Featured post

Why This Site

ralph-028-crop-974x280.jpg

Why This Site?

This site exists for the purpose of resolving the many issues of current piano study and music theory that are illogical, incorrect, and extremely harmful. These issues are…

  1. All music theory texts lack sufficient definitions. Definitions that are included are unilateral without the possibility that words may have more than one meaning depending on context. This is not indicated in theory texts or music dictionaries.
  2. The study of music theory has little implication for the piano student. Theory exists only for itself. Theory therefore, is really not an issue. The pianist should be focused on what is there in relation to musical elements of patterns, functions, characteristic intervals, etc.
  3. The piano student has one method of piano practice; practice and memorize… practice and memorize. There is no learning in practice. Memorizing a piece of music is not learning, but ‘parroting’ by playing over and over until it is eventually memorized but with no understanding or awareness of what is there.
  4. Music is a language with its syntaxes, vocabulary, and harmonic qualities. It is regarded mostly as a concatenation of notes and little else. Notes are memorized, notes are read, and notes are corrected. Patterns, modulations, and functions, etc. are foreign concepts to the pianist, piano student, and piano teacher. Yet they are the basis of understanding the language of music. No theory text exists to address those concepts.
  5. The combination of theoretical agents is one the most glaring issues of music theory. The combination of functional symbols along with figured bass indications is one of the worst. ‘V7’ is an example that combines harmonic function with figured bass that results in the error of regarding it as a symbol of harmonic quality, which it is not. Combining quality with function is another problematic issue, viz, ‘ii-V-I’. These combined agents may be currently fashionable, but lead to greater misunderstanding than they inform.
  6. Symbols for all harmonic qualities have been created except for the dominant that has no symbol, viz, M, m, o, and Ø. There is none for the dominant. None whatsoever. And there lay the greatest enigma in all of music theory. Is it somehow necessary that the dominant exists without a symbol for its quality? One can only wonder.
  7. Without a qualitative symbol for the dominant, theorists have devised a way around the missing symbol; the ‘secondary dominant’. If a chord with a dominant quality is located on other than the fifth note of a major scale, it is ‘secondary’, with ‘V7’ unfortunately denoting harmonic quality; V7/V. Tonicization, the corollary of the secondary dominant is completely illogical as it looses its primary function. The theory of the secondary dominant must be eliminated from all future theoretical texts as irrelevant.
  8. Providing a functional symbol for a diminished seventh chord lacks even the most basic scholarship for these chords. A diminished seventh chord per se is not functionable except as a dominant sans root. The notes in the diminished seventh are equal distance apart, therefore no one note may lay claim as the root. A diminished triad however, is functionable under certain circumstances.

Articles, lectures, demonstrations, and performances will correct the many errors as enumerated above. They will be forthcoming in this site on a regular basis. Look for them.

Modulation

Modulation: moving from one key area to another key area; CM to FM, EbM to GM, for two such examples. The process of doing this is the subject of this article.  First of all, avoid indications of modulations whenever possible.  If there are several measures of a new key, then a modulation may be in order.  However, if there are only a few chords that seem to be in a different key, avoid the indication of a modulation.  The example below is from the JS Bach ‘Kleine Praeludium’ nr 2 in ‘C’ major.  The first measure is on the tonic (I).  When a ‘B-flat’ is added, it becomes a dominant (x) on the tonic (I).  ‘C’ may move to ‘F’ in the following measure, but it is short lived. An indication of a modulation is not appropriate. Also, an indication of a ‘secondary dominant’ is completely inappropriate.  When one adopts ‘x’ as the symbol for the dominant identity, a dominant (x) on the tonic (I) works well. The tonic moves to the sub-dominant in the second measure. There are no conflicts.

The following is from the same Bach Preludes.  Nr 3 in ‘C’ minor shows a true modulation after the tenth measure. The Prelude begins in the key of ‘C’ minor, and continues in the key of ‘C’ minor for the first ten bars.  However, at the eleventh measure it modulates to the key of the Mediant (III), ‘G’ minor, and stays in the key of ‘G’ minor for the remaining of the Prelude. The symbol, ‘ox’ indicates a dominant (x) without a root (o).[1]  ‘F-sharp’ is the major 3rd, and ‘C’ is the minor 7th of the dominant (x) on ‘D’, the missing root. The major 3rd and minor 7th are the characteristic intervals of the dominant (x) with or without a root.  ‘D’ is the dominant of the new key of ‘G’ minor.  The root, ‘D’ does finally appear at the end of measure 12.

