Skip to main content

Recently I visited a diary that had nothing to do with any of my interests or skills (probably a familiar experience for most of us).  In this case, it was the diary title that caught my eye; this diary's title is an hommage.

That diary was about video aspect ratios, and it prompted a bemused comment from me, which was met with puzzlement from the diarist.  This diary is about frequency ratios and their importance in music theory; perhaps it will prompt a few bemused comments from video nerds.

When I started learning music seriously I was a teenager.  I'll turn 54 in a couple of weeks, and I'm still figuring out all this stuff, despite (or perhaps because of) being a professional musician and music teacher for three decades.  Having a 7-year-old daughter is an enormous help.

In high school I took my first music theory class.  The teacher's name was Mr. March, which should have been a clue.  The first day, he said to the class, "I'm going to test your musical ears."  He told us to take out a piece of paper.  Then he said, "I'm going to play two intervals on the piano.  You write down which is bigger, the first or the second."

Then he turned his back to us and pressed some keys on the piano.

I did not have a freakin' clue what was going on.

I did not recognize that he was hitting two keys simultaneously.  What I heard was a series of sounds.  What did he mean by "which one is bigger"?  I'm pretty sure I just gave up on the exam.

Mr. March was operating under some default assumptions that were never stated.  This is not uncommon in teaching, and it's practically a given in music teaching, where teachers are distressingly likely to start where they are, rather than where their students are.

Here's what I tell students who want to learn about music theory.

Musical sound concerns itself with vibration within the frequency range that our ears can perceive.  Vibrations outside that range don't get picked up by our ears, so we won't talk about them.

Some vibrations have periodicity.  Others do not.  An example of the first kind is a tone played on a flute; an example of the second kind is crumpling a sheet of paper.

While musical performance uses both types of sounds, the study of harmonic relationships is only concerned with periodic sounds — the ones with identifiable frequencies, usually measured in cycles-per-second.  Sounds with identifiable frequencies are called tones.  If you take a series of rhythmic impulses and speed them up, they will turn into tones.

If you have two tones with the same frequency, they are in a very specific relationship.  Their numbers match; they are in a 1-to-1 ratio.  The musical term for this relationship is unison.

If you and I sing the exact same note, our vocal chords are vibrating at the exact same speed, and we are singing in unison.  If we're almost but not quite at the exact same speed, the frequency ratio between our voices changes from 1:1 to something more complicated.  189.235147 : 193.772121 is almost the same as 190:190 (which reduces to 1:1) but it's a more complex relationship — and it's perceived by our ears as "out of tune."  Obviously there are a lot more ways to be out of tune than to be in tune!

If you have two tones in the frequency ratio 2:1, their numbers no longer match, but their relationship is still simple.  One vibration moves twice as fast as the other.  The musical term for this relationship (in Western musical tradition) is octave.  

Notice that the term "octave" means "eight," which has absolutely nothing to do with the actual mathematics involved.  

To our ears, the frequency of any power of 2 seems to have the same "quality" as any other.  Notes an octave apart are given the same name in nearly every world musical system that goes so far as to name the notes in the first place.  This means that experientially, 2:1, 4:1, 8:1, 16:1... are all identical 1:1.


Musical intervals can be quantified in various ways.

Keyboard or melodic distance simply measures how far you have to move your finger to get from one member of an interval pair to the other.  From the lowest A on the piano to the highest is a finger distance of about a meter and a half.  From "middle C" to the C-sharp immediately above it is a finger distance of about a centimeter.  By this measure, the first interval is significantly "bigger."

Ratio size just addresses the distance between the two numbers, and it maps nicely onto the melodic distance measure.  From the lowest A to the highest is a ratio of 128:1; from middle C to the adjacent C# is a ratio of 16:15 (n.b., if you know this already, you also know that on the piano, thanks to the baffling miracle of equal temperament, this statement is untrue.  Bear with me for the purposes of discussion, 'k?).  128 to 1 is a bigger jump than 16 to 15, so the first interval is significantly "bigger."

Harmonic distance, on the other hand, measures the complexity of the ratio involved.  From the lowest A to the highest is a ratio of 128:1; from middle C to the adjacent C# is a ratio of 16:15 — but 128:1 reduces to 1:1, and 16:15 doesn't reduce.  An eight-octave jump has a harmonic distance of zero, while a "semitone" has a much greater harmonic distance.  So when we use this measuring system, the second interval is "bigger."


All harmonic intervals can be described as frequency ratios.  Here are some of the ones we use most often:

3:2 is described in Western musical terms as a "fifth."

Notice that the Western term describes the scalar or melodic distance (Do-Re-Mi-Fa-Sol / 1-2-3-4-5), which has nothing to do with the actual mathematics involved.

4:3 is described in Western musical terms as a "fourth."

Notice that the Western term describes the scalar or melodic distance (Do-Re-Mi-Fa / 1-2-3-4), which has nothing to do with the actual mathematics involved.

5:4 is described in Western musical terms as a "Major Third."

Notice that the Western term describes the scalar or melodic distance (Do-Re-Mi / 1-2-3), which has nothing to do with the actual mathematics involved.

5:3 is described in Western musical terms as a "Major Sixth."

Notice that the Western term describes the scalar or melodic distance (Do-Re-Mi-Fa-Sol-La / 1-2-3-4-5-6), which has nothing to do with the actual mathematics involved.


As my little videos demonstrate, rhythmic impulses turn into pitch when you accelerate them.  If you record yourself tapping 2-against-3 for an hour, then accelerate the recording by multiple orders of magnitude, you'll wind up with two tones a fifth apart.


You don't need to know about frequency ratios to use them effectively (just listen to the Beatles and you'll hear some dynamite frequency ratios rendered with exquisite fidelity by people who never gave the math a moment's thought).  Most composers don't know.  Most musicians don't know.

So why bother?

Speaking personally, I can say that learning all this has transformed my experience of music.  I can spend a long time perfecting the tuning of a single interval — precisely because I have learned to perceive it as a source of deep experiential insight into simple mathematical relationships.  Why bother?  Because it's cool; because it's beautiful; because it's universal.

Okay, that's all for today.



Originally posted to WarrenS' Blog on Tue May 08, 2012 at 09:05 AM PDT.

Also republished by An Ear for Music, DKOMA, J Town, Protest Music, and Community Spotlight.



37%50 votes
44%59 votes
6%8 votes
11%15 votes

| 132 votes | Vote | Results

Your Email has been sent.
You must add at least one tag to this diary before publishing it.

Add keywords that describe this diary. Separate multiple keywords with commas.
Tagging tips - Search For Tags - Browse For Tags


More Tagging tips:

A tag is a way to search for this diary. If someone is searching for "Barack Obama," is this a diary they'd be trying to find?

Use a person's full name, without any title. Senator Obama may become President Obama, and Michelle Obama might run for office.

If your diary covers an election or elected official, use election tags, which are generally the state abbreviation followed by the office. CA-01 is the first district House seat. CA-Sen covers both senate races. NY-GOV covers the New York governor's race.

Tags do not compound: that is, "education reform" is a completely different tag from "education". A tag like "reform" alone is probably not meaningful.

Consider if one or more of these tags fits your diary: Civil Rights, Community, Congress, Culture, Economy, Education, Elections, Energy, Environment, Health Care, International, Labor, Law, Media, Meta, National Security, Science, Transportation, or White House. If your diary is specific to a state, consider adding the state (California, Texas, etc). Keep in mind, though, that there are many wonderful and important diaries that don't fit in any of these tags. Don't worry if yours doesn't.

You can add a private note to this diary when hotlisting it:
Are you sure you want to remove this diary from your hotlist?
Are you sure you want to remove your recommendation? You can only recommend a diary once, so you will not be able to re-recommend it afterwards.
Rescue this diary, and add a note:
Are you sure you want to remove this diary from Rescue?
Choose where to republish this diary. The diary will be added to the queue for that group. Publish it from the queue to make it appear.

