Today I was teaching Mathematics to my two younger brothers. We were working two-step reading problems.  The problems were something like the following example: 
         

           What is the cost of 1 1/4 lbs. of cheese at $2.16 per lb.

While working out examples for them on the blackboard, I somehow wandered from Arithmetic to Reasoning.  I have been giving much thought of late on the subject - particularly in the area of Apologetics.  Little wonder!  :)

I will replicate for you what my Math students turned their attention toward for the remaining twenty minutes of my class period. 

Problem:

            1 1/4 multiplied by $2.16 = ?

Circular Reasoning:
               
                1 1/4 multiplied by $2.16 = ?
 >             1 1/4 =  5/4 = 4 divided by 5 = 1 1/4
 >             1 1/4 multiplied by $2.16 = ?

We are left with the same premise/conclusion and data points that we began with when using this method of reasoning.  Remaining are also the same questions unanswered.  (In my class, I demonstrated this point by working the circle over again until they reacted.)

I am thinking namely of the presuppositional method of argumentation in apologizing for the existence of God, the inerrancy of scripture, etc.  The conclusion is essentially a restatement of the original data and/or thesis.  We are left with the problem unanswered.

 

I left the room briefly.  While I was away, my little opportunist took the liberty to take a sneak-peak at the teachers manual that lay open on my desk.  When I came back, he immediately informed me that he had the answer.  He did have the answer, but could give no rational defense for it.

So it is with the “confessional method” of argumentation.  If we do not subscribe to the “faith-without-reasoning” belief, we can not be satisfied with the answers that this argument gives.

To find a solution for our problem, we had to consider the mode of the numeral (i.e. decimal, improper faction, etc.).  We also had to take into account that we were working with two separate systems (monetary verses nominal). Had we neglected these factors, we would have ended with an alternate incorrect solution.   Alternate Solution:
                      1 1/4 times 2.16 =

    >              1.25 times 2.16 =
    >              368.00

Rather pricy?!  Pay if you will; I’ll bypass that method.

Correct Solution:

1 1/4 multiplied by $2.16 =

>              4 divided by 2.16 = .54
>              $2.16 + .54 =
>              $2.70

Above we have utilized in our correct solution both deduction and induction.  Moreover, we were able to come to the answer not by the given data solely, but also due to the fact that we could transfer one system of calculations over to another system, thus solving for inferred information.

We can be intelligent.  We can come to and give forth intelligent answers. After all, were we not created in the likeness of an intelligent Creator?  Whether we are discussing the weather or other things more important, let us not forget this fact.  :)

 

   

     

Musical Temperaments: Now and Then

Because music of the time is interrelated with the temperament of the time, historical developments are central to discussions of temperament. In 2500-2001 BC, the Chinese developed the five-note (pentatonic) scale. Two hundred to three hundred years later, we have record of five-tone and seven-tone scales in Babylonian music. We credit Pythagoras for his Pythagorean Temperament at approximately 550 BC, in which the chromatic scale was generated by tuning in perfect 5ths, using the circle of 5ths. He is said to have introduced the octave around this time.

Although we appreciate the concepts he promoted, we must recognize that his thoughts were imperfect. For example, when tuning in perfect 5ths, there is much to be desired with the 3rds. Moreover, although the Pythagorean temperament results in a scale with perfect 4ths and 5ths, at the last it ends in very poor dissonance. Due to this factor, we see the development of the Equal Temperament (ET) scale not but one hundred or so years after the invention of the Pythagorean Temperament. (If we were to tune by contracting each 5th by 23/12 cents, we would end up with exactly one octave and that is one way of tuning an ET scale.)

Since the introduction of the Pythagorean temperaments, all following temperament have been improvisions of the named temperament. Thus, we have the Meantone Temperament (MT), in which the 3rds were made just instead of the 5ths. I can see the sense in this idea because of the prominent role that 3rds play in certain music, especially during an age when music made greater use of 3rds. * The flaw presented by the MT is of a greater degree worse than that of its predecessor.

*{To illustrate, play a simple 4/4 timing song that changes chords only on new phrases. I noticed today, that as I played the tonic chord in my right hand [while still leaving the melody at top], and played the 3rd as the bass line for my left hand, it actually sounded more consistent in the Dorian mode. (When ending in the Dorian, I felt the need to play the tonic. I think perhaps that my ear is trained that way however, because it was not at all erratic to end on the 3rd.) Moreover, I played first in the Dorian mode, and without pausing switched to D Major to play the song anew. It was necessary to play the tonic in this key; and I would naturally have it no other way except for in passing tones, etc.}

Well Temperaments (WT) struck a compromise between Meantone and Pythagorean. Additionally, WT opened the door for not only good 3rds, but also good 5ths. It appears by Bach’s harmonies in his compositions that essentially all the details of tempering were already worked out by Bach’s time (before 1700). (Example: His "Well Tempered Clavier").

Over the past one-hundred years or so, ET has been the excepted temperament while the other temperaments are labeled "historical". The musical freedom presented by ET and our current trends toward increasing dissonance are two primary reasons for this preference. The advantages for the piano tuners are numerous. Moreover, ET reduces many lurking wolves that the various other temperaments allow to creep in with indiscretion. We do harm to the purity of intervals by providing security from these beasts by sacrificing the very motivation for chromatic scales (i.e. pure intervals).

I think that Bach would have us shed the muddy, tasteless water of ET for the pure, sweet tones of the WT. Yet what of the wolf tones? And what are we to do with the complexities of such the WT system? The quantry is age old. Unfortunately, I do not know much at all on this vast topic. Yet I would love to hear about you explorations and brainstormings on the topic...