AAM-HW-3

Problem 1.

i. Logarithm.

  • Neutral water's H⁺ concentration is around mol/L, which means .
  • One Bel difference corresponds to 10 times the sound power.
  1. When value decreases by 2.5, how many times higher/lower is the hydrogen ion concentration in the solution?

Ans:

When the pH decreases by 2.5, the change in can be represented as:

increases by times.

Therefore, the hydrogen ion concentration in the solution is 316.23 times higher.

  1. Earplug A reduces the sound level by 33 dB; what percentage of sound power does it reduce?

Ans:
The relationship between the sound level in decibels (dB) and the ratio of power is:

Where represents the original sound power and represents the reduced sound power after attenuation.

For a reduction of 33 dB:

Divide by :

Raise both sides to the power of :

This means the output power is about 0.0501% of the input power. Therefore, the sound power reduction is:

So, the earplug reduces approximately 99.95% of the sound power.

  1. Earplug B reduces 98% of sound; what is that in dB?

Ans:
If 98% of the sound power is reduced, then only 2% of the original power remains. The power ratio is:

Use the formula for dB:

Substitute the ratio:

This corresponds to a reduction of approximately 17 dB.

ii. Irrational numbers.

Prove that is an irrational number, by contradiction:

  • Suppose for some natural numbers and .
  • What contradiction will you get?

Ans:

To prove is irrational by contradiction:

  1. Assume , where .
  2. Then . Raising both sides to the -th power gives .
  3. is a power of 2, and is a power of 3. These have distinct prime factorizations, so is impossible.
  4. This contradiction shows that is irrational.

iii. Simple continued fractions.

Recall that every rational number can be written as a finite simple continued fraction, by the Euclidean algorithm. This can be extended to any (real) number , possibly with infinite terms:

  • Example and notation:

  • [Look up the steps to perform this computation using either a calculator or a computer code.]

  1. Briefly explain (in words, with pictures, or with proof) why
    (periodic), and whether is rational or irrational.

Ans:

  • To write as a continued fraction:
    This process repeats, producing (periodic).
  • is irrational because it has an infinite continued fraction (finite ones correspond to rational numbers).
  1. Calculate the first few terms (up to ) of continued fractions for .

Ans:

Approximating , the continued fraction process yields:

iv. Best rational approximations.

Find a way (for example, by hand, by linear recursion, or by sympy continued_fraction_convergents) to compute the first few convergents of a continued fraction.

  • Theorem: If is a convergent (of the continued fraction) of a real number , then it is the best rational approximation of with denominator at most .
  1. Find the best rational approximations of with denominator at most 10, and at most 20.

Ans:
At most 10: best rational approximations:
At most 20:best rational approximations:

  1. Write down all convergents of (the continued fraction for) up to denominator 53.

Ans: , , , , , , .

Problem 2.

Part 1: See Week_5_2.pdf

Problem 2. A=440, 12-tone equal temperament: the standard modern tuning system.

  • 2:1 ratio in frequency corresponds to one octave interval.

    • Each octave is divided equally into 12 semitones.
    • Each semitone is further divided into 100 cents.

  • A4, the first "A" above middle C ("A0"), in the fourth octave is defined to have frequency 440 Hertz.

(i) Logarithm and exponentiation: going between multiplicative ratios and additive differences

  1. How many octaves/semitones/cents is the pitch interval?
    If two pitches have a frequency ratio , the pitch interval is:

    Octave interval formula:

    Semitone interval formula:

    Cent interval formula:

  2. What is the frequency if a pitch is semitones above A4?
    Its frequency is calculated as:

  3. What frequency ratio is the interval of 50 cents? 14 cents?
    From the formula for cents:

    • A 50-cent interval corresponds to:
    • A 14-cent interval corresponds to:

(ii) Fill in the frequencies of all 12 semitones in the 4th octave

  • The general formula for the frequency of a semitone above C4 is:

  • Using , calculate the following table:

Pitch in the 4th octave C C#/D D D#/E E F F#/G G G#/A A A#/B B
Semitones above C 0 1 2 3 4 5 6 7 8 9 10 11
Frequency ratio with C 1.000 1.059 1.122 1.189 1.260 1.335 1.414 1.498 1.587 1.682 1.782 1.888
Frequency (Hz) 261.63 277.18 293.66 311.13 329.63 349.23 369.99 392.00 415.30 440.00 466.16 493.88

(iii) Major scale (大调音阶), equal temperament vs just intonation

  • Focus on the 7 white keys: C, D, E, F, G, A, B:
    • Calculate their frequency ratios with C in decimals:
      • Equal temperament ratio:
      • Just intonation: Use the table provided under "just interval".
Note Equal temperament ratio Just interval ratio Approx. difference in cents
C 1.000 1.000 0
D 1.122 1.125 (9:8) ~3.91
E 1.260 1.250 (5:4) ~13.69
F 1.335 1.333 (4:3) ~1.96
G 1.498 1.500 (3:2) ~(-1.96)
A 1.682 1.667 (5:3) ~19.55
B 1.888 1.875 (15:8) ~11.73
  • Convert the just intervals into cents:
    Use the formula:

(iv) Chords in just intonation:

  • Major triad (大三和弦): Cmaj (C-E-G)

    • Ratios:
    • Expressed as integers:

  • Minor triad (小三和弦): Amin (A-C-E; A is one octave lower)
    .

