„πµ“√“ß∑’Ë 3
Table 2. Formants of the Thai notes played by a middle pitch Thai flute.
One octave of
Frequency (Hz)
Thai musical notes
DFT
STFT
AR
MD
do
rae
mee
fa
sol
la
tee
do
457.14
514.29
557.14
600.00
657.14
714.29
814.29
900.00
457.14
500.00
557.14
600.00
657.14
714.29
814.29
900.00
442.86
557.14
657.14
650.00
700.00
785.71
864.29
914.29
465.39
511.17
556.95
602.72
663.76
717.16
816.35
892.64
'
187
Magnitude (unitless)
Magnitude (unitless)
«“√ “√‡∑§‚π‚≈¬’ ÿ√π“√’ ªï∑’Ë 11 ©∫—∫∑’Ë 3, °√°Æ“§¡-°—𬓬π 2547
Frequency (Hz)
Magnitude (unitless)
Magnitude (unitless)
Frequency (Hz)
Frequency (Hz)
Frequency (Hz)
Soectrogram (uniless)
Soectrogram (uniless)
Figure 3. Formants of the notes (a) “do” (b) “mee” (c) “sol” (d) “do'” obtained from DFT
(original sound played by a middle pitch Thai flute).
Frequency (Hz)
Soectrogram (uniless)
Soectrogram (uniless)
Frequency (Hz)
Frequency (Hz)
Frequency (Hz)
Figure 4. Formants of the notes (a) “do” (b) “mee” (c) “sol” (d) “do'” obtained from STFT
(original sound played by a middle pitch Thai flute).
188
Magnitude (unitless)
Magnitude (unitless)
°“√«‘‡§√“–Àå√–¥—∫‡ ’¬ß¥πµ√’‰∑¬
Frequency (Hz)
Magnitude (unitless)
Magnitude (unitless)
Frequency (Hz)
Frequency (Hz)
Frequency (Hz)
Magnitude (unitless)
Magnitude (unitless)
Figure 5. Formants of the notes (a) “do” (b) “mee” (c) “sol” (d) “do'” obtained from AR
(original sound played by a middle pitch Thai flute).
Time (ms)
Time (ms)
Frequency (Hz)
Magnitude (unitless)
Magnitude (unitless)
Frequency (Hz)
Time (ms)
Frequency (Hz)
Time (ms)
Frequency (Hz)
Figure 6. Formants of the notes (a) “do” (b) “mee” (c) “sol” (d) “do'” obtained from MD
(original sound played by a middle pitch Thai flute).
189
«“√ “√‡∑§‚π‚≈¬’ ÿ√π“√’ ªï∑’Ë 11 ©∫—∫∑’Ë 3, °√°Æ“§¡-°—𬓬π 2547
Pitch interval (cent)
