The Harmonic Series and Equal Tempered Scale

[EstroArmonico]

It's not a question of accuracy. In fact, you might as well use 1.06 for all practical purposes. If you don't explain where that number comes from it just seems random, which is more confusing than a touch of high school math IMO.

Just a thought.

[imperialbari]

Has the concept of leading notes been touched here?

From http://www.dolmetsch.com/musictheory27.htm

Clavichord maker Peter Bavington, explains:
The truth is that in 'just intonation' with pure major thirds D# would be below Eb (the discrepancy is about 35.69 cents), but in what Barbieri calls 'Pythagorean-expressive' intonation it is the other way round: D# lies above Eb (by the Pythagorean comma of 23.46 cents). He documents the change, occurring around 1750 and the years after: at that time teachers still thought of the whole tone as consisting of nine parts ('commas' as they called them) whereas before 1750 the chromatic semitone (D-D#) was regarded as having four and the diatonic (D#-E) five. Later people began to put them the other way round so that D-D# has five commas and D#-E only four - an example of the raising of the leading note. Similarly Eb-E natural now had five commas, so as to bring the Eb closer to D.
I remember attending a lecture at Morley College ages ago on microtonal music, where the lecturer convincingly demonstrated with recordings that (modern) violinists playing, say, D-Eb-D narrowed the semitone to about 1/3 of an equal temperament whole-tone. I suspect they do this sort of thing all the time when playing melodies, but at cadences and in slow passages revert to the older 'harmonic' intonation where they have a sustained note making a third or sixth with the bass (which I think is actually not all that often).
So, to summarize, D# is sometimes above and sometimes below Eb, depending on style and many other things. It is only on our inflexible keyboards that we are forced to choose!

[sloan]

It's not a question of accuracy. In fact, you might as well use 1.06 for all practical purposes. If you don't explain where that number comes from it just seems random, which is more confusing than a touch of high school math IMO.

Just a thought.

Right - there's no use in being irrational.

[mark38655]

It's not a question of accuracy. In fact, you might as well use 1.06 for all practical purposes. If you don't explain where that number comes from it just seems random, which is more confusing than a touch of high school math IMO.

Just a thought.

Right - there's no use in being irrational.

The 12th root of 2 is something that I don't understand well enough yet. I'm familiar with the terminology from college days, but for me to write about it, I need to have a better understanding of it myself first. Could you, in layman's terms, and in the least complicated math possible, explain what the 12th root of 2 is and how it is used to divide the octave, or ratio of 1:2 into 12 equal parts. I'll gladly give you credit for being my source.

I know this is a problem I (might have) had to solve in 8th grade math but help me understand how to convert the twelfth root of two into the number "1.05946309435929..."

(I do see what you are saying that it appears random from the way I put it in the text.)

[mark38655]

After further review....

I think I will simply state that the number 1.05946309435929... is arrived at using the 12th root of 2, and that although it is an irrational number, and therefore not exactly precise, it is about as close you will ever need to get.

[sloan]

It's not a question of accuracy. In fact, you might as well use 1.06 for all practical purposes. If you don't explain where that number comes from it just seems random, which is more confusing than a touch of high school math IMO.

Just a thought.

Right - there's no use in being irrational.

The 12th root of 2 is something that I don't understand well enough yet. I'm familiar with the terminology from college days, but for me to write about it, I need to have a better understanding of it myself first. Could you, in layman's terms, and in the least complicated math possible, explain what the 12th root of 2 is and how it is used to divide the octave, or ratio of 1:2 into 12 equal parts. I'll gladly give you credit for being my source.

I know this is a problem I (might have) had to solve in 8th grade math but help me understand how to convert the twelfth root of two into the number "1.05946309435929..."

(I do see what you are saying that it appears random from the way I put it in the text.)

Simple:

a) there are 12 equal steps in an octave (look at a piano keyboard and count the keys from C to C
b)frequencies are related by MULTIPLICATIVE factors (for example, raising a pitch by an octave requires DOUBLING the frequency).
So, successive octaves are at f, 2f, 4f, 8f, 16f, ... [edited to correct gross error]
c)other intervals work the same way. Raising the pitch by a given interval requires that you MULTIPLY the frequency of the lower pitch
by a fixed factor
d) combining these ideas - to raise the pitch by a semi-tone (one of the 12 equal steps in the equal tempered scale, you multiply by some
factor - call that factor 'c'. Now...what is the value of 'c'?
e) you can go up an octave by doubling the frequency (from f to 2f), or you can go up an octave by taking 12 steps: f to cf to ccf to cccf to ccccf ...
all the way to ccccccccccccf (that's 12 'c's multiplied together - and then multiplied by 'f' - the frequency of the lower octave.
f) The Princeton Band
g) Therefore, c is the value which when multiplied by itself a total of 12 times (cccccccccccc) equals 2.
2f = ccccccccccccf, so 2 = cccccccccccc
h) exercise - use your calculator to find the value of c. Do this by guessing a value (say, 1.5), calculating 1.5*1.5*...*1.5 (there are 12 1.5's there)
That value will be too high. Guess a smaller value and try again. repeat, getting closer and closer to 2.0000.
You will find that c - 1.06 is pretty darn close (just a little bit too big - but 1.05 is much too small)

