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Number-names — Intervals — page 2



Non-Major scale intervals — the remaining interval number-names

The Major scale’s intervals (all of them Major or perfect) and the unqualified numerals (one through seven) used to represent them are the starting point of music’s standard system of formula-number symbology. All other intervals, meaning those that do not occur naturally above the tonic of a Major scale (e.g. minor, augmented, and diminished intervals), specifically the remaining five non-Major tones, are symbolized with numbers by adding accidental marks to the Major scales unqualified interval numerals. The differences of those non-Major-scale intervals are then seen and compared relative-to the Major scale’s (Major and perfect) intervals — the standard referents. For example; the minor-third interval (which does not occur above the tonic of a Major scale) is symbolized b3. In simple language, that number-symbol flat-third means; “a little less than a Major-third”. The referent is the Major-third (i.e. the number 3 with no flats or sharps), and the flat symbol is the qualifier, meaning “narrowed by one half-step or semitone”. Figure 9 shows more examples of how standard procedure numbers these non Major-scale intervals. Notice that all of these interval number-symbols (potential formula-digits) are qualified with flats or sharps.



Figure 9






For a quick introduction to the method of numbering and qualifying non-Major tones, and to confirm that said method is common to both system components (letter-names and number-names) compare these two renderings of a chromatic scale (Figure 10). Note, the designations #1, #2, and #6, are possible, but because of infrequent use I omitted them here.



Figure 10






Enharmonic number-names

Approximately one-half of all available intervals are either Major or perfect — they are natural to the Major scale. The remaining intervals are either minor, augmented, or diminished. But due to standard procedure’s habit of assigning enharmonic names to those non Major-scale intervals (alternate but synonymous letter-names and number-names assigned to any single tone, e.g. #4 = b5 and 6 = bb7) the apparent quantity of non-Major-scale intervals doubles, at least. In other words, it will seem as if there are twice as many non Major-scale intervals than there really are, and there will be twice as many non Major- scale interval names (in fact) to learn about (and then translate). [Thankfully, there are no enharmonic numbers in the Cipher System. The Cipher System treats each pair of standard procedure’s enharmonic alternates as a single tone (which they are in the equal tempered chromatic scale used on all pianos and guitars) and assigns a single (real, chromatic) number-value to each such tone. Any of the Cipher System’s single number designates, then, will always encompass any and all of standard procedure’s enharmonic alternates for the given tone.]

Conflicting (or clashing) accidentals

Even though our two most fundamental system components, letter and number names, share much in design and methodology their wedding, in practice, is less than harmonious. The dual use of accidental marks, simultaneously qualifying both the letter and number names of any given tone (we’ll call it a name-pair), as a rule, results in mixed messages. They are visually and conceptually confusing. For example, one of the two members of a given name-pair might use a flat while the other member uses a sharp or no accidental at all. Conflicting letter and number names occur everywhere in music theory except the key of C Major. That is, they're present in all other Major and minor scales, their respective chords, etc.

For a sampling of clashing accidentals, examine the key of G# minor natural (Figure 11). Scan vertical pairs of names comparing letter and number names. Three distinct combination types are encircled, but the name-pairs of every tone are clashing.



Figure 11






Neutral qualifying prefixes

To compensate for those ubiquitous conflicting names, a full set of number-qualifying prefix-words, whose meanings are nearly equivalent (or analogous) to the accidental marks was developed: Major, minor, perfect, augmented, and diminished. They provide neutral vocabulary that function well regardless of environment. That is, irrespective of the accidental marks present among both letters and numbers. Incorporated within the interval number-names, the neutral qualifying prefixes lend greater precision to communication by operating above the level of variables. They can mean only one thing, no matter which or how many accidentals are needed to make it so. While the accidental marks and prefix words do not fall into exact and exclusive one-to-one correspondence, their parallel function and near correspondence is strong enough to make point of and examine.
First, review the functions of the five accidental marks (Figure 12):



Figure 12






Here then are the (nearly) corresponding (and neutral) number-qualifying prefixes (Figure 13):



Figure 13






The precise meanings of the neutral qualifying-prefixes are defined and graphically illustrated next (Figure 14 and 15). Again, these names apply regardless of the accidental marks needed to achieve the states (i.e. interval widths) defined.



Figure 14






Figure 15






Note; the two types of qualifier-prefixes, words and accidental marks, cannot be brought into exact and exclusive one-to-one correspondence. The word-qualifier diminished, for example, may correspond to either of two different accidental marks: the flat [b] and double-flat [bb]. e.g. diminished-fifth intervals use single flats but diminished-sevenths use double flats.

The two variations of interval number-name — summary

In standard procedure the number-name of any interval is made-up of two parts: an Arabic numeral plus a qualifying prefix. The prefix can appear either as a word [e.g. Major, minor, augmented, or, diminished], or a symbol, i.e. an accidental mark, e.g. . The meanings and use of these two types of qualifier are similar but not identical. In any event, we must be able to understand, link, (and ultimately translate) both kinds of prefixed numbers. I’ve included separate interval number-name translation-tables on this web site for each of the two types of prefixes (word and accidental-mark) and a third summary table combining both.

The two forms of interval number-name, employing either word prefixes or symbol prefixes, are used for different purposes. Word prefixes, like Major, or minor, communicate the particulars of intervals while remaining neutral with regard to specific combinations of accidental marks present at any given time. Word prefixes, then, are collective and neutral qualifiers. For example; the following pairs of lettered tones are all Major third intervals. They all span the same distance, 2 whole-steps: B-D#, Cb-Eb, B#-Dx,  C-E,  C#-E#,  Db-F. Each spelling of this one interval displays a different combination of accidental marks (or none), yet all of them can share the same collective number-name (Major third) because that name incorporates the neutral qualifying word Major rather than some single and specific accidental mark.

Interval number-names that incorporate accidental-mark prefixes (e.g. b3 and #5) are more compact than their word-prefixed counterparts and they’re also more precise. This form of interval number-name is used primarily as a kind of musical notation, a written shorthand, and it’s the preferred form for cataloguing musical formula. For example; in standard procedure, chord formula are typically written like this: R, b3, 5, b7 = Minor-Seventh chord. Each digit of such formula (with or without accidental marks) represents and names an interval of a specific width in whole-steps or half-steps. Those digits, those interval number-names, are the things we must translate to real (counting) numbers, i.e. half-step or semitone value numbers.

Non-Major intervals: minor, diminished, and augmented.

With our completed set of prefixes, all twelve intervals can finally be named (Figure 16). The five non-Major tones can be addressed by either of two enharmonic number-names, e.g. aug.1st = min.2nd. Two (otherwise) Major intervals receive enharmonic names as well; (P-5th = dim.6th), and (Maj.6th = dim.7th).

The significant difference in the number-names of Major-scale and non Major-scale intervals (simple or compound) is this; the number-names of non Major-scale intervals are expressed as variants of and relative to the Major and perfect reference intervals and their plain (unqualified) number-names. For example, chromatic tones neighboring the tone called 6 — that is, lying one half-step lower or higher in pitch — would be called (b6 or min.6th) and (#6 or aug.6th) respectively. That is, they’re qualified sixths. The numeral 6 is appropriated and incorporated within the number-names of those non Major scale intervals. So it is throughout music theory. Non Major-scale intervals do not have truly independent and distinct names, whether we use letters or numbers. The Major scale bias is all pervasive.



Figure 16









Compound Intervals (next page)






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© Copyright 2002   Roger Edward Blumberg


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