The following is from the Chopin Prelude Op 28 nr 3 in G major.  Measure 17 shows a modulation to the key of the Sub-Dominant (VI), ‘C’ major with a dominant (x) on the dominant (V).[2]  The key of ‘C’ major continues for the next six measures.

Measure 23 modulates back to the key of the Tonic (I), ‘G’ major, as ‘F-sharp’ is no longer cancelled.  ‘C’ is the sub-dominant of ‘G’ major.  The passage is taken from the ‘C’ major scale, the sub-dominant of ‘G’ major.  Since it contains an augmented 4th, ‘F-sharp’, the scale is Lydian.

As a principle stick with functional indications of the prevailing key, unless there are several measures in a new key, then a modulation might be appropriate.  Always try for differing analytic solutions so you don’t get stuck in a ‘one-way’ street.

Ralph Carroll Hedges, B.Ed., B.Mus., M.M.

[1] 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, (see ‘Tartini’s tone’ in Wikipedia), also see, ‘sum and difference tones’.  Diminished seventh chords such as ‘F#o’ are not functionable under any circumstance, such as a ‘#IVo’ since all intervals are of the same size; step and a half.  Therefore, no one note may lay claim to a root.

[2] There are two definitions of ‘dominant’.  They are; 1. The fifth note of any scale.  2. A unique chord quality with the major 3rd and minor 7th as its characteristic intervals, with ‘x’ as its unique symbol.

Characteristic Intervals Define Chords

‘Harmony’ is defined as, ‘Any simultaneous combination of tones’. (from ‘Dictionary’ on the web).

‘Identity’ is the thing, person, scale, interval, or chord itself without reference to a function.  Scales, intervals, and chords must have their own unique identifiers; M (major), m (minor), x (dominant), Ø (half diminished), and ‘o’ (diminished) in order that they be identified.[1]

‘Function’ is position; nothing more, nothing less, and expressed with numbers; 1-2-3, etc., and I-II-III, etc. Identity and function must not be confused nor combined into a compound symbol.[2]

The major scale and the major chord may be identified by the major 3rd, 6th, and 7th intervals.

The minor scale and chord may be identified by the minor 3rd, 6th, and 7th intervals; referred to as ‘normal’. The major scale is a concatenation of steps and half steps. She is the ‘mother’ of all our music.  From ‘her’ we derive all scales, intervals, and chords that are her ‘children’, each of whom has their own unique identity, regardless of any functional consideration. Jazz harmony utilizes these characteristic intervals in voicings often built with fourths instead of thirds.

The major chord is identified by the major 3rd and the major 7th.  These are its characteristic intervals that give the chord its special, identifiable quality. The minor chord is identified by the minor 3rd and the minor 7th.  These are its characteristic intervals that give the chord its special, identifiable quality.

The dominant chord is identified by the major 3rd and the minor 7th.  ‘V7is not its identifier.  ‘V7’ is a functional indication of a chord built on the fifth note of a scale, and may be of any quality.  ‘V7’ for example is a normal minor chord in a minor key. The dominant as a unique sound quality must have its qualitative identifier in order that it may be hear for what it is; a dominant chord that has no necessary relationship to the fifth note of a scale. All words, for example have more than one definition.  ‘Wall’ board has an entirely different definition then ‘Wall’ street.  ‘Dominant’ the function (V), and dominant the identity (x) have no relationship whatsoever.  ‘Dominant’ must have two different definitions.  One is a position within a scale, the other a sound quality that is identifiable as a unique chord.  The ‘dominant’ as a unique sound quality may be found anywhere in the diatonic or chromatic scale.  The ‘dominant’ as a function (position) is a chord located on the fifth note of any scale and indicated as, ‘V’. Adding ‘7’, as in ‘V7’ adds nothing to its function (position).  Also, ‘V7’ is a compound symbol made up of two numbers.  Numbers do not indicate in any way the identity (quality) of a chord, only its position.

