The Why and the Wherefore of the European War: Bible Study (Classic Reprint)

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The simple-ratio intervals upon which JI is based are what the human auditory system recognizes as consonance, if it ever has the opportunity to hear them in a musical context. 1 The significance of whole-number ratios has been recognized by musicians around the world for at least 2,500 years. You have asked for a scenario in which only one of wherefore or therefore will work and they aren’t interchangeable. Here’s a good one:

Wherefore stood originally as one of a series of very tightly related word-pairs differing only in whether they began with wh- or they began with th-: I sat down at the table, closed my eyes, and took my first deep breath in what seemed like a year or so. The old woman looked at me and didn’t ask any questions, wherefore I gave her no answers. I really wished you were here, Kiera, because I felt the need to confess and to have some help sorting out what had just happened. How do the intervals of twelve-tone equal temperament (12TET), the predominant tuning system in the West over the past 250+ years, compare with the simple-ratio intervals I just described? The basic premise of temperament (any temperament) is that the number of pitches required to play in different keys can be reduced by compromising the tuning of certain tones so that they can perform different functions in different keys, whereas in JI a slightly different pitch would be required to perform each function. In other words, temperament compromises the quality of intervals and chords in the interest of simplifying instrument design and construction and playing technique. 12TET takes advantage of the fact that the sum of twelve perfect fifths (312:212) is slightly greater than seven octaves (27:1). The difference is the Pythagorean comma (531,441:524,288, about 23.5 cents). Each fifth in 12TET is flatted (narrowed) by 1/12 of the Pythagorean comma (approximately 1.96 cents) causing the fifths to form a closed circle. To achieve this, the starting frequency is multiplied twelve times in succession by the twelfth root of two (12√2), an irrational number that is approximately 1.05946. This ensures that none of the resulting intervals except the octave will be simple ratios. The problems with 12TET are not limited to its fifths being slightly narrow; chaining fifths, whether perfect or slightly flatted, does not result in good thirds or sixths. And, of course, 12TET, being a closed system, makes no provision for the admission or understanding of any additional intervals. It provides “acceptable” approximations of those intervals that most musicians in Europe c. 1750 considered useful and no more. Attempts by some twentieth-century composers and musicians to expand the resources of 12TET by subdividing it into smaller arbitrary intervals, such as quarter tones, third tones, etc., failed to solve its fundamental psychoacoustic problems.There it is used like the conjunction and so; using therefore in that position would have caused a comma-splice error. To have used therefore, a semicolon not a comma would have been required. After only a brief study of intervals it becomes clear that there are two ways of composing with them: 1) arranging them in a fixed mode, or gamut, & then composing within that structure. This is the Strict Style, & is the vastly predominant world method. However, another way is possible—2) to freely assemble, or compose, with whatever intervals one feels that he needs as he goes along. This is the Free Style… 7

Now Sauron prepared war against the Eldar and the Men of Westernesse, and the fires of the Mountain were wakened again. Wherefore seeing the smoke of Orodruin from afar, and perceiving that Sauron had returned, the Númenóreans named that mountain anew Amon Amarth, which is Mount Doom. The phrase is an interesting instance of the way English speakers can turn one part of speech into another without breaking stride. Conventional grammar would say that why is an adverb, but here it lurks in the guise of a noun, meaning “reason or explanation”. Likewise wherefore. Here are a few more examples of wherefore from some of Tolkien’s more commonly read works. He uses it a total of eight times in the the posthumously published Silmarillion assembled by his son but only just once in The Lord of the Rings; he does not use it at all in The Hobbit. Beats take place between two simple tones whose frequencies are near unison. These beats—regular variations in loudness—occur at a rate that is the difference in Hz between the two generating frequencies. Beats can be perceived clearly when the difference is less than 20–25Hz, but as the difference increases beyond this point the beats blend together, giving rise to a general sensation of roughness. This roughness gradually decreases as the difference increases, persisting until the difference exceeds an interval called the critical band, which, for most of the audio range, falls between a whole tone and a minor third. You won't encounter this word really very much in modern writing. However, there is one place where it is common, and that is in the idiom "whys and wherefores" or sometimes just "wherefores" as a noun. This means "an intent or a why". (See for example in wiktionary above.)

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When composing in 12TET, you never need ask yourself, “which Bb do I want here?” or “am I sure that I want a G# here rather than an Ab?” (and which G# or Ab?). In JI, such questions are meaningful and often critical. Referring to the lattice in Figure 5, you will see that there are three varieties or “flavors” of “Bb” in close proximity to C 1/1. B7b 7/4 is the harmonic seventh of C. Bb– 16/9 is a perfect fifth below F 4/3 and a 9:8 whole tone below C. Bb 9/5 is a minor third above G 3/2 and a smaller 10:9 whole tone below C. Any of these tones could be used melodically in relation to C, though with different effects. Harmonically, their uses are quite distinct. B7b 7/4 can be used to form a consonant dominant-seventh type chord with C, E, and G or a subminor triad with G and D. Bb 9/5 is the minor third of a minor triad on G 3/2. Bb– 16/9 is the root of a triad including D– 10/9 and F 4/3. All of these tones also perform other roles in tonalities more distant from C 1/1. One need not look far on the lattice to see many other pairs of tones near C 1/1 that are represented by a single tone in 12TET, such as D 9/8 and D– 10/9 or A 5/3 and A+ 27/16 or C 1/1 and C7+ 63/32. Recognizing such distinctions and understanding their melodic and harmonic implications is an essential aspect of understanding and working with just intonation. F. T. Bullen Log Sea-waif 149 ― The ill-used crew promptly refused to do any more in her, and were, of course, clapped in jail therefor. He’s asking out of what earlier reason he was born, not to what future purpose. He’s just asking why; you can’t start a question with therefore. Hooker Eccl. Pol. v. lxiii. §1 ― The true reason wherfore Christ doth loue belieuers is because their belief is the gift of God. As you see, those are all location or direction words, with the wh- ones alone useable as the start of a question. Wherefore connects to the past and therefore to the future. Wherefore therefore means “why” or “which is why”; sometimes it means “and so”.