The Cochlea

The cochlea is the name given to the organ in the inner ear that is primarily responsible for converting physical sound waves into electrical nerve signals that our brains can understand. It is what allows a series of pressures to be perceived as a sound or a note.

As a sound wave enters the ear, the outer ear condenses the wave and increases the power (which affects perceived loudness in dB) of the wave. The wave then causes the eardrum to fluctuate inwards and outwards in response to the changing pressures; a pressure higher than that of the inner ear causes the drum to push inwards; a pressure lower than that of the inner ear causes the eardrum to pull outwards. The eardrum then moves a series of bones (named the "hammer," "anvil," and "stirrup.") which, through leverage, further amplify the sound. The last bone, the "stirrup," is connected to a small membrane called the "oval window" and pulls and pushes it back and forth which sets up the wave that entered the ear in the cochlea.

So, the series as a whole goes as follows: A pressure series enters the ear and causes the eardrum to move. The eardrum takes the energy contained in the pressure and converts it to mechanical energy which is transferred through a series of bone levers. The last bone lever causes the oval window membrane to move which converts the mechanical energy of the motion of the bones back into a series of pressures.

The cochlea itsself is a coiled tube which gets narrower towards the end and is divided in half lengthwise by a basilar membrane. The basilar membrane extends the entire length of the cochlea except for a small opening at the narrow end of the tube which allows the two halves to be connected. The basilar membrane contains many thousand audiocilia hair cells which run the length of the membrane--different cells are activated when different frequencies enter the ear and send electrical nerve signals to the brain which are interpreted as sound.

Because the cochlea is less flexible near the wide end and more flexible near the thin end, the sound waves that enter the ear do not retain their exact amplitude along the entire length of the cochlea; this is what causes different parts of the basilar membrane to be receptive to different frequencies. More information can be found in this link.

Why we like Diatonic Scales.

I first read about this while reading Rameau's A Treatise on Harmony which is an 18th century text; this isn't new knowledge by any means. In fact, the harmonic series, which I'll get to shortly, was first discovered by Pythagoras using a "monochord"--a one-stringed instrument used for examining properties of waves--in ancient Greece. Although this explanation doesn't sit well with me (I'll get to that later), it is considered by most music theorists to be the correct explanation.

Let me first take a moment to demonstrate how to "read" a keyboard and how to construct a major scale. Below is an illustration of an octave, starting and ending on C, on a keyboard for you to refer to.




Each key represents one half step out of the 12 half steps which make an octave. As you can see, the white keys are given the names C,D,E,F,G,A,B, & C, respectively. These names practically never change. The black keys, however, can be assigned two names... one with a sharp, and one with a flat. Remember that "flat" means one half-step lower, and "sharp" means one half-step higher. Because each key is a half step higher than the one previous, and conversely a half step lower than the one following,  and because sharp denotes being a half-step higher and flat denotes being a half-step lower, it would make sense that the black key between C and D can be called either "C#" or "Db", the one between F and G is "F#" or "Gb". A slight problem arises between E & F and B & C because there is no black key between these notes. In this case, "E#" is actually the same as F, and "Fb" is the same as E; "Cb" is the same as B, and "B#" is the same as C.

By looking at this keyboard, it is easy to see the construction of the major scale as a series of whole- and half-steps. The C major scale is C, D, E, F, G, A, B, C as has been stated before in other posts. You can see by looking at the keys on the keyboard, that to get to D from C you have to move up 2 keys, or 2 half-steps i.e. one whole-step. From D to E is another whole-step; E to F, a half-step; F to G a whole-step; G to A a whole-step; A to B a whole-step; and B to C a half-step. In short, the formula is: W-W-h-W-W-W-h, where "W" denotes a whole-step and "h" a half-step.

Notes like "E#" and "Cb" are used instead of writing "F" or "B", respectively, for this reason: When writing a diatonic scale, each letter of the musical alphabet (A through G) must be used once. So if you start on the note F#, use each letter once, and also follow the W-W-h-W-W-W-h pattern, you get this: F#-G#-A#-B-C#-D#-E#-F#. 

So why do we like this arrangement of notes? The reason lies in what is called the harmonic- or overtone series. The harmonic series is a series of intervals given off by any natural, vibrating object; a pure sine wave does not give off a harmonic series. When a natural tone (called the fundamental tone) is played, it gives off overtones in multiples of its frequency. For instance, if a fundamental tone of 500Hz is played, then overtones of 1000, 1500, 2000, 2500Hz, etc. will also resound.  These overtones diminish in loudness as they get farther and farther away from the fundamental. If  you would notate these out musically, you would get the following progression of notes (using C as the fundamental):

C, C, G, C, E, G, Bb, C, ...

If you spell this out as intervals between the two consecutive notes (not between the fundamental C and the note) it would look like this:

perfect Octave, perfect Fifth, perfect Fourth, major Third, minor Third, major Second. 

Remember that a major 6th is an inversion of a minor third, and a minor 7th is an inversion of a major second. When you include these two intervals, and rearrange the other intervals to suit, you get this:

Unison, major 2nd, major 3rd, perfect 4th, perfect 5th, major 6th, minor 7th, perfect octave. With the exception of the 7th (which would be a Bb), this is a diatonic major scale. C-D-E-F-G-A-B-C. 

If you take the same pattern W-W-h-W-W-W-h, and shift the pattern to start on the sixth note (in our case A) and then circulate through, you get: W-h-W-W-h-W-W. This is the A minor scale. It is also known as "A aeolian" or the "sixth mode of C". A 'mode' is when you take the notes of a major scale, and start on a different scale tone. In the case of the A minor, you use the same notes as a C major scale, but start on A instead of C. 

Now we have to go back and reexamine the overtone series. When you take the intervals and arrange them as was previously done, but this time include the minor 7th, the following scale emerges: C-D-E-F-G-A-Bb-C.  This is actually the mode "C mixolydian" or the "fifth mode of F", because it uses the same notes as an F major scale (F-G-A-Bb-C-D-E-F), but starts on a C, which is the fifth note of the F major scale. 

So in review of the information, here are some things that I find interesting: 

1) If naturally the mixolydian mode is formed instead of the regular diatonic scale, why do we not prefer this mode over any other mode or scale? (the mixolydian mode is the most commonly used mode, though; besides the aeolian, which is the same as the diatonic minor scale.)

2) Why does the major scale on a piano that uses all the white keys (i.e. the "simplest" one) start on C and not A-- after all, A is the beginning of the alphabet and would make the most logical sense? What this suggests to me is that either the piano was intended to 'naturally' play in the key of A minor for whatever reason, or that we as humans naturally prefer the minor scale over the major scale. In other words, was the piano (and musical scales in general) named thusly because it was "intended" to play in A minor rather than C major because we humans liked the minor scale the best, or was it just arbitrary?

Tone Clusters

A tone cluster, or chord cluster, is a group of 3 or more notes separated by half-steps, whole-steps, or occasionally one-and-one-half-steps... i.e. Minor 2nds (1 semitone), Major 2nds (2 semitones), and Augmented 2nds (3 semitones). Examples would be [C, C#, D, D#] or [C, D, E, F#]. To form a tone cluster, consecutive notes from a chromatic scale (a scale containing all 12 semitones in order), a diatonic scale (major or minor scale), or occasionally a pentatonic scale (this is when augmented 2nds come into play), are played simultaneously. While a a chord cluster obviously contains a great amount of dissonance, since minor- and major-2nds are the 2 most dissonant intervals, the cluster also has a recognizable quality present in it--just like a regular, consonant chord or interval.

