Course:PHYS341/2018/Calendar/Lecture 19

Phys341 Lecture 19: Summary and web references

2018.02.26

Textbook 14.1-14.12 (going light on the math)

Intervals and Scales

Dividing up the audio frequency spectrum
- Most humans can hear fairly well sound with frequencies 100 Hz -10 kHz.
- Why should we even bother trying to divide the range up into discrete frequencies?
- (We don’t for speech communication)
- Physics reasons:
- The human vocal tract does not produce single frequencies; it naturally produces tones which are harmonic series (as does any air column, or string).
- Two tones sound pleasant together if partials coincide: the beginning of harmony.
- Two tones will only sound in harmony if their fundamental frequencies are in simple integer ratio with the first one.
- The simpler the integer ratio, the more harmonic partials coincide.
Where to start
- The simplest integer ratio is 2:1
- If two tones have fundamental frequencies in this ratio, then every second partial of the lower series will coincide with every partial in the upper series.
- In Western music, the interval between these two tones is called an octave (for reasons which will become apparent).
- We can’t make much music with octaves: need to divide up further to produce a scale.
Pitch (an aside)
- Pitch is a matter of human perception rather simple physics.
- For a harmonic series, pitch can be identified with the fundamental frequency.
- When the fundamental frequency is so low that either the instrument does not radiate it, or we cannot hear it, or both, pitch can be discerned from the pattern of harmonic partials.
- For non-harmonic series (e.g. the sound of bells), it can be difficult to understand why we hear the pitches we do.
How many notes in a scale?
- There are limits to how many musical notes we can reasonably fit into a scale.
- Physiological limitation:
- Finite number of nerve connections to our basilar membranes, therefore:
- Our pitch discrimination is limited (although it can be improved by training) to 1% or so in frequency. (Jumping ahead, a semitone is a 6% change in frequency).
- Memory limitation:
- We can only hold a certain number of pitches in memory.
- (Jumping ahead, a three-note triad is easier to recall than an seven-note diatonic scale, which in turn is easier than a 12-note semitone scale).
- Physics limitation:
- The less simple the integer ratio, the less musical two notes sound together (the fewer partials overlap). If too close the tones cause beating.
Marquis yi’s tomb - https://en.wikipedia.org/wiki/Tomb_of_Marquis_Yi_of_Zeng
Nomenclature - see p.172 of text
- Diatonic (7 note) major scale of C: C-D-E-F-G-A-B-C
- Intervals
  - Minor second (semitone) E-F, B-C
  - Major second (whole tone) C-D, D-E etc.
  - Minor third E-G, A-C etc.
  - Major third C-E, F-A etc.
  - “Perfect” fourth C-F, G-C etc.
  - “Perfect” fifth C-G, F-C etc.
  - Major sixth C-A etc.
  - Octave C-C
Guitar demonstration
Pythagorean scale
- Build a scale with simple ratios – 1:2 (octave), 2:3 (fifth), 3:4 (fourth)
- A fifth and a fourth make an octave – 3/2 × 4/3 = 2
- Go up a fifth and down a fourth, i.e. C-D (major second) – 3/2 × 3/4 = 9/8
- Go up a fifth from D, i.e. C-A (major sixth) – 9/8 × 3/2 = 27/16 (not so nice!)
- Go up a major second from D, i.e. C-E (major third) - 9/8 × 9/8 = 81/64 (not nice at all!)
- Alternative method: “Circle of Fifths” going up in fifths and down in octaves
- Used in China and the West
- Similar problems, even the octaves are not perfect factors of two
- Circle of fifths - https://en.wikipedia.org/wiki/Circle_of_fifths
Just temperament
- Band-Aid solution: force intervals to be small-integer ratios
- Many possibilities, e.g. Just Diatonic Scale (a.k.a. Ptolemy’s Intense Diatonic Scale)
- Asymmetric heptatonic scale - mix of slightly unequal tones (T) and semitones (S-T)
- Basis of Western diatonic scale
- F-A-C, C-E-G, and G-B-D be just major triads; A-C-E and E-G-B are just minor triads.
- Thus the semitone interval (E to F, B to C) is 16/15 = 1.0667
- N.B. Prior to 12-tone equal temperament, Chinese heptatonic scale was aimed at equal temperament
Why just temperament needed fixing
- As music became more complex harmonically it became necessary to play in different keys:
- Cannot use just temperament C-scale to move to key of D:
- 5/4 divided by 9/8 = 10/9, (the ratio for E to D) which is not quite the same as 9/8 (the ratio for D to C), so the interval ratios are different.
- This is not a problem for unfretted string instruments, where slight adjustments can always be made with a roll of a finger.
- Ditto for wind instruments: good players can bend notes by varying the vocal tract.
- It is a problem for keyboard instruments and fretted instruments (and harps), where note “bending” is difficult or impossible, so ensembles involving these instruments need the interval ratios to be the same for all musical keys.
The fret problem
- As music became more harmonically complex, many string instruments acquired more and more frets.
- Position of frets determine the frequency of the string – where to put them?
- For example, the pipa 琵琶:
- Tang dynasty (early 7th – early 10th C): 4 frets
- Qing dynasty: (mid 17th C – early 20th C) 10-16 frets (~ semitone)
- 20th C: 24 frets (= semitone)
- Similarly for the Arab oud, Western lute etc.
The new mathematics
- Divide the octave into twelve such that adjacent frequencies are in the same ratio.
- Mathematically adjacent frequencies must all have ratios of the twelfth root of two:
- 2^1/12
- Because twelve such steps recover the octave’s factor of two
- ( 2^1/12)¹² = 2
- How to calculate? Easy now, impossible before the late 16th century.
Need to calculate the square root of the square root of the cube root of 2.
- Try doing that with Roman numerals and/or integer fractions
“System Requirements”
- Concept of zero
- Positional numeral system
- Babylonian (sexagesimal)
- Chinese counting rods (decimal)
- Hindu-Arabic numerals (decimal)
- Decimal fractions
- China 1st C BC
- Arabia (Abu'l-Hasan al-Uqlidisi 10th C)
- Europe (Immanuel Bonfils, France c.1350)
- Square root of two and three
- Babylonian (√2 to three sexagesimal places c.1700 BC)
- Chiu chang suan shu (Nine Chapters on the Mathematical Art, c.100 CE)
Babylonian numerals - http://www-history.mcs.st-and.ac.uk/HistTopics/Babylonian_numerals.html