You must be a member of a group to use this feature.

Add a quick update to your diary without changing the diary itself:
Are you sure you want to remove this diary?
(The diary will be removed from the site and returned to your drafts for further editing.)
(The diary will be removed.)
Are you sure you want to save these changes to the published diary?

Comment Preferences

  •  i was vaguely aware of this (23+ / 0-)

    and loved this diary.  and it does contribute to a progressive utopia when we increase understanding and beauty.  

    i have an intuitive grasp of color theory.  sometimes i have trouble explaining it.  but acquiring the vocabulary of shades, tones, tints, hues, helps me explain to others what i see in my mind.  

  •  Republished to the goup (14+ / 0-)

    from a bright young conservative: “I’m watching my first GOP debate…and WE SOUND LIKE CRAZY PEOPLE!!!!”

    by Catte Nappe on Tue May 08, 2012 at 09:18:07 AM PDT

  •  Wonderful post, Warren. (12+ / 0-)

    I've loved music almost my whole life, took piano for a couple of years when I was a kid (which was a very long time ago; I've got a few years on you), and never knew any of this.  Still spend a very good amount of my time and energy interacting with music across a pretty broad spectrum of styles and genres.

    I love this community.  It is such a source for learning, and sharing, and participating, and growing.  Kinda like contributing to a progressive utopia increasing in understanding and beauty.

    Yeah, that's the ticket.

  •  I studied music for years. (7+ / 0-)

    And never really "got" the music/mathematics connection until adulthood. Thanks for the great post.

    There are, in every age, new errors to be rectified, and new prejudices to be opposed. ~Samuel Johnson (1709-1784)

    by slksfca on Tue May 08, 2012 at 09:26:02 AM PDT

  •  I'm a 6:7 kinda guy. (6+ / 0-)

    Love that crazy 7th.

    These capitalists generally act harmoniously and in concert, to fleece the people... -Abraham Lincoln

    by HugoDog on Tue May 08, 2012 at 09:28:36 AM PDT

  •  A man after my microtonal heart. (15+ / 0-)

    I always snigger when (lay) people say things like "but math is the heart of music!" because, yeah, sure, but not in any way they understand. Hee hee.

    I'm also a composer, and a sound designer, and I've been working with alternate tunings and intonations since going to school with a girl from India who could sing, as I recall, 36 discrete intervals per octave. What my western ear heard as a mistuning, she heard as a proper note. It blew my mind, and got me to start playing with other tonalities and microtunings.

    Great diary, Warren!

    "'club America salutes you' says the girl on the door/we accept all major lies, we love any kind of fraud"--The Cure, "Club America"

    by Wheever on Tue May 08, 2012 at 09:39:16 AM PDT

  •  nearly everything turns into math (10+ / 0-)

    if you study it long enough

    Be ashamed to die until you have won some victory for humanity. Horace Mann (and btw, the bike in kayakbiker is a bicycle)

    by Kayakbiker on Tue May 08, 2012 at 09:47:31 AM PDT

  •  I'll be leaving... (8+ / 0-) a few minutes — I play guitar for a homeschoolers' chorus, and after that I direct a class for homeschooling parents and their kids on the music/math connection.

    I'll catch up with comments again later today!

    Freedom isn't "on the march." Freedom dances.

    by WarrenS on Tue May 08, 2012 at 09:56:42 AM PDT

  •  Intervals Is All We've Got in Celtic Trad (10+ / 0-)

    since there is no accompaniment in the old tradition other than maybe bones or hand drum.

    Thanks for this diary, a great overview of the principles.

    When Celtic trad was first recorded in the wax cylinder and 78 rpm days the bosses would force a piano onto the tradders, something they never worked with, operated by people who only perceived 3-4 chords and whose timing phrasing is done backwards compared to trad.

    One development I really welcomed as trad evolved into Celtic Music was the arrival of jazz-informed guitarists who had the chops to pick up on the often micro-short chord changes and fit heretofore exotic and often much more appropriate chords in between drones (real or implied) and melody.

    But if we're going to talk pitches, we have to hear from the most dramatic pitch machine western or maybe any music ever devised. From the Queen Mother's funeral, "The Mist-Covered Mountains."
    This one piece is an encyclopedia on different ways of handling pitch and melody.

    We are called to speak for the weak, for the voiceless, for victims of our nation and for those it calls enemy.... --ML King "Beyond Vietnam"

    by Gooserock on Tue May 08, 2012 at 10:02:50 AM PDT

    •  The Highland bagpipes are an interesting beast. (1+ / 0-)
      Recommended by:

      They came to the Gaelic-speaking world in the early-mid sixteenth century and immediately were put to work playing harp music.

      Gaelic harp music at the time was dominated by the old Mixolydian scale (with a flattened 7th), using dual-tonic music based on the I and VII. The art music of Gaeldom (ceol mòr) frequently would have a passage consonant with the tonic, paired with a phrase dissonant with it, and then returning to consonance.

      When the pipes began to gradually take over the harp music (a process that concluded with the reprisals after the 1745 rebellion), they based their tuning around this music.

      Because pipe harmony involves drones set to octaves of the tonic, pipes remained in a just tuning, and never switched to equal temperament - so that all notes would be a harmonic of the drones, or vice versa.

      One major difference was the subdominant, which was 3/4 below the pitch of the flattened subtonic, rather than 4/3 above the tonic, leading to a scale that went:

      VII: 8/9
      I: 1/1
      II: 9/8
      III: 5/4
      IV: 27/20
      V: 3/2
      VII: 5/3
      VII: 9/5
      8ve: 2/1

      Notice the bottom VII is anomalous, and isn't quite an octave below the high VII.

      The scale of the instrument also happens to be the major scale relative to the subdominant, so as the old Gaelic art music became increasingly confined to scholars and career competitors, tunes based on the subdominant came to dominate the instrument. Of course, this was a problem with the tonic being noticeably sharp relative to the rest of the scale, so the subdominant eventually dropped from 27/20 to 4/3.

      O Chi, Chi Mi Na Mòr-bheanna (The Mist Covered Mountains) takes advantage of the pipes' ability to also do a relative minor scale on the supertonic, using a ii-I chord progression. Lovely tune, set to several sets of both Gaelic and English words, including a lovely and sad set from Canada about the Clearances, called "Hush, Hush".

      This speaks to Highland pipes only. The Irish Uilleann pipes derive from the pan-British pastoral bagpipes of the mid-18th century, and probably did standard just intonations on the major scale from the start. Notably, the old pastoral pipe had a foot that could often be set to a flattened or sharpened subtonic, but that foot was removed when the pastorals became the Uilleann pipes.

      There were numerous pipes played in the Germanic Lowlands of Scotland from the late 15th century on, but they seem to have always had a major scale, again, just intoned. A close relative of the earliest of these pipes survives in Spain as the Galaician gaita.

      Non enim propter gloriam, diuicias aut honores pugnamus set propter libertatem solummodo quam Nemo bonus nisi simul cum vita amittit. -Declaration of Arbroath

      by Robobagpiper on Wed May 09, 2012 at 03:58:04 AM PDT

      [ Parent ]

    •  And it's not quite true that there's no tradition (1+ / 0-)
      Recommended by:

      of accompaniment in Celtic trad.

      That's certainly largely the case in Irish trad, but Scottish trad has a history of fiddle & cello (fiddle 's bus) for their dance tunes going back to the era of Neil Gow and before. Eventually, the harpsichord & piano took over for the cello; and finally, in smaller dance bands, the accordion took over for both melody and bass line - to the point where fiddlers in dance bands were there mainly for decoration, since they couldn't be heard over the accordion.

