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當白努利提出了一個理論來解釋『聖彼得堡悖論』時,就開啟了『效用』 Utility 的大門︰

邊際效用遞減原理】:一個人對於『財富』的擁有多多益善,也就是說『效用函數U(w) 的一階導數大於零 \frac{dU(w)}{dw} > 0;但隨著『財富』的增加,『滿足程度』的積累速度卻是不斷下降,正因為『效用函數』之二階導數小於零 \frac{d^2U(w)}{dw^2} < 0

最大效用原理】:當人處於『風險』和『不確定』的條件下,一個人『理性決策』的『準則』是為著獲得最大化『期望效用』值而不是最大之『期望金額』值。

Utility』依據牛津大字典的『定義』是︰

The state of being useful, profitable, or beneficial:
(In game theory or economics) a measure of that which is sought to be maximized in any situation involving a choice.

如此『效用』一詞,不論代表的是哪種『喜好度』 ── 有用 useful 、有利 profitable 、滿足 Satisfaction 、愉快 Pleasure 、幸福 Happiness ──,都會涉及主觀的感覺,那麼真可以定出『尺度』的嗎?『效用函數』真的『存在』嗎??

170px-Pakkanen
溫度計
量冷熱

魯班尺
魯班尺
度吉凶

一九四七年,匈牙利之美籍猶太人數學家,現代電腦創始人之一。約翰‧馮‧諾伊曼 Jhon Von Neumann 和德國-美國經濟學家奧斯卡‧摩根斯特恩 Oskar Morgenstern 提出只要『個體』的『喜好性』之『度量』滿足『四條公設』,那麼『個體』之『效用函數 』就『存在』,而且除了『零點』的『規定』,以及『等距長度』之『定義』之外,這個『效用函數』還可以說是『唯一』的。就像是『個體』隨身攜帶的『理性』之『溫度計』一樣,能在任何『選擇』下,告知最大『滿意度』與『期望值』。現今這稱之為『期望效用函數理論』 Expected Utility Theory。

由於每個人的『冷熱感受』不同,所以『溫度計』上的『刻度』並不是代表數學上的一般『數字』,通常這一種比較『尺度』只有『差距值』有相對『強弱』意義,『數值比值』並不代表什麼意義,就像說,攝氏二十度不是攝氏十度的兩倍熱。這一類『尺度』在度量中叫做『等距量表』 Interval scale 。

溫度計』量測『溫度』的『高低』,『理性』之『溫度計』度量『選擇』的『優劣』 。通常在『實驗經濟學』裡最廣泛採取的是『彩票選擇實驗』 lottery- choice experiments,也就是講,請你在『眾多彩票 』中選擇一個你『喜好』 的『彩票』。

─── 《物理哲學·下中…

 

歲末逢春之時,天寒地鳴之際,如何『度量』心情★

咀嚼

響度

響度(loudness又稱音響音量),是與聲強相對應的聲音大小的知覺量。聲強是客觀的物理量,響度是主觀的心理量。響度不僅跟聲強有關,還跟頻率有關[1]:47

響度級

為了在數量上估計一個純音的響度,可以把這個純音和1000Hz的某個聲強級[注 1]的純音在響度上作比較。這兩個聲音在聽覺上認為是相同的響度時,就可以把1000Hz純音的這個聲強級規定為該頻率純音的響度級。響度級的單位為(Phon)[2]:67

舉例來說,一個純音的頻率1000Hz,若希望其響度能達到40方,根據等響度曲線圖,其聲強級就必須達到40dB SPL。

等響度曲線

右圖中,橫坐標為頻率,縱坐標為聲壓級,波動的一條條曲線就是等響度曲線(equal-loudness contours),這些曲線代表著聲音的頻率和聲壓級在相同響度級中的關聯。

不同頻率的純音,在和1000Hz某個聲壓級純音等響時,其聲壓級也不相同。這樣的不同聲壓級,作為頻率函數所形成的曲線,稱為等響度曲線。改變這個1000Hz純音的聲壓級,可以得到一組等響度曲線[2]:67。最下方的0方曲線表示人類能聽到的最小的聲音響度,即聽閾 ;最上方是人類能承受的最大的聲音響度,即痛閾

最常用的等響度曲線是弗萊徹-蒙森曲線。之後又出現了羅賓遜-達德森曲線,但使用度仍不及前者。2003年,在更加國際化的調查研究基礎之上,國際標準化組織發布了作為國際標準的等響度曲線的ISO 226,其結果更接近羅賓遜-達德森曲線,而弗萊徹-蒙森曲線在低頻部分偏差很大。