Õ—µ√“ à«π§«“¡∂’Ë∑’Ë°≈à“«∂÷ßπ’ȧ”π«≥‰¥â‚¥¬π”§à“ · ¥ß§à“§«“¡∂’Ë¢Õ߇ ’¬ß‡ªï¬‚π „π™à«ß∑’Ë„°≈⇧’¬ß
§«“¡∂’¢Ë Õ߇ ’¬ß∑’µË ¥‘ °—π¡“À“√°—𠇙àπ ¡’ : ‡√ ‡ªìπµâπ °—∫§«“¡∂’Ë¢Õߢ≈ÿଇ撬ßÕÕ∑’Ë«‘‡§√“–À剥⠮– —߇°µ
®“°π—Èπ§”π«≥√–¬–摵™å µ“¡ ¡°“√ (16)
‰¥â«à“√–¬–摵™å‡¡◊ËÕ‡ ’¬ß‚¥‡∑’¬∫°—∫‡ ’¬ß‚¥ ¡’§à“
‡∑à “ °— ∫ 1,200 ‡´πµå æ Õ¥’ ∑’Ë ‡ ªì 𠇙à π π’È ‡ æ√“–
(16)
√–¬–摵™å = K log 2 f1
f2
‡∑§‚π‚≈¬’°“√ √â“߇§√◊ËÕߥπµ√’ “°≈„πªí®®ÿ∫—π
¡’
√–¬–摵™å®–¡’Àπ૬‡ªìπ‡´πµå (cent) ‡¡◊ÕË K = 1,200 °“√„™â‡§√◊ÕË ß¡◊Õ«—¥≈–‡Õ’¬¥™à«¬„π°“√ª√—∫µ—ßÈ ‡ ’¬ß
(Wood, 1975) f1 ·≈– f2 §◊Õ§«“¡∂’Ë∑’Ë®–π”¡“ ·≈–¬— ß · ¥ß„Àâ ‡ ÀÁ π «à “ ¥πµ√’ “°≈·∫à ß ÕÕ°‡ªì π
§”π«≥‡ª√’¬∫‡∑’¬∫°—π ¡’Àπ૬‡Œ‘√µ´å (Hz) ´÷Ëß ÀⓇ ’¬ß‡µÁ¡ ·≈– Õߧ√÷Ë߇ ’¬ß™—¥‡®π (À¡“¬∂÷ß
f1 / f2 „π∑’Ëπ’È°Á§◊Õ§à“Õ—µ√“ à«π§«“¡∂’Ëπ—Ëπ‡Õß √–¬– Õ—µ√“ à«π‡∑à“°—∫ 1.12 ·≈– 1.06 √–¬–摵™å‡∑à“°—∫
' ¬∫°—∫‡ ’¬ß‚¥ (À√◊Õ‡¡◊ËÕ§√∫ 200 ‡´πµå ·≈– 100 ‡´πµå µ“¡≈”¥—∫)
摵™å‡¡◊ËÕ‡ ’¬ß‚¥‡∑’
Àπ÷Ëß∑∫‡ ’¬ß) „π∑“ß∑ƒ…Æ’®–‡∑à“°—∫ 1,200 ‡´πµå
æÕ¥’
¢âÕ¡Ÿ≈„πµ“√“ß∑’Ë 3 · ¥ßÕ—µ√“ à«π§«“¡∂’Ë
·≈–√–¬–摵™å∑’˧”π«≥‰¥â®“°¢âÕ¡Ÿ≈øÕ√å·¡π∑å∑’Ë
‡ªìπº≈≈—æ∏å¢Õß«‘∏’ MD Õ—µ√“ à«π§«“¡∂’¡Ë §’ “à ‡∑à“ Ê
'
°—π ª√–¡“≥ 1.10 ¡’‡æ’¬ßÕ—µ√“ à«π¢Õß ∑’ : ≈“
´÷ßË ‡∑à“°—∫ 1.14 ·≈–√–¬–摵™å¡§’ “à ‡©≈’¬Ë ‡∑à“°—∫ 161.08
‡´πµå Õ¬à“߉√°Áµ“¡ √–¬–摵™å∑’˧”π«≥‰¥âª√“°Ø
Õ“°“√·°«àß®π‰¡àÕ“®¬Õ¡√—∫„π ¡¡µ‘∞“π§à“√–¬–
æ‘™µå§ß∑’ˉ¥â (Morton, 1976) πÕ°®“°π—Èπ√–¬–
摵™å‡¡◊ÕË §√∫Àπ÷ßË ∑∫‡ ’¬ß¡’§“à ‡∑à“°—∫ 1,127.57 ‡´πµå
´÷ßË √Ÿª∑’Ë 7 · ¥ß°“√‡ª√’¬∫‡∑’¬∫√–À«à“ߧà“√–¬–摵™å
One octave of Thai musicalnotes
∑’˧”π«≥®“°º≈¢Õß«‘∏’ MD ·≈–§à“µ“¡ ¡¡µ‘∞“π
¢Õß Morton ∑’Ë°”Àπ¥„Àâ‡∑à“°—∫ 171.43 ‡´πµå Figure 7. Pitch intervals analyzed by MD
(original sound played by a middle
‡¡◊ËÕæ‘®“√≥“Õ—µ√“ à«π§«“¡∂’Ë·≈–√–¬–摵™å¢Õß
pitch Thai flute ana a metal tenor
‡ ’¬ß¢≈ÿଇ撬ßÕÕ‡∑’¬∫°—∫¥πµ√’ “°≈ ´÷Ëßµ“√“ß∑’Ë 4
xylophone).
Table 3. Frequency ratios and pitch intervals by MD (middle pitch Thai flute).
