Number-names — Intervals — page 1 Page   1    2   3   4         This large tutorial on intervals (and later their inversions) is the key to getting up to speed, getting your foot in music theory’s door, and understanding the origins of the Cipher System’s semitone value numbers. While it is very methodical and thorough, this is not an “everything is fine and dandy with music theory” style of tutorial. Its something of an exposé. In its full original context (in the book), this discussion of intervals and their numbers directly follows a lengthy discussion of music’s letter names and tone spelling system — another face of music’s primary nomenclature wrought with problematic design,  resulting in confusion and making it difficult for novices to understand and communicate fundamentally simple ideas and patterns. Both of music theory’s two primary systems, it’s letters and numbers, share the same types of problematic design. They’re cut from the same cloth — the seven thing, seven tone, diatonic cloth. By the time we’re done here, we’ll have permanently woven some new thread into the fabric of our musical nomenclature, a larger wardrobe, one better suited to the 21st Century, and one more capable of fulfilling the communications needs of our students, our string instuments, our isomorphic chromatic keyboards, and our teachers alike. If you want to know where we’re headed here, see Figure 31 on page 4 of this Intervals Primer (that link will open in a new browser window). See the line of numbers 0° through 24° at the top of the chart. Those are the semitone numbers, the half-step values of all intervals (the new thread) the expanded nomenclature of The Cipher. Number-names assigned to musical tones Western music theory uses two kinds of alpha-numeric markers to name tones: letters and numbers. Elsewhere we covered the letter-names assigned to musical tones (but only briefly). Now we'll turn to numbers, in depth. You should know up front that music theory's system of interval number-names evolved in a manner that resembles in both logic and methodology the development of it's allied system of letter-names: both systems share the fundamental limitation of having too few core designates - seven letters and here seven numbers for twelve tones. both systems were forced to use qualifying symbols (superscripts) in the form of accidental marks (flats and sharps) to enable representation of the five non-Major scale tones and many other spelling and numerating requirements. both systems generate enharmonic names e.g. the interval spanning eight half-steps can be called either a "sharped-fifth" or "flatted-sixth". both system components share a built-in bias for the Major scale. That is, the type and placement of virtually every accidental mark employed in music theory or its notation, affixed to either letter or number, is dictated relative to the stuff that is either natural to the C Major scale or not natural to it. In the case of numbers, the stuff that occurs naturally above C of C Major (alone, specifically) is the dictating factor. [Remember, the Major scale bias per-say is not the problem. The fact that the Major scale is a seven tone thing is. Any seven tone scale used to underpin and contain twelve tones would generate an equal amount of disorder.] alas, both standard system components, letters and numbers, render clear instruction and ready comprehension nearly impossible in music education Numbers that are not numbers In music, most instructions and all musical formula (e.g. the formula for intervals, scales, chords, and progression root-movements) are communicated by way of numbers (see Figure 1).     Figure 1       But the numbers used in music theory are not numbers in the normal sense of the word. They’re not counting numbers. They’re merely a set of referents, i.e. names. Just like letters are used as names in music theory with no regard for literal alpha-numeric one-to-one correspondence [remember; seven letters (only) for twelve tones], music theory’s numbers (as they stand today) are more like generic symbols, stripped of common meaning, that have been re-purposed and reused to provide a second set of tone names. For example, the interval called a seventh can be either nine, ten, or eleven things away (meaning semitones, half-steps, frets, or black and white keys on the piano), but never seven. In the real world (since the mid 1600’s), whether you’re looking at the chromatic piano keyboard or the chromatic guitar fretboard, there’s very little “seven” about it. So the true meanings of the numbers used in music theory have little to do with the face values of the numerals used. This situation results from the same practice we saw in Part One (in the book) with music’s letter-names. That is, music theory uses only seven numerals to name twelve tones. Interval numbers are the key Excluding time signatures and other rhythmic notations where numbers are used; every number used anywhere in music, single or series, Arabic or Roman, refers to and is an interval number — an interval is the measured distance or pitch separation between any two tones. Understanding intervals and their number-names, then, is the key to understanding all musical number-formula, e.g. scale-formula and chord-formula, hence all of music theory generally. Music theory’s interval numbers are (presumably) used to name the measured distances between tones. I say presumably because, in reality, music theory’s interval numbers do not measure, name, nor communicate anything about the distances between tones. They are in fact position numbers, describing the relative positions of tones (first, second, third, etc.) within seven tone scales, not distance numbers. That’s why they fail to communicate mensurally useful information to us (in and of themselves). Because intervals and their number-names are the key to understanding all musical number-formula we need to know what those numbers were intended to communicate. So again, we’ll start at the beginning and try to make some sense of this mess. Intervals are usually introduced in the context of notation fundamentals, that is, as they appear (black dots) written among the lines and spaces of the staff. That method implies