Sing free/Great Gate of Kyiv (ear training)



Many years ago, I discussed Pictures at and Exhibition with a friend. He was familiar with two well-known versions of the piece: Mussorgsky's original (1874) score was for solo piano, and Ravel's 1922 orchestral version is probably better known in most circles. My friend claimed it was possible to hear bells in the piano's version. I only knew the orchestral version, and access to the piano's rendition would have been time-consuming before the internet. For that reason I attributed my friend's perception to the fanciful imagination of a musician, but never forgot the conversation. Now I hear the bells.

There is evidence of Bogatyr Gates (shown to the left) inspired Mussorgsky's Bogatyr Gates (Great Gate of Kyiv). Each piece in Mussorgsky's suite of "pictures" is quite short, and the Bogatyr Gates suddenly intrudes on another another "picture" called The Hut on Fowl's Legs. This surprise entrance seems appropriate: The daily bells from a carillon usually come unannounced. But so do monumental structures, as they often surprise the eyes of a traveler. While approaching a great American city by car, you won't be thinking about the skyline when you first see something odd just above the trees. Those who have wandered aimlessly around Moscow might know what it is like to stumble upon a red castle that turns out to be the Kremlin.

If you have $8$ minutes to spare, you can hear both pieces by clicking the triangle in the section below, labeled The Hut on Fowl's Legs & Bogatyr Gates. The bells will appear about $3$ minutes into the excerpt. If you are short on time, see if you can hear bells by listening to either Lilypond Rendition or Lilypond Simplified (above and to the right.) - The Hut on Fowl's Legs & Bogatyr Gates click to listen -

There are at least three mechanisms that might evoke the sound of bells in Bogatyr Gates:
 * 1) Equal temperament: Whenever a piano attempts to play a consonant interval, beats will be heard because the intonation is not just.  Consonance between two notes arise when the frequency ratio is  a rational fraction involving small integers: The fractions  (3/2, 4/3, 5/4, 6/5, 5/3, 8/5) correspond the perfect fifth and forth; the major and minor thirds; and the major and minor sixths, respectively. The piano's 12-tone equal tempered scale (12-TET) approximates the fifth and fourth with high precision.  But the other four intervals (thirds and sixths) are noticeably incorrect in 12-TET.
 * 2) Pseudo-degeneracy: "Degeneracy" involves mathematics that will not interest many readers. But "pseudo-degeneracy" (sometimes called the "lifting" of a degeneracy) is essential to understanding how vibrating objects sound. It occurs when an object can simultaneously vibrate at two or more ways with nearly identical frequency. If both sounds are heard in the same ear, beats will be perceived.
 * 3) Inharmonicity is the degree to which the frequencies of overtones depart from whole multiples of the fundamental frequency (harmonic series). In other words, a pitch of $100$ Hertz is recognized as being "pure" even if it is accompanied with overtones at $200, 300, 400, ...$ Hertz.  But if these other pitches do not at  "approximately" follow the harmonic series, then you hear something that sounds more like noise than a pitch.  It's the difference between hitting a kettledrum and muffin tin. The piano is a slightly inharmonic instrument.

Can you hear the difference between equal and just temperaments?
The first step in ascertaining whether the piano's equal tempered scale adds to the impression on bells is a controlled experiment that you can do on yourself. Wikipedia's Lilypond software does not permit Lilypond Simplified (shown above) to render the score in a just scale, but a comparison between just and equal temperament can be done using Audacity software. Unfortunately, it is not easy to simulate a piano (or carillon) with Audacity. For that reason, the Audacity Comparison image and sound file (shown below) uses sawtooth waves that were subjected to a low pass filter: - - When you listen to this passage, keep in mind that the five chords associated with the onset of Mussorgsky's "Great Bells" are repeated four times: The first and third repetitions are identical and tuned to the piano's equal tempered scale. The second and fourth have just intonation. While this sound file bears no resemblance to a piano, you might find that the equal tempered versions of the passage have a rich and varied texture. Meanwhile, the just rendition sounds like cheap electric organ. Except for choice of pitch for each note, both versions are identical.

