Life is messy.
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FRACTIONAL PYRAMIDS
THE 13 BASIC DYADS
WITHIN AN OCTAVE
(12 with unison added)
(including unison)
Since 100 cents is the distance between half-steps in the 12 tone equal temperament (12-TET) scale, every possible just fraction between Unison and Octave must reside within 50 cents (either above or below) of one and only one of the 12 half-steps.
In this "first look" listing, every possible just fraction is thus parsed out, assigned to one of the 12 half-steps, with an additional 50 cents added at the top and bottom for Unison and Octave. (13 including Unison. The Unison page reveals extended denominators from 32 to 40 in order to show where its fractions within 50 cents begin, because no fractions with denominators below 31 are within 50 cents of Unison.)
(more info)
These fractional maps limit the "top-note window" to a 50-cent width to either side of the top-note fundamental frequency. Every irreducible fraction within 50 cents of either side of the top not is included. One useful exercise is the visual comparison of intervals keeping their traditional consonance comparisons in mind.
All 12-TET Dyads
Every irreducible fraction between 1 and 2 is theoretically included in this pyramid. (or between 1/1 and 2/1)
In other words, every just interval possible within an octave is here. This example is limited to a denominator of 20, but the pyramid goes on forever.
The gray bricks are reducible fractions.
A dyad's dominant fraction is the fraction near the top of the fractional field closest to the fundamental frequency of the top note.
It is usually a small-number fraction, and generally the smaller its denominator, the more consonant a dyad is considered to be.
Dominant
Fraction
The vertical isolation index is an indicator of negative fractional space and, like the horizontal isolation index, suggests how much independence a dominant fraction enjoys while acting on the peaks of the fundamental frequencies of both notes. The index is obtained by subtracting the denominator of the dominant fraction from the denominator of the next nearest (under 50 cents) fraction. A negative number means other near fractions have lower denominators, and is followed by the number of such fractions.
Vertical
Isolation
Index
---
Unison
"Unison is two or more musical parts that sound either the same pitch or pitches separated by intervals of one or more octaves, usually at the same time."
A dyad's dominant fraction is the fraction near the top of the fractional field closest to the fundamental frequency of the top note.
It is usually a small-number fraction, and generally the smaller its denominator, the more consonant a dyad is considered to be.
Dominant
Fraction
1
1
The vertical isolation index is an indicator of negative fractional space and, like the horizontal isolation index, suggests how much independence a dominant fraction enjoys while acting on the peaks of the fundamental frequencies of both notes. The index is obtained by subtracting the denominator of the dominant fraction from the denominator of the next nearest (under 50 cents) fraction. A negative number means other near fractions have lower denominators, and is followed by the number of such fractions.
Vertical
Isolation
Index
34
minor 2nd
"The minor second occurs in the major scale, between the third and fourth degree, (mi (E) and fa (F) in C major), and between the seventh and eighth degree (ti (B) and do (C) in C major). It is also called the diatonic semitone because it occurs between steps in the diatonic scale. The minor second is abbreviated m2 (or −2). Its inversion is the major seventh (M7 or Ma7)."
A dyad's dominant fraction is the fraction near the top of the fractional field closest to the fundamental frequency of the top note.
It is usually a small-number fraction, and generally the smaller its denominator, the more consonant a dyad is considered to be.
Dominant
Fraction
18
17
The vertical isolation index is an indicator of negative fractional space and, like the horizontal isolation index, suggests how much independence a dominant fraction enjoys while acting on the peaks of the fundamental frequencies of both notes. The index is obtained by subtracting the denominator of the dominant fraction from the denominator of the next nearest (under 50 cents) fraction. A negative number means other near fractions have lower denominators, and is followed by the number of such fractions.
Vertical
Isolation
Index
-1; 1
Major 2nd
"A major second (sometimes also called whole tone) is a second spanning two semitones A second is a musical interval encompassing two adjacent staff positions (see Interval number for more details). For example, the interval from C to D is a major second, as the note D lies two semitones above C, and the two notes are notated on adjacent staff positions."
A dyad's dominant fraction is the fraction near the top of the fractional field closest to the fundamental frequency of the top note.
It is usually a small-number fraction, and generally the smaller its denominator, the more consonant a dyad is considered to be.
Dominant
Fraction
9
8
The vertical isolation index is an indicator of negative fractional space and, like the horizontal isolation index, suggests how much independence a dominant fraction enjoys while acting on the peaks of the fundamental frequencies of both notes. The index is obtained by subtracting the denominator of the dominant fraction from the denominator of the next nearest (under 50 cents) fraction. A negative number means other near fractions have lower denominators, and is followed by the number of such fractions.
