Miro Quartet at Orcas Island

Michelle and I flew in for the Orcas Island Chamber Music Festival this year and caught the Miro Quartet playing with Aloysia & Jon on Tue Aug 13. Our last-minute decision afforded stage seating, stage right behind the musicians. We really liked this. The experience and sound is different and quite wonderful, reminding me of my own weekly chamber music rehearsals years ago.

Miro opened with the Mozart quartet K 458 The Hunt. Their sound struck me like a velvet hammer: big, round, smooth, rich and fat yet detailed. A huge grin spread across my face and the back of my neck tingled. I especially noticed their dynamics, micro and macro, and their tight timing playing off each other handing the lead back & forth every few bars like a great jazz ensemble, yet with all the musical refinement that Mozart demands. The menuette bounced and the adagio soared, breaking tradition as they came in that order. The allegro set it on fire and summed it up.

Kevin Puts entered the stage and introduced his piece, Arcana for solo cello and string quartet. He described how watching the sun rise over a volcano on Maui inspired him to write this impressionistic piece. Julian Schwarz (son of Gerard Schwarz, prior conductor for Seattle Symphony, who was also visiting the OICMF this year) and Aloysia Friedmann joined Miro to play the lead cello and supporting violin, respectively.

The guest musicians left the stage and Miro played Schubert’s Death and the Maiden. More specifically, the andante which is an absolute classic of the chamber music repertoire and structured as a theme and variations. It ranges widely from lyricism to flaming virtuosity giving each musician a showcase and the Miro quartet just nailed it. The piece had a few moments in the lyrical sections when Ching (lead violin) sounded just slightly off in timing or intonation, but it could have been my own ears.  That’s part of the character and expressive joy of live music performance: every piece is unique rather than perfect in the robotically sterile way that recordings sometimes can be, and this enhances the experience. A robust standing ovation delayed the intermission.

Upon returning, Jon Kimura-Parker was scheduled to play a Clara Schumann piece, but instead played Schubert Impromptu Op. 90 # 3, one of my favorites of the solo piano repertoire. He played with a depth, delicacy and refined power that perfectly suits this piece. The performance reminded me of Radu Lupu’s style, but Jon made it his own. For me, this piece was the highlight of the concert in terms of emotional intimacy.

Last yet certainly not least, Miro joined Kimura-Parker on stage to perform the famous Brahms piano quintet in F minor Op. 34. A few years ago when Michelle and I last attended an OICMF concert they also played this piece, so I knew we were in for a treat. We were not disappointed. We were sitting just behind Kimura-Parker so close we could have reached out and touched him. I was reading his tattered and heavily annotated (in different colors!) sheet music as he played and his daughter turned pages for him. We could hear and feel the power and wonderful woody resonance of the Steinway Model D in the FFF sections. The strings were no less in the game as they brought the piece to its fiery and satisfying conclusion.

Headwinds & Tailwinds

It seems obvious that head and tail winds are equally likely. That is, assuming the direction of the wind and your flight are both random, head and tail winds should be equally likely. But it’s wrong.

Of course, even if head and tail winds were equally likely, you would spend more time flying in headwinds, simply because they slow you down. But that’s not the reason I’m talking about here.

The reason is simple. When the wind is 90* to your direction of flight, you have to turn toward it slightly to maintain your desired direction of flight, so it slows you down. Visualize the entire 360* circle that the angle of the wind can have relative to your direction of flight. If wind at exactly 90* slows you down, then more than half of the range of the circle slows you down. A wind from the side must be slightly behind you in order for the loss of speed turning toward it, to be countered by the gain in speed it adds pushing you along. In other words, when the wind is from the side, it must be slightly behind you to break even.

Of course, the same applies to boats. But not cars, because you don’t need to steer into a crosswind when driving (well you do, but it requires so much less correction as to be insignificant).


Spins can be a contentious topic among some pilots because not all airplanes are approved for intentional spins, the maneuver is no longer required for any pilot certification except becoming a CFI, and the reason the FAA removed the spin requirement was because spin training was causing more deaths than than the training was preventing.

However, I believe spins still make valuable flight training, when approached from a careful perspective. Entering & recovering from spins is quite simple, not requiring the kind of stick and rudder finesse of other maneuvers like chandelles. But spins do something that those maneuvers don’t: they familiarize the pilot with aggressive airplane behavior under unusual attitudes, and reinforce a methodical response rather than panicked reaction.

