**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?