Vertical
Isolation
Index
-1; 1
minor 3rd
"...a minor third is a musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions (see: interval number). The minor third is one of two commonly occurring thirds. It is called minor because it is the smaller of the two: the major third spans an additional semitone."
A dyad's dominant fraction is the fraction near the top of the fractional field closest to the fundamental frequency of the top note.
It is usually a small-number fraction, and generally the smaller its denominator, the more consonant a dyad is considered to be.
Dominant
Fraction
6
5
The vertical isolation index is an indicator of negative fractional space and, like the horizontal isolation index, suggests how much independence a dominant fraction enjoys while acting on the peaks of the fundamental frequencies of both notes. The index is obtained by subtracting the denominator of the dominant fraction from the denominator of the next nearest (under 50 cents) fraction. A negative number means other near fractions have lower denominators, and is followed by the number of such fractions.
Vertical
Isolation
Index
1
Major 3rd
"...a musical interval encompassing three staff positions ...and spanning four semitones. Along with the minor third, the major third is one of two commonly occurring thirds. It is qualified as major because it is the larger of the two: the major third spans four semitones, the minor third three."
A dyad's dominant fraction is the fraction near the top of the fractional field closest to the fundamental frequency of the top note.
It is usually a small-number fraction, and generally the smaller its denominator, the more consonant a dyad is considered to be.
Dominant
Fraction
5
4
The vertical isolation index is an indicator of negative fractional space and, like the horizontal isolation index, suggests how much independence a dominant fraction enjoys while acting on the peaks of the fundamental frequencies of both notes. The index is obtained by subtracting the denominator of the dominant fraction from the denominator of the next nearest (under 50 cents) fraction. A negative number means other near fractions have lower denominators, and is followed by the number of such fractions.
Vertical
Isolation
Index
3
Perfect 4th
"...a musical interval encompassing four staff positions in the music notation of Western culture, ...and is the fourth spanning five semitones (half steps, or half tones). For example, the ascending interval from C to the next F is a perfect fourth, because the note F is the fifth semitone above C."
A dyad's dominant fraction is the fraction near the top of the fractional field closest to the fundamental frequency of the top note.
It is usually a small-number fraction, and generally the smaller its denominator, the more consonant a dyad is considered to be.
Dominant
Fraction
4
3
The vertical isolation index is an indicator of negative fractional space and, like the horizontal isolation index, suggests how much independence a dominant fraction enjoys while acting on the peaks of the fundamental frequencies of both notes. The index is obtained by subtracting the denominator of the dominant fraction from the denominator of the next nearest (under 50 cents) fraction. A negative number means other near fractions have lower denominators, and is followed by the number of such fractions.
Vertical
Isolation
Index
7
tritone
"...In music theory, the tritone is defined as a musical interval composed of three adjacent whole tones (six semitones).[1] For instance, the interval from F up to the B above it (in short, F–B) is a tritone as it can be decomposed into the three adjacent whole tones F–G, G–A, and A–B. According to this definition, within a diatonic scale there is only one tritone for each octave. For instance, the above-mentioned interval F–B is the only tritone formed from the notes of the C major scale."
A dyad's dominant fraction is the fraction near the top of the fractional field closest to the fundamental frequency of the top note.
It is usually a small-number fraction, and generally the smaller its denominator, the more consonant a dyad is considered to be.
Dominant
Fraction
7
5
The vertical isolation index is an indicator of negative fractional space and, like the horizontal isolation index, suggests how much independence a dominant fraction enjoys while acting on the peaks of the fundamental frequencies of both notes. The index is obtained by subtracting the denominator of the dominant fraction from the denominator of the next nearest (under 50 cents) fraction. A negative number means other near fractions have lower denominators, and is followed by the number of such fractions.
Vertical
Isolation
Index
2
Perfect 5th
"...In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so.
In classical music from Western culture, a fifth is the interval from the first to the last of five consecutive notes in a diatonic scale.[1] The perfect fifth (often abbreviated P5) spans seven semitones, while the diminished fifth spans six and the augmented fifth spans eight semitones. "
A dyad's dominant fraction is the fraction near the top of the fractional field closest to the fundamental frequency of the top note.
It is usually a small-number fraction, and generally the smaller its denominator, the more consonant a dyad is considered to be.