Review: what is a spin? It’s when an aircraft is stalled, and one wing is more stalled than the other. You can’t spin unless you first stall, and for a stall to become a spin, the angle of attack must be different on each wing. That means your flight must be uncoordinated. Conversely, if you maintain coordination you will never spin, even if you stall the airplane while banked in a turn. Spins are sometimes confused with a tight spiral dive, especially in airplanes like the 172 that are spin resistant. That is, the pilot attempts to spin the airplane, but he fails to overcome the airplane’s spin resistance and the stall becomes a tight spiral. Videos like this are common on youtube (“Hey, watch this spin!” “That’s not a spin, it’s only a spiral dive. Looks like you don’t really know what a spin is.”).

Before spinning an airplane, here are some regs to consider:

  • Aerobatic flight: FAR 91.303
    • Away from congested areas, assembly of people, airways, controlled airspace
    • Above 1500′ AGL
    • At least 3 miles visibility
  • Parachutes: FAR 91.307
    • Never required unless you have passengers (non crewmembers)
    • Bank > 60*, pitch > 30*
    • Exception for maneuvers required for certification, including spins, when given by a CFI
    • For more detail, see PS at the bottom of this page.
  • Airplane limitations: the POH
    • Airplane must be approved for intentional spins
    • The POH may specify W&B necessary for spins: follow it

Further thoughts on parachutes. A chute is not a magic talisman that wards off danger. It doesn’t help you if you can’t get out of the airplane. And it’s difficult nigh impossible to get out of an airplane in flight. Even at slow airspeeds there’s so much air pressure on the doors it’s hard to open them enough to depart the airplane (especially if said airplane is not in straight & level flight, but spinning or otherwise flailing around the sky). That’s why skydiving airplanes have a door entirely removed, or a special door designed to open in flight. Also, a chute doesn’t help you if you don’t know how to use it. Woe betides you if the first time you ever have to use a chute for real, is a bona fide emergency. So if you’re doing aerobatics and wearing a chute, believing you’re both legal and safe, you may be deluding yourself. Ensure you have a way to depart the aircraft while in flight, and do a few jumps to familiarize yourself with the chute.

Here’s how I interpret this flying my 1980 C-172 Superhawk.

  • Spins are approved only in utility class, which means GW < 2100# and CG < 40.5″.
    • With 180# solo pilot, fuel must be 35 gals or less (full fuel is 40 gals).
    • With 2 people up front, any amount of fuel can be used so long as total weight < 2100#. With full fuel, that’s up to 380# in front seats.
  • Empty back seat, baggage, tail; clean airplane with no FOD.
    • Last thing you want is loose items flying around the cabin, possibly obstructing the controls or lodging in the tail cone, moving the CG rearward.
  • If you’re doing spins solo or with a CFI, you don’t need parachutes (and they wouldn’t be much help in a 172, unless you removed a door before flight). If you have a passenger, you do need them.
  • Altitude: at least 6,000 AGL for spins up to 2-turns (higher for more).
  • Spin entry: Cessna 172 L and later models don’t like to spin and will only spiral unless the spin is entered with an aggressive stall. So:
    • Start with a partial power-on stall, so you have a higher nose angle and a crisp breaking stall.
    • Just as the stall breaks (not before), briskly pull the yoke all the way back and stomp full rudder in the direction you want to spin.
      • NOTE: it will spin L easier than R due to engine torque, but you can spin it in either direction.
    • If the airplane goes into a spiral instead of a spin, immediately release the pro-spin inputs, level the wings, climb back to altitude and try again.
      • NOTE: how to tell if you’re in a spin, or just a tight spiral?
      • A spin is a low-G maneuver: the airplane is mostly unloaded and you don’t feel much G force (even though you’re spinning around). If you feel significant Gs, you’re probably in a spiral not a spin.
      • In a spin, the stall horn squeals loud and hard constantly. If the stall horn is silent, or is just barely squealing, you’re probably in a spiral not a spin.
      • In a spin, the airplane rotates quickly: it literally spins. If the airplane is “flying” around in tight circles, you’re in a spiral not a spin.
    • While spinning, hold these pro-spin control inputs at full maximum.
  • Spin recovery: Cessna 172s follow the standard PARE (power, aileron, rudder, elevator) recovery. They are spin-resistant and will recover instantly as soon as you reduce pressure on pro-spin controls. However, best practice is to firmly apply proper anti-spin controls:
    • Throttle to idle (pull all the way back)
    • Ailerons neutral
    • Briskly stomp and hold full opposite rudder
      • If you’re not sure which way you’re spinning, look through the windshield at how the Earth is rotating:
        • If CW, stomp R rudder (you’re spinning to the L)
        • If CCW, stomp L rudder (you’re spinning to the R)
    • Just after the rudder hits the opposite stop, briskly push the yoke forward
    • Hold these anti-spin inputs until rotation stops
      • NOTE: as soon as rotation stops, the spin becomes a steep dive. You must take the next step quickly to avoid over-speeding or over-stressing the airframe.
    • Neutralize rudder and smoothly pull out of the dive
      • If you pull too hard, you may exceed airframe G limits
      • If you pull too gently, you may exceed airframe Vno speed