      Non enim propter gloriam, diuicias aut honores pugnamus set propter libertatem solummodo quam Nemo bonus nisi simul cum vita amittit. -Declaration of Arbroath

      by Robobagpiper on Wed May 09, 2012 at 04:13:24 AM PDT

      [ Parent ]

      •  Damn you, autocorrect! Don't you know I'm typing (1+ / 0-)
        Recommended by:


        That should be "fiddle & cello (fidhle 's bus)"

        Non enim propter gloriam, diuicias aut honores pugnamus set propter libertatem solummodo quam Nemo bonus nisi simul cum vita amittit. -Declaration of Arbroath

        by Robobagpiper on Wed May 09, 2012 at 04:14:20 AM PDT

        [ Parent ]

      •  That said, the tradition of piano accompaniment (1+ / 0-)
        Recommended by:

        to the Scottish Gaels of Cape Breton, NS is a much later addition - the Gael immigrants that came over from the Highlands and Western Isles were generally in the "melody and no accompaniment" tradition associated with smaller rural communities, and accompaniment developed there in line with the scenario you describe, which also applies to Irish trad.

        Non enim propter gloriam, diuicias aut honores pugnamus set propter libertatem solummodo quam Nemo bonus nisi simul cum vita amittit. -Declaration of Arbroath

        by Robobagpiper on Wed May 09, 2012 at 05:12:18 AM PDT

        [ Parent ]

  •  Next Up: Musical Frequency and Tempered Tuning (10+ / 0-)

    Nicely done post on the theory of musical frequencies.

    But there's theory and then there is practice (and every musician knows that practice is the way to get to Carnigie Hall!).

    In practice, the exact mathematical ratio you describe are not used, but are slightly altered in order to accomadate the perceptions of the listener.

    Tuning a musical instrument to exact frequencies dictated by mathematics results in intervals of fifths and thrids sounding strange and inharmonious when played in different keys.  Through the centuries, various ways of tuning a musical instrument have evolved to accommadate the differences between exact mathematical relationships and the tastes of the listeners.  

    The current practice of "well temperment" or "circular temperment" began in the 15th and 16th century, in part because of the work of JS Bach in promoting this system of tuning.  His famous work "The Well-Tempered Clavier" was written in part to show off how a well-tempered insturment could sound good even when played in all the different keys, a feat that would be impossible on an instrument tuned to strict mathematical frequency ratios.

    "The fool doth think he is wise: the wise man knows himself to be a fool" - W. Shakespeare

    by Hugh Jim Bissell on Tue May 08, 2012 at 10:22:40 AM PDT

    •  Is that why my digital turner is whacked? (3+ / 0-)
      Recommended by:
      raincrow, Larsstephens, WarrenS

      FDR 9-23-33, "If we cannot do this one way, we will do it another way. But do it we will.

      by Roger Fox on Tue May 08, 2012 at 11:13:46 AM PDT

      [ Parent ]

    •  It's not to accomodate the taste of the listeners- (10+ / 0-)

      it's a compromise so that you can modulate from key to key and each key will be equally off from those ratios. There were a number of competing tuning systems around and just previous to Bach's time, all of which were compromises which became necessary because 1. people were used to the "pure" ratios, 2. keyboard instruments such as harpsichord, clavichord and piano became popular, and 3. Western music began incorporating lots of key changes within a single piece, so that intervals that result in simple ratios in one key do not in another.

      It took some time for the 12 equal interval per octave tuning to catch on, because it sounded out of tune to people. Simple ratios sound out of tune to most Westerners now, because we're used to 12 equal. The brain becomes "hard-wired" to hear the tuning you grow up with as in tune and everything else as out, unless you spend enough time with other tunings to break down that wiring.

      Pineapples don't have sleeves.

      by ubertar on Tue May 08, 2012 at 11:51:52 AM PDT

      [ Parent ]

      •  the significance of "2." above is (5+ / 0-)

        that keyboard instruments are of fixed pitch and can't retune on the fly. With fretless stringed instruments such as violins and cellos, or a slide trombone, it's not (as much of) an issue.

        Pineapples don't have sleeves.

        by ubertar on Tue May 08, 2012 at 11:54:15 AM PDT

        [ Parent ]

      •  One advantage of today's digital insturments (7+ / 0-)

        One of the advantages of today's digital insturments is that you can hear these different tuning systems at the touch of a button.

        I play piano and own a digital insturment.  The digital piano has no strings or hammers, but instead creates the piano sound using electronics.   In truth, the sound is not as nice as a well-tuned acoustic insturment, but one feature available to me is I can change the tuning system at the touch of a button.

        With one button press, I can switch from well temperment, to equal temperment, to Pythagorean  tuning, a couple of different Werckmeister tunings, Kirnberger temperment, and so on.

        Some of these are hauntingly beautiful, and give me a broader understanding of how music has evolved to be what we understand as music today

        "The fool doth think he is wise: the wise man knows himself to be a fool" - W. Shakespeare

        by Hugh Jim Bissell on Tue May 08, 2012 at 01:15:43 PM PDT

        [ Parent ]

        •  Some analog instruments are tunable too-- (5+ / 0-)
          Recommended by:
          JayC, raincrow, Larsstephens, WarrenS, madhaus

          I make a variety of electroacoustic instruments that can be tuned to any scale, microtonal or otherwise. Here's an album I did recently using tunings like 13 equal, 15 equal, etc.

          Pineapples don't have sleeves.

          by ubertar on Tue May 08, 2012 at 01:39:04 PM PDT

          [ Parent ]

        •  At some levels, that really matters. (1+ / 0-)
          Recommended by:

          The truth is, though, that few instruments play a note and stay at a single pitch frequency.  They tend to wiggle in a range.  Like the way guitars after you pluck the string always go a little flat.  Pianos too.  So "sort of close" should be good enough for government work in the twelve-tone scale.  There will be some place in there where the two notes are in sync.

          When I'm tuning my guitar, it drives me nuts at times, because the note sounds correct to me when I pluck-pluck-pluck it... But when I play it side by side with the next string, they don't sound right, because they both detune at different rates.  Or maybe that's just my curse for using a cheap guitar.

          •  The reason pianos wiggle in range. (5+ / 0-)

            Many of the notes on a piano are actually created using multiple strings. For instance, higher notes have three strings which are simultaneously struck by the hammer. Most people know this, but one thing that is not well known is that piano tuners will purposefully introduce slight tuning differences among the three strings to give the sound more character!

            •  I suppose that's true. (1+ / 0-)
              Recommended by:

              Our family piano, the one we had when I was growing up and that my brother and I took apart and nearly destroyed many times, in our curiosity, had only one string, I think, but it wasn't a grand piano, just a cheap upright.

              I think the "wiggle" comes from the deformation of the string as it is struck, just as the wiggle of the guitar comes from the deformation of the string as it is plucked and released.  

              Here's an online guitar tuning app.  Click the little orange dots on the bottom to pluck the strings.  Maybe you can hear how the note changes.  I suspect if we drew a graph of the frequency of the note over time, (especially on the lower frequency strings), we'd get a graph that peaked shortly after the pluck, oscillated briefly, then started to become more even but slowly decline.  

        •  digital pianos use samples of real pianos (1+ / 0-)
          Recommended by:

          to create the piano sounds using electronics.

    •  Tuning, & different tunings, can be interesting (7+ / 0-)

      ...especially in terms of how instruments work "on the ground".

      One can see this when comparing keyboard instruments (usually equal temperament, except for certain old-fashioned harpsichords) with strings ("just" intonation) — which also differ in that hand-held strings simply need to be tuned more often.

      Say, for example, you've got a bass guitar & you're checking its tuning. In the standard tuning, EADG, the 5th fret of the low E should equal the open (unfretted) A, right?

      Yet, this particular time, when you play them together you hear "Womwomwomwomwom..." The notes "beat" against each other, meaning that they're out of tune. No worries, just raise (or lower, but usually raise) the pitch or the other a bit... "Waaaaoum-waaaaoum-waaaaoum..." Still not quite there, so turn the peg again until: "Bmmmmmmmmmmm..." ...they might as well be the same string.