Additionally, when the first and last note of the cluster are spaced a consonant interval apart (like a perfect 4th), the cluster tends to be more agreeable to the human ear. This further deepens the 'mystery' of why chord clusters, despite overwhelming dissonance, have various distinct qualities and have varying degrees of agreeableness. Why is it that humans find these clashing notes agreeable and harmonious?

The research that I have done so far suggests 2 things about our appreciation of music:

a) There are musical principles that we innately find appealing. Certain scales, chords, etc. just naturally sound good to us. This is evidenced by the fact that certain scales and chords have been dominant in music since melodic music's conception several thousand years ago.

and

b) There are musical principles which seem to have appeal to us only because we are used to it. It is an established cultural aspect which we have been exposed to since birth and we now accept it as being "Okay" and often times even find it appealing. This is evidenced by the fact that infants, who have not yet been heavily exposed to our musical culture, very strongly prefer consonance over dissonance, and yet we find vast amounts of dissonance (even though it is almost always resolved to consonance, with the exception of experimental avant-garde music) in basically all modern music around the world.

We may find it interesting, then, that these tone clusters are a very recent addition to music. The first published song which utilized chord clusters was Heinrich Biber's "Battalia a 10" in 1673, but for the next 250 years, tone clusters would virtually disappear from music only appearing a handful of times and never more than once or twice in an entire piece. In the early 1900's and 1910's chord clusters had a resurgence and became heavily present in jazz and ragtime styles of music. Since then, avant-garde composers have been using them extensively in many different styles of music, from jazz to classical to neo-folk. At first tone clusters were rejected as being unharmonious, however over the decades they have become much more accepted. It would seem, then, that these clusters have only developed and become accepted because of cultural influence--as they become more predominant in music, we have become more accepting of them--however, what, then, inspired musicians to utilize them in music in the first place, and why can we appreciate the qualities that each have? There seems to be a mix of both innate and acquired taste in this.

Over the next few days I plan on investigating various aspects of these chord clusters such as the 1st, 2nd, and 3rd tier Tartini tones that are produced and how these tones may affect the cluster or if these tones match up the "regular" chords which share the same qualities.

Thoughts about future research.

I received a message today from someone who read this blog, and they pointed out some things that they would like to see in this blog.

First of all, I want to point out that I have to do this blog piece-by-piece... it's a very deep topic which spans across 3 different subject areas, 6 continents, and 5000 years of history. So far I've been focusing on the simple things, such as intervals and triads and basic harmonies and basic chord qualities, and on Western music, especially that which originated in Greece.

I will most likely never investigate extended chords, at least not past 4-note harmony, or entire songs any deeper than I have now as a casual observation. This would just get far too in-depth, and I don't plan on devoting decades of my life to investigating this, despite how interesting I might find it.

I do, however plan on delving deeper into the following areas of study:

Cluster chords--Dissonance is heavily present, and yet distinct chord qualities can be heard.
Eastern/Middle Eastern musics--Hebrew music is supposedly the basis for Western/Mid-East music, but what about Japanese or Chinese music? Are there common scales or chords or theory principles between East/West despite originating completely independently?
"Chord Ladders"--Some chords, based on their placement in the scale and their distance from the root note naturally become somewhat dissonant to the chord progression as a whole--even though in and of themselves the chords are entirely consonant. They want to be "resolved"... if a song ended on these chords, we would be left with a very unsettling feeling. Why?

Additionally, I want to delve deeper into many of the topics that I've already discussed in my blogs posts to date. Some key points that I want to find out as a whole are:

Why do we perceive some intervals/harmonies as being dissonant or consonant?
What about qualities of chords besides just "happy" and "sad"?
Why do certain scales come easy to us or sound pleasant, while other scales are harder to learn or just sound bad?

I'm sure that along the way countless other questions will arise, some which might be answered in part, many which will never be answered, and none which will be answered fully.

P.S. to notes about music.

In light of my previous posts regarding the overall "mood" of a song being largely based on the first and last chord, I feel that it's necessary to clear some things up. When I say that a song's mood is based upon these chords, I'm talking about the overall, general positive/negative, happy/sad, major/minor feeling that the chords convey. But it is very important to remember how many other things make up a song as a whole. The various instruments which lend their timbres to a song, the multiple rhythms and tempos which provide "groove" (yes, that is a legitimate musical term) to a song, and nowadays even the multimedia such as music videos that lend visual feeling to a song, all shape the overall feeling of a song very greatly. Additionally, there are some exceptions to this effect. Just because a certain chord progression or musical sequence has become culturally established as belonging to a certain genre, doesn’t mean that that chord progression must be used for that genre. The opposite is also true.

Below are 2 videos that demonstrate what I mean:

The first video shows parts of 36 different songs all played on the same 4 chords repeating over and over again--3 of the 4 chords are major, including the start and end chords, but one of the chords in the middle is minor. As you'll notice, the songs all sound generally major or at least not sad. None of them sound truly depressing or sad. But, there are some very great differences between the more intricate feeling of the song. Just compare Journey's "Don't Stop Believin'" to Men at Work's "Land Down Under" ... they both sound major, but in very very different ways.

**at a about 4 minutes into this clip there is some profanity. Please don't watch it after the 4-minute mark.

The next video, "Blue" by Yngwie Malmsteem, is a song in the style of the the Blues genre, but uses no typical "blues" chord progressions in it.

P.S. I'm not really a fan of cheesey 80's music or classical-music-inspired guitar shredding, but these video clips are relevant to the topic and make good points.

P.S. I'm not really a fan of cheesey 80's music or classical-music-inspired guitar shredding, but these video clips are relevant to the topic and make good points.

HSS PowerPoint

Here's a presentation which I am delivering tomorrow to a small group of middle- and high-school students and teachers. It outlines the main points of Hypersonic Sound technology and also hints at the extensions into music of the physics principles that make HSS possible.

Hypersonic Sound

View more presentations from eikejmaas.

Some more notes about music.

This is post basically a short collection of after-thoughts and P.S.'s for the last post. It is also a documentation of the though process which led me to "discover" and realize a vital part of the connection between physics & music, numbers & aesthetics.

I mentioned in the last post about how the chords in the middle of a chord progression really don't significantly affect the overall "positive" or "negative" quality of the progression. I drew a connection between this and how similarly you can have very dissonant and negative intervals in an individual chord and yet they have basically no affect on the overall "positive" or "negative" quality. They absolutely do change the quality of the chord some... but they don't truly affect the overriding positive/negative, consonant/dissonant polarity of the chord.

I thought about this "phenomenon" a little while after that last post and now I have to draw another connection; this time between a chord progression and "stacked" chords vs. "arpeggiation". A stacked chord is a regular chord--3 or more notes played at the same time. An arpeggio is when you take the notes of a chord and play them in sequence rather than simultaneously. It is a way of taking a chord--the harmony--and making it into the "leading" line--the melody; arpeggios can be played as a harmony though, despite their being played in a sequence like a melody line. The reason why?...When you play an arpeggio, the quality of the chord is still retained--despite the notes not sounding all at the same time!

