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Vowels + Music

Vowels + Music. March 25, 2014. From a Tuesday Point of View. Course Project Report #4 is due! Course Project Report #5 guidelines to hand out! On Thursday, I’ll give you guidelines for the final term paper/presentation. Let’s wrap up vowels today…

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Vowels + Music

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  1. Vowels + Music March 25, 2014

  2. From a Tuesday Point of View • Course Project Report #4 is due! • Course Project Report #5 guidelines to hand out! • On Thursday, 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. 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 (1821 - 1894) • 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 = 1046.5 Hz • etc.

  22. Frame of Reference • The central note on a piano is called “middle C” (C4) • Frequency = 261.6 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.)

  25. Implications • Are there any potential problems with singing this high? • F1 (the first formant frequency) of most vowels is generally below 1000 Hz--even for females • There are no harmonics below 1000 Hz for the vocal tract “filter” to amplify • a problem with the sound source •  It’s apparently impossible for singers to make F1-based vowel distinctions when they sing this high. • But they have a trick up their sleeve...

  26. Singer’s Formant • Discovered by Johan Sundberg (1970) • another Swedish phonetician • Classically trained vocalists typically have a high frequency resonance around 3000 Hz when they sing. • This enables them to be heard over the din of the orchestra • It also provides them with higher-frequency resonances for high-pitched notes • Check out the F6 spectrum.

  27. more info at: http://www.ncvs.org/ncvs/tutorials/voiceprod/tutorial/singer.html How do they do it? • Evidently, singers form a short (~3 cm), narrow tube near their glottis by making a constriction with their epiglottis • This short tube resonates at around 3000 Hz • Check out the video evidence.

  28. Singer’s Formant Demo

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