ADVANCED STUDENTS ONLY: consider a scale where the individual steps are not all the same size. Call them c1, c2, ..., c12. Keeping octaves "in tune"
just means that c1 * c2 * c3 *...* c12 = 2.0000. Consider the interval of a fifth. What constraint keeps fifths "in tune". Now consider the interval of a major third. What constraint keeps those "in tune". Can you satisfy all of these constraints simultaneously? In particular, does c1 = c2 = c3 = ... = c12 satisfy
all of these constraints simultaneously? Is it "close enough for jazz?"

[mark38655]

Simple:

a) there are 12 equal steps in an octave (look at a piano keyboard and count the keys from C to C
b)frequencies are related by MULTIPLICATIVE factors (for example, raising a pitch by an octave requires DOUBLING the frequency).
So, successive octaves are at f, 2f, 3f, 4f, 5f, ...
c)other intervals work the same way. Raising the pitch by a given interval requires that you MULTIPLY the frequency of the lower pitch
by a fixed factor
d) combining these ideas - to raise the pitch by a semi-tone (one of the 12 equal steps in the equal tempered scale, you multiply by some
factor - call that factor 'c'. Now...what is the value of 'c'?
e) you can go up an octave by doubling the frequency (from f to 2f), or you can go up an octave by taking 12 steps: f to cf to ccf to cccf to ccccf ...
all the way to ccccccccccccf (that's 12 'c's multiplied together - and then multiplied by 'f' - the frequency of the lower octave.
f) The Princeton Band
g) Therefore, c is the value which when multiplied by itself a total of 12 times (cccccccccccc) equals 2.
2f = ccccccccccccf, so 2 = cccccccccccc
h) exercise - use your calculator to find the value of c. Do this by guessing a value (say, 1.5), calculating 1.5*1.5*...*1.5 (there are 12 1.5's there)
That value will be too high. Guess a smaller value and try again. repeat, getting closer and closer to 2.0000.
You will find that c - 1.06 is pretty darn close (just a little bit too big - but 1.05 is much too small)

ADVANCED STUDENTS ONLY: consider a scale where the individual steps are not all the same size. Call them c1, c2, ..., c12. Keeping octaves "in tune"
just means that c1 * c2 * c3 *...* c12 = 2.0000. Consider the interval of a fifth. What constraint keeps fifths "in tune". Now consider the interval of a major third. What constraint keeps those "in tune". Can you satisfy all of these constraints simultaneously? In particular, does c1 = c2 = c3 = ... = c12 satisfy
all of these constraints simultaneously? Is it "close enough for jazz?"

Thanks--

That fits my understanding after doing a little more reading since the post I made here.
Would you agree that the way I am wording it in my LAST post ("After further review....") would be sufficient?

[EstroArmonico]

12th root of 2 means that the octave (1:2) is divided so that the ratios between the adjacent notes are equal (approx. 1:1.06 for each half-step). A chromatic scale between A 110 and A 220 (110:220 = 1:2) would have approximate frequencies of 110, 116.5, 123.5, 130.8, 138.6, 146.8, 155.6, 164.8, 174.6, 185, 196, 207.7, and 220hz. You get these consecutive numbers by multiplying the previous number by ~1.06 aka 1+2^(1/12) (12th root of 2).

You can see here that we cannot just add a certain number to the frequency and expect the same interval each time. The first half-step (A to Bb) increases the frequency by 6.5hz while the last (G# to A) adds 12.3hz even though the interval is the same (meaning 110:116.5 ≈ 1:1.06 and 207.7:220 ≈ 1:1.06).

Equal intervals (as in equal temperament) change the frequency exponentially rather than linearly, hence the need for all the maths.

You could actually divide the same octave into 12 linearly equal parts where the ratios are 12:13:14:15:16:17:18:19:20:21:22:23:24. The frequencies would be 110, 119.2, 128.3, 137.5, 146.7, 155.8, 165, 174.2, 183.3, 192.5, 201.7, 210.8, 220. Here, the intervals become progressively smaller (the first interval being 12:13 - approx. 138.6 cents - and the last being 23:24 - approx. 73.7 cents) although we are increasing the frequency by the same amount each time (9.16666...). This could be called an overtone scale and is theoretically (see discussion of instrument design, etc.) what you would get if you played between the 12th and 24th partials of a brass instrument.

Notice that the math for the latter scale is much simpler and that each ratio can be expressed by integers. These are "pure" aka "just" intervals even though the consecutive notes would be dissonant. Notice that pure, consonant intervals are found within this scale, such as 12:16 = 3:4, which is a pure perfect 4th or 16:20 = 4:5, a pure major 3rd.