      Congratulations, you're on your way to tuning your instrument — with itself, as they say. Typically if you're playing with a keyboard instrument, though, your 1st major step in tuning will likely be to ask for an E (or whatever note you're checking at the time), match that note, then tune everything else by that — which is better than a tuner for a jam session, since that gets you right in tune with the keyboard as well as yourself.

      On the other hand, tuning a piano is a major operation in the life of the instrument...

      Tell Congress: DON'T BREAK THE INTERNET! Fight CISPA! Stop Cyber Spying!

      by Brown Thrasher on Tue May 08, 2012 at 11:58:50 AM PDT

      [ Parent ]

      •  Sounds like your bass needs a setup. :) (5+ / 0-)

        If the intonation is set correctly, you shouldn't have that problem. What you're describing has nothing to do with the difference between just and equal tempered tuning. Getting the intonation exactly right on a fretted stringed instrument can be tricky, though. Most can only be adjusted at the bridge. There was a company a while back that made guitars that could be adjusted at the nut as well for more accurate intonation. There are currently some companies out there that use a fanned-fret system to improve intonation as well. There are also guitar makers who make just-intonated guitars, but that's a whole other thing entirely.
        I make a lot of instruments, including electric guitars with movable frets, for playing all kinds of microtonal scales, which have nothing to do with 12-equal or just intonation.

        Pineapples don't have sleeves.

        by ubertar on Tue May 08, 2012 at 12:08:23 PM PDT

        [ Parent ]

        •  "Just" & "equal" instruments having to co-exist (4+ / 0-)
          Recommended by:
          ubertar, WarrenS, raincrow, Larsstephens

          ...was pretty much the point I was clumsily trying to make.

          Since you mentioned setup, though, what's something that can be done about how an instrument like that reacts to cold weather? Though it's usually a tuning job (they tend to go sharp), the worst case usually involves hiking up the action at the bridge — or just thinking ahead & using a bass with better-behaved wood!

          Tell Congress: DON'T BREAK THE INTERNET! Fight CISPA! Stop Cyber Spying!

          by Brown Thrasher on Tue May 08, 2012 at 01:14:23 PM PDT

          [ Parent ]

          •  If it's a problem of intonation, (5+ / 0-)

            so long as it's an intonatable bridge, you should be able to adjust the distance of each saddle in relation to the nut so the octave lines up properly with the 12th fret. In theory, everything else should then line up as well. In practice, that's not necessarily true and is just the cost of playing a fretted instrument. As I mentioned above, there are people who have designed instruments that solve this problem; on the other hand, some people like the quirkiness of fretted instrument intonation. They say it's part of the charm. I'm a movable fretted guy, so it's a non-issue for me most of the time. I play in stuff like 13 equal or Wendy Carlos' gamma.

            Pineapples don't have sleeves.

            by ubertar on Tue May 08, 2012 at 01:35:38 PM PDT

            [ Parent ]

            •  That's pretty much what I do in practice. (2+ / 0-)
              Recommended by:
              WarrenS, ubertar

              The bridge lets me adjust individual strings, so I can raise a string (& retune) if it buzzes on the frets or anything like that.

              Movable frets? Like a sitar?

              Tell Congress: DON'T BREAK THE INTERNET! Fight CISPA! Stop Cyber Spying!

              by Brown Thrasher on Tue May 08, 2012 at 06:32:07 PM PDT

              [ Parent ]

              •  you don't have to raise the saddle up or down (2+ / 0-)
                Recommended by:
                WarrenS, Brown Thrasher

                to change intonation-- that also changes the action, which you may or may not want to do. You can adjust it forward or backward (closer or further from the nut).

                Movable frets sort of like a sitar. They're my own design. I'm thinking about patenting it. I've used a number of different fret mechanisms over the years and this is the best yet. My main instrument these days is more like a Persian setar than an Indian sitar, but the frets are maybe more sitar-like. Besides being movable, they're also removable. I generally keep to one scale within a piece, so I only use the frets I need. Having no superfluous notes makes concentrating on musical form and composition during an improvisation so much easier.

                Pineapples don't have sleeves.

                by ubertar on Tue May 08, 2012 at 07:34:16 PM PDT

                [ Parent ]

          •  get a different bass (4+ / 0-)

            Fenders tend to need a setup seasonally.  I love the sound, not much I can do about the quirkiness.

            Rickenbackers . . my Ric was last set-up in 1993 or so.  Neck is as straight as a ruler still, and I have taken that bass from Colorado where it was setup to Indiana.

            Spectors are rock-solid too.  I read a guy took his Spector to be plek'd (look up the term, but it is basically a computerized setup rig) and it said "no adjustments necessary."  As the owner of a Spector myself, I can validate that.  I bought mine 12 years ago.  Still perfect.

            •  That's pretty much what I did this past winter. (1+ / 0-)
              Recommended by:

              A friend of mine gave me a bass that seems to be sturdier (not sure what kind of wood or finish), & I used that one rather frequently.

              The way you describe Fenders seems to apply to Ibanezes & Deans too — though personally I don't really find the quirkiness to be a problem, & in most circumstances my usual bass holds tune very well.

              Tell Congress: DON'T BREAK THE INTERNET! Fight CISPA! Stop Cyber Spying!

              by Brown Thrasher on Tue May 08, 2012 at 06:59:05 PM PDT

              [ Parent ]

              •  it is seasonal (0+ / 0-)

                My guitar player just got a Strat to add to his collection, and twice now the weather has put a nice bend in the neck.  One time right before a show.

                He went back to the Les Paul.

                It's the composition of the wood, and the design of the truss rod, I think.  Fender doesn't change those of course because then the sound, feel, look, and everything else about those classic instruments would change, and people would stop buying them.  

        •  On the acoustic bass (classical), (6+ / 0-)

          We're taught to tune by making the octave+5th harmonic on one string match the second octave harmonic on the next lower string.

          •  the difference between a tempered fifth (3+ / 0-)
            Recommended by:
            Larsstephens, Brown Thrasher, WarrenS

            and a "pure" one is two cents, IIRC. Three cents is about as small as we can really perceive the difference of. On a fretless instrument where intonation depends on your fingers and ears, that two cents means even less, if it mattered before at all. Nothing at all wrong with tuning that way.  

            Pineapples don't have sleeves.

            by ubertar on Tue May 08, 2012 at 04:44:50 PM PDT

            [ Parent ]

            •  You are mixing apples and oranges (1+ / 0-)
              Recommended by:

              You're talking about JND (Just Noticable Difference), which is a melodic difference. If the spectra is reasonably rich, we can here harmonic detuning far smaller, as small as 0,1 cents quite easily, and under ideal circumstances, as little as 0,005 cents!

    •  Watch the "Pianomania" documentary... (5+ / 0-)

      ...for a tuner-centric good time :)  It's on Netflix streaming, as is another amazing piece “Note by Note: The Making of Steinway L1037”.

      If you're into more modern old school, there's always

      Stop government funding of religion.

      by here4tehbeer on Tue May 08, 2012 at 01:18:44 PM PDT

      [ Parent ]

    •  Well vs. Equal (4+ / 0-)
      Recommended by:
      WarrenS, raincrow, Larsstephens, ubertar

      It's my understanding that Bach wanted to promote a type of temperament that is not exactly the same as the equal temperament that is now common. You may be on the same page here (or perhaps not!) but I just wanted to clarify that there is a difference between these terms because sometimes you'll see people suggest that Bach was promoting equal temperament when, to my knowledge he wasn't.

    •  Last year I did a diary (5+ / 0-)
      Recommended by:
      Eric Nelson, ubertar, WarrenS, madhaus, IreGyre

      On Bach HERE and I included the following pic of the original front page of Bach's Well-Tempered Clavier.

