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This page sets out to address the question of what is so special about the notes that we have chosen for making up the TTsTTTs intervals of the well-tempered major scales that are used in Western music. The Chinese, Japanese, Hawaiians, Arabic cultures, for example, use other systems (a number of these scales, as found described elsewhere on the Internet, are listed, for any given tonic, in this Excel spreadsheet.) So, the major mode is cultural, and learned. It only sounds normal to Western ears because it is the one that we have been brought up on. So, this begs the question of how we have come to settle on this one.
The Major scale maps on to the chromatic scale of 12 well-tempered semitones within each octave. It suffers from two major problems:
The Wikipedia page, Harry_Partch's_43-tone_scale, discusses one particular alternative system, and gives cross references to pages that explain why two notes whose frequencies do not bear a simple integer relationship to each other sound discordant. Along with other pages, such as 12Tone.htm, cross-references are given to other systems (with octaves that contain 15, 19, 22, 24, 31, 34, 41, 53 or even 72 smaller intervals), each of which seeks to address one or both of the above problems.
Unfortunately, though, they all suffer, in turn, from two other great disadvantages:
These four bullet points can be summarised and condensed down to two:
Our chromatic scale, using 12 well-tempered semitones, is considered to give, by far, the best compromise between these.
But this still does not explain why we group them in a TTsTTTs pattern, since, as far as the well-tempered scale is concerned, each semitone in an octave is of equal importance. This can be confirmed by looking at a piano with the keyboard removed: all the strings lie in a continuous sequence beside each other, with a set of hammers all arranged in a featureless line.
Going back to the desire for simple ratios in the frequencies of the two notes being sounded in a chord, the simplest ration is that of 2:1. This is the interval of the octave. Doubling the frequency of a note is considered not really to change the note, but merely to take it to a related version of the same note.
The next simplest ratio is 3:1, and consequently 3:2, too (since the division by two merely involves dropping down an octave). An assertion made earlier on this page is based on the approximation 27/12 ≈ 3/2, which says that augmenting by 7 twelfths of an octave (that is, by 7 semitones) is roughly the same as augmenting by a perfect fifth.
Repeating this stepping up by 7 semitones twelve times causes us to end up back at the original note, albeit 19 octaves higher (where 19=12+7). Thus, the well tempered scale is based on the approximation that 312≈219 (531441≈524288).
In order to avoid ending up 19 octaves higher up, we normally drop down the octaves as we go (needing to do this more often when starting from a note in the later part of the octave than when starting from a note in the earlier part). The following assumes an octave starting at F, namely F-F#-G-G#-A-A#-B-C-C#-D-D#-E-F':
Sometimes we "multiply by 3, divide by 2, and divide by 2 again", and sometimes we just "multiply by 3, divide by 2". Given that there are twelve "multiply by 3", and twelve "divide by 2", there must be seven "and divide by 2 again".
Suppose that we were to call the first five notes "red notes", and the last seven "green notes". Re-grouping the statements into a chain of 12 augmentations by a fifth, we would get the following pattern:
The arithmetic causes the operations to group into a (3:2 3:4) (3:2 3:4) (3:2 3:4) (3:4) (3:2 3:4) (3:2 3:4) (3:4) pattern; so, we see the XXXyXXy pattern starting to emerge. This is simply the most uniform way that 5 distinctive cases can be distributed over a total of 12 cases.
Traditionally, of course, we call the first seven notes "white notes", and the last five "black notes". Using this colouring, the keyboard ends up as a combination of pale green, dark green, pale red and dark red notes.
More significantly, we see that both arrangements of the twelve operations give the same sort of pattern of green and red, or of white and black. We do notice that there is a mirror reversal in the pattern, though: first seven notes and then the last five, versus first five and then the last seven; and XXXyXXy versus TTsTTTs. (Incidentally, changing to augmenting by fourths instead of fifths does not give a satisfactory resolution of this discussion).
Finally, we get to the mathematical way in which we can transpose key signatures up and down from any key to any other, just by adding the appropriate number of sharps and flats (and naturals, when sharps and flats cancel each other out) with them always being added to, or removed from, the key signature in the same order. (This leads to the table on "Modulation" shown in the insert on the right of this page.) This property would hold no matter where you start the scale, and it is arbitrary where you choose. Historically, Bach and his predecessors chose F (thereby placing C conveniently in the register).
It seems, therefore, that Western composers have settled on the 12-semitone chromatic scale as being one that approximates fairly accurately most of the usable harmonics from our instruments; and that they have then settled on the TTsTTTs pattern of the major mode (and hence also on the minor and other modes) because of the way that the arithmetic lines up the patterns of notes, and the consequences for consistent representations for, and operations on, the key signatures.
Well Tempered Scale
(Roughly, 774 nm corresponds to 387 THz, and vice versa.)
Intervals, chords and inversions
Each of the eight intervals can be referred to either by name, or by Roman numeral. A triad whose root is the mediant in a piece in C-major, will be a chord in E-minor (E-G-B).
