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?