The main thing that intrigues me about this is this: Tartini tones do seem to have an affect on the quality of intervals... the perfect intervals produce Tartini tones--even at the 2nd and 3rd levels of Tartini tones--that match up with the 2 tones of the interval, just at different octaves--essentially only those 2 original tones are being played, just at different octaves; conversely, the Tritone produces several different Tartini tones at the 2nd and 3rd and 4th levels, and in the end you hear 5 or 6 different tones being reproduced, rather than just the original two. So, logically, it would seem that these Tartini tones are what cause dissonance or allow for consonance. And yet, when the notes are played in succession, the quality of the chord is maintained, despite Tartini tones clearly not being produced. But, the consonance or dissonance of the chord disappears... the "mood"-quality is maintained, the consonant/dissonant-quality is removed.

I believe that a connection can, and must, be drawn here in order to better understand the connection between physics and music. Somewhere in the human neurophysiology a process happens which converts the dissonance and consonance caused by the physics of clashing Tartini tones into a certain perceived "mood" to relate to the chord... even more compelling: this process is independent of temporal context to a certain degree--the brain is somehow able to deliver the same mood to a chord played in sequence which contains NO Tartini tones, as a chord played simultaneously which contains SEVERAL Tartini tones. This is, perhaps, the basis of what I call the "Human Connection".

Some notes about music.

Here are some interesting observations related to intervals (2 simultaneous notes), triads (chords with 3 notes), extended chords (4 or more notes), and chord progressions:

**For the purposes of this blog, I'll be basing all intervals, triads, extended chords, and chord progressions on the base note and scale of "C".

As you'll remember, some intervals are commonly perceived as dissonant, and some as being consonant; additionally, some intervals are given qualities such as "dark," "sinister," or "brooding," some are given qualities such as "happy," "positive," "anticipatory," and some (the perfect intervals) aren't really given a strong quality at all. Combine these intervals, though, and the qualities change quite drastically.

Because in a previous post we learned that the Major 3rd is one of the "happiest" sounding intervals, it would make logical sense that combining two of these intervals would give a very happy-sounding chord. So, to create this chord, you would simultaneously play the notes C-E-G#. This quality of triad is called an "augmented" triad; play it, and you will notice that it is incredibly dissonant, and not at all "happy" sounding.

In fact, the "happiest" triad in music is considered to be the combination of a major third, and a minor third. C-E-G. This quality of triad is called "major"--named so because of the major third contained between the tonic ("tonic" means the "base" note--in this case: C) and the next note of the triad.

It may follow next, logically, that the "saddest"-sounding triad would be the combination of two Minor 3rd intervals since the Minor 3rd is considered to be the saddest interval. If, however, you play two minor thirds, you get the notes C-Eb-Gb. This triad quality is called a "diminished" triad, and if you play it, you will notice that it is extremely dissonant.

In fact, the "saddest" triad in music is considered to be the combination of a minor third and a major third. C-Eb-G. This quality of triad is called "major"--named so because of the minor third between the tonic and the next note of the triad.

Here is where triads get (more)interesting:

If you take the 2nd or 3rd note of a triad and place it on the bottom, you create what is called an "inverted" triad. For example: take C-E-G, and invert it to be E-G-C. Notice now the the intervals, rather than being Major 3rd/Minor 3rd, are Minor 3rd/Perfect 4th. Now you might expect that this triad would sound minor... the minor 3rd is a sad interval, and the perfect 4th has no quality. BUT, instead, this inverted triad retains the exact same quality as the original triad. The other inversion, G-C-E, Perfect 4th/Major 3rd, also retains the quality of the original chord. This "phenomenon" works with all triads--diminshed, augmented, major, and minor--and also with all other types of triads and chords, with some very minor exceptions (not exceptions of which chords can be inverted... just some exceptions of which notes within those chords can be inverted into which positions.)


Next: extended chords. An extended chord occurs when you add another, higher note to any of the 4 qualities of basic triads. In this blog, I'll specifically be focusing on the "Dominant 7th" quality of chord. The notes for this chord are C-E-G-Bb. Adding this 4th note causes the "happy" C-E-G Major triad to become a positive-sounding and consonant "anticipatory" chord which wants to lead into another chord and sounds "unresolved" (because it wants to lead into the chord which is its resolution).

When you dissect this chord, you will notice this: the "stacking" of intervals goes Major 3rd/Minor 3rd/Minor 3rd, and from E to Bb is a Tritone (diminished 5th), which is considered one of the most dissonant and "sinister" intervals in music. Despite two minor intervals and the tritone, this extended chord is one of the least "sad" chords in music, and the quality has virtually no unsettling dissonance.


Lastly: chord progressions. A chord progression is a progression of chords. A chord progression can be as short as 2 different chords (although this is rarely called a chord progression), and has virtually no end as to the maximum number of chords it can contain (although typically a chord progression will cycle through about 7 or 8 chords at max until it returns to the starting chord). Within one key (the scale that the song is based in), there are 7 different triads and countless extended chords. These triads are based on each tone of the scale which the key is in, once again, I'll base them off of C. These 7 triads, besides being based on the 7 tones of this scale, also use only the notes from this scale to form the other intervals in the triad. For the entire scale, this would go as such:

C-E-G (Cmaj)
D-F-A (Dmin)
E-G-B (Emin)
F-A-C (Fmaj)
G-B-D (Gmaj)
A-C-E (Amin)
B-D-F (Bdim)
C-E-G (Cmaj)

So, a song, or a part of a song, which remains in one key, can use any of these triads and remain in that key. What this means is that a song can be in the key of C Major, and still use D minor, E minor, A minor, and B diminished chords. In fact, a progression could be composed mostly of these minor chords and yet still overall sound major if it starts and ends on a major (typically the C major) chord. The same goes for songs in a minor key: so long as they start and end on a minor chord, you can throw in virtually as many major chords as you want without it sounding major as an overall progression.

... Notice the pattern?

Triads and chords can contain many different qualities of intervals and yet those individual intervals have little bearing on the overall quality of the chord.

Chord progressions can contain many different qualities of chords, and yet those individual chords have little bearing on the overall quality of the progression.

Research Update #3

First off: dealing with the 12 chromatic intervals and the Tartini tones produced, I've found some interesting correlations. In fact, if you'd like to know more, there is a book written by Paul Hindemith titled The Craft of Musical Composition. I haven't read it, but I came across the title while checking my math against what others have already written. It definitely now seems as though consonance and dissonance are related to the Tartini tones, as I expected. Here's why:

As a starting point, it should be pointed out that the Tartini tones produced do follow a pattern that is similar to the Harmonic series; it isn't a perfect fit, however... in fact, some notes are off by as much as 49 cents (50 cents is a quarter tone--get it: 100 cents = one half tone; one-hundred one-hundredths (i.e: 1) = one whole of the West's smallest intervals, which are half-tones) and most are off by more than 5 cents (the smallest perceivable difference between tones when they are played consecutively rather than simultaneously--because when consecutive they don't produce interference beats).

The amount off from the harmonic series, however, is actually insignificant at it's base level; the Perfect fourth is almost as far off as the Tritone--although the tritone does have the greatest difference at 49 cents. Where it gets hairy is when you look at the 2nd-, 3rd-, and 4th-order Tartini tones. The Perfect intervals' higher orders continue to produce low octaves, just repeating itsself over and over again; however, the Tritone produces 4 different tones in the first 4 orders--where the perfect intervals, with higher-order Tartini's included, continue to only resonate in 2 tones which blend together well, the Tritone produces 4 tones which, because they sound simultaneously, clash horribly. I haven't investigated the other intervals yet (i.e. the 2nd's, 3rd's, 6th's, and 7th's), but I will do so shortly and examine how they may add to the consonance or dissonance that is produced in their respective intervals.