The following is taken from the song, ‘When Sunny Gets Blue’ by Marin Fisher.  The pick-up is on the dominant (V). The dominant (V) usually moves to its tonic (I), but here goes ‘back’ to the super-tonic (II), then to the dominant (V). The second bar shows a minor (m) on the sub-dominant (IV).  The sub-dominant in a major key is normally major (M).  But here it’s a minor chord that moves (progresses) to the dominant (x) on the lowered leading-tone (bVII), that progresses up a step to the tonic (I).[3]  The characteristic intervals of each chord are present in this arrangement. Note that functional indications (positions) are below the staves (I-II-III), etc., and chord qualities (identities) are between the staves.  It works, and works beautifully.  Traditional theory texts cannot handle this.  This is an important illustration of chord function and chord identity.

The following is from the Chopin Nocturne op 9, nr 2.  Measure 20 contains chord functions below the staves, and chord identities between the staves.  I shudder to think how this would be analyzed with traditional harmony and its ‘secondary dominant’ theory.  The second beat of measure 21 contains a dominant sans root (oxm9) over an ‘E-flat’ pedal.  The ‘o’ indicates a missing  root with a minor 9th, ‘m9’.   The diminished chord on ‘D’ represents the upper four notes of the dominant sans root, the major 3rd ‘D’, the 5th ‘F’, the minor 7th ‘A-flat’, and the minor 9th ‘C-flat’.[4]

Ralph Carroll Hedges, B.Ed., B.Mus., M.M.

[1] For some unexplained reason, theorists refuse to acknowledge ‘x’ as the symbol for the identity of the dominant.  Their use of ‘V7’ is a functional indication, not a symbol of the dominant quality.

[2] See ‘Compound Symbols’ by the author.

[3] Note that ‘IV-bVII’ is a Circle progression, and in this case, a ‘half-cadence’, i.e. intransitive.  ‘Cadence’: a place to breath.  The following measure is a new ‘sentence’.  The two must be separated.

[4] 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, (see ‘Tartini’s tone’ in Wikipedia), also see, ‘sum and difference tones’.

Ralph Carroll Hedges, B.Ed., B.Mus., M.M.

[1] For some unexplained reason, theorists refuse to acknowledge ‘x’ as the symbol for the identity of the dominant.  Their use of ‘V7’ is a functional indication, not a symbol of the dominant quality.

[2] See ‘Compound Symbols’ by the author.

[3] Note that ‘IV-bVII’ is a Circle progression, and in this case, a ‘half-cadence’, i.e. intransitive.  ‘Cadence’: a place to breath.  The following measure is a new ‘sentence’.  The two must be separated.

[4] 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, (see ‘Tartini’s tone’ in Wikipedia), also see, ‘sum and difference tones’.

Tonal Music

Tonal music, or ‘Western’ tonal music is based on the major scale, and the major scale in turn is derived from the ‘key circle’, aka ‘circle of fifths’, or simply the ‘Circle’.  The major scale is constructed with steps and half-steps. But these steps are different in music composed before the middle of the 18thcentury, and music thereafter.  

In music prior to the middle of the 18thcentury the major 3rdwas made up of two steps with the mean of their mathematical sum, creating a ‘mean-tone’ system whereby the major 3rdwas a ‘just’ major third, i.e. a ‘true’ major 3rd.  ‘Perfect’ 4ths and 5ths were also ‘perfect’ in their mathematical ratios.  This system created beautiful harmony in keys not more than two, maybe three sharps or flats. Harmony in keys of four and more sharps or flats created harmonies that were so ‘out of tune’ that they ‘howled’. Modulations therefore, were seldom if ever employed.

It was JS Bach that realized that the 5ths and 4ths must be ‘tempered’, i.e. the 4ths made a very small amount ‘wide’ (sharp), and the 5ths a small amount flat in order that all keys might sound ‘in tune’. He was one of the first to tune his keyboard instrument in ‘the tempered’ scale, hence ‘well-tempered’.  This system replaced the ‘mean-tone’ system of the 17thcentury.  To test his theory of the ‘tempered’ scale he created one of the world’s great masterpieces with each set of pieces in all keys.  His ‘Wohl Tempierte Clavier’ was published in the middle of the 18thcentury; around 1750. 

