1. Vowels + Music
March 18, 2013

2. Monday, Monday… Course Project Report #4 is due!
Course Project Report #5 guidelines to hand out!
On Wednesday, I’ll give you guidelines for the final term paper/presentation.
Let’s wrap up vowels today…
And then get into sonorant acoustics for the rest of the week.
First: fun videos + clips!
Mumford and Sons
T-Pain
Auditory Scene Analysis

3. Theory #2 The second theory of vowel production is the two-tube model.
Basically:
A constriction in the vocal tract (approximately) divides the tract into two separate “tubes”…
Each of which has its own characteristic resonant frequencies.
The first resonance of one tube produces F1;
The first resonance of the other tube produces F2.

4. Open up and say...
For instance, the shape of the articulatory tract while producing the vowel resembles two tubes.
front tube
back tube
Both tubes may be considered closed at one end...
and open at the other.

5. Resonance at Work An open tube resonates at frequencies determined by:
fn = (2n - 1) * c 4L
If Lf = 9.5 cm:
F1 =
35000 / 4 * 9.5
= 921 Hz

6. Resonance at Work An open tube resonates at frequencies determined by:
fn = (2n - 1) * c 4L
If Lb = 8 cm:
F1 =
35000 / 4 * 8
= 1093 Hz
 for :
F1 = 921 Hz
F2 = 1093 Hz

7. Switching Sides
Note that F1 is not necessarily associated with the front tube;
nor is F2 necessarily determined by the back tube...
Instead:
The longer tube determines F1 resonance
The shorter tube determines F2 resonance

8. Switching Sides

9. Switching Sides

10. A Conundrum
The lowest resonant frequency of an open tube of length 17.5 cm is 500 Hz. (schwa)
In the tube model, how can we get resonant frequencies lower than 500 Hz?
One option:
Lengthen the tube through lip rounding.
But...why is the F1 of [i]  300 Hz?
Another option:
Helmholtz resonance

11. Hermann von Helmholtz (1821 - 1894)
Helmholtz Resonance
A tube with a narrow constriction at one end forms a different kind of resonant system.
The air in the narrow constriction itself exhibits a Helmholtz resonance.
= it vibrates back and forth “like a piston”
Hermann von Helmholtz ( )
This frequency tends to be quite low.

12. Some Specifics
The vocal tract configuration for the vowel [i] resembles a Helmholtz resonator.
Helmholtz frequency:

13. An [i] breakdown Helmholtz frequency: Length(bc) = 1 cm
Area(bc) = .15 cm2
Volume(ab) = 60 cm3

14. An [i] Nomogram
Helmholtz resonance
Let’s check it out...

15. Slightly Deeper Thoughts
Helmholtz frequency:
Length(bc)
Area(bc)
Volume(ab)
What would happen to the Helmholtz resonance if we moved the constriction slightly further back...
to, oh, say, the velar region?

16. Ooh!
The articulatory configuration for [u] actually produces two different Helmholtz resonators.
= very low first and second formant F1 F2

17. Size Matters, Again Helmholtz frequency:
What would happen if we opened up the constriction?
(i.e., increased its cross-sectional area)
This explains the connection between F1 and vowel “height”...

18. Theoretical Trade-Offs
Perturbation Theory and the Tube Model don’t always make the same predictions...
And each explains some vowel facts better than others.
Perturbation Theory works better for vowels with more than one constriction ([u] and )
The tube model works better for one constriction.
The tube model also works better for a relatively constricted vocal tract
...where the tubes have less acoustic coupling.
There’s an interesting fact about music that the tube model can explain well…

19. Some Notes on Music
In western music, each note is at a specific frequency
Notes have letter names: A, B, C, D, E, F, G
Some notes in between are called “flats” and “sharps”
261.6 Hz
440 Hz

20. Some Notes on Music
The lowest note on a piano is “A0”, which has a fundamental frequency of 27.5 Hz.
The frequencies of the rest of the notes are multiples of 27.5 Hz.
Fn = 27.5 * 2(n/12)
where n = number of note above A0
There are 87 notes above A0 in all

21. Octaves and Multiples Notes are organized into octaves
There are twelve notes to each octave
 12 note-steps above A0 is another “A” (A1)
Its frequency is exactly twice that of A0 = 55 Hz
A1 is one octave above A0
Any note which is one octave above another is twice that note’s frequency.
C8 = 4186 Hz (highest note on the piano)
C7 = 2093 Hz
C6 = Hz
etc.

22. Frame of Reference
The central note on a piano is called “middle C” (C4)
Frequency = Hz
The A above middle C (A4) is at 440 Hz.
The notes in most western music generally fall within an octave or two of middle C.
Recall the average fundamental frequencies of:
men ~ 125 Hz
women ~ 220 Hz
children ~ 300 Hz

23. Harmony
Notes are said to “harmonize” with each other if the greatest common denominator of their frequencies is relatively high.
Example: note A4 = 440 Hz
Harmonizes well with (in order):
A5 = 880 Hz (GCD = 440)
E5 ~ 660 Hz (GCD = 220) (a “fifth”)
C#5 ~ 550 Hz (GCD = 110) (a “third”)
....
A#4 ~ 466 Hz (GCD = 2) (a “minor second”)
A major chord: A4 - C#5 - E5

24. Extremes Not all music stays within a couple of octaves of middle C.
Check this out:
Source: “Der Rache Hölle kocht in meinem Herze”, from Die Zauberflöte, by Mozart.
Sung by: Sumi Jo
This particular piece of music contains an F6 note
The frequency of F6 is 1397 Hz.
(Most sopranos can’t sing this high.)