      People long have wondered exactly what tuning Bach used to make his keyboards "Well-Tempered," because there are many custom tunings to try to get sweeter sounding ratios.  And then somebody realized that ALL THOSE FUNNY CURLICUES at the top of the page are a keyboard tuning guide, one that his students would have recognized and understood.

      Spacy stuff, eh?

  •  I am a former music teacher. Oboe is my (9+ / 0-)

    instrument.  I learned most of this before and in college, but the mathematical ratios were never discussed.

    This does put the math of music in a different and very informative light for me, and I am hotlisting this diary so that I can listen to the musical examples and Gooserock's Youtube later at home (work computer does not allow these).

    I remember one theory teacher had the class sing a scale by half-steps from middle C, and when we were done we were NOT in tune with the piano's octave C.  That exercise gave me at least a clue what "well-tempered" was all about!

    Great evil has been done on earth by people who think they have all the answers. - Lynn W Andrews 1987

    by Spirit of Life on Tue May 08, 2012 at 10:32:40 AM PDT

  •  Great diary! (6+ / 0-)

    Frequency ratios--harmonics--also seem to be at the core of reality itself.

    For example, the brain-wave frequency of the conscious human brain is an harmonic of the frequency of Earth' orbit around the sun. And the brain-wave frequency of a sleeping human brain is an harmonic of the moon's orbit around Earth.

    The notes of the musical scale are also harmonics of the color spectrum.

    Makes ya wonder, doesn't it?

    There are two types of Republicans: millionaires and suckers.

    by Phil T Duck on Tue May 08, 2012 at 11:09:47 AM PDT

  •  Actually that's true of just intonation, (6+ / 0-)

    not of the contemporary standard of 12 equal intervals to the octave, where the ratios are more complex. Recent neuroscience studies show the brain becomes "hard-wired" to the tuning a person is exposed to by a certain age (the wiring isn't quite so "hard" though; you can train yourself to hear other tunings as correct) so the simple ratios of just intonation sound out of tune to most Westerners.
    Once you add in overtones, the relationships between the overtones in one note and another, taking into account the differences in amplitude of each overtone in a particular timbre, things get a lot more complicated. More so when you deal with non-linear timbres (timbres with an overtone series other than the harmonic series). Indonesian gamelan uses mainly non-linear instruments. William Sethares wrote a lot about this. Recent research suggests it doesn't really matter though, in terms of subjective perception of consonance and dissonance. It turns out to be more culturally than physically determined.
    Tuning is fascinating stuff, with a long and sometimes controversial history. Thanks for the diary.

    Pineapples don't have sleeves.

    by ubertar on Tue May 08, 2012 at 11:42:17 AM PDT

    •  So what's this "recent research"? (2+ / 0-)
      Recommended by:
      WarrenS, Larsstephens

      Would like to know, really. Re: subjective perception, yeah it's programmed, from before birth, as the Queen's University of Belfast study shows. BUT, and it's a very big but, once you learn how to REALLY listen to the actual quality of an interval (harmonic spectra with overtones, that is), pure tuning is an unmistakable sound which can never be forgotten, and the probable origins of consonance/dissonance (before all that mucking-about with the affliction known as temperament) becomes obvious. Sort of like riding a bicycle or swimming. Or good wine. Or sex...

      So yeah, we can be taught to stop listening, and those who have been taught to be blind indeed cannot see. Ditto hear...

      •  it's actually from a paper (2+ / 0-)
        Recommended by:
        WarrenS, Larsstephens

        that hasn't been published yet. My business is making and selling hexaphonic guitar pickups (separate output for each string) and one of my customers brought up an interest in microtonal music, and we got talking. It turns out he is a neuroscientist, and is working on the same issues of perception and tuning I've been interested in for years. The results of their studies upend a lot of what's commonly accepted, and I'm pleased to find that it totally supports what I've found through years of working with microtonal music with kids. I'd love to upload the paper for anyone to read, but it's not mine to put up and hasn't been published yet, so I apologize for that.

        That said, I have to disagree with you about the quality of pure tuning. There are a lot of other tunings I prefer. Quite a bit of just intonated music just sounds awful to me. To each his or her own, though. If you like it, more power to you. There's a lot of woo surrounding just intonation, going back to Kepler, Pythagoras and Ptolemy (in the Western tradition). A lot of people get sucked into that stuff, but I think it's as hokey as religion or new-agey nonsense. Again, to each his or her own...

        Pineapples don't have sleeves.

        by ubertar on Tue May 08, 2012 at 01:51:50 PM PDT

        [ Parent ]

      •  do you have a link to the (2+ / 0-)
        Recommended by:
        WarrenS, Larsstephens

        Queens U of Belfast study? I'm curious as to what they're saying is programmed before birth. If it's a specific tuning system, that goes against anything I've read. I recall a mention in Oliver Sacks' book Musicophilia about how there's a critical period when the sense of in or out of tune becomes set (the pathways for "in tune" intervals become optimized). He doesn't really get into questions of tuning though.

        Pineapples don't have sleeves.

        by ubertar on Tue May 08, 2012 at 01:56:04 PM PDT

        [ Parent ]

        •  'ere ya go, mate! (2+ / 0-)
          Recommended by:
          WarrenS, ubertar

          Not about tuning per se, but it shows how the musical indoctrination begins at the prenatal phase. Of course, the size of the intervals is part of the process of cultural programming, which is why western white folk have such poor practical pitch discrimination (not talking about JND) whilst your average Arabic or Indian musician can run circles around them. Western harmony is like English cooking. Unless, of course, you listen to a lot of Partch...

          •  I see. In this case, we're talking about the (1+ / 0-)
            Recommended by:

            results of prenatal experience, not genetics, so this only supports what I said earlier about it being cultural rather than physical. We start absorbing culture before we are even born.

            Pineapples don't have sleeves.

            by ubertar on Wed May 09, 2012 at 05:33:28 AM PDT

            [ Parent ]

    •  In equal-temperament... (7+ / 0-)'s actually incorrect to say the "ratios are more complex."

      E-T isn't done with ratios at all; the elements of tempered intervals are incommensurate.

      But our hearing of 12-tone ET tends to push the tones in the direction of the appropriate Just tuning.  I have heard this demonstrated very convincingly on a piano.

      The demonstrator (W.A. Mathieu) noodled for a minute in the key of C, then began tapping a high A on the keyboard.  With his left hand he played a D minor chord followed by an F major chord (ii - IV in C, a very generic progression).

      As the A 3 octaves above went from being the "Fifth of D" to being the "Third of F," everybody in the room (over 300 people) distinctly heard it drop in pitch.

      It's impossible, of course...but our ears really really really wanted those two different interval values in their respective places (27:16 to 5:3).

      Freedom isn't "on the march." Freedom dances.

      by WarrenS on Tue May 08, 2012 at 02:14:50 PM PDT

      [ Parent ]

      •  When first studying counterpoint... (5+ / 0-)

        ...I remember how surprising it was in fourth species to hear a suspended note also seem to change it's frequency much as you describe above. Very enlightening and cool!

      •  perhaps it's misleading to put it the way I did (2+ / 0-)
        Recommended by:
        Larsstephens, WarrenS

        but it's not incorrect. The intervals in 12 equal aren't derived by calculating ratios, but any interval can be represented as a ratio.

        Interesting story. I wonder how much depends on context-- in this case, the context of Western classical harmony. It would be interesting (though difficult) to set up an experiment to test whether something similar would happen in an alternative system, once the test subjects were sufficiently immersed in the system.

        Pineapples don't have sleeves.

        by ubertar on Tue May 08, 2012 at 04:05:05 PM PDT

        [ Parent ]

        •  Actually, in equal tempering, (3+ / 0-)
          Recommended by:
          WarrenS, Brown Thrasher, madhaus

          there are NO whole number ratios, because the intervals are all based on 12th root of 2, which is irrational.  