PS: The regulations may allow more than just a CFI as an exception to parachutes rule. This postscript develops this topic.

FAR 91.307 requires parachutes for aerobatic flight only when there are non-crewmembers on board. Specifically:  no pilot of a civil aircraft carrying any person (other than a crewmember) may execute any intentional maneuver that exceeds…

Here’s how 14 CFR 1.1 defines a crewmember: Crewmember means a person assigned to perform duty in an aircraft during flight time.

Note that 14 CFR 1.1 does not specify what that duty must be. And FAR 91.307 does not say “required crewmember”. By this definition, a crewmember is anyone the PIC (pilot in command) says is a crewmember. As PIC, I can say to the right front seat passenger, “Please look out the window and tell me when you see other airplanes.” If she agrees, she’s been assigned a duty during flight making her a crewmember.

Having done this, we can legally do spins and other aerobatic maneuvers without wearing parachutes because everyone on board the airplane (other than the pilot)  is a crewmember, which complies with the exception in FAR 91.307.

Now, whether this is a good idea, or whether you want to test this interpretation with the FAA, is entirely up to you. I mention it only out of academic interest in studying the regulations.

SRAM Bike Brake Stiff Lever

The brake levers on my MTB have been gradually getting stiffer to operate, more friction in the brake pull with a weaker return upon release. I bought this bike in late 2014 and have bled the brakes and replaced the pads. The lever stiffness has been gradually increasing. On my most recent ride on Tiger Mtn, the brakes were dragging pretty hard because the levers wouldn’t return. This was an incredible PITA on the steep uphills, and risks overheating the brake pads & rotors.

At Tiger summit, one of the other riders mentioned this was a known problem with SRAM hydraulic brake levers. When I got home I checked it out and found that was indeed true. Some people had returned their levers to have SRAM replace under warranty. But they said it was a PITA and took a long time because SRAM support dragged their heels not wanting to admit there was a problem. So I figured it was worth at least trying to fix it myself.

There are several YouTube videos about this. Here is one I found useful: https://www.youtube.com/watch?v=Ex882BIH-Fo

Here’s a summary of the problem and fix. Each brake lever has a small master cylinder inside, a piston with rubber seals. The piston is made of plastic and the cylinder is metal. Inside the master cylinder there is also a spring that pushes back against the piston to help it return to the neutral position. When the entire assembly gets warm/hot, the piston expands more than the cylinder, scuffing against the inside of the cylinder, increasing friction and getting stuck. It gets stuck so hard that the spring can’t push it back.

The solution is to remove the master cylinder piston and use fine (600#) emery paper to scrub off edge material (gently, smoothly, evenly), making it slightly smaller in diameter. To do this you must remove the brake lever from the hose, drain the brake fluid from the lever, disassemble the lever, remove the piston and its rubber seals, sand it down until it freely slides back & forth in the cylinder, clean everything up, reassemble it, then re-bleed the brakes. The procedure is tedious manipulating some tiny parts, and requires an experienced touch sanding down the pistons. But it doesn’t require any special tools, just the usual stuff: torx wrenches, brake bleed syringes, fresh DOT 4 or 5.1 fluid, etc.

The procedure was successful; my brakes are like new again. This took me almost all day, but I hadn’t done it before. I could do it again in less than half a day.