Dominant
Fraction
3
2
The vertical isolation index is an indicator of negative fractional space and, like the horizontal isolation index, suggests how much independence a dominant fraction enjoys while acting on the peaks of the fundamental frequencies of both notes. The index is obtained by subtracting the denominator of the dominant fraction from the denominator of the next nearest (under 50 cents) fraction. A negative number means other near fractions have lower denominators, and is followed by the number of such fractions.
Vertical
Isolation
Index
11
minor 6th
"...a minor sixth is a musical interval encompassing six staff positions...and is one of two commonly occurring sixths (the other one being the major sixth). It is qualified as minor because it is the smaller of the two: the minor sixth spans eight semitones, the major sixth nine."
A dyad's dominant fraction is the fraction near the top of the fractional field closest to the fundamental frequency of the top note.
It is usually a small-number fraction, and generally the smaller its denominator, the more consonant a dyad is considered to be.
Dominant
Fraction
8
5
The vertical isolation index is an indicator of negative fractional space and, like the horizontal isolation index, suggests how much independence a dominant fraction enjoys while acting on the peaks of the fundamental frequencies of both notes. The index is obtained by subtracting the denominator of the dominant fraction from the denominator of the next nearest (under 50 cents) fraction. A negative number means other near fractions have lower denominators, and is followed by the number of such fractions.
Vertical
Isolation
Index
2
Major 6th
"....a [Major] sixth is a musical interval encompassing six note letter names or staff positions, and the major sixth is one of two commonly occurring sixths. It is qualified as major because it is the larger of the two. The major sixth spans nine semitones."
A dyad's dominant fraction is the fraction near the top of the fractional field closest to the fundamental frequency of the top note.
It is usually a small-number fraction, and generally the smaller its denominator, the more consonant a dyad is considered to be.
Dominant
Fraction
5
3
The vertical isolation index is an indicator of negative fractional space and, like the horizontal isolation index, suggests how much independence a dominant fraction enjoys while acting on the peaks of the fundamental frequencies of both notes. The index is obtained by subtracting the denominator of the dominant fraction from the denominator of the next nearest (under 50 cents) fraction. A negative number means other near fractions have lower denominators, and is followed by the number of such fractions.
Vertical
Isolation
Index
4
minor 7th
"...a minor seventh is one of two musical intervals that span seven staff positions. It is minor because it is the smaller of the two sevenths, spanning ten semitones. The major seventh spans eleven."
A dyad's dominant fraction is the fraction near the top of the fractional field closest to the fundamental frequency of the top note.
It is usually a small-number fraction, and generally the smaller its denominator, the more consonant a dyad is considered to be.
Dominant
Fraction
9
5
The vertical isolation index is an indicator of negative fractional space and, like the horizontal isolation index, suggests how much independence a dominant fraction enjoys while acting on the peaks of the fundamental frequencies of both notes. The index is obtained by subtracting the denominator of the dominant fraction from the denominator of the next nearest (under 50 cents) fraction. A negative number means other near fractions have lower denominators, and is followed by the number of such fractions.
Vertical
Isolation
Index
-1; 1
Major 7th
"... seventh is a musical interval encompassing seven staff positions..., and the major seventh is one of two commonly occurring sevenths. It is qualified as major because it is the larger of the two."
A dyad's dominant fraction is the fraction near the top of the fractional field closest to the fundamental frequency of the top note.
It is usually a small-number fraction, and generally the smaller its denominator, the more consonant a dyad is considered to be.
Dominant
Fraction
17
9
The vertical isolation index is an indicator of negative fractional space and, like the horizontal isolation index, suggests how much independence a dominant fraction enjoys while acting on the peaks of the fundamental frequencies of both notes. The index is obtained by subtracting the denominator of the dominant fraction from the denominator of the next nearest (under 50 cents) fraction. A negative number means other near fractions have lower denominators, and is followed by the number of such fractions.
Vertical
Isolation
Index
-1; 1
Octave
"...an octave (Latin: octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music," the use of which is "common in most musical systems." The interval between the first and second harmonics of the harmonic series is an octave."
A dyad's dominant fraction is the fraction near the top of the fractional field closest to the fundamental frequency of the top note.
It is usually a small-number fraction, and generally the smaller its denominator, the more consonant a dyad is considered to be.
Dominant
Fraction
2
1
The vertical isolation index is an indicator of negative fractional space and, like the horizontal isolation index, suggests how much independence a dominant fraction enjoys while acting on the peaks of the fundamental frequencies of both notes. The index is obtained by subtracting the denominator of the dominant fraction from the denominator of the next nearest (under 50 cents) fraction. A negative number means other near fractions have lower denominators, and is followed by the number of such fractions.
Vertical
Isolation
Index
17