          Suppose X is the frequency of Do.  Then to go from Do to Sol, you would multiply the frequency of Do times 2 to the 5/12 power.   Because Do and Sol have a 2/3 ratio, you would expect Sol to be 1.5 times X, but  in equal tempering, you get a weird long-ass number which has no real harmonic ratio.

          HOWEVER, it's close enough for most people's ears, if they aren't absurdly picky about it.  And tuning that way lets you change key without it sounding bizarrely out of tune.

    •  Wow interesting stuff... (6+ / 0-)
      Recent neuroscience studies show the brain becomes "hard-wired" to the tuning a person is exposed to by a certain age (the wiring isn't quite so "hard" though; you can train yourself to hear other tunings as correct)
      As nutso as this will sound: I was born with perfect pitch discovered at age four.  I could even discern at that young age whether a pitch was "off a little" or "off in the middle" - a 4yr olds way of describing perhaps quartertones of being offkey, either sharp of flat.   My first instrument was piano.  Then I started playing clarinet at age 11 and it set up this wild conflict in my head.

      Since clarinet is in Bflat, piano is in C - I started to have to transpose things back to C in my head and go from there to be able to correctly identify what key I was in.  I thought I was losing my pitch but based on what you just shared my head seems to simply have gotten re-wired to the key of Bflat.

      When everybody talkin' all at once no one can hear the wise one speak, So just be still and silence will provide the wisdom that you seek - by Tori del Allen

      by Dumas EagerSeton on Tue May 08, 2012 at 02:28:48 PM PDT

      [ Parent ]

      •  Baroque recorder would drive you nuts, then. (2+ / 0-)
        Recommended by:
        WarrenS, Brown Thrasher

        I've worried before that the weird tuning of the baroque recorder might be messing up my head for other instruments.

      •  the conflict you describe (2+ / 0-)
        Recommended by:
        WarrenS, Brown Thrasher

        is still within the same tuning system. It's more a semantic issue than a tuning perception issue. A visual parallel would be if all the color names were switched to refer to different colors. That would be confusing, but it's different from a scenario in which the range of light frequencies covered by a color name shifted slightly. For example, what had been blue-green is now included within the category of blue, and some frequencies on the other end of blue toward the indigo range were no longer part of blue, and so on. The number of color names might become more or less.
        A musical example would be a scale that divides the octave into 7 equal intervals. This scale and variations of it are used in parts of Southeast Asia and Africa. To people in those cultures, music in 12 equal or just intonation sound out of tune. In your case, notes that were in tune didn't become out of tune-- they just had a different letter name assigned to them.

        Pineapples don't have sleeves.

        by ubertar on Tue May 08, 2012 at 04:14:59 PM PDT

        [ Parent ]

  •  Is that Pythagorean tuning? (1+ / 0-)
    Recommended by:

    I'm inclined to think so by the precise intervals you mention.

  •  When you read music theory (7+ / 0-)

    from before the 18th century, there are pages and pages and pages of ratios.  I could never get into them, because math gives me hives, but you can tell that the people working on at least some of the treatises (Vincenzo Galileo, Marin Mersenne) had a similar experiential relationship to the the reams of abstract pages they produced.

    If religion means a way of life, and life's necessities are food, clothing, and shelter, then we should not separate religion from economics. - Malcolm X

    by dirkster42 on Tue May 08, 2012 at 01:02:25 PM PDT

    •  Galileo... (4+ / 0-)
      Recommended by:
      dirkster42, WarrenS, madhaus, IreGyre

      Came from a family of Italian musical instrument makers, and he was in line to become another one, but for his fascination with mathematics.  Mathematics was never very interesting to the ruling classes of the Renaissance and other times, seeing it as something related to the work of lower level craftsmen.  It shouldn't be surprising, then, that many of history's great mathematicians had musical origins.  Math was THEIR job.

  •  Thank You - N/T (4+ / 0-)

    "Upward, not Northward" - Flatland, by EA Abbott

    by linkage on Tue May 08, 2012 at 01:24:14 PM PDT

  •  Love the diary but couldn't vote in the poll (3+ / 0-)
    Recommended by:
    WarrenS, Larsstephens, Brown Thrasher

    I'm sure your poll questions were just meant in humor, but I wish there had been an option between the extremes of "I knew nothing", "I already knew everything", and "I find this totally irrelevant". Don't mind me, though. I just get grumpy when there's no poll option for me because I have fun voting in these silly polls. Well, in truth I could have voted for "pie" because I almost always want pie, but that wouldn't have really reflected my appreciation for all the work you put into doing this diary.

    For the record, the poll options I could have voted for include:

    - I knew some of this and thank you for expanding my understanding.

    - How much longer until football season starts up again?

    Thanks again for this diary. I'm bookmarking it so I can keep it as a reference.

    If "elitist" just means "not the dumbest motherfucker in the room", I'll be an elitist! - David Rees from "Get Your War On".

    by Oaktown Girl on Tue May 08, 2012 at 01:28:01 PM PDT

  •  Really enjoyed this! (5+ / 0-)

    Thanks for taking the time to put it all together.  Right now I'm a sponge for music theory ... I studied piano and organ from age 5-18, and played trombone in HS band.  Then I just stopped playing.  A lot of it's still in there, it's just covered with 20+ years of rust and dust.  

    So I don't know what came over me, but I just bought myself a bass guitar for my 44th birthday.  DH is a professional musician (keys and sax) and it's always bothered me that for all our shared love of music, I don't have the skills to actually jam with him.  He's an improv master and I'm a play-exactly-whats-on-the-page kinda gal, so this has potential...

    Until I start lessons next week, I've been all over the web and the Guitar Center refreshing my memory on the basics.  This is a welcome addition to my bookmarks, please keep writing!

    "as long as there last name is not obozo, i am voting for them." -- some wingnut blogger

    by SteelerGrrl on Tue May 08, 2012 at 01:57:59 PM PDT

  •  Don't know anything about this stuff, but (3+ / 0-)
    Recommended by:
    WarrenS, raincrow, Larsstephens

    I really do appreciate your letters that you write daily.
    Those I do understand, and I really am glad that you persist.

  •  An analogy to color? (4+ / 0-)
    Recommended by:
    WarrenS, raincrow, Larsstephens, ubertar

    Three color receptors, centered at different wavelengths.

    In language, as cultures add in more names for distinct colors, the split tends to correspond to receptor wavelengths. At first split, color names get centered approximately at a receptor wavelength. At second split, color names get centered approximately between a pair of receptors.

    I wonder if cultural notions of scale and tone work similar. As a near universal: at first split, tones with names correspond to fifths and fourths.

    And that, as with color, these splits approximately correspond to a physical/mathematical/biological basis, rather than mirror the mathematical basis precisely.  

    •  I think there might be a difference (2+ / 0-)
      Recommended by:
      Garrett, WarrenS

      that makes musical ratios more "universal" and less bound to the human condition than colors are.  The receptors in the eye are tuned for different frequency, perhaps, but that doesn't seem to be how the human ear distinguishes what is musically harmonic, which is based on the pure objective mathematical ratio of sound waves.

      An alien species, trying to understand our art, might be able to tell that we had color receptors in this, this, and this range by our choices in colors, ranges that would seem arbitrary and human-specific.  From our music, it might be able to tell what our natural hearing range is, from top to bottom, which is also human-specific.  But the actual harmonic ratios in our music would be something they would understand, even if only on paper, even if they couldn't hear sounds themselves, but could only look at oscilloscope printouts.  That would be universal, something not a happenstance of our being born a certain way.

      •  I think Garrett is on to something (2+ / 0-)
        Recommended by:
        Garrett, WarrenS

        and that harmonic relationships are overblown. If you look at music in cultures that feature non-linear instruments such as metallophones, wood xylophones, etc. in which the overtone series does not line up with the harmonic series, tunings and scales are radically different from harmonic ratios.