The problem is definitely not about the piston’s rubber seals. I removed those before sanding it, and the piston was super-tight in the cylinder even with the rubber seals removed and the cylinder cleaned. I sanded the piston until it was loose in the cylinder, easily sliding back & forth from gravity just tilting the assembly up and down.

The piston’s rubber seals are tight and one-directionally facing. Remove with care, ensuring you don’t scratch or score them, and ensure they’re facing the right direction when you reinstall. Before reassembling, make sure everything is scrupulously clean. You don’t want sanding dust from the piston or other crud inside your brakes!

I can’t figure out how or why this problem took 4-5 years to manifest. The piston was not deformed in any obviously visible way. Why didn’t this happen during the first year of ownership?

Magnepan/Dipole Speaker Setup

Having owned Magnepan 3.6/R for 20 years and set them up in 3 very different listening rooms, I’ve learned a few things. I want to capture the important things here.



  • Front wall: in front of the listener, behind the speakers.
  • Rear wall: behind the listener, in front of the speakers.
  • SBIR: speaker boundary interference response
    • The total response at the listener position includes sound reflected from the front and side walls near the speaker.
    • This response depends on the distance and angle of the speaker to these walls, and the treatment of those walls.
  • LBIR: listener boundary interference response
    • The total response at the listener position includes sound reflected from the rear and side walls near the listener.
    • This response depends on the distance and angle of the listener to these walls, and the treatment of those walls.
  • Speed of sound: 1130 f/s at sea level and 70*. Slower when cold, faster when warm.

All speakers are sensitive to room setup, but planars are dipoles which are more sensitive than conventional speakers. This is both a blessing and a curse. The blessing: iIf something isn’t right you can often fix it with simple rearrangement. The curse: for ideal sound, the speakers are going to be further into the room away from the walls, than conventional speakers.


All speakers (even forward-firing cones) propagate both forward and back. But a dipole’s back wave has inverted amplitude. This is often called inverted phase or 180* out of phase, which is technically inaccurate; it would have that effect for a steady-state signal like a sin wave. Yet for a constantly changing musical signal, inverted amplitude is different from being 180* out of phase.

Example 1: consider a speaker parallel to the front wall, 3′ away, which is 1/4 wavelength of 94 Hz. The back wave hits the front wall, reflects and as it passes the speaker it has traveled 1/2 wavelength, so it is 180* out of phase with the direct (non-reflected) wave from the speaker. This attenuates 94 Hz. But if the speaker is a dipole, it does the opposite (boosts) because the back wave started out with inverted amplitude, so shifting it 180* out of phase brings it back in-phase.

Conclusion: due to SBIR, dipoles boost the 1/4 wavelength frequency.

Example 2: consider what that same speaker does at 188 Hz (twice the frequency, half the wavelength). Now the 3′ distance is 1/2 wavelength, so the distance traveled is a full wavelength. A conventional speaker will boost this frequency because it’s in phase. A dipole will cut this frequency.

Conclusion: due to SBIR, dipoles cut the 1/2 wavelength frequency.

Direct vs. Reflected

Dipoles have a flatter impedance vs. frequency curve, without the strong Q resonances that conventional speakers have. This makes them a near-resistive load which is easy for amps to drive and gives them flatter phase response and group delay with a big, open, transparent sound. Conventional speakers sound thick and muddy in comparison.

With all speakers, the sound you hear is a mix of direct and reflected. With dipoles this mix has relatively more reflected, less direct. This can make them sound big and phasey in underdamped rooms. With dipoles your room typically needs more damping than it does with conventional speakers.

One way to tackle this is to damp the walls behind the speakers to reduce reflection. How much damping you need depends on the room size, shape, materials, and your personal preference. Too much damping and the dipole will sound thick & muddy like a conventional speaker.

Conclusion: in small to medium sized rooms, you will need to damp the wall behind dipoles to some extent, but not entirely. This damping must be effective down into bass frequencies, so it can’t just be acoustic foam; it must be tube traps, bass traps, etc.


This topic doesn’t at first appear to be unique to dipoles, but it turns out to have an important difference. Consider a listener 3′ in front of the rear wall. Sound from the speakers reflects from the rear wall and comes forward, having traveled 6′ when it reaches the listener again. At 94 Hz, this is half a wavelength, so it attenuates that frequency. At 188 Hz this is a full wavelength, so it boosts that frequency.