Secondly: other research I've done since the last update. This has more to do with music history and development than physics. Eventually, hopefully, I'll be tying the history and the physics together--music has become what it is for a reasons both cultural and scientific. It's hardly arbitrary.

First I'll be posting some random incohesive notes I've taken which I'll tie together later. Then I have some information which I have already begun connecting related to Hexachords, Tetrachords, Solmization, and the rebirth of flats and sharps (accidentals) in the Middle Ages.

So here are random notes:

• Semitic music/theory most likely influenced both Arabian and Greek music.



• Arabian theory was very much influenced by the science of music. Al-Farabi wrote texts in the 800's AD motivated by both mathematical predictions and his aesthetic knowledge of music as a performer.



• Al-Farabi describes fretted string-instruments which utilized quarter tones.



• Greek music was more influenced by aesthetic, and traditional/cultural practice rather than mathematics/science of music. They did, however, understand the science and math and had knowledge of the overtone series--tradition would not allow them to develop many new music styles though.



• Earliest records of melodic music are from 3rd millenium BC in Mesopotamia



• A lyre (stringed instrument) found in Mesopotamia was tuned to a heptatonic (7-tone--just like western) scale. I don't know if it used half tones or quarter tones or what the scale quality was. But, it did divide the octave seven times.



• Hindu music theorists divided the octave into 22 "Shruti"--these are just barely larger than a quarter tone, which would have 24 divisions to the octave.



• In 1953, research done by Constantin Brailoiu suggests that Pentatonic scales (5 tones to the octave with intervals of a Major 2nd or Minor 3rd between consecutive tones) can be found in the indigenous music of all 5 continents.



• East-Asian theorists list pentatonic scales which contain intervals of almost-semi-tones (remember a semi- or half-tone is a Minor 2nd)(perhaps "Shruti"?), whole tones (remember a whole tone is a Major 2nd), and minor thirds. (In fact, if you play a regular, Western pentatonic scale, you'll most likely think it sounds "Asian").



• Some pentatonic scales actualy divide up the octave entirely equally rather than in variations of Minor 2nd's, Major 2nd's, and Minor 3rd's. This is common practice in some Southeast-Asian, Asian/Pacific Island (Java), and African cultures. In Javanese these are called "Slendro" modes ('mode' is just another word for 'scale').

Now then ("now then"--what a stupid, oxymoronic phrase...), notes which I have begun piecing together: These notes relate to the development of music, specifically in the 1000's to 1200's AD. Later, I will go further back, concentrating on the Greeks in the 1000's BC to 100's AD, and the Mesopotamians in the 2- and 3-thousands BC. Eventually, I will also be connecting Eastern musics, Semitic music, and perhaps even African music, to modern physics. It is my belief that music has developed the way it has because of the reaction of acoustic physics with our neurology. For some reason, almost all cultures have decided to divide up the octave into 12 tones; some cultures have chosen 24 tones, some 22, 5, or some other number--but all of these divisions have a reason for occuring (especially 12, since it is the most common). It isn't human nature to do things arbitrarily... especially for 4- or 5-thousand years.

Hexachords:

The word "hexachord" comes from the greek words "hexa"--meaning 'six'-- and "chorda"--meanging 'string'. So 'hexachord' literally means "six-string". But, a hexachord doesn't have six strings; a 'hexachord' is a scale which consists of 6 notes--specifically: the first 6 notes of the Western diatonic ("regular") major scale. The reason it is named "six-string" is because in Ancient Greece scales were played on instruments called 'lyres'; lyres were stringed, but unfretted, instruments (like a harp), and so each string could only be assigned one note. If a musician wanted to play 6 different notes, he would need a lyre with six strings. As I've said, I'll be posting more on lyre's, tetrachords (Grecian four-note scales), Grecian music, kitharas, auloses, and the monochord--which is a one-stringed instrument used to study the physics of sound, and is not a scale. (All these "chords" start to get confusing, don't they? Especially when you add in Major and Minor and other qualities of chords which are neither stringed instruments nor scales...)

**In the following few paragraphs, I'll be using some music notation which I haven't explained before--this notation relates to the octaves of notes, but doesn't use subscripted numerals, this is called "Helmholtz Pitch Notation". Pay attention to the comma (,) and prime (') symbols which may be placed after a note name. Here's how it works: The lowest C is notated as [C,,] (capitalized, 2 commas), the next higher octave: [C,] (capitalized, one comma), next: [C] (capitalized, no marks), then: [c] (lower case, no marks), then: [c'] (lower case, 1 prime), then: [c''] (lower case, 2 primes), and finally: [c'''] (lower case, 3 primes). So, to recap and simplify: [C,, C, C c c' c'' c''']. All 12 notes between [C,,] and [C,] get two commas. I.e: the note "D" which is just higher than [C,,] is notated: [D,,] . So the entire octave (excluding notes with flats or sharps for the sake of simplicity) would be [C,, D,, E,, F,, G,, A,, B,, C,] next octave: [C, D, E, F, G, A, B, C] then: [C D E F G A B c] etc. Sorry for the ridiculously confusing punctuation. **Note that in actual notation, no brackets were used. I'm just using them to hopefully make it somewhat less confusing which letter receive which punctuation.

**Keep in mind that music in the time period of the hexachord's invention was primarily for writing melodies with no accompaniment or background harmonies-- no chords, counterpoints, etc. Most musics in the Dark Ages were either Alleluias (one-line church "songs") or monks' chantings.

So a hexachord is a scale of 6 notes, the first 6 notes of the major scale, and was first introduced into written music theory in the early 1000's AD by Guido of Arezzo. Guido decided to start on the note [G], and from there created the first hexachord: [G A B c d e]. He then jumped to the fourth note of that scale, and created a 2nd hexachord: [c d e f g a], and then jumped to the fourth note of that scale, and made a third and final hexachord: [f g a bflat' c' d'] , lastly, he took these same 3 scales an octave higher and added them to the mix: [g a b c' d' e'], [c' d' e' f' g' a'], and [f' g' a' bflat'' c'' d'']

At this point, I will point out 2 things:

1) These 3 complimentary scales, all used to construct one single melody, contain in them a B and a Bb despite being constructed from complimentary major scales and despite their supposedly being entirely consonant to form a melody which would please the ear.

2) Some of the notes in these 3 complimentary, yet separate, 6-tone scales occur in more than one scale. [g], for example, occurs in the [c]-hexachord, the [f]-hexachord, and the octave of the [G]-hexachord. [d] occurs in both the [G]-hexachord and the [c]-hexachord. If you look, you'll realize that several notes occur in 2 or 3 scales.

By having both the B and the Bb in the 3 (6 counting the octaves) usable scales, Guido opened up the door for "mutation" or in modern terminology "modulation"--this is the process by which a melody and with it it's harmony change keys (remember, the key is the 8-tone scale that the song is based in). This opened up a world of new possibilities to composers as they no longer had to use just 8 tones to create a song, and was certainly a great step forward in the development towards modern Western music.

One more thing (and inadvertently perhaps the most important thing) that Guido of Arezzo did was that he "Solmized" the music; he called each note by a syllable, rather than by a letter (ut re mi fa sol la si ut). By doing this, he made the music entirely relative--a concept which in modern music is very common, but in medieval music was unheard of due to variances in intonations (I'll talk about the specifics of that in another post). Now, Guido didn't entirely solmize the music--there were still letters associated with the notes--but what he did do was use solmization to distinguish between say, the [c] in the [c]-hexachord and the [c] in the [G]-hexachord. I won't go into the details of how he went about doing this, though, because the main point here is that he made his scales solmized; this will alter the way Western music progresses considerably.