The concepts of ‘consonance’ and ‘dissonance’ was well established up to Bach’s time.  Major and minor 3rds, and perfect 4ths and 5ths were ‘consonant’ because the 3rds were ‘true’ and 4ths and 5ths were ‘perfect’.  But with the tempered tunings from Bach’s time to today, 3rds are not ‘true’ and 4ths and 5ths are not ‘perfect’.  Therefore, these intervals are not ‘consonant’, theoretically. This has resulted in a re-evaluation of the theory of consonance and dissonance, and the concepts of what is beautiful. Even the theory of the ‘figured bass’ has been re-evaluated by the 20thcentury.[1]  

‘Figured bass’ was developed during the Baroque period (1600-1750) for the keyboardist to ‘realize’ an accompaniment for the Baroque orchestra.  Roman numerals were not used, therefore the system was based purely on intervals. By the 19thcentury theorists began combining figured bass with Roman numerals, resulting in a theory using compound indications of chords that confuses function and identity.  The ‘dominant-seventh’ (V7) is such an indication that confuses function, the numbers, with the aural identity of the unique sound of the dominant.[2]

Analysis of music of the ‘Western’, or ‘European’ based music must now be done with harmonic function and harmonic identity separate so the compound figures do not confuse the reader, …or the writer for that matter.


Ralph Carroll Hedges, B.Ed., B.Mus., M.M.

[1]See L. Ulehla, ‘Contemporary Harmony’ and the section ‘Figured bass re-evaluated’.

[2]The unique identity of the dominant is made up of the characteristic intervals of the major 3rdand minor 7th, not to be confused with the function, ‘V7’.  Its symbol therefore, is not ‘V7’, but ‘x’, as used by many theorists, John Mehegan of the Juilliard School for one.  ‘x’ must be used to replace ‘V7’ as the identity of the dominant so that musical analyses be thereby demystified. 

The Tritone: its elemental functions

The Tritone: its Elemental Functions[1]

The tritone is one of the more important intervals in music.  Theory text books barely mention it. The interval of a tritone is the distance of three steps, hence ‘tri’- tone. It is the characteristic interval of the dominant with and without a root. It also is made up of two possible enharmonic[2] intervals; the augmented 4thand diminished 5th. The tritone is the fulcrum upon which the dominant (x), with or without a root is identified with its characteristic intervals of a major 3rdand a minor 7th..

In the line below, the first measure starts with a perfect 4th, ‘F-Bb’. ‘Bb’ is raised to create an augmented 4th(#4).  The second measure starts with a perfect 5th. ‘C’ is lowered creating a diminished 5th.  The augmented 4thand diminished 5thare enharmonic, ergo[3]the same tritone, just spelled differently.

Tritones that are inverted remain the same tritone, but change their name; #4/o5.  The third measure shows and augmented 4thinverted to another augmented 4th. They, again are the same tritone.

In the following line the dominant (x) contains a tritone from the major 3rd to the minor 7th. In the first measure the dominant (x) is spelled with an augmented sixth, ‘B-sharp’. The tritone is an augmented fourth, ‘F-sharp’ to ‘B-sharp’. In the second measure the dominant (x) is spelled with a minor seventh, ‘F-sharp’ to ‘C’. The two dominants and the two tritones are identical albeit spelled differently. The third measure shows a ‘D’ dominant (x) spelled with a minor 7th. The same dominant is then spelled with an augmented 6th, ‘B-sharp’. The tritone is ‘B-sharp’ to ‘F-sharp’. The two dominants and tritones are identical while spelled differently.  ‘A rose is a rose by any other name’ (Gertrude Stein).

A tritone inverted remains the same tritone even if it’s spelled enharmonically. The first measure below shows identical tritones…all of them.  The second measure shows a tritone that is the minor 7thand major 3rdof the dominant (x). The third measure shows the same tritone, but functions as the major 3rdand augmented 6thof the dominant on ‘C-flat’. (Please note that accidentals remain good within a measure).  

