Page   1   2   3   4     Number-names — Intervals — page 4     Half-step values or true intervalic distances Establishing meaningful vocabulary is necessary to the development of any body of knowledge. In music theory, numbers are the predominant units of communication. It’s not an exaggeration to say that all of music theory is discussed by way of numbers. That is, the tools, words and symbols, traditionally reserved for counting are employed throughout. But using the word numbers when referring to words and symbols used in music is a bit of a stretch. The impetus to name intervals of pitch, one assumes, was utilitarian. That is, to aid communication by defining the (at least initial) units of measure necessary to the study of any physical medium. The additional set of referents, combined with existing letter-names, should allow the framing of precise yet brief statements and questions. Properly used, number designates would also provide the tools necessary to conceptual thinking, i.e. pattern recognition. Unfortunately, the standard number-names assigned to intervals of pitch are ineffective as instruments of (mensural) communication. Their face values have nothing to do with measured increments of pitch. Even their whole-step number values (the standard unit of measure) are cryptic and unrevealing. Half-step or semitone value numbers alone, once you have them, are the only real numbers to be found in music. They’re the only island of sanity in the whole of Western music’s nomenclatura (until you get to a physics class). The problem is, no-one ever uses them. Figure 27 shows six kinds or variations of data types that one might find used to describe a Perfect Fifth interval (and that’s excluding examples of staff notation). For my money, only the semitone or half-step value numbers (last line) communicate immediately useful information. But they’re the one data type usually omitted as as a rule! You’ll only rarely see them, and even if you do, you’ll never be encouraged (or permitted) to use them.     Figure 27       Using clear language, the interval commonly called a perfect fifth represents the pitch distance of seven half-steps or semitones. Notice that the no-nonsense measurement (seven) is not reflected in the intervals name “fifth”. There is however, a difference of five alphabetically ordered letters between the letter-names of the two tones involved. In other words, a supposed measurement of pitch is actually a letter count. The existence and total of half-steps or whole-steps spanned, the measurement, is neglected entirely. The word interval means (explicitly) a point to point measurement. Yet, all of the “numbers” used in music theory are of the type exemplified here by the word fifth. They’re not numbers in the arithmetical nor utilitarian (counting and measuring) sense of the word. So all of music theory’s so called numbers, it’s diatonic position numbers and whole-step distance numbers fail to communicate any mensurally usable information. You can’t count with them. As I said earlier, 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. This is why the Cipher System uses semitones or half-steps, i.e. counting numbers, rather than whole-steps to define and describe intervals. In the Cipher System, a Perfect-Fifth interval equals seven half-steps. The number seven used here has clear meaning. It’s a real number, a counting number, like seven fingers or seven frets. It’s information that you can use. [End of rant, back to our story.] Identifying half-step values (of interval number names) To determine the true intervalic widths (in half-steps) of the Major and perfect (reference) intervals we return again to the C Major scale. Shown in Figure 28 is approximately one and one-half octaves of C Major scale material, including: lettered tones, interval number-names, and the Major scale whole-step/half-step formula. The latter dictates the intermediate true values of the “would-be” interval number names.     Figure 28       Adding up the whole-steps and half-steps (left to right) from (C 1st) to any Major scale tone provides us (first) the whole-step value of the given interval — a number representing the sum of whole-steps spanned. e.g. a perfect-fifth (C to G) equals 3½ whole-steps. Finally, convert the whole-step totals to equivalent half-step values; multiply the whole number portion of the whole-step sum by two, then add the ½ unit (if any) counted as one half-step. Following that procedure; 3½ whole-steps equals 7 half-steps. So the interval commonly called a perfect fifth spans seven half-steps. The table in Figure 29 summarizes the conversion of all Major scale intervals, simple and compound, to their equivalent half-step (or chromatic) number values. Their true intervalic widths are finally revealed. To conclude, we must fill in the gaps. That is, convert the number names of the non-Major intervals (i.e. chromatic degrees) to their equivalent half-step values, and insert them among the Major and perfect intervals (see Figure 30).       