        Pineapples don't have sleeves.

        by ubertar on Tue May 08, 2012 at 05:22:07 PM PDT

        [ Parent ]

  •  This former organ tuner knows this well (4+ / 0-)
    Recommended by:
    WarrenS, raincrow, Larsstephens, IreGyre

    In the organ world, pitch is expressed as the length of an ideal pipe, which closely corresponds to the actual length of a real pipe of the diapason family. Half the length = twice the frequency.

    Not counting the pedal organ, which is based at 16', the main diapason "chorus" consists of the following overtones of the 8' foundation tone

    4' Octave
    2 2/3' Quint
    2' Super octave
    1 3/5' Tierce
    1' ... etc.

    Used all together, they produce a bright, sometimes "crunchy" tone. Selective use produces special effects, e.g the Quint & Tierce (together called "Sesquialtera" make a reedy, oboe-like sound.

    Don't let millionaires steal Social Security.
    I said, "Don't let millionaires steal Social Security!"

    by Leo in NJ on Tue May 08, 2012 at 02:29:39 PM PDT

  •  I'm a drummer (8+ / 0-)

    and what is this?

    We already have death panels. They're called insurance companies.

    by aztecraingod on Tue May 08, 2012 at 02:34:36 PM PDT

  •  I'm not sure how much of this will sink in (3+ / 0-)
    Recommended by:
    WarrenS, Larsstephens, Brown Thrasher

    but I certainly enjoyed participating. I've made more than one attempt to learn something about music theory; the results were uniformly depressing. I appear to be constitutionally incapable of grasping the theoretical aspects of it all even though I managed to gain some minimal dexterity as a bass player when I was young. It was all intuitive for me; if I tried to think about it the result was nonsense.

  •  Syncopation is timbre, only much slower. N/T (3+ / 0-)
    Recommended by:
    raincrow, Wheever, WarrenS
  •  So, in answer to your question (4+ / 0-)

    Yes, please. Write more along these lines.

    Join the 48ForEastAfrica Blogathon for the famine in east Africa: Donate to Oxfam America

    by JayC on Tue May 08, 2012 at 03:17:57 PM PDT

  •  notation vs. sound (3+ / 0-)
    Recommended by:
    raincrow, WarrenS, Larsstephens

    Great post!  I gave up music theory as a child when I couldn't read notes fast enough to keep up.  I wish we'd have started with just sounds then, instead of all the confusing stuff.  I wouldn't have waited until my 40s to start again if so :)

    kfractal on soundcloud

  •  Donald Duck explains it all... (4+ / 0-)

    Starting at 2:30 in the clip, which is where the clip starts, if I set it up right.  The whole film is there, about 30 minutes.  I did a diary on this and ratios called The Physics of Music a couple of years ago that embedded the Donald Duck film, but apparently that clip has been since removed.  I remember watching Donald Duck in MathMagicLand during every rainy day session when I was in elementary school.  I loved it.

    Isn't it interesting that the machinery of the human ear is capable of distinguishing harmonic relationships like this, vibrations that are happening so quickly in mathematical ratios?  And not just with simple examples like two notes a fifth apart, but chords of three, four, five notes piled on top of each other.

    Tritones are interesting as well, notes that are between a fourth and a fifth (like Do and the note between Fa and Sol).  They aren't in any ratio.  Their relationship is "irrational" in the mathematical sense of the word, meaning there is no ratio of two whole numbers possible to express it.  It's one of the most dissonant interval of two notes in music, and yet it can be exciting and interesting just because of that.

    •  Just bought this for my grandson (and me)!!! (3+ / 0-)
      Recommended by:
      Dumbo, WarrenS, Larsstephens

      One of greatest short films evar, IMO.

    •  Sorry... (4+ / 0-)

      ...but this:

      Tritones are interesting as well, notes that are between a fourth and a fifth (like Do and the note between Fa and Sol).  They aren't in any ratio.  Their relationship is "irrational" in the mathematical sense of the word, meaning there is no ratio of two whole numbers possible to express it.  It's one of the most dissonant interval of two notes in music, and yet it can be exciting and interesting just because of that.
      is incorrect.

      The tritone's most common appearance is as the major third of a dominant chord on the second scale degree.  D7 in the key of C, functioning as the Dominant of the Dominant.

      That F# is a 5:4 ratio away from the D, which is a 3:2 from G, which is a 3:2 from C.  Two perfect fifths and a major third — ratio 45/32.  A very beautiful interval when sung precisely.

      Freedom isn't "on the march." Freedom dances.

      by WarrenS on Tue May 08, 2012 at 04:48:58 PM PDT

      [ Parent ]

      •  45/32 (2+ / 0-)
        Recommended by:
        WarrenS, Geenius at Wrok

        is still way out there, the largest denominator in the table, I think, which is fine.  Some note has to be the baddest ass in the scale.  Why not F#?  

        Calculated it out just now...  F# (relative to C) is NOT 5/12 or 7/12 but 6/12, so I was off both times.  It's the square root of 2 in equal tempering.  square root of 2 divided by 45/32 gives you an difference of more than one half of one percent, which is still a pretty big difference.

        I did an interesting experiment back in my twenties.  I had an Apple II+ and a thing called the Mountain Computer Music Synthesizer System, a set of boards you could stick in your Apple that gave it composing capabilities, but probably wasn't more sophisticated than the typical sound card chip on motherboards today.  I played around with it quite a bit.  One of the things I made a tape of was of just going up the scale, 2 to the 1/12, over and over again, with all eight voices overlapping.  Not very impressive until it got to the integer overflow point, at which point it turned into weird outerspace music.  

  •  If anyone had told that music was math, (5+ / 0-)

    I'd be unemployed today - and for a good chunk of my life.  Thankfully, they waited until I was totally hooked on music to call my attention to the math involved.  I used to have fun with my students, showing them how science and math are so integral to music.  Of course, most of them could take it farther than I could because I'm a math-phobe.  :-)

    You visual effects are really helpful in illustrating the concepts.  Thanks!

    -7.62, -7.28 "Hold fast to dreams, for if dreams die, life is a broken winged bird that cannot fly." -Langston Hughes

    by luckylizard on Tue May 08, 2012 at 04:25:06 PM PDT

  •  My first partner was a mathematician and a painter (3+ / 0-)
    Recommended by:
    WarrenS, Eric Nelson, Larsstephens

    and said that she loved math because it expressed the kinds of forms and relationships that painting permitted her to put into physical, material manifestation.

    Thanks for this great diary.  I learned a lot!

    That's one more thing to add to my long list of small problems. --my son, age 10

    by concernedamerican on Tue May 08, 2012 at 04:36:15 PM PDT

  •  Equal temperment. a metronome and counting beats.. (3+ / 0-)
    Recommended by:
    WarrenS, Larsstephens, IreGyre

    ..what fun.
    "A well-tempered Clavier" - Bach

    So music is the perfect example of compromise - even between one similar instrument to the next (eg. piano to piano.)

    I can say that learning all this has transformed my experience of music.  I can spend a long time perfecting the tuning of a single interval — precisely because I have learned to perceive it as a source of deep experiential insight into simple mathematical relationships.  Why bother?  Because it's cool; because it's beautiful; because it's universal.
    Yes, after tuning attempting to tune my first piano (and a trip to the library) I realized that all intervals are a perfectly imperfect balancing of compromise. Some a bit flat, some sharp, and even more intiguing is the higher octaves gradually sharpen whereas the bass notes flatten - it's beautiful.