a dependency — interval numbers to notation (the staff). The relationship implied is actually a matter of design consistency — the staff being the latter development. Intervals can be illustrated and explained using the familiar C major (or C chromatic) letter-line with no loss of meaning, less stress, and less confusion. So that’s what we’ll use instead. Because the method used in the Cipher System is based on the half-step or semitone values buried within the standard system, the (ulterior motive) objective of this tutorial on intervals is to provide and examine the source material, standard interval numbers, from which the Cipher System draws context (springs from, integrates with, defers and refers to). This overview of music theory's standard interval number-names and their respective half-step or semitone values will ease any apprehension, skepticism, or fear about a “new” system. You'll see the origins of the Cipher system and understand that it's neither foreign nor contrary to every day presentations of music theory. The Cipher system merely redirects our attention and concentrates on the half-step values and semitones — which have been with us, though under-utilized and unexploited, all the while. While reading, then, be attentive of the half-step values shown in any upcoming illustrations. It'll take some time before we to get to them however.   The Major scale’s interval numbers – numbering the Major scale’s tones Major and perfect intervals Music-theory and its many subsystems is centered around one thing, the Major Scale, and the C Major scale in particular. Intervals are no exception. In fact, intervals, their names and numbers, is the place in music theory where the Major scale’s influence first takes hold. The number-names of intervals, and the rules of writing musical number-formula (i.e. which numbers and accidental marks to use when and where) are all dictated by the Major scale. The interval widths that occur naturally above the tonic of the Major scale are used as reference-intervals in music theory. Likewise, the number-symbols assigned to the Major scale’s intervals are the starting-point of music’s standard system of formula-number symbology. Given that music’s interval number-names and symbols are all based on the Major scale’s intervals we’ll begin there and progress to the number-names and symbols of non Major-scale intervals. The C Major scale and it’s lettered tones is the preferred reference or model scale spelling due to it’s lack of flats and sharps. To obtain the eight (initial, stock) interval number-names used in music theory, number the lettered tones of a C Major scale with uniform Arabic numerals one through seven (Figure 2). The last tone, the octave of the first, can be numbered as either eight or one.     Figure 2       These uniform numbers give the appearance of symmetry or equidistant (sonic) spacing between successive tones. No hint is given of the actual distances involved, i.e. the whole-steps and half-steps between tones. While music theory recognizes twelve tones per octave, the above numbering scheme makes no allowance for the remaining five non-Major or chromatic tones. Standard procedure numbers the tones of any Major scale 1 through 8. Those same numbers are then used as interval number-names — the foundation reference-set of interval number-names (Figure 3).     Figure 3       All of the intervals natural (innate) to the Major scale were called either Major (type) or Perfect (type) intervals. In chord-formula and scale-formula numerology then (meaning the mechanics of music’s standard number-formula symbology), the intervals that occur naturally above the tonic of any Major Scale are depicted and numbered with plain unqualified Arabic numerals 1 through 8 — free of accidental marks (flats and sharps). They’re the same numbers normally used to number-mark the scale-degrees of any Major scale, and they refer to exactly those same Major scale tones. So any time you see an unqualified numeral used in any standard number-formula (interval, scale, or chord) you’ll know that the interval it represents belongs to (is a member of) a Major scale — i.e. it occurs naturally above the tonic of any Major scale. You’ll also know that any unqualified interval number (formula digit) will always and only represent a Major or perfect type interval, see Figure 4.     Figure 4             It takes two numbers to fully describe any interval So far, we still don’t know anything about the distances between tones that these interval numbers supposedly represent. By definition, the word interval means a measurement of time or space. So music theory’s interval numbers should communicate the measured distances between tones. But of course they don’t, and that’s the problem with standard procedure’s numbers generally. Standard procedure uses the Major scale’s scale-degree position-numbers (1 through 8) as interval number-names, and then as digits in it’s stock number formula for intervals, scales, and chords). Because those numbers have nothing to do with the distances between tones they should not rightly have been called interval numbers, but they are. The fact is, it takes a total of two entirely different numbers to fully describe any one interval: a diatonic (seven thing) position number (or letter-count) and a chromatic (twelve thing) distance number. Numbers allow us to think about musical materials and patterns in collective and neutral terms, free of specific and changing letter-spellings. A single number-formula can represent all spellings of a given interval, scale, or chord, in any key, or from any tonic or root. Beyond that general function, numbers are used in music theory for two specific reasons. There are two things we need to know about all tones at all times: their scale degree positions (place-numbers or letter-count) e.g. the 3rd tone of seven (Major scale tones) or a three letter difference between some diatonic scale tone and another and more importantly, their true measured distances from their tonic or root. Whole-step formula The device used in music theory to calculate and depict distance numbers is called a whole-step formula (Figure 5). Such formula are made up of units called whole-steps and half-steps or alternatively tones and semitones (often abbreviated as  T and S). These steps, whether whole-steps or half-steps, can really only be understood by looking at a chromatic scale and it’s twelve uniform divisions otherwise called semitones or “half-steps”. And we will do that shortly. If you want to peek ahead, see Figures 30 and 31 on page 4 of this Intervals Primer (those links will open in a new browser window). Look for “Half-Step” values and the lines of numbers that read 0°, 1°, 2°, 3°, 4°, 5°, 6° etc. Those are the chromatic scale semitone or half-step values of all intervals. One half-step or semitone equals one increment (one step or one tone) of a chromatic scale. Two half-steps (two increments of a chromatic scale) equal one whole-step. For example, the distance from the 1st to the 3rd tone of a Major scale equals 2 whole-steps or alternately 4 half-steps.     