Figure 29           Figure 30       For some icing on the cake; try this version of the All Intervals Number-Name Summary we first saw on the previous page. This improved version has a row of numbers 0°–24° near the top of the chart. For my money, without those numbers, that chart is only half done. What do you think? Are chromatic numbers valuable tools — or should we say invaluable?     Figure 31       Conclusion Given that true intervalic distances are not communicated via staff notation nor any of music theory's standard nomenclature; letter-names and number-names, employing half-step or semitone value numbers, the chromatic distances between tones, is the key to understanding music theory and and the key to transmitting musical instructions effectively. Knowing that an interval C to G spans seven semitones (and therefore seven frets-worth of pitch) is information that anyone can use. The statement has meaning because the number seven means something. It represents an correct use of mensural numbers. It’s an accurate measurement. Conversely, calling that interval a fifth is essentially meaningless (mensurally). It bears no relation to the chromatic world we live in, so it cannot communicate mensurally useful information. Therefore, it’s of limited practical instructive value*. *It is annoying (in and of itself) to hear the ubiquitous claim (or parrot) that “music is mathematics”. But then to be told that a given tone is a “ninth” from some other tone, when it is “fourteen” things away ... and wonder .... why you might have difficulty recognizing any mathematics in it! Until you have enough information (and confidence) to conclude for yourself about that matter (music theory = math) your only safe assumptions are these; if music theory is in the realm of mathematics, those things in common among them are certainly these: • Both, involve the study of patterns • Both, are usually taught poorly • You’ll need to study some physics (acoustics), e.g. the harmonic series and calculation of intervals, before formulating your opinion. [By the way, if you ever do take that physics class, you’ll recognize the Cipher numbers immediately. In acoustics, "Calculation of intervals" is most often done using a unit of measure called “cents”. When you calculate intervals using cents, you use the numbers 0-1200 for one octave (0 is the tonic, 1200 is the first octave of the tone 0). Using cents units, 100 cents equals one (equal tempered) semitone (as a reference or jumping off point). So 700 cents is equivalent to 7 semitones aka a Perfect Fifth. In other words, the 0-based chromatic numbers used in The Cipher System will prepare you for working with tuning and temperaments (real math and science, the real music=math stuff). You'll will be right at home with concept and the arithmetic. If you know that an equal tempered 5th equals 700 cents, when you get a result that reads 712 cents or 694 cents you'll know you're dealing with tone very near 700 cents (one conceptual baseline for a P-5th).] The completed list of intervals with their corresponding half-step values in Figure 30 is the primary bit of translation needed to understand the whole of music theory. Those half-step numbers are the numbers used in the Cipher System. And that’s why the Cipher System is 100% compatible with standard presentations of music theory — yet so much easier to use and understand. So the Cipher System was created to compensate for music theory’s cryptic and uninformative numbers. It’s charge is to translate all of the numbers and number-formula used in music theory to equivalent chromatic or half-step values and then provide the means of applying those numbers to chordable instruments, the guitar, other string instruments, and isomorphic chromatic keyboards. That step alone is rectification enough to provide everyone on earth access to music theory fundamentals. Spacifically, that 99% of humanity who can not and never will read music. I hope that the irreverent tones I’ve used here (and elsewhere) didn’t sound too arrogant, or insult and alienate anyone, and that you understand that I’m an ally*, a friend. If I didn’t love music so much, I wouldn’t have given the lions share of my adult life to this work. I just believe it’s more productive to be up-front and honest with students, to admit existing problems, sympathize and commiserate when appropriate, establishing trust, so that we can all get on with the tasks at hand: teaching, learning, and making music. * Merriam-Webster's Main Entry: 1 al·ly Function: verb 1 : to unite or form a connection between : ASSOCIATE 2 : to connect or form a relation between (as by likeness or compatibility) : RELATE Main Entry: 2 al·ly Function: noun 1 : a sovereign or state associated with another by treaty or league 2 : a plant or animal linked to another by genetic or taxonomic proximity 3 : one that is associated with another as a HELPER : AUXILIARY             Next up . . . Inversion of Intervals See also: Chart —  All Intervals on the Guitar Fretboard Tables — Interval Number-Name Translation Tables (4 tables) and the explanatory text to accompany them. Next logical topic — Inversion of Intervals, on paper and on the guitar fretboard.       Page   1   2   3   4         Up to Top of page