    Thx for the lessons WarrenS

  •  loved the diary (3+ / 0-)

    more please

    My grandmother said that when she got her music degree in college in the 30's it was taught in the math department. is America's Blog of Record

    by WI Deadhead on Tue May 08, 2012 at 05:25:59 PM PDT

  •  we are all the same, then- (1+ / 0-)
    Recommended by:
    Notes an octave apart are given the same name in nearly every world musical system that goes so far as to name the notes in the first place.
    I often wonder about things like that, whether the differences or things that are the same are due to culture or pure physics, or what combination of the two.
  •  THIS IS WONDERFUL!!! (2+ / 0-)
    Recommended by:
    Brown Thrasher, WarrenS

    I have swum in a sea of music, internal and external, all my life, but it's always been heart and gut - not something I could wrap my head around.

    I haven't made it through this...I don't know if I understand what I've read so far, but thank you for it.  Most sincerely.


    by chmood on Tue May 08, 2012 at 07:06:49 PM PDT

  •  Warren thanks for this diary (2+ / 0-)
    Recommended by:
    Brown Thrasher, WarrenS

    I remember you posting in my music diaries oh so long ago.  Glad to see you doing some great ones.

    Best. Tagline. Ever. #WithPerfectHashtag

    by madhaus on Tue May 08, 2012 at 07:29:27 PM PDT

  •  I can approach from both sides (2+ / 0-)
    Recommended by:
    Brown Thrasher, WarrenS

    scientific and musical, and there is a paradox.  Sometimes the most emotionally profound music I create is from a completely mechanical framework; often my emotions will get in the way of competent performance...

    While I don't try to let theoretical constructs guide my music, what I know comes in handy A LOT, and it allows me to take a random melodic thought and develop it quickly and cogently.  It also allows me to mage a good guess at scales to improvise on when I know the chord progression of a new tune.

    But 9 in 10 musicians I've met in the rock world have been ignorant of the intellectual side of most of the time I keep what I know to myself.

  •  next week: (2+ / 0-)
    Recommended by:
    WarrenS, tampaedski

    dodecaphonic blue meanies

    and I wait for them to interrupt my drinking from this broken cup

    by le sequoit on Tue May 08, 2012 at 07:51:29 PM PDT

  •  funny thing, (2+ / 0-)
    Recommended by:
    WarrenS, TrueBlueMajority

    I've played cello, guitar, percussion, clarinet, learned the major minor system inside and out, but I still notate a heard melody with phantom fingerings on the trumpet.

    and I wait for them to interrupt my drinking from this broken cup

    by le sequoit on Tue May 08, 2012 at 08:04:01 PM PDT

  •  I love this stuff, and you did it proud. (1+ / 0-)
    Recommended by:

    Great diary.

    Long ago, in the early 80's a music teacher, good friend (who was his son), personal computers, assembly code, ratios, cycle counts and that piano all came together to punch the tempered scale home!

    ***Be Excellent To One Another***

    by potatohead on Tue May 08, 2012 at 09:40:57 PM PDT

  •  This diary is a good refresher course for me ... (1+ / 0-)
    Recommended by:

    ... as I took a class during my undergraduate years in the early 1970s titled "The Acoustics of Music." It covered much of what you wrote about, WarrenS, and lots more. I earned a solid "A" mostly due to a detailed paper I wrote on the designs of concert venues. It was great to think again about this stuff, since I love music and science. One of the most wonderful aspects of the college course I took was that one of the texts was "A Guide to Electronic Music" -- a two-fer of text and vinyl records released on the Nonesuch label by Paul Beaver and Bernie Krause. I still have my worn copy, with its instruction on pitches, frequencies, tones, etc. I need to fire up my turntable and spend a day reviewing Beaver and Krause's magnum opus. Thanks again for taking the time and effort to write this diary!

    Ah, my friends from the prison, they ask unto me, "how good, how good does it feel to be free? " And I answer them most mysteriously, "are birds free from the chains of the skyway? " (Bob Dylan)

    by JKTownsend on Tue May 08, 2012 at 10:21:08 PM PDT

  •  "Well" temperaments: there is no such thing! (1+ / 0-)
    Recommended by:

    I haven't quite been able to run down who is responsible for this blunder, but this popular modern-day term is a grammatical abomination which has no historical basis. "Wohl" in German is an adverb, just as "Well" in English. If your work is well-done, you have done a good job, not a well job, unless of course you work in the petroleum industry. Thus a keyboard well-tempered has been given a good temperament. Werckmeister called his three rational circulating temperaments "correct" (richtig), but this was only to contrast them 1/4 comma meantone, which he called "incorrect" (unrichtig). Neidhardt called his many experimental systems Quinten Circuls, and Sorge simply called them "good", if and when he called them anything other than just "a temperament". Furthermore, to hang the whole history of this family of temperaments on a single collection of works of Bach, who had nothing whatsoever to do with the move to employ them (primarily it was to address problems of orchestral instruments and organs being tuned at two or three different pitch levels), represents a distortion of the historical reality. Careful scholars will avoid this term, which is why you won't find it in quality musicological writing, like the Grove's article on temperament, which I now edit, having taken over from the master of modern temperament literature, Mark Lindley.

    I have a low-key non-challenging blog:

  •  Pythagoras lives!! (1+ / 0-)
    Recommended by:

    very cool!

    also...a 3:2 ratio also describes the aspect ratio of a normal 35mm film image

    PLEASE donate to a global children's PEACE project: Chalk 4 Peace

    by RumsfeldResign on Wed May 09, 2012 at 06:26:30 AM PDT

  •  Fun Stuff (2+ / 0-)
    Recommended by:
    WarrenS, Seneca Doane

    I teach music theory (I, II, III -- right through chromatic modulation, augmented 6th chords, and extended/altered V chords, stopping at the dissolution of common practice with Tristan) and composition at Wayne State University in Detroit.

    One thing I try to impress on my beginning students is the sloppiness of most of our language when talking about music. In our first class meeting, I'll play a low C and an Eb four octaves to the right on the keyboard, and then ask my students -- who are all music majors, mind you -- which note is "higher."

    Inevitably they will all say the second one.

    At which point I get out my measuring tape, make a bit of a show about measuring from the piano to the floor, and then say, "Nope. They're both the same distance from the ground."

    Once the sloppy language begins to get demolished, we're finally able to have a useful conversation about pitch as frequency, along with the concomitant issues of amplitude, timbre, attack velocity envelopes, and the rest.


    Let us love one another while the light lasts.

    by ProvokingMeaning on Wed May 09, 2012 at 07:11:17 AM PDT

  •  ahhhhh warren dearest ... (2+ / 0-)
    Recommended by:
    WarrenS, Seneca Doane

    i am heading out just now to yoga right now ....  and missed this posting  yesterday .... can you send me an email when you post? Cause you know how much I love anything you have to say.

    but just this point for starters is SO YOU and so what every teacher & SME should preface every teaching experience with

    Musical sound concerns itself with vibration within the frequency range that our ears can perceive.  Vibrations outside that range don't get picked up by our ears, so we won't talk about them.
    I am just starting the study of Qigong and last night watched and interview with Mingtong Gu and Rick Hanson: "A Qigong Mind and A Benevolent Brain" which so profoundly taps into this concept of where we start from in our study of the potentials of the human brain and how potentially infinite the breadth of information we cannot access.

    Returning to read the rest of your post later today.

    I have so missed you ...

  •  This was great - both the subject and the pedagogy (1+ / 0-)
    Recommended by:

    My newly teenage daughter, a talented budding singer, is going to study music theory at home this year.  (Alas, she doesn't get it in school.  Budget cuts, you know.)  So anything you can throw out there I'll want her to read -- and if you have suggestions to others' work, I'll want that as well!

    Pro-Occupy Democratic Candidate for California State Senate, District 29 & Occupy OC Civic Liaison.

    "I love this goddamn country, and we're going to take it back." -- Saul Alinsky

    by Seneca Doane on Wed May 09, 2012 at 11:35:28 AM PDT

Subscribe or Donate to support Daily Kos.

Click here for the mobile view of the site