What’s different about dipoles: the LBIR and SBIR distances, when equal, negate each other’s effects. With conventional speakers, they exaggerate each other. That is: if the speakers are 3′ from the front wall and the listener is 3′ from the back wall, the dipoles give flat frequency response: SBIR cuts the same frequencies that LBIR boost. Conventional speakers give a double-sized cuts and boosts at the same frequencies.

Conclusion: when setting up dipoles in a small to medium sized rooms, try to make the LBIR and SBIR distances roughly equal.

Planar Speakers

More specifically, why I like planar magnetic speakers (and headphones!).

Sound quality: this one is subjective, yet important. When set up properly, planars sound more natural, open, and transparent than conventional speakers. They’re perfect for acoustic music across all genres from small to large ensemble classical, jazz, vocals, etc. Solo piano, vocals and chamber music are particularly good on planars.

Low distortion: Measuring total distortion in Room EQ Wizard, my  Magnepan 3.6/R measure about -60 dB (0.1%) in the treble, -50 dB (0.3%) in the midrange, and -40 dB (1%) in the bass. That’s lower than most conventional speakers, even lower than most headphones. And it is an uncorrected figure, including the distortion in the microphones, amplifier, and DAC; the actual distortion from the speakers alone is even lower. The Audeze LCD-2 headphones (planar magnetic) have 0.1% total distortion throughout the entire frequency spectrum, even in the bass. No conventional headphone matches that, not even the Sennheiser HD-800.

Why is planar distortion so low? I can think of 2 reasons. First, each Mag 3.6 panel spans the area of about a dozen 12″ woofers, and its ribbon tweeter is 5′ long. The drivers are physically large, so it only takes very small movement/excursion to produce the same sound level. And the distortion that a driver produces is related to its excursion. Second, the drivers don’t have as strong Q resonances as conventional drivers do, both mechanical and electrical.

Linear phase: The 3.6/R have a relatively flat impedance curve: 4.2 ohms in the bass, to 3.3 ohms in the treble. They don’t have the big impedance vs. frequency swings that conventional speakers have. This promotes linear phase and flat group delay.  The 3.6/R measure group delay of a flat zero through most of the frequency range, and only exceeds 10ms in the bass (below 80 Hz).

Easy load: Because planars have relatively flat impedance vs. frequency, they are primarily resistive loads that are easy for amplifiers to drive, despite their lowish impedance.


Planars are dipoles, so they radiate equal energy front and rear, and the rear energy has inverted phase. This makes them more sensitive to room setup than conventional speakers. This can be a blessing or a curse, depending on your situation.

Planars tend to be inefficient, so they require more power for the same listening level. However, their dispersion is line-source (rather than a point-source), so the volume does not drop with distance as quickly as with conventional speakers.

Planars are difficult to measure because near-field, you can’t “hear” all the drivers from a single microphone position. And far-field, what you measure is as much the room as it is the speakers.

Planar drivers are side by side (the panel and the ribbon tweeter). They can’t be aligned vertically like conventional speakers, so the midrange to treble timing and impulse response depends on the angle between the speakers & listener. More specifically, the speakers should be angled so the panels are closer to the listener than the ribbon tweeters.

Planars usually require a big room, and sound best when placed well into the room away from the walls. This leads to a low wife-approval-factor, unless you have a dedicated audio room.

While planars have taut, low distortion bass, they usually don’t reproduce the lowest octave. The larger ones, like the 3.6/R, are good down to about 30 Hz, which is fine for most music. But if you want that room-shaking 20 Hz rumble for movies with explosions and such, you’ll need a subwoofer.

Meier Audio “FF” Frequency Adaptive Feedback

Meier Audio has a feature in their amps called “FF” or Frequency Adaptive Feedback. Jan Meier describes it here. His article is detailed yet long and can be hard to understand exactly what it does, and why. Here I give a simpler explanation. FF is based on 3 key concepts.

If my explanation here makes sense, go back and read Meier’s and you’ll get an even deeper understanding.

Musical Hearing

When it comes to human perception of sound and music, all frequencies are not created equal. The ear is most sensitive to frequencies from around 100 to 2000 Hz. And, most music (at least voices and acoustic music) is concentrated in this range.