Modulation to Related Keys
The related keys appear in the following table as the five keys neighbouring the current key. For example, if the piece is presently in the key of G-major (with +1 sharp in its key signature), it would be conventional for it to modulate to D-major, C-major, E-minor, B-minor or A-minor).
Writing for transposing instruments
The Major (or Ionic) mode is one of seven related modes. The Minor mode (Melodic Minor mode, descending) is based on the Aeolian mode.
The chord notation C min maj 13 indicates a chord whose root is C, with each of the odd intervals, up to the one indicated (13th in this case) placed above it, but with the 3rd flattened.
In this way, C dim 9 indicates a chord whose root is C, with a flattened 3rd, a flattened 5th, a double-flattened 7th (flattened once by the presence of dim, and flattened again by the omission of the maj) and a flattened 9th.
When a small even number is used as the main interval: 2 is equivalent to 9, 4 is equivalent to 11, and 6 is equivalent to 13; but with the omission of the 7th possibly implied in each case. When it is explicitly stated as a 5, it implies the omission of the 3rd.
Implementation in Excel
In the same Excel work-book as was referred to in the main part of this page, there is a second spreadsheet, called "Chord notation". This does not pretend to be perfect, or completely polished, but is offered here, in case it is of interest, or even of use, to someone.The idea is that you can type values into the green cells, and the spreadsheet gives you answers in the pink cells, based on intermediate working in the yellow cells. (All but the green cells are protected from being changed, but there is no password, if you wish to unprotect the others.)
The naming scheme can be set to 1, 2, 3, 4 or 5. 2 is a good scheme for Germanic-based lettering in a sharp key (F#), 3 for the same in a flat key (Gb); 4 is a good scheme for Italian-based naming in a sharp key (Fa#), and 5 for the same in a flat key (Sol.b); 1 is for naming the notes by counting the number of semitones from C.
The next green cell is used to select the key, but for the present requires the user to specify it in terms of the number of semitones from C. Thus 5 indicates the key of F, with the value in the pink box confirming the key that has been selected.
Normally, the user will then type in the array of chords, for the chosen piece of music, in the big green box, specifying the Root, Quality, Maj/Dom and Main Interval for each. The first couple of dozen of chords in the spreadsheet, though, are presently just set up to illustrate some of the combinations described next. Normally, the contents of this box contain a very much simpler set of chords.Unfortunately, the Root presently has to be specified in terms of the number of semitones from the tonic. Thus a C, in the key of F, is specified as 5.
The quality of the chord can be specified as min, aug, dim, sus2, sus4, sus, or can be left blank. The major/dominant cell can be maj, for major, or left blank for dominant. The Main interval cell can be left blank for a simple triad, or can be set to 7, 9, 11 or 13 for an extended chord.
It is common to omit the 5th from a 9th chord, and the 3rd from an 11th chord. The spreadsheet does this, so, if the user disagrees with these omissions, the simplest option is to set the main interval to 13, so as to be able to see all of the notes that are on offer in the chord. In the same vein, a guitarist might need to omit other notes, too, or to choose any of the possible inversions of the chord.
Messages of the well-tempered scale approximation
The well tempered scale is based on the approximation that 312≈219 (531441≈524288, which is a better approximation than one used routinely by computer scientists, that 210≈103).
312≈219 says that repeatedly going up by an interval of an octave-fifth twelve times approximately brings us back to the original note, but 19 octaves higher. Equally, this involves starting at one note, taking its third harmonic to find the fundamental of the new note, then taking its third harmonic, and so on, twelve times.
Rearranging this as (3/2)12≈27 says that repeatedly going up by an interval of a perfect fifth twelve times approximately brings us back to the original note, but 7 octaves higher.
Rearranging this further as (3/2)≈27/12 says that going up by an interval of a perfect fifth is approximately the same as going up by an interval of seven well-tempered semitones (a well-tempered fifth).
If multiplying the frequency of a note by (3/2) signifies going up by an interval of a perfect fifth, then multiplying it by (2/3) signifies going down by an interval of a perfect fifth; then going up from there by an octave means that we end up having gone up by a perfect fourth. This means that multiplying the frequency of a note by (4/3) signifies going up by an interval of a perfect fourth.
Rearranging the earlier expression as (4/3)12≈25 says that repeatedly going up by an interval of a perfect fourth twelve times approximately brings us back to the original note, but 5 octaves higher.
Rearranging this further as (4/3)≈25/12 says that going up by an interval of a perfect fourth is approximately the same as going up by an interval of five well-tempered semitones (a well-tempered fourth).
Lastly, we note that an interval of a perfect fourth, multiplying the frequency by (4/3), corresponds to no harmonic on a brass instrument, so can only be obtained using the valves.
Miscellaneous other pages
Durance-Luberon project (en Français)
K315 project (in English)
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