Over the next 100 or 200 years, from 1000 AD to 1100 or 1200 AD, polyphonism started to gain a larger role in Western music. And most western music had adopted the use of solmized hexachords as the basis for writing melodies. When solmized notes and polyphonism tried to combine, however, a problem occured. In the harmonies, there was a need to place a note a perfect fifth above and below the B's and Bb's; however the note a perfect fifth above B is F# (remember # means "sharp"... sharp means "plus one half step"), and the note a perfect fifth below Bb is Eb. If you look at the 3 hexachords, you will notice very quickly that there are no Eb's or F#'s. At first, composers used what was called "musica ficta" or "musica falsa"; in other words, they 'invented' notes (Eb and F# were known notes at the time, they just didn't fit the scales--hexachords-- in use). Over the years, this became a very common practice since it was musically necessary to produce real harmonies. In modern music, we call these kinds of notes--notes which aren't in the 'proper' scale--"accidentals", and they are commonplace in all music except for the most basic elementary studies.

As you can see, due to hexachords and solmization, Western music made a huge advancement towards its modern state.

Research Update #2

Using this website from the University of New South Wales, I measured my audio response to various frequencies. This process is known as "audiometry", and the results, when compiled into a graph, are called a "Loudness Contour".



The point of such a task is to measure where a person's (in this case mine) audio sensitivity peaks. The loudness is measured in negative decibels (dB)... the closer to zero, the louder the sound. (-3 dB is MUCH louder than -90 dB).



The decibel scale is a logarithmic scale which is set up so that an increase of 3 dB sounds to the human ear as a doubling in loudness. Ex: -27 dB is twice as loud as -30 dB; -24 dB is twice as loud as -27 dB and thus four times as loud as -30 dB; -21 dB is twice as loud as -24 dB, four times as loud as -27 dB, and eight times as loud as -30 dB... etc. Remember, this is how it "should" sound to a human... to me, personally, I can't really tell when a sound is "twice as loud"... I just perceive louder and quieter relatively (i.e. much louder, a little louder, a little quieter, a lot quieter, etc).



At the bottom of the chart that's posted right under this paragraph, I have several notes related to some of the figures that I got. Make sure you read them. Also, remember that if you do this same test, at the same website, but using a different computer and different speakers or headphones, your results could be drastically different even if your ears are exactly as good (or bad) as mine. Things like speaker response, room acoustics (if using speakers rather than headphones or earphones), and your computer's soundcard all affect the results of this. The test is relative to yourself--you can see where your peaks and valleys are, but don't compare this test person-to-person unless they are using the same equipment in the same environment. Below this chart, which reports my results, is the chart of a friend's results; he took the test in a different environment, on a different computer, using different hardware and software.


Hz ; dB level when heard


30 ; -3*
45 ; -12*
60 ; -21*
90 ; -33
125 ; -45
187 ; -51
250 ; -57
375 ; -57
500 ; -60
750 ; -63
1k ; -72
1.5k ; -75
2k ; -81
3k ; -87
4k ; -90
6k ; -90
8k ; -78 (-90)**
12k ; -72 (-90)**
16k ; -52***


*These very low numbers, especially for 30 and 45 Hz, are probably due to the low-end limit of the earphones I was using. (iPod headphones)

**These two tones did something odd: at the listed numbers, -78 and -72 respectively, I could hear the listed tone. BUT, from -90 dB up to -78 or -72 dB, I could hear an octave undertone. I'm not sure of the reason for this other than something happening with the earphones as they maybe approached their upper limit, a problem with the computer's soundcard not being able to properly process the signal (I don't know much about computers, but I do know that soundcards have an effect on the performance on this test.), non-linear effects in my ear somehow producing a Tartini tone an octave lower, or a neuro-processing 'error'.


***This frequency is actually about the pitch that my ears were ringing at. So, until the dB level reached -52 dB, I was unable to tell if it was my ears ringing, or the speakers playing. I thought this was a funny phenomenon.


Here are my friend's results:

30 ; -30

45 ; -27

60 ; -30

90 ; -33

125 ; -37

187 ; -33

250 ; -30

375 ; -48

500 ; -54

750 ; -45

1k ; -54

1.5k ; -63

2k ; -51

3k ; -75

4k ; -69

6k ; -66

8k ; -78

12k ; -66

16k ; -39

This unit of research has raised a new question:

4) How could the human acuity to hearing certain frequencies alter the way humans hear and neurally process chord intervals and Tartini tones at different octaves?

Research Update #1

After learning about Tartini/difference tones, I was interested to learn how these could affect harmonies in music--I'm trying now to make a connection between the rigid, quantitave mathematics of physics, and the opinionated, qualitative aesthetics of music. Accordingly, in this post I'll be using some music-based ideas such as the qualitative 'dissonance' and 'consonance' as well as "chord qualities" such as 'Major', 'Minor', and 'Perfect'.

**One important note is that although I call dissonance and consonance qualitative, they are, in fact, biologically different. When the brain processes consonant harmony, neurons fire at different rates; when the brain processes dissonant harmony, neurons fire simultaneously. I guess what I mean by qualitative is the level of dissonance--some harmonies are obviously dissonant to everyone (such as the minor and major 2nd), and some harmonies which are classified as dissonant simply sound excessively dark and brooding to some people (such as the diminished 5th a.k.a. the Tritone). The book This is Your Brain on Music: the Science of a Human Obsession is where I read the information about dissonance and consonance being actual differences which occur in the mind. A bibliographical entry for this book is available at the end of the post about Hypersonic Sound; also, I'll be publishing a full bibliography for the information in this post in one of the next few posts.

**Chord qualities are not always meant to convey a certain mood--major is not always positive and minor is not always negative; however, often they are said to have a certain mood attributed to them. This depends greatly on whether or not the chords are consonant or dissonant and will be explained further in the chart which compares the 2 sounding tones and their difference tone.

**Mathematically, to find the pitches of each note you would take the appropriate logarithmic scale which passed through 130.8 and 261.6 Hz and divide the x-axis into 12 parts including these tones. The y-coordinate is the frequency of the tone x number of half-steps above C3. This is based on the "A=440" scale which means that the "A" above "Middle C" is at a frequency of 440Hz. The "3" placed next to the "C" denotes it's octave relative to a piano's range. The space between 2 subsequent divisions is called a "half tone".

So, because there is an octave between 130.8 and 261.6 (i.e. 261.6 has twice the frequency of 130.8) and this space has been divided into 12 parts which are each a half tone apart, the octave contains 12 half tones (the upper octave tone is the 12th half tone, and the lower octave is not counted in this case). These are named (starting on C) "C-Dflat-D-Eflat-E-F-Gflat-G-Aflat-A-Bflat-B-C". From now on, i'll be using "b" to indicate "flat" as this resembles the actual musical notation for a flat symbol -- Eb = Eflat.