等響度曲線反映了響度聽覺的許多特點[2]:68

  • 響度受聲強制約,聲強級提高,響度級一般也要提高。
  • 響度不僅和聲強有關,還和頻率有關,聲強級相同的純音,頻率不同,響度級也不相同。
  • 不同頻率的純音有不同的響度增長率。響度級都從聽閾提高到100方,1000Hz的純音需要提高100dB,20Hz的純音只需要提高不到60dB。

響度的數量標度

一個響度級60方的純音,其響度一定大於一個40方的純音。但是不能說前者的響度是後者的1.5倍。響度級是漸強標度,不能說明兩個純音的響度之間的數量關係。

響度的數量標度單位是Sone)。響度為2宋的聲音比1宋的聲音響兩倍,1宋的聲音又比0.5宋的聲音響兩倍。1宋等於頻率為1000Hz、聲壓級為40dB的純音的響度。

純音的響度(宋)和對應的響度級(Phon)之間成非線性關係[2]:68

響度與時長

聽閾上的聲音如果頻率和聲強都不變、時長增加,它的響度也可能隨之增大。下圖中有三條等響度曲線,線上各點的響度分別相當於聲強級為20、50和80dB的1000Hz純音的響度,它們的時長是500ms。隨著聲音時長的增加,所需的等響度的聲強級減少;在80ms之前變化最大,之後漸趨緩慢。

響度和語言

語言言語中,可以改變音高、音長和/或音強等,從而改變某些音節或語音成分的響度,即為輕重音。不同的語言中,改變響度時所側重的要素不同。語言中,響度不僅跟音強有關,跟音高音長的關係更加密切[3]:44

漢語普通話中的響度較小的輕聲,主要是靠縮短時長來實現的。英語中的重音則主要靠提高音高。

 

內容多遍!莫能釋懷『宋』與『方』★☆

僅邀讀者自觀 hyperphysics 『字詞網』☆

Loudness

Loudness is not simply sound intensity!
Sound loudness is a subjective term describing the strength of the ear’s perception of a sound. It is intimately related to sound intensity but can by no means be considered identical to intensity. The sound intensity must be factored by the ear’s sensitivity to the particular frequencies contained in the sound. This is the kind of information contained in equal loudness curves for the human ear. It must also be considered that the ear’s response to increasing sound intensity is a “power of ten” or logarithmic relationship. This is one of the motivations for using the decibel scale to measure sound intensity. A general “rule of thumb” for loudness is that the power must be increased by about a factor of ten to sound twice as loud. To more realistically assess sound loudness, the ear’s sensitivity curves are factored in to produce a phon scale for loudness. The factor of ten rule of thumb can then be used to produce the sone scale of loudness. In practical sound level measurement, filter contours such as the A, B, and C contours are used to make the measuring instrument more nearly approximate the ear.

Since “loudness” is a subjective measurement of perception, one must be careful about how much accuracy you attribute to it. But though ff is much louder than p in dynamic level, it is not 1000x louder, so one must attempt to develop a scale of loudness that comes closer to mapping your ear’s perception. The “rule of thumb” for loudness is one way to attempt that.

……

Annotated Equal Loudness Curves

Click on any of the highlighted text for further details about the equal loudness curves.
………

Phons

Two different 60 decibel sounds will not in general have the same loudness
Saying that two sounds have equal intensity is not the same thing as saying that they have equal loudness. Since the human hearing sensitivity varies with frequency, it is useful to plot equal loudness curves which show that variation for the average human ear. If 1000 Hz is chosen as a standard frequency, then each equal loudness curve can be referenced to the decibel level at 1000 Hz. This is the basis for the measurement of loudness in phons. If a given sound is perceived to be as loud as a 60 dB sound at 1000 Hz, then it is said to have a loudness of 60 phons.

60 phons means “as loud as a 60 dB, 1000 Hz tone”
The loudness of complex sounds can be measured by comparison to 1000Hz test tones, and this type of measurement is useful for research, but for practical sound level measurement, the use of filter contours has been commonly adopted to approximate the variations of the human ear.

Sones

The use of the phon as a unit of loudness is an improvement over just quoting the level in decibels, but it is still not a measurement which is directly proportional to loudness. Using the rule of thumb for loudness, the sone scale was created to provide such a linear scale of loudness. It is usually presumed that the standard range for orchestral music is about 40 to 100 phons. If the lower end of that range is arbitrarily assigned a loudness of one sone, then 50 phons would have a loudness of 2 sones, 60 phons would be 4 sones, etc.

 

Dynamic Level
Phons
Sones
fff
100
64
90
32
f
80
16
70
8
p
60
4
50
2
ppp
40
1

 

參讀

Olson, Harry F. (February 1972). “The Measurement of Loudness” (PDF). Audio: 18–22.

Richard Cabot and Ian Dennis. “Understanding & Verifying Loudness Meters” (PDF). Retrieved 2011-03-10.