Theory texts describe ‘non-harmonic’ or ‘non-essential’ tones as those that do not belong to the prevailing harmony. This is false, as all tones are harmonic. All notes produce harmony, and not relegated to a position of non-essential, or non-harmonic.
‘Harmony’ is defined as, ‘Any simultaneous combination of tones’. (from Dictionary.com). Look it up. The following example is from the Chopin in E minor, paraphrased. The second measure shows a minor 11thchord (m11), followed by a dominant (x) on the dominant (V), spelled with an ‘E-flat’ instead of ‘D-sharp’. Enharmonic notes do not change the identity of chords. The following measure shows another dominant 11th(x11) on the lowered super-tonic (bII). The 4thbeat shows a dominant without a root, ‘E’ with ‘G-sharp’ the 9thand ‘C’ the 13th. The third measure shows fourth beat as a dominant m9th (oxm9), without a rooton the sub-dominant (IV). The distinctive and characteristic intervals of the dominant (x) are the major 3rdand minor 7th, ‘C-sharp’ and ‘D’ respectively. ‘B-flat’ is the minor 9th.
The next line is taken from the Chopin Polonaise Op 40 nr 1 in ‘A’ major. The first measure is mostly on ‘A’ major, but the last sixteenth is a ‘B’ minor chord, not a ‘passing’ chord as theorists might call it. Its characteristic interval is the minor 3rd, ‘B’ to ‘D’. It must be recognized, and heard. The first chord in the second measure is a dominant (x) with the missing root, ‘B’. The tritone of this dominant is ‘A’ to ‘D-sharp’, with ‘A’ the minor 7thand ‘D-sharp’ the major 3rd. ‘B-sharp’ is the minor 9thenharmonic with ‘C-natural’. It may also be identified as a diminished chord.
At measure 31 of the Chopin C# minor Nocturne (1830), there is a scale in the treble over a dominant in the bass. In its present form, it cannot be deciphered. The same measure transposed with enharmonic notes is shown below the original.
With the transposed notes, we may begin to see a ‘G’ dominant (x) in the bass followed a half-step higher with a ‘G-sharp’ dominant, with its root, ‘G-sharp’. The treble scale now may be seen as an altered ‘G’ major scale with the ‘G-sharp’ and ‘A-sharp’ as minor 9thand augmented 9th, respectively. ‘G-sharp’ and ‘F-sharp’ on the second eighth note of the second beat is the minor seventh of the ‘G-sharp’ dominant on the third beat to which it becomes. Listen!
Scales are also harmonic both as the seven-note scale and partial scales. A 3-note to a 5-note major scale is characterized by the major 3rd. along with the step pattern of 1-1-1/2-1. When these notes are in close proximity a harmony is discerned as major. The minor scale is characterized by the minor 3rd, 6th, and 7thdegrees. The melodic form of the minor scale is characterized by the minor 3rd, and the major 6thand 7thdegrees. These scales are harmonies only when heard or conceived in their entirety as identities. No single note creates ‘harmony’.
Below is measure 15 from the same Nocturne that shows an F-sharp harmonic minor scale over a C-sharp dominant harmony. This scale is ‘relative’ to no other scale. It is its own identitywith the characteristic intervals of a minor 3rd, ‘A’, a minor 6th, ‘D-natural’, and a major 7th, ‘E-sharp’. It is derived from its own ‘mother’ scale of ‘F-sharp’ major, from which the characteristic intervals are obtained. It has no relation to the ‘C-sharp’ harmony of the bass. Theory texts will describe the scale as a dominant (Mixolydian) scale, 5-5 from the root, ‘C-sharp’. The scale, however, is not a dominant scale but a minor scale with its own root of ‘F-sharp’. As a ‘C-sharp’ dominant scale, ‘A’ would be sharped, the major 6thof the scale. ‘A’ however, is the minor 3rdof the ‘F-sharp’ minor scale.
Scales may be named according to their functional position name; a ‘tonic’ scale, a ‘super-tonic’ scale, dominant scale, etc. A major scale like a major chord is identified by its intervals. A sub-dominant scale is identified by the augmented 4th. Below is an example from the JS Bach Fantasy in F minor. The scale begins on ‘C’, the major 3rd of ‘A-flat’. The second note of the scale is ‘D’, an augmented 4thof ‘A-flat’. Therefore, this is not a major scale per se, but a Lydian major scale. An ‘A-flat’ major scale would have ‘D-flat’. Scales create harmony because of the proximity of their notes and also because of pedal use that combines sounds into harmony. Scales are constructed with steps, but that does not contribute to their harmony. Characteristic intervals within scales do.
According to traditional theory texts, minor scales ascend as melodic minor and descend as ‘normal’ minor. This is misleading as it isn’t necessarily the case in music. Each scale has its own identity and must be heard as such. Scales are not ‘relative’ to other scales, nor do they ascend one way and descend another. All harmonies, intervals, scales, and chords have their own identities consisting of their characteristic intervals and with their own identifiers. Identifiers are expressed with letters and symbols, ‘M, m, x, Æ, o. Functions are indicated with numbers, Arabic or Roman, 1-2-3 or I-II-III. The two must not be mixed lest the concept of identity and function be confused. As may be noted above, harmonic identifiers are place between the staves, and functional indicators in Roman numerals below the staves.
Ralph Carroll Hedges, B.Ed., B.M, M.M.
The Paderewski edition of the Preludes does, in fact change ‘E-flat’ to ‘D-sharp’, the 3rdof the dominant.
The theory of the ‘missing root’ (sum and difference tone) may be found in Helmholtz, ‘On the Sensations of Tone’ pg 152 ff, and Ulehla, ‘Contemporary Harmony’ pg 114 ff, and Giuseppi Tartini, “Trattato di musica secondo la vera scienza dell’armonia'” (Padua, 1754), and on the web under ‘missing roots’ or ‘sum and difference tones’.