Figure 5       In the standard diatonic (Major scale oriented) mechanics of music theory, where twelve tones are forced to occupy only seven places or positions, a given tone’s scale-degree position-number (e.g. the 3rd tone is numbered 3) is not equal to and does not reveal its true distance from its tonic/root (in whole-steps and half-steps). With regard to its diatonic materials then, music theory must use two different numbers to fully describe any tone or interval. Whole-step values of the Major scale’s (Major and Perfect) intervals Music theory has always used a combination of diatonic and chromatic numbers to help name and describe its fundamental materials, intervals. But it’s implementation of the idea is lame and indirect, and the resulting numbers (still) fail to enlighten us. For example, the diatonic interval number-name perfect-fifth (e.g. from C up to G) is quite meaningless until and unless you are told that that interval spans 3½ whole-steps. That latter number, 3½, though less than ideal, is the key. The number-term 3½ whole-steps is a compressed and disguised form of chromatic number-information. 3½ whole-steps really means 7 half-steps or seven tones of a chromatic scale (or seven frets-worth of pitch on the guitar fretboard). The expressions 3½ whole-steps and 7 half-steps mean essentially the same thing, and both could serve the same descriptive purpose. Both are a form of chromatic number-information, and both are examples of the other number needed to describe what a perfect-fifth is. The reason that whole-step or half-step numbers are needed to help define any given interval is that they alone give the true measured distance (the interval) between  tones using some kind of measured increment. To Illustrate this, we’ll use the lettered tones of the C Major-scale, spelled: C, D, E, F, G, A, B, C.  In the C Major scale, counting from left to right (being the ascending direction), tone F is the fourth tone. So the number 4 is F’s scale-degree position number (Figure 6). You can also think of this as a letter-count. There are four (Major scale letter) letters difference from C up to F [C, D, E, F = 4 letters spanned]. Figure 7 shows the Major scale’s whole-step formula again — the device used in music theory to calculate and depict diatonic distance numbers. F’s ascending distance from C is 2½ whole-steps. So the two numbers needed to describe tone F of C Major are 4 and 2½ — position number (or letter-count) and distance number, respectively.     Figure 6           Figure 7           So whole-steps and half-steps (the chromatic data numbers) are the true basic units of pitch measurement in Western music. One increment (degree or step) of a chromatic scale equals one half-step (or one semitone). Two half-steps (two degrees of a chromatic scale) equals one whole-step (otherwise called a Tone). Whole-step distance values (numbers) are determined by adding up the 1’s and ½’s (whole-steps and half-steps) between any two tones. In this case we’re adding up the 1’s and ½’s from the tonic/root of a scale up to any other scale tone. The whole-step formula’s individual digits are the raw or un-summed distance numbers. [Note, whole-steps and half-steps themselves are intervals — Major-second and minor-second intervals respectively]. For purposes of initial demonstration, intervals are most often measured from the lower pitched tone up to the higher pitched tone. The starting tone of any interval (any measurement) is called the tonic or root. The determined number-name of the other tone will become the name of the interval itself. That number-name is determined by first noting the upper tone’s position or letter-count number. This will tell you, for example, that you have some kind of fifth — which isn’t much help given that there are three different kinds of fifths, each having different widths. Ultimately you must compare and match the intervals distance (in whole-steps or half-steps from the given tonic/root) with the known distance values of existing (Major scale) reference intervals and their common names — 3rd, 4th, etc. Such a matching table (whole-step value to common reference-interval name) is shown in Figure 8. If the interval’s width (in whole-steps and half-steps) does not match the value of one of the reference intervals, the number-name of the closest matching reference interval will be used, but it will be qualified in some way, i.e. with an accidental mark or word qualifier.     Figure 8           All of these so called numbers however, diatonic position numbers and whole-step distance numbers, are still essentially meaningless. They fail to communicate any truly usable information. As we said earlier, you can’t count with them. If you have an interval called a 5th spanning 3½ whole-steps, you can’t look at the (chromatic) piano keyboard nor the (chromatic) guitar fretboard and count 5 somethings nor 3½ somethings to get to where you want to go. The Cipher System, on the other hand, uses half-steps, counting numbers, rather than whole-steps, to define and describe intervals. Using half-step value numbers is crucial to communicating about music effectively. But more of that in a minute. Non-Major scale intervals — the remaining interval number-names (next page)       Page   1    2   3   4         Up to Top of page