I realize this is a total digression but I strongly believe you should thoroughly understand the subject before explaining any of it to students. Getting it "pretty much correct" could just confuse the matter further. Being a teacher myself, I understand the dilemma of deciding how much and in what manner to teach the theory of intonation. This topic usually isn't explored until college, which is, in my opinion, when it makes sense to study it in this way.

In my opinion, understanding the theory doesn't actually help anyone to play in tune so I am definitely in agreement with the those advocating the study of solfege and singing. I teach intonation (in a group/sectional setting of beginning to intermediate players) by first having the students put the tuners away and then giving three basic rules:

1. if it sounds bad, it's out of tune,
2. it it's out of tune, move your pitch (by hearing it, not with the embouchure), and
3. if it gets worse, go the other way.

Too simple? Maybe. But the truth is that intonation can only be learned with the ears, not by doing calculations to find cents deviation from equal-temperament or studying the history of intonation systems. Like Arnold Jacobs taught: seek the products and don't worry about the methods. A scientist knows far more about the human body than an athlete, but does that help him to throw a ball?

~Luke Storm

[EstroArmonico]

Sloan -

f, 2f, 3f, 4f, 5f, etc... are NOT octaves. That would be the harmonic series with decreasing intervals of P8, P5, P4, M3, m3, etc.

Octaves are: f, 2f, 4f, 8f, 16f, 32f, 64f, etc.

[sloan]

Sloan -

f, 2f, 3f, 4f, 5f, etc... are NOT octaves. That would be the harmonic series with decreasing intervals of P8, P5, P4, M3, m3, etc.

Octaves are: f, 2f, 4f, 8f, 16f, 32f, 64f, etc.

You are, of course, correct. My typo. [original edited to fix this gross error]

[iiipopes]

I know this is a problem I (might have) had to solve in 8th grade math but help me understand how to convert the twelfth root of two into the number "1.05946309435929..."

Back up. We have a definitional problem that needs clarification for what a "root" is, mathematically speaking. A "root" is a number, which, when multiplied by itself a defined number of times, gives the answer that you're wanting to find the root of. For example, if I want to find the 2nd root, or "square" root of four, I do the math and find that 2 is the 2nd root of four, meaning if I multiply 2 by itself two times, I get four. Likewise, if I want to find the 3rd root, or "cube" root of 27, then doing the math I find that three, multiplied by itself three times, equals 27, and the 4th root of 625 is 5, because if I multiply 5 by itself 4 times, I get 625.

For the octave and the 12th root of 2, doing the math, the number that is approximated 1.0594309... , when multiplied by itself twelve times, equals 2. 2 is the most important number in tonal Western music (from @ the year 1600 to 1900), because it is the defined octave, the musical interval on which all other subdivisions, whatever tuning or tempering system is used, is based (pun probably intended). Again, in tonal Western music, this octave has been by definition divided into twelve semitones to the octave, hence the 12th root being the number considered for equally tempered tuning.

For "natural" tuning, the fractions are used: 2/1 is the octave, 3/2 is the fifth, 4/3 is the fourth, 5/4 is the major third, 6/5 is the minor third, etc.

For tempered tuning, including all the variants of "mean tone" and their ilk, these mathematical rational intervals are adjusted to a specific degree of impurity to make more than one key usable, at the expense of others.

For equally tempered tuning, in order to play with the same "key color" in all keys, the math is used to arbitrarily make each semitone equal, which means as a practical matter equally out of tune for "fixed intonation" instruments, which includes keyboards and, arguably, fretted instruments to a degree.

So, all of the "natural" and other "tempered" systems of tuning have their references based on what kind of playing is preferred and are derived from "natural" tuning; "equally tempered" tunings use the mathematical 12th root of 2.

[iiipopes]

The reason tuba players are assigned whole notes is so we have time to work our calculators and adjust our tuners.

Yeah, my other favorite musical hobby is 18th century English pipe organ music, because organs of that era, for the most part, don't have pedals, or at the most an octave or so of "pulldowns" to the lowest notes of the manual (my feet are not that coordinated -- I can't play a trap set of drums, either), and there were ongoing debates about the appropriate tunings, even up into the 1840's and 1850's. I have a 1st edition copy of the Hopkins and Rimbault from that era, hand annotated by another pipe organ enthusiast as myself, about his contemporary impressions of the organs of the day, including some hand-written notes from personal observations on specifications of intermediate rebuilds of some famous instruments that are otherwise now lost to the present day.

BTW -- last week was the 155th anniversary of the birth of Hertz, for whom the unit of measurement of alternating waveforms, whether sound, electronic, etc., was named for his work in defining electromagnetic wave theory. He was a pupil of Hemholz, whom I have also read and have cited as the primary reference for his treatise, "On the Sensation of Tone," for those who wish to study such as this thread discusses, as he discussed them first.