  • Suspended fourth chord (Csus4): C-F-G
    Ratios:

  • Diminished triad (减三和弦): Bdim (B-D-F; B is one octave lower)
    Ratios:

Problem 3.

Controlling for other variables, the fundamental frequency of a vibrating string or air column is inversely proportional to its length.

i. “Harmonic” and “Inversion”:

For a chord with frequency ratio 4:5:6, what are the string length ratios, in the smallest integer ratios?

Ans:

  • For Frequency Ratios 4 : 5 : 6:
    Length Ratio =

For a chord with frequency ratio 10:12:15, what are the bamboo pipe's length ratios?

Ans:

  • For Frequency Ratios 10 : 12 : 15:
    Length Ratio =

ii. 三分损益法 (Sanfen Sunyi):

From the antiquities, across various cultures including Mesopotamia, China, and Greece, one method of generating the 5/7/12 tones are as follows:

  1. Start from a standard bamboo pipe, call this pitch C / 黄钟 / 宫.
  2. Take a new bamboo pipe whose length is 2/3 of the previous one.
    However, if this new pipe would be less than 1/2 of the standard one:
    Instead, increase the new length to 4/3 of the previous one.
  3. Repeat step 2.
  4. End if the new pipe would be within 2% difference from the standard one; discard it.
  5. Order the pipes in descending length order.

Question: What do the length ratios 2/3 and 4/3 mean, according to Problem 4.iii?

Ans:

The length ratios and represent fundamental musical intervals in the construction of the scale:

  1. shortens the pipe length to generate a pitch a perfect fifth higher.
  2. extends the pipe length to generate a pitch a perfect fourth lower.

These ratios reflect the cycle of intervals used to construct the twelve-tone scale, forming relationships between adjacent notes.

Table Completion:

Shí-èr lǜ 十二律 黄钟 (C) 林钟 (G) 太蔟 (D) 南吕 (A) 姑洗 (E) 应钟 (B) 蕤宾 (F#) 大吕 (C#) 夷则 (G#) 无射 (D#) 中吕 (A#) 清黄钟
Order of generation 0 1 2 3 4 5 6 7 8 9 10 12
Length ratio (2^n/3^m) 1 2/3 8/9 16/27 64/81 128/243 4/9 32/81 128/729 256/729 1024/2187 1/2
Length (decimal) 1.000 0.667 0.889 0.593 0.790 0.527 0.444 0.395 0.175 0.351 0.469 0.500
Frequency ratio 1.000 1.500 1.125 1.687 1.265 1.897 2.250 2.532 5.731 2.847 2.132 2.000
Length if first = 81 81.00 54.00 72.00 48.00 64.00 42.67 36.00 32.00 14.18 28.44 37.96 40.50
Order of descending 1 3 2 5 4 7 6 9 8 11 10 12

Arranging in descending length order, the odd numbers are the 六律, and the even numbers are the 六吕; list them in that order and grouping.

Groupings:

  • 六律 (Odd numbers): 黄钟 (C), 太蔟 (D), 姑洗 (E), 蕤宾 (F#), 夷则 (G#), 中吕 (A#).
  • 六吕 (Even numbers): 林钟 (G), 南吕 (A), 应钟 (B), 大吕 (C#), 无射 (D#).

iii. After the calculations, compare with:

  • Link 1 (《淮南子·天文训》)
  • Link 2 (《史记·律书》), and/or other classic texts.

The 十二律 (Twelve Tones) and 十二吕 (Twelve Modes) are ancient Chinese musical scales, with the 律 representing the fundamental tones and the 吕 representing the derived tones. The 律 are the primary notes, while the 吕 are the secondary notes derived from the 律.

iv. If A is set to 432 Hz instead of 440 Hz, what are the frequencies of 五音? Play music on desmos.com, where tone(f) plays a sinewave tone at frequency f.

Ans:

If , the frequencies of the pentatonic 五音 (C, D, E, G, A) are calculated by scaling relative to . Using the 12-TET system:

  1. Frequencies:

    • 宫
    • 商
    • 角
    • 徵
    • 羽

  2. Play on Desmos:
    Use tone(f) for each frequency:

tone(256.87), tone(288.00), tone(324.00), tone(384.87), tone(432.00)
Table Of Contents
AAM-HW-3
Problem 1.
i. Logarithm.
ii. Irrational numbers.
iii. Simple continued fractions.
iv. Best rational approximations.
Problem 2.
Problem 2. A=440, 12-tone equal temperament: the standard modern tuning system.
(i) Logarithm and exponentiation: going between multiplicative ratios and additive differences
(ii) Fill in the frequencies of all 12 semitones in the 4th octave
(iii) Major scale (大调音阶), equal temperament vs just intonation
(iv) Chords in just intonation:
Problem 3.
i. “Harmonic” and “Inversion”:
ii. 三分损益法 (Sanfen Sunyi):
iii. After the calculations, compare with:
iv. If A is set to 432 Hz instead of 440 Hz, what are the frequencies of 五音? Play music on desmos.com, where tone(f) plays a sinewave tone at frequency f.