**Also, notice that there is no "Fb" or "Cb"... this is because an "Fb" is the same pitch as an "E" and a "Cb" is the same pitch as a "B". The reason for this is simply because although the scale is broken into 12 half-tone pitches, naming conventions which have developed over the centuries use 7 letters rather than 12 or 6. [In more advanced music and in music theory, sometimes Fb and Cb are notated depending on the scale and chord which is being used; also, sometimes Csharp (C#) would be written instead of Db for the same reason (this goes for other notes too, such as G# instead of Ab)... this is a naming convention issue that doesn't affect the sound of a chord... regardless of name, an Fb sounds the same as an E, and a C# sounds the same as a Db.]

Although at this point I'm not sure, I think that the decision to use 7 pitches instead of 12 has something to do with how a typical major scale is constructed: it uses 7 different tones. The C-major scale contains the notes C-D-E-F-G-A-B-C; notice that it contains no flats or sharps. I find it interesting that the only major scale which contains no flats or sharps starts on the letter "C" rather than "A" (which would make logical sense since "A" starts the alphabet); however, the only minor scale which contains to flats or sharps is the A-minor scale, which contains the notes A-B-C-D-E-F-G-A. Perhaps the reason for this is that in musical history the minor scale developed prior to the major scale or had more importance than the major scale. Then, maybe, from these scales the other ones were named and 'adding' a flat or sharp to a note was easier to remember than trying to remember which letters were in each scale. i.e: it was easier to write "G, A, B, C#, D, E, F#, G," rather than naming each half step "A, B, C, D, E, F, G, H, I, J, K, and L," and trying to construct the same scale from that based on the number of half-tones between each scale tone, which would lead to confusion as some letters would be entirely excluded. In our major and minor scales, every letter A through G is used once, and then depending on the scale a certain number of sharps or flats are added to some notes.

**Interference beats occur at a rate which is equal to the difference between the two sounding tones. The number of beats per second between the two tones 200Hz and 203Hz is 3. Because humans cannot hear pressure fluctuations below 20Hz, when two tones which are less than 20Hz apart sound simultaneously, it is usually perceived simply as a fluctuation in sound volume; over 20Hz, and a Tartini tone is heard with a frequency of the difference between the two sounding tones. The Tartini tone resulting from 200Hz and 250Hz is an audible tone at 50Hz. As you should remember, this is the basis for HSS.

Initially I created a chart which compared the musical tones between and including 130.8 Hz to 261.6 Hz--C3 to C4. The chart compared the base tone C3 to each of the 12 subsequent chromatic (going up by half-step) scale tones. I then included in this chart the frequency and note name of the resulting Tartini tone. My hypothesis was: If the difference in the two sounding tones is below 20Hz (because humans would perceive this as a rapid volume fluctuation rather than as an added third tone (the Tartini tone)), then the resulting harmony will be recognized as being dissonant.

Using the major 2nd, a very dissonant interval, as an example, I noted that between C3 and the note one whole step higher (two half steps), D3, the difference in pitches was a mere 16Hz--Slow enough to not produce an audible Tartini tone, and yet fast enough to not really be perceived as a volume fluctuation. According to my hypothesis, this was the reason for the dissonance--the fact that it didn't produce a Tartini tone and yet was more than just a volume fluctuation.

What you should quickly realize, however, is that as you move up in octaves, keeping this same musical (or mathematical if you're following the same logarithmic scale) interval of a Major 2nd (to reiterate: 2 half steps), the pitches move farther and farther apart frequency-wise. Between C3 and D3 there are 16 Hz of separation, however between C7 and D7 there are 256 Hz of separation. And yet, C7 and D7 are still extremely dissonant despite producing a substantial Tartini tone. So, my initial hypothesis was shown to be wrong.

**One note which I must make is that in music, chords which are in the lower octaves do tend to sound more "muddled" than chords in higher octaves. It is standard music compositional practice to allow about a Perfect 4th (5 half steps) between the lowest tone and the next tone above it--sometimes more or less depending on how deep the lowest tone is. I'm almost sure that this has to do with the tones being so close in frequency combined with the human ear not being able to differentiate such relatively small differences as easily as larger differences.

Here is the chart that I made comparing C3 to all of it's possible half tone harmonies up to C4:

Note ; # of 1/2 steps ; Hz difference ; harmony name ; Note name of Tartini tone (approx)
C3......... 0.................. 0............. Perfect unison.................... --

Db........ 1.................. 7.78.......... minor second................... C--

D.......... 2................ 16.02.......... major second ...................C0

Eb........ 3............... 24.75........... minor third...................... G0

E ...........4............... 34.00......... major third ...................Db1

F ...........5 ...............43.80 .........Perfect fourth ..................F1

Gb........ 6............... 54.19 ..........Tritone............................. A1

G.......... 7............... 65.19.......... Perfect fifth ......................C2

Ab........ 8............... 76.84.......... minor sixth..................... Eb2

A.......... 9............... 89.19.......... major sixth ..................F2/Gb2

Bb .......10............. 102.27 .........minor seventh................. Ab2

B......... 11.............. 116.13 ..........major seventh................. Bb2

C4 .......12 .............130.81 .........Perfect octave ..................C3

**I'll be using the term "inversion" in the next few paragraphs. An inversion is when you take two notes, such as a C and an E, and raise the lower note one octave. Ex: The base (not bass) tone is C3 and the harmonizing tone is E3--there are 4 half steps between these two tones and the interval is a major third. The C3 is raised to C4 so that now the base tone is E3--There is a distance of 8 half steps between these two notes and the interval is a minor sixth.

**"Perfect" harmonies are considered to be entirely consonant, however have basically no 'mood' associated with them. The Major 3rd is considered a "happy" or "positive" chord--the Minor 3rd a "sad" or "negative" chord. Both the minor and major 2nd are considered very dissonant tones. The major 6th, while considered fairly positive, is not as "happy" as a major 3rd. This most likely stems from the Major 6th being an inversion of the Minor 3rd. The minor 6th is exactly the opposite; additionally, the minor sixth is considered slightly dissonant and dark or brooding. The minor seventh is typically a slightly uplifting interval. This is most likely due to it "wanting" to resolve to--or lead into--another note; in other words, it creates anticipation which is normally interpreted positively. The Major seventh, being an inversion of the minor second, is fairly dissonant.

After making this chart I saw some interesting correlations between dissonance, consonance, and chord quality, and the tartini tone produced. Some of the relations made sense... some didn't:

First of all: dealing with the Perfect intervals, the unison produces no Tartini tone because the pitches are the same; the octave produced a Tartini tone equivalent to C3 because it's a doubling of that note and thus the frequency difference is equal to the base tone. The perfect 4th (F3) produced a Tartini tone of F2, and the perfect 5th (G3) produced a Tartini tone of C2. This is interesting because, as can be seen, all of the other intervals produced tones besides just the 2 harmonizing tones. Another note to make, unrelated to the Tartini tone, is that a perfect 4th is an inversion of a perfect 5th and thus has a similar, although just slightly more dissonant, quality.

Secondly: dealing with the thirds and sixths, the minor third (Eb) produced a Tartini tone of G0, which forms a perfect harmony with C and thus, having relatively no dissonance or other quality, shouldn't affect this chord at all--and yet the minor third is considered a "sad"-sounding harmony. The major third (E), however, was interesting in that the Tartini tone produced by it was Db1. This note is dissonant to the C when transposed upwards a few octaves to be more relative to C3, and forms a minor third with the E when transposed upwards. And yet, despite the dissonance with the C and the "sad", minor quality with the E, the Major third is widely considered a "happy" interval. The minor sixth (Ab) produced an Eb as a Tartini tone. This is interesting as a minor third is considered to have an ominous or "minor" sound, and is fairly dissonant. It is interesting to note, however, that the Eb does form a perfect interval with the Ab. Remember that a MINOR 6th is the inversion of a MAJOR 3rd, and a MAJOR 6th is the inversion of a MINOR 3rd.