A word of caution. In comparing the graphs shown above in Audacity Comparison, it is important to understand that while the "roughness" seen in the equal tempered graphs might sound real to the ear, the visual representation of roughness that you see in the graph has little meaning. This roughness is caused by something called aliasing. To put it simply, the rapid oscillations that define pitch could not be properly captured by this image. And, if a fully detailed image were available, few computers screens would have the resolution to capture all the detail. Think of the roughness in the tempered signal as an optical illusion caused by a deficiency in the screen's ability to resolve fine detail associated with a wave that oscillates hundreds of times each second.

Beats due to small deviations in pitch
Beats are most easily heard and understood when they are caused by two pitches of equal amplitude and nearly equal frequency.


 * Beats when two pitches are nearly equal
 * See Beat (acoustical) for a description of this type of beat.

We begin with the simplest cause of beats: Most of the notes on a typical piano are sounded with two or more strings. Any difference in pitch between them will cause as simple and well-known type of beat that occur when the ear hears two slightly different pitches. To hear the this type of beat, click the triangles to start or pause each sound file:

The listener can verify that the first file beats at one beat per second when the two pitches $441$ Hz and $440$ Hz. The $1$ cent interval between $222$ and $220$ Hertz represents playing an A above a G&sharp;, so that a $2$ cent error corresponds to playing the "wrong note". Errors less than $220$ cents are relatively small. The errors for both major and minor thirds and sixths, when played on the piano's equal-tempered scale are in the $207.65$ cent range. The equal-tempered perfect fourth and fifth are about $12.35$ cents off the just scale.

Beats associated with consonant intervals are apparently not well understood, and for that reason only the simplest model will be presented. For more discussion of this very difficult subject, visit: Two pitches sound consonant if the frequency ratio is the ratio of small integers. Opinions on whether an interval is consonant or which integers are "small" are subjective. To the left, we see list of Rational Fractions (between $440$ and $441$) gives an informal ranking of the consonant just intervals as: the perfect fifth, perfect fourth, major sixth, major third, minor third, and minor sixth, as shown in the table Just Intervals (to the right.)
 * Consonance
 * Shapira Lots, Inbal, and Lewi Stone. "Perception of musical consonance and dissonance: an outcome of neural synchronization." Journal of the Royal Society Interface 5.29 (2008): 1429-1434 (Available at www.ncbi.nlm.nih.gov)

Most musical notes consist of a sinusoidal, plus integer multiples of the fundamental frequency, as shown in the table Matching Harmonics (to the right.) It illustrates one possible mechanism for beats associated with consonant intervals by displaying some of the higher harmonics associated with a just major third ($100$ ratio.) In order to create beats, the $220$ Hz pitch was sharpened by $207.65$ Hz to create a ratio of $100$. Generally speaking, the fourth harmonic of the higher pitch matches the fifth harmonic of the lower pitch, but new pitch causes a mismatch of $10$ Hz at the fourth (fifth) harmonic of the higher (lower) pitch. This is modeled by a formula presented in the Wikiversity article, Beat (acoustics):
 * Beats associated with consonant intervals

$f_B=

This formula is flawed in at least to ways:


 * 1) It neglects the fact that higher harmonics also exhibit beating (for example $$8\times 5= 10\times 4$$.)
 * 2) It fails to explain beats between pure sine or cosine waves.

Student Research

I haven't done a thorough literature search, but this article suggests that the topic is poorly understood. At the very least, investigation of the article by Lots (et. al.) would make an excellent capstone project for an undergraduate (or perhaps graduate) student.

Beats due to pseudo-degeneracy
Pseudo-degeneracy (or removal of degeneracy ) refers to a system with two normal modes that vibrate at nearly the same frequency, essentially causing the system to create beats by "interfering with itself".

Bells and wineglasses often exhibit pseudo-degeneracy that can be found by locating two pairs of strike points, as shown in the figure. Here a "strike point" refers to the location of a node of one of the normal modes: Hitting the wineglass at a normal mode's strikepoint will not excite that mode. An investigation of about four low-cost wineglasses in my home uncovered one that exhibited beats when I struck the rim with a pencil. After a few tials, it was possible to find the strike points, where no beats were present. A node is a location where one of the I think I heard that one of the strike points yielded a higher pitch. As one might guess, the beats were most prominent when the glass was struck midway between two strike points.