The following two lines are taken from the Chopin Prelude nr 15. Measure 9 shows a dominant (x) on the dominant (V) on the 4thbeat.  The tritone of this dominant is ‘C-flat’ to ‘F’.  All well and good.  But measure 13 shows the same tritone without its root (D-flat).  Theory texts do not and cannot handle this since they do not subscribe to the tritone and the characteristic intervals of the dominant (x).

The following line is taken from the JS Bach Prelude nr 3 from the set of twelve.  Measure 23 contains a dominant (x) on the sub-mediant (VI).  The tritone of this dominant is ‘C-sharp’ to ‘G’, a diminished 5th.  Theory texts will describe this chord as an ‘augmented-six’ chord since ‘C-sharp’ is the augmented sixth above the root, ‘E-flat’.  And, augmented sixths are to resolve upward according to these texts. But here it resolved down to ‘C-natural’.  Did Bach make a mistake? Or is the issue of enharmonic intervals meaningless, theoretically?

The following is taken from the Chopin Fantasy Op 49 in F minor. Measure 65 shows a passage based on the dominant (x) on the sub-mediant (VI), a ‘D-flat’ dominant harmony. This moves to a major triad on the dominant (V), a ‘C’ major harmony.  But like the Bach example above there is a problem here. The tritone of the dominant (x) is ‘C-flat’ to ‘F’.  The minor 7th, ‘C-flat’ resolves up to ‘C’.  According to theory texts shouldn’t ‘C-flat’ be written as its enharmonic, ‘B’, an augmented-sixth of this dominant?  Did Chopin also make a mistake? First off, there is no such thing as an ‘augmented-six’ chord. An augmented-six is an interval, not a chord. Secondly, the chord is a ‘D-flat’ dominant chord, and the scale is a ‘D-flat’ dominant scale employing the characteristic interval of a minor 7th, ‘C-flat’.  If the chord should have been spelled with an augmented sixth (B-natural), shouldn’t the scale also have been called an ‘augmented-six’ scale?  Mmmm, one wonders! The point is… it really doesn’t matter how you spell it because its identity does not change with either spelling. For example;  Sue and Susan, Bob and Robert. Different spellings, same persons.

Music theory texts are full of nonsense that piano students neither understand, nor remember once they graduate.  To quote Dr. Leopold Mannes, founder of the Mannes Music School in New York…

“To the gifted and experienced musician, music is a language—to be understood in sentences, paragraphs and chapters.  The student who is still struggling with letters and words, so to speak, needs the guidance that will reveal to him the larger meanings of the musical language.  Theory, as it is called, has always been upheld as the promised gateway to this broad understanding, but there are thousands upon thousands of eager young musicians as well as disappointed older ones who will testify to the seemingly unbridgeable gap between their theoretical studies and the living experience of music itself.”  … from the Preface of Dr. Felix Salzer’s book, ‘Structural Hearing’.

Ralph Carroll Hedges B.Ed., B.M., M.M.


[1]Elemental

[2]Enharmonic

[3]Ergo

Chopin Nocturne C# minor (1830)

Nocturne in C# minor (1830)

…an analysis

©2018 Ralph Hedges

Conventions Used in the Analysis of the Nocturne

Notes are enclosed with singe quotation mark to differentiate between a note vs. an article; ‘A’ major triad on…, vs.  ‘A major triad is …’  Accidentals are usually spelled out; ‘A-flat’, ‘B-natural’, ‘C-sharp’, etc.  Chords are indicated with Roman numerals in Times Roman, 12pt; ‘VI, bII, V’, etc. …below the bass staff. The key the work is in is located above the treble staff and in Times Roman 14pt bold ‘I’ ‘VI’, etc.  Modulations are also indicated in Times Roman 12pt bold, ‘III’, ‘bVI’ etc.  When a key is described it is done with the word starting in a capital letter, ‘The key of the Tonic’ (I), ‘The key of the Super-Tonic’ (II), etc., in Times Roman 14pt, not in bold.