Thirdly: the sevenths and seconds. The minor second (Db) produced the C three octaves below the base tone C3, which is an inaudible frequency. What is interesting, is that although this is exactly what happened with the non-dissonant perfect intervals, the minor second is one of the absolute most dissonant intervals in music. The major second (D)also produced a C, this time only two octaves below C3--which is still inaudible (although at higher frequencies this interval would produce an audible Tartini tone.)-- and once again, although this is exactly what the perfect intervals did, the major second is one of the most dissonant intervals in music. The minor seventh (Bb), and inversion of the major second, is less dissonant than the major second by far, and, as I've said, is considered to be positive probably due to it's tendency to create anticipation. This interval produced a Tartini tone at Ab-- a tone that is very dissonant with the Bb, and also somewhat dissonant with the C. In fact, depending on the inversion, an Ab can form a minor third with a C which, as you should remember, is considered a "sad" interval--and yet despite this, the minor 7th is considered a very "positive"-sounding interval! The major 7th (B), being an inversion of the minor 2nd, is still very dissonant, although not quite as much as its inverted counterpart. It produced a Tartini tone of Bb2; although a Bb is not very dissonant with a C, when a C, B and the Tartini tone Bb are all played at the same time, 3 tones which are all only one half step apart are sounding and, as I've already said, the minor second (half step) is one of the most dissonant tones.



Lastly: The tritone (Gb). This is considered one of the most dissonant intervals in music--most likely due to the fact that it splits the octave evenly in half (6 half tones). While personally I don't consider it the most dissonant, it does sound very dark, ominous, and still has a lot of dissonance. In medieval times, musicians were not allowed to write this interval into music or play it because of a superstition that playing it would call the Devil to come forth from Hell (Sadly, I'm not making this up...). The reason for this dissonance is unclear, while the Tartini tone A does form a minor third with the Gb, this is not a particularly dissonant interval--more just "sad"-- and also the A is not very dissonant with the C itself, forming a major sixth.

Because there doesn't seem to be any real correlation on the surface between the Tartini tone produced and the amount of dissonance or the quality produced by an interval--especially because the Tartini tones are sometimes audible and sometimes inaudible depending on the octave--I plan on investigating further what, both in physics and in neurology, could be causing humans to perceive such a distinct difference between these characteristics.


Another thing that I want to investigate, which is related to the topics of quality and dissonance, is the development of musical scales over the past several millenia. For instance, while all of the Western scales, and several Eastern scales as well, are broken into a combination of the 12 half tones, many Eastern, Middle Eastern, African, and Pacific (most people lump all of these categories simply into "Eastern" scales) scales are broken into combinations of quarter tones instead. To most western ears, this would sound very dissonant and out-of-tune--most would probably perceive it as a mistake by the musician.


So:
1) Why do humans perceive consonance and dissonance? Is it physics, neuroscience, or a combination of the two?



2) Is there a deeper correlation that can be found between the half-step intervals and their dissonance/quality in relation to the Tartini tone which is produced? (One interesting thing that I plan on investigating soon is the Tartini tones' relation to the overtone series of a fundamental C... at a very quick glance, the progression of tones seems to follow this series fairly closely.)


3) Why were the various Western and Eastern scales developed the way they were? An exploration of the origins of melodic music, evolution of instruments, and spread of musical styles/genres will be conducted.



3.b) Was there a "base scale"-- a type of fundamental scale which most ancient cultures seemed to use from which the other scales were possibly derived as embellishments and minor changes were made? If so, why were those embellishments or changes being made the way they were?

Terms which I will be investigating so far are:

Aulos, Kithara, Lyre, Tetrachord, Quarter Tone, Pentatonic Scale, Terpander, Constantin Brailoiu, Al-Farabi, and Hebrew Music (which is what it is presumed most ancient music stemmed from).

Hypersonic Sound

Hypersonic Sound (HSS) is the term used to describe the process by which audible sound waves can be produced using ultrasonic sound waves that are free from non-linearity. The first attempts at hypersonic sound were made in the 1960’s using underwater sonar. In the 1970’s it was proven that mathematically HSS could be produced in air, but by the 1980’s the technology was abandoned because of problems with distortion. In the late 1990’s HSS was again researched because of advances in sound production technology and in 1998 the first working, commercial prototypes were made under the name “Audio Spotlight”.

The advantage of using ultrasonic sound is that sound transmissions can be focused into a narrow, far-reaching beam that resists diffusion and attenuation; therefore, the beam can be transmitted over greater distances with pinpoint accuracy. Additionally, this sound beam can be targeted to only a single object or person, leaving the surrounding environment free of noise pollution. Already, this technology is being put to use in the advertising and automobile industries, and the United States military.

First, it is important to understand what a sound wave is. A sound wave is a series of alternating high (condensation) and low (rarefaction) pressures created by some object disturbing the environment through which the sound wave is traveling. This pressure wave, then, is received by the eardrum which converts it through the inner ear into an electric signal which the brain can process. The key thing to recognize in the case of hypersonic sound is that each of these small pressure changes is a different micro-environment; the small portions which are low-pressure have different densities (atmospheric density is related to pressure) than those that are high-pressure. This is extremely important to note when dealing with the transmission of a sound wave across distances.

Next, it is important to understand the terms diffusion and attenuation, which describe the behavior of a sound wave over time. Diffusion is the process by which a sound wave expands outward, and attenuation is the process by which a sound’s intensity diminishes. These two characteristics of a sound wave are very interrelated; as a sound wave expands and increases its area occupied, its intensity (which is inversely proportional to area occupied) decreases. Additionally, a sound wave’s absorption into the surrounding environment as well as its reflection off of objects and particles in the environment decreases its intensity and thus contributes significantly to its attenuation.

Next, it is important to understand what it means to be non-linear and how or why a sound wave is non-linear. Non-linearity simply means that as the wave advances through the environment and time elapses, the conditions of the environment in which the wave exists do not remain constant. Explaining how or why a sound wave is non-linear is a little more complicated and requires the piecing-together of some facts which have already been noted. Because a wave’s frequency depends on the speed of sound, and the speed of sound depends on the density of the environment through which the wave is traveling, and the density of a fluid (fluids are gasses and liquids) environment depends on the pressure—which is fluctuating due to the nature of the wave—of the fluid, a wave’s frequency depends greatly, although transitively, on the pressure of the fluid. As a wave moves through various pressures, its frequency and speed change. Because the wave’s speed changes, the rate of diffusion changes as a result of its rate of expansion changing. Because of both the rate of diffusion changing, and because of the amount of particles to reflect off of (because of the compression and rarefaction, where lower densities have fewer particles and vice-versa) changing, the rate of attenuation changes. All of these factors are even further affected as the sound wave travels outwards because of the diminished intensity and conversely the diminished compression, rarefaction, attenuation, and diffusion. Thus, a sound wave is non-linear both in small segments (from one micro-environment to the next) and as an entire segment (as its intensity diminishes from the source at point A to the target at point B).