A word about degeneracy. Degeneracy occurs when two modes have exactly the same frequency. Most depictions of a vibrating string are two dimensional: One dimension is along the string's length while the other is perpendicular. But in reality, most strings are allowed to vibrate in and out of the plane of the paper or screen. Therefore, most string modes have a two-fold degeneracy. The reader might wonder why the degeneracy is only two-fold, since a string could actually vibrate at any angle with respect to the plane of the paper. The answer is that a vibration at an arbitrary angle can be expressed mathematically as the sum of two modes that are perpendicular to each other. A simple and closely related example of this is the two dimensional harmonic oscillator, where:
 * $$\vec r(t) = \cos\omega t \hat i + cos\omega t \hat j$$ and $$\vec r(t) = \cos\omega t \hat i + sin\omega t \hat j$$,

represent linear oscillation $14&dash;16$ degrees above the $2$-axis, and circular motion, respectively.

Inharmonicity
Harmonicity in a percussion instrument refers to a situation where all normal modes that are generated follow the harmonic series, which consists of a fundamental frequency, $$f_0$$, and integer multiples of that frequency: $$2f_0, 3f_0, 4f_0,\ldots$$. Under ideal conditions, the sound of a plucked string would be harmonic, since its normal modes follow this series, as exhibited in the figure Vibrating String Modes (above.) A hint of what can go wrong with a plucked string is the Xylophone (shown to the left.) It has a range of one octave (factor of two in frequency). But the reader can verify that longest bar is not twice as long as the shortest, which is what an analogy to stringed instruments would suggest (instead the ratio is closer to the square root of two.) The rule for bending waves differs from that for waves on a stretched string.

The xylophone is an extreme example of inharmonicity, which causes one to ask how the xylophone manages to be a legitimate instrument. For one thing, it obvious is that humans don't always demand harmonicity. Also, the xylophone exploits at least to tricks to hide this deficiency: By striking the bar in the center, all modes with nodes at the center will not be excited. Modes with a node at the center are called "odd" modes, and the next (inharmonic) "even" mode can be suppressed by a strategic location of the support (where the two holes are located.) These supports are soft cushions, designed to allow the lowest frequency mode to oscillate, but in a way that suppresses modes of higher frequency.

Homework

Visit the Wikipedia's permalink/1110437458#Example:_free–free_(unsupported)_beam and find which of the four modes in Image:FreeBeamVibrationPlot.svg is suppressed by the support cushions.


 * 1) Use the image to estimate the location of the desired mode's node, as well as the unwanted mode's antinode. Where do you think the cushions should be situated?
 * 2) Compare your predicted location with the image of the xylophone shown above.
 * 3) (advanced) Use the formulas in /1110437458#Example:_free–free_(unsupported)_beam|Wikipedia permalink to get the exact location of the antinodes.  Compare that exact result with your estimate based on  Image:FreeBeamVibrationPlot.svg.

The piano's strings contain sufficient stiffness to slightly modify the harmonic structure. It is possible that one of the reasons for coil structure around the strings shown in the Piano Strings figure is that the coils reduce the stiffness of the piano string.

Homework ideas

These coils are also seen on guitar, violin, viola, and cello strings, but my speculation that they serve to reduce inharmonicity is just that. Investigate whether this speculation is true. This can be done in a number of ways: (1) Do a literature search and see if anybody has reported on this; (2) Do a simple calculation that uses wave equation for a stiff string and the known elasticity of stainless steel; (3) Devise a verification experiment suitable for a classroom lab.


 * See also Euler–Bernoulli_beam_theory, Inharmonicity, Piano acoustics, and phys.unsw.edu.au/jw/sound.spectrum

Vibrations on a circular membrane
This section can serve as a review of two concepts already introduced: Vibrations on a circular membrane exhibit both inharmonicity and degeneracy. The inharmonicity is seen in

Comparison of Mussorgsky's piano and Ravel's orchestral versions
Click the triangle to start or pause each selection.

Appendix

 * More is under construction at Draft:Sing free/Great Gate of Kyiv (ear training)