Lower case for minor ‘ii’  ‘iii’, etc. are never be used as they appear to be an alteration from the norm. ‘II’ and ‘III’ areminor.  They are normal in a major key and need no special indication to show them as minor. In any case, vii is not minor. ‘V/V’ or ‘VofV’ for a super-tonic dominant is an aberration. It completely loses the super-tonic function, and its form cannot be used on other ‘secondary’ dominants.  A super-tonic triad in a minor key is a normal diminished triad and is functionable (II).   A diminished-seventh chord is not functionable under any circumstance.  It is considered a dominant with a missing root (sans root), and will be indicated with a ‘o’ in front of the x; ‘ox’.  Thus, a ‘B-diminished’ seventh chord will be indicated as a ‘G’ dominant.  

A distinction must be made between ‘function’ and ‘quality’.  Dominant function (V) is not to be confused with dominant quality (x).  ‘V’ is only a function and may be minor, major, or dominant quality.  Chord quality indications are placed between the staves, chord functions below, and modulations in Roman numerals above the staves. Figured bass is not included here as its use is limited. Elements of theoretical indications must be kept separate to avoid any possibility of confusion; ‘ii’ combines function with quality, ‘V7’ combines function with figured bass and is erroneously regarded as a symbol of a harmonic quality, ‘V7/V’ combines and confuses function with quality, and with figured bass. 

Analysis of the Chopin Nocturne in C# minor (1830)

Analysis of these measures might look like the line below.  Note that functions I, IV, etc are combined with identities, Im, VI#6, and VM.  ‘VI#6’ is particularly disturbing since its identity is not given, only the function ‘#6’.  An ‘augmented-six’ chord is a ‘dominant’[1], and since it doesn’t fall on ‘V’ it should be regarded as a ‘secondary dominant’, according to the theory.

It could be designated as a ‘secondary dominant’ .  However, that doesn’t do the trick because an ‘augmented-six’ chord isn’t a ‘dominant-seventh’…or is it?[2]


When harmonic functions (I, IV, V) are separate from harmonic identities (m, M, x), a more logical analysis presents itself. [3]

The same line below shows the first chord of the second measure as a super-tonic dominantninth; II and x9.  You will notice that the treble notes are the same as the original by Chopin.  Whereas the ‘F-double-sharp’ is a major 3rdhere, it was the augmented-six in the original, VI#6. This, however, is not the main point. Characteristic intervals are. They define harmonies, and in the case of the dominant (x), the major 3rdand the minor 7thdefine the dominant (x); they are the characteristic intervals. And, the forth measure shows the original chord.


The following lines are fairly self-evident, except that there is a major chord on the tonic, measure 7, followed by a dominant chord (x), then the super-tonic chords in the following measure are normal half-diminished chords.  With no identifying symbol for the dominant, theorists would have to call this dominant a ‘secondary dominant’; ‘V7/IV’. But there is no ‘IV’ for it to be ‘V’ of. In addition, its function as a tonic (I) would be lost. Really stupid.

The scale in measure 15 could be analyzed as a minor scale with a major 7th, E# and a minor 6th, D-natural, except that it is over a C# root, making it a dominant scale with a major 3rd, E# and a minor 7th, D-natural.

Without this vital information, ear training of chord identities, especially of the dominant is impossible. One does not hear, as a dominant, an ‘augmented-six’ chord.[4]  In addition, ‘secondary dominants’ only indicate functions of functions (V7/V) and not the identity of the chord.  It’s identity is not there, having been replaced by a ‘function of a function’.  These two issues provide ample reason to remove the theory of the ‘secondary dominant’ from all music theory textbooks.

Measure 17 shows a ‘D’ major chord with the 3rdin the bass, a ‘Neapolitan’. Measure 18 shows a dominant (x) on the leading-tone, VII, with ‘B’ as the root, followed by a dominant (x) on the sub-dominant, IV with an augmented-sixth (minor 7th) in the bass leading up to ‘G’ the tonic (I).

The key of the Sub-Dominant (IV), ‘A’ major is well established by the number of V-I movements in the line. 


Measure 31 show a dominant (x) on ‘bV’ where there is a tritone ‘F-B’, the characteristic interval of the dominant on ‘G’, the following note after the tritone. The half-diminished chords on the Super-Tonics (II) are normal.