Because of sound’s non-linearity, it is extremely difficult to project a sound across long distances, and when a sound is projected across long distances, it becomes extremely distorted. So, logically, to counteract these effects, sound has to be given a linear quality. To do this, ultrasonic sound waves are used. Ultrasonic sound is sound that is above the human range of hearing (20,000Hz); in HSS, frequencies in the hundreds-of-thousands of hertz are used both to improve the linearity of the sound and to prevent harm to animals whose range of hearing exceeds that of humans. Because the frequency of ultrasonic waves is hundreds or even thousands of times faster than audible waves, and frequency is a measure of number of pressure fluctuations per second, the pressure fluctuates between rarefaction and compression hundreds or even thousands more times per second. Because the micro-environments are now hundreds or thousands of times smaller than with audible sound, the effects of the micro-environments on the propagation of the sound wave become negligible and thus the non-linear characteristics which were present in audible sound waves are not present in ultrasonic waves; additionally, because of the new nature of the pressure differences, the air through which it is traveling loses its non-linearity (because it is essentially “part of the wave”) and thus fails to make the sound “audible” (being “audible” would cause the sound to lose intensity and attenuate). This is what enables HSS to travel over incredibly long distances without losing intensity, becoming distorted, or propagating spherically outward rather than in a straight line. When the wave then hits an object that is non-linear, such as a wall or a human, the wave disturbs that object (because a wave is a disturbance of the surrounding environment and the object is its new environment through which to propagate) and uses that object to once again become “audible”.

In the case of these ultrasonic sound waves, though, “audible” does not really mean audible (ultrasonic is by definition inaudible); rather, what it means is that the wave once again becomes non-linear so that a theoretical human ear capable of hearing over 20kHz would be able to decipher it. To make the ultrasonic waves audible, a phenomenon known in music as the “Tartini tone” or in physics as the “difference tone” is employed.

A difference tone is a frequency that is generated when two other frequencies interact (in music: form a chord). This phenomenon is a result of both physical interaction between the two frequencies and neurological processing of the two frequencies. This phenomenon is almost like interference beats, which are caused when two frequencies of the same pitch are sequenced out-of-phase and thus cause the amplitude to fluctuate. A difference tone, however, is caused when two pure tones (perfect sine waves) are played in-phase but at different frequencies. When done at differences in frequency of over about 100Hz and not including the pitch which is at twice the frequency of the lower tone (in music: the octave)—which would be inaudible when using ultrasonic tones anyway--, a new tone with the frequency of the difference between the two original tones is created. For example: a tone at 440Hz (in music: the note “A”) and 660Hz (in music: the note “E” a perfect fifth higher) would produce the tone of 660Hz – 440Hz, which is the tone 220Hz (in music: an “A” an octave lower than the original “A”). What would not work is playing the tone 440Hz and it’s doubling at 880Hz, because the resultant difference tone would have the same frequency as the original 440Hz tone. Due to the capabilities of the human ear and aural processing centers in the brain, frequency differences which are almost a doubling of the original tone and frequency differences that are so small that the two tones are almost the same tend not to work; the ideal differences in tones are from 5:4 to 3:2 (in music, from a major third to a perfect fifth). This is because when the two tones are as such, the brain simply processes the sounds as being very dissonant, rather than allowing the difference tone to become clear.

Because the frequencies of ultrasonic tones are so high, however, a ratio of 5:4, when the original tones are at 400,000Hz and 500,000Hz, would still produce a difference tone of 100,000Hz--5 times the highest tone that is recognizable to a human. However, due to the temporary non-linearity of ultrasonic sounds, the ratios can be shrunken so that the difference of the tones can be as little as about 200Hz, producing a tone well within the human hearing range. The only disadvantage of this method of sound transfer is that bass tones (lower than about 200Hz) are unable to be reproduced. In applications like music, this would affect the harmonies and could make a performance sound “top-heavy”. In speech, the absence of bass tones, while not detracting from the decipherability of what is said (which is primarily affected by the mid- and high-range tones), would alter the timbre of a voice, which would affect the audience’s recognition of familiar voices and their ability to judge which person is speaking when 2 or more speakers are present (such as in a stage performance).

Hypersonic Sound technology is already being used in advertising displays so that sound can be projected only at one person without disrupting people who are not in the beam’s path, by the US military both to convey messages over long distances and, in a more powerful form, as a sound stun gun (LRAD). Future uses for this technology could include installation into automobiles—each passenger could hear their own music; concert halls—true surround sound projected from a central location and reflected off of the walls--sound projections could even move around the room in real-time; laptop computers—listen to podcasts, videos, or music without disrupting anyone else; and even in megaphones—whisper a message to one person instead of yelling over an entire crowd. HSS, now only in its relative infancy, will soon become a technology that will be seen frequently in a myriad of applications as prices shrink and popularity grows.

Equations:

Frequency (f) (measured in Hz) equals wavelengths (λ) per second (s) through a certain point:

f = λ/s

Speed of sound (v) equals wavelength (λ) times frequency (f):

v = λf

Speed of sound through an ideal gas (v) equals the square root of {[(the ratio of specific heats at a constant pressure {γ}) times (Boltzmann’s constant {k}) times (temperature in Kelvin {T})] divided by mass of one molecule of the gas (m)}:

Intensity of spherically radiating sound (I) equals power (P) (measured in watts) divided by surface area of a sphere (4πr):

Bibliography + Annotations:

Feynman, Richard P., Robert B. Leighton, and Matthew Sands. The Feynman Lectures on Physics. Reading, Mass.: Addison-Wesley, 1963.

Chapter 47 of this book explores the topic of sound waves and eventually their relation to electromagnetic waves and atomic harmonics. Equations are given in calculus format.

Kock, Winston E. Sound Waves and Light Waves. Garden City, NY: Anchor Books, 1965.
This book provides the fundamentals of sound- and light-wave motion and delves into the topic of propagation and dissipation of waves.

Levitin, Daniel J. This Is Your Brain on Music : The Science of a Human Obsession. New York: Plume, 2007.

This is an excellent book about how the brain processes music and sound. In addition, a section of the book is devoted to explaining the basics of music notation and jargon.

"Sound Attenuation." Sound Attenuation. 4 May 2002. Silex Exhaust Systems. 6 Feb. 2009 . This PDF file discusses various issues related to sound, sound transfer, and sound dampening (forced attenuation). Equations and explanations are provided here.

Wolfe, Joe. "Interference Beats and Tartini Tones." Music Acoustics. University of New South Wales. 2 Feb. 2009. .

This website from the University of New South Wales's physics department provides an excellent discussion of interference beats and difference tones. Audio examples are also provided here.

Links:

A link to a video of Woody Norris, a pioneer in popularizing Hypersonic Sound, provided by TED. Woody talks about the development of HSS technology, demonstrates the technology, and talks about the future of the technology. 15 minutes.

http://www.ted.com/index.php/talks/woody_norris_invents_amazing_things.html

YouTube video of Woody Norris demonstrating HSS as featured on the May 10, 2006 television show "Future Weapons." Unlike in the TED demonstration, in this video the technology is audible. 2 minutes.

http://www.youtube.com/watch?v=5imaJwfJMZ8

The Wikipedia article on "Sound from Ultrasound."

http://en.wikipedia.org/wiki/Sound_from_ultrasound

University of New South Wales's physics department's website pertaining to interference beats and difference tones.

http://www.phys.unsw.edu.au/jw/beats.html#Tartini


The official website of Audio Spotlight.

http://www.holosonics.com/


The official website of Woody Norris's Hyper Sonic Sound.

http://www.atcsd.com/site/content/view/34/47/