One octave of
Thai musical notes
Frequency
(Hz)
Frequency ratio
(unitless)
Pitch interval
(cent)
do
rae
mee
fa
sol
la
tee
do
465.39
511.17
556.95
602.72
663.76
717.16
816.35
892.64
1.10
1.09
1.08
1.10
1.08
1.14
1.09
162.44
148.49
136.73
167.01
133.96
224.27
154.67
'
190
°“√«‘‡§√“–Àå√–¥—∫‡ ’¬ß¥πµ√’‰∑¬
”À√— ∫ º≈Õ’ ° à « πÀπ÷Ë ß ‡ªì π ¢Õß°√≥’ ° “√
«‘ ‡ §√“–Àå ‡ ’ ¬ ß√–π“¥‡Õ°‡À≈Á ° °“√¥”‡π‘ π ß“π
°√–∑”„π≈—°…≥–‡¥’¬«°—π°—∫°√≥’°“√¥”‡π‘πß“π
«‘‡§√“–Àå‡ ’¬ß¢≈ÿଇ撬ßÕÕ µ—Èß·µà°“√∫—π∑÷°‡ ’¬ß
π”¢âÕ¡Ÿ≈¡“ºà“π°“√°√Õß —≠≠“≥ µ≈Õ¥®π‡∑§π‘§
∑’Ë„™â„π°“√«‘‡§√“–Àå º≈°“√«‘‡§√“–Àå„π¢—Èπµâπ
ª√“°Ø≈—°…≥–∑’˧≈⓬§≈÷ß°—π°—∫°√≥’¢≈ÿଇ撬ßÕÕ
®÷ß𔇠πÕº≈∑’ˉ¥â®“°«‘∏’ MD ·≈–º≈°“√§”π«≥
∑’Ë —¡æ—π∏å°—π¥—ß· ¥ß‰«â„πµ“√“ß∑’Ë 5 ®– —߇°µ‰¥â
«à“Õ—µ√“ à«π§«“¡∂’¡Ë §’ “à ‡∑à“ Ê °—πÕ¬Ÿ∑à ª’Ë √–¡“≥ 1.11
√–¬–摵™å¢Õ߇ ’¬ß°Áª√“°ØÕ“°“√·°«àß™—¥‡®π √–¬–
' ¬∫°—∫‡ ’¬ß‚¥¡’§“à ‡∑à“°—∫ 1,214.15
摵™å‡¡◊ÕË ‡ ’¬ß‚¥‡∑’
‡´πµå ·≈–®“°¢âÕ¡Ÿ≈„πµ“√“ß∑’Ë 5 ¬—ßæ∫«à“ §à“‡©≈’¬Ë
¢Õß√–¬–摵™å‡∑à“°—∫ 173.45 ‡´πµå Õ¬à“߉√°Áµ“¡
º≈®“°¢âÕ¡Ÿ≈°“√∑¥≈Õß∑’Ëπ”¡“§”π«≥√–¬–摵™å
¥—ß∑’Ë· ¥ß¥â«¬°√“ø„π√Ÿª∑’Ë 7 ∫àß™’È«à“√–¬–摵™å
¢Õ߇ ’¬ß‚πâµ¥πµ√’‰∑¬ ¡’·π«‚πâ¡≈¥≈ß„π™à«ß
¢Õ߇ ’ ¬ ß‚¥∂÷ ß ø“ µà Õ ®“°π—È π ·≈â « √–¬–æ‘ µ ™å ¡’
Õ“°“√·°«àߢ÷Èπ≈߉ª®π‡ ’¬ß‚¥' πÕ°®“°π—Èπ °“√∑’Ë
§«“¡∂’Ë¢Õ߇ ’¬ß√–π“¥‡Õ°‡À≈Á°‰¡à‡∑à“°—πæÕ¥’°—∫
§à “ §«“¡∂’Ë ¢ Õ߇ ’ ¬ ߢ≈ÿà ¬ ‡æ’ ¬ ßÕÕ ‡æ’ ¬ ß·µà ¡’ §à “
„°≈â ‡ §’ ¬ ß°— π ‡π◊Ë Õ ß¡“®“°°“√¥”‡π‘ π ß“π„π
°√–∫«π°“√º≈‘ µ à « πÀπ÷Ë ß ·≈–§«“¡∂’Ë ‡ ’ ¬ ß∑’Ë
µà“ß°—π‡æ’¬ß‡≈Á°πâÕ¬‡™àππ’È °Á‡°‘𧫓¡ “¡“√∂„π
°“√‰¥â¬‘π¢Õß¡πÿ…¬å∑’Ë®–·¬°·¬–§«“¡·µ°µà“߉¥â
·≈–πÕ°®“°π—È π ‡§√◊Ë Õ ß¥πµ√’ ∑—È ß Õ߬— ß ¡’ ∞ “π
°”‡π‘¥‡ ’¬ß∑’˵à“ß°—π °≈à“«§◊Õ‡ ’¬ß√–π“¥‡Õ°‡À≈Á°
‡°‘¥®“°°“√°√–∑∫°—π¢ÕߢÕß·¢Áß §◊Õ‰¡â√–π“¥
°—∫≈Ÿ°√–π“¥∑’ˇªìπ‚≈À– ∑”„À⇠’¬ß∑’ˇ°‘¥¢÷Èπ¡’
§«“¡‡¢â¡¢âπ ¡Ë”‡ ¡Õ ¡“°°«à“‡ ’¬ß¢≈ÿà¬∑’ˇ°‘¥®“°
°“√ —Ëπ¢Õß≈”Õ“°“»„π°√–∫Õ°¢≈ÿଠՓ°“»π—Èπ¡’
°“√‡ª≈’ˬπ·ª≈ßµ“¡Õÿ≥À¿Ÿ¡‘·≈–§«“¡™◊Èπ‰¥âßà“¬
Table 4. Frequency ratios and pitch intervals (piano).