The dominant (x) in measure 34 may be regarded as a super-tonic (II) on root, ‘D-sharp’, in which case ‘A’ in the bass is a o5 of the chord, or with ‘A’ as the root with ‘D-sharp’ as the +4/o5 of the dominant, ‘Vl+6’. These are the same chords found in the second measure of the piece.  ‘bV and II’ are inversion of one another as both share the same tritone, ‘F##-C#’

The scales in the following measures look like and are E major scales over half-diminished harmonies, and as such contain the same characteristic intervals of the minor 3rd, minor 7th and diminished 5th. A half-diminished super-tonic is normal within the key, and the scale is also normal within the key.

Ralph Carroll Hedges, B.Ed., B.Mus., M.M.


[1]In addition, the word ‘dominant’ has only one definition in a musical sense, that of the fifth note of a scale. Period.  There is no contrasting definition.  (see two definitions of ‘dominant’).

[2]To help the reader understand the resolution to this problem, refer please to ‘The Tritone Its Function’.

[3]The reason theory texts are so convoluted and irrational is because the PhD’s who write the texts refuse to provide the dominant with an identity apart from its function, and they insist that a function is an identity, which mixes apples and oranges, so to speak. A function is a number, a position, I, II, III, etc.  An identity is a symbol or letter, M, m,  V7.  Doesn’t that look strange, however?  Why is ‘V7’, a function included with harmonic identities?  That, my good reader, is the enigma that has confounded theorists for the past 300 years!  They simply will not allow the dominant an identity.  The solution is to provide the dominant with the identifier, ‘x’.  ‘x’ does not conflict with any other harmonic symbol, and it solves most of the problems of music theory.  Now a complete roster of harmonic identities may be listed as; M, m, x,  and o …major, minor, dominant, half-diminished, and diminished, respectively.

[4]Refer to the article “There is No Such Thing as an Augmented-Six Chord” by the author.

No Such Things as Non-Harmonic Tones

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’[1]. 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 root[2]on 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.


[1]The Paderewski edition of the Preludes does, in fact change ‘E-flat’ to ‘D-sharp’, the 3rdof the dominant.

[2]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’.

Functions in Music

‘Function’ refers to the positionthat a note occupies in a scale, interval, or chord. It is expressed with Arabic numerals; 1, 2, 3, etc. for note functions.  It may also refer to the position a chord has within a key and expressed with Roman numerals; I, II, III, etc.  Thus, ‘E’ is the 3rdof the ‘C’ major chord, i.e. it functions as the 3rdof the ‘C’ major chord, etc. The terms ‘triad’, and ‘seventh-chord’ are used, but must be used sparingly.  A ‘triad’ is defined as a three-note chord.  But the chord ‘B-F-G’ is not a triad.  A seventh-chord is defined as a four-note chord with a 7thadded.  But the chord ‘B-F-G’ is not a four-note chord, and where is the number 7?  These are but a few of the inconsistencies found in music theory.

‘Scale-tone seventh chords’ are chords built on the notes of the major or minor scale.  Each chord has its own identity depending upon its position (function).  Thus, a seventh-chord built on the first degree of the ‘C’ major scale is indicated as ‘I’.  It is a major seventh chord.  A chord built on the fifth degree of the ‘C’ major scale is indicated ‘V7’. It is a dominant seventh chord.  However, these indications are in no way indicative of their chords identity except coincidentally. ‘I7’ is not the symbol for ‘major’ any more than ‘V7’ is for ‘dominant’.[1]  These indications are functional indications only, with their identities merely coincidental within a major or minor scale. 

Lower case Roman numerals – ‘ii’, ‘vii’, etc. are an unfortunate effort to indicate both a chord’s function and its identity.  But ‘vii’ isn’t minor, and ‘ii’ in a minor key isn’t minor, and ‘V7’ isn’t major. There are so many inconsistencies in music theory that it is difficult to enumerate all of them.  

Ralph Carroll Hedges, B.Ed., B.Mus., M.M.


[1]The word ‘dominant’ must have two definitions; 1. The fifth note of a scale indicated with a number five.  2. A chord identity with characteristic intervals of a major 3rdand a minor 7thindicated with a symbol ‘x’.