One octave of
Thai musical notes
do
rae
mee
fa
sol
la
tee
do
'
Frequency
(Hz)
523.25
587.33
659.26
698.46
783.99
880.00
987.77
1,046.50
Frequency ratio
(unitless)
Pitch interval
(cent)
1.12
1.12
1.06
1.12
1.12
1.12
1.06
200
200
100
200
200
200
100
Table 5. Frequency ratios and pitch intervals by MD (metal tenor xylophone).
One octave of
Thai musical notes
Frequency
(Hz)
Frequency ratio
(unitless)
Pitch interval
(cent)
do
rae
mee
fa
sol
la
tee
do
465.39
518.80
572.20
625.61
694.27
762.94
846.86
938.42
1.11
1.10
1.09
1.11
1.10
1.11
1.11
188.09
169.61
154.49
180.28
163.29
180.66
177.73
'
«“√ “√‡∑§‚π‚≈¬’ ÿ√π“√’ ªï∑’Ë 11 ©∫—∫∑’Ë 3, °√°Æ“§¡-°—𬓬π 2547
191
§«“¡∂’Ë ¢ Õ߇ ’ ¬ ߢ≈ÿà ¬ ‡æ’ ¬ ßÕÕ®÷ ß Õ“®µà “ ߉ª®“°
π§√√“™ ’¡“, Àπâ“ 5-47.
§«“¡∂’¢Ë Õ߇ ’¬ß√–π“¥‡Õ°‡À≈Á°Õ¬Ÿ∫à “â ß·µà‰¡à¡“°π—° —π∑—¥ µ—≥±π—π∑å. (2542). ∫—π∑÷°‡æ≈߉∑¬‡ªìπ
·≈– àߺ≈„Àâ√–¬–摵™åÀπ÷ßË ∑∫‡ ’¬ß¢Õߢ≈ÿ¬à ‡æ’¬ßÕÕ
‚πâµ “°≈Õ¬à“߉√. „π: §√Ÿ¥πµ√’¢Õß·ºàπ¥‘π,
¡’§à“µà“߉ª®“° 1,200 ‡´πµåÕ¬Ÿà∫â“ß
¡™“¬ √—»¡’. §√—Èß∑’Ë 1. ∂“∫—π√“™¿—Ø
∫â“π ¡‡¥Á®‡®â“æ√–¬“, °√ÿ߇∑æœ, Àπâ“ 6-18.
Õÿ
∑
‘
»
𓧠«— ¥‘Ï. (2514). ∑ƒ…Æ’·≈–°“√ªØ‘∫—µ‘
√ÿª
¥πµ√’‰∑¬. ®—¥æ‘¡æå„π°“√©≈Õߪﰓ√»÷°…“
°“√¥”‡π‘πß“π«‘‡§√“–Àå‡ ’¬ß¥πµ√’‰∑¬„π∑∫‡ ’¬ß
√–À«à “ ß™“µ‘ 2513. ‚√ßæ‘ ¡ æå §ÿ √ÿ ¿“,
‡æ’¬ßÕÕ ¥â«¬‡∑§π‘§ª√–¡«≈º≈ —≠≠“≥¥‘®µ‘ Õ≈π—πÈ
°√ÿ߇∑æœ, Àπâ“ 6-11.
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941-981.
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Õ“°“√·°«àß„π™à«ß‡ ’¬ß´Õ≈∂÷ß‚¥' ‡ ’¬ß¥πµ√’‰∑¬
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°‘µµ‘°√√¡ª√–°“»
¢Õ¢Õ∫§ÿ≥¡Ÿ≈π‘∏‘ ‚∑‡√ ‡æ◊ÕË °“√ à߇ √‘¡«‘∑¬“»“ µ√å
ª√–‡∑»‰∑¬∑’‰Ë ¥â π—∫ πÿπ∑ÿπ«‘®¬— ·°à‚§√ß°“√«‘®¬— π’È
‡Õ° “√Õâ“ßÕ‘ß
∫√‘…—∑ √’¥‡¥Õ√å ‰¥‡® ∑å (ª√–‡∑»‰∑¬) ®”°—¥.
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