Consequently, this is the most critical range for reducing distortion. You might not hear 1% distortion at 30 Hz, but you can definitely hear it at 1000 Hz.

Analogy: Dolby B

Readers with a few grey hairs remember cassette tapes and Dolby B noise reduction from the 1970s and 80s. Dolby B was brilliant in its simplicity. Tape hiss has a wide frequency spectrum but it’s most noticeable in the treble. If you cut the treble during playback, it reduces hiss but it also dulls the music. So when recording, boost the treble. Then during playback, cut the treble by the same amount you boosted it. You get the same hiss reduction without any reduction in treble, because you’re only cutting exactly what you boosted earlier. The music has flat frequency response and sounds cleaner with higher S/N ratio.

Amplifier Feedback

Solid state amplifiers have a negative feedback loop that reduces distortion and increases stability.

What exactly is negative feedback? A portion of the output signal is inverted, attenuated, and fed back into the input. Imagine what happens when you do this. Because it’s inverted, each distortion tone becomes its mirror-image opposite. As this passes through the amplifier, it opposes the distortion tones that the amp produces. The distortion tones oppose and cancel each other.

Frequency Adaptive Feedback

Combine these 3 ideas and you have Meier Audio’s FF. Start with the musical signal.

  • Step 1: boost the critical frequency range (100 Hz to 2000 Hz)
    • Alternately, attenuate frequencies outside this range. This can be a better approach since attenuation means no chance of clipping.
    • This is the first thing you do when the signal enters the amp.
  • Step 2: pass the signal through the normal amp / feedback stage
    • The signal being amplified and in the feedback loop has the critical frequency range exaggerated.
  • Step 3: attenuate the critical frequency range
    • Do the reverse of what you did in step 1.
    • This is the last thing you do before the signal leaves the amp.

In step 2, because the critical frequency range is exaggerated, the feedback loop reduces distortion in this range more effectively.

In step 3, when you attenuate the critical frequency range back to its original level, this has the side effect of attenuating any residual distortion in that range. This improves the S/N ratio in this frequency range.

In summary, FF does to distortion what Dolby B does to tape hiss. It’s based on the same concept.

Incidentally, the Redbook CD specification has something called “emphasis”, which boosts high frequencies (flat up to 1 kHz, increasing to +10 dB @ 20 kHz). CD players are expected to attenuate those frequencies on playback. Similar to FF, this improves the S/N ratio through the mids & treble where the ear is most sensitive.

Musical Energy vs Frequency

The energy in music (and most other sounds) is not evenly spread across frequencies. Most of the energy is in the bass, and energy drops by about 6 dB per octave into higher frequencies. This is true for most music, from chamber music to rock.

However, human hearing is most sensitive in the midrange and treble. Since these are at lower levels than the bass, they’re closer to the noise floor. This means recording gives us the opposite of what we really need. We get high S/N ratio in the bass, where we don’t need it, and we get reduced S/N ratio in mids and treble where we need it most.

Concept: boost the midrange & treble when recording, then cut it on playback. Alternately, cut the bass on recording and boost it on playback. Either of these approaches optimizes the S/N ratio by frequency to better match our perception.


Here we’ll play some devil’s advocate.

If distortion is already below audibility, then FF is a solution looking for a problem – what is the point? In fact, the cure could be worse than the disease! FF requires filters on the input and output to shape the frequency response. These filters cause their own distortions (such as phase distortion from analog filters or minimum phase digital filters). The overall effect is a trade-off between the benefits of FF and the drawbacks of having this extra signal processing.

FF actually increases distortion outside the critical frequency range! With FF you will have higher distortion at the extreme low frequencies (because FF attenuates them in the feedback loop). But you’ll have lower distortion in the midrange and treble. FF shapes distortion to match the sensitivity of our hearing: less distortion where our hearing is most sensitive, at the cost of higher distortion at frequencies where we can’t hear it.

Fractional Octaves

I’ve been working with parametric EQ settings lately; here’s a quick cheat sheet.


We perceive the frequencies of sounds logarithmically. Each doubling of frequency is an octave. Thus, the difference between 40 and 80 Hz sounds the same as the difference between 4000 and 8000 Hz. Even though the latter difference is 10 times greater, it sounds the same to us. This gives a range of audible frequencies between 9 to 10 octaves, which is much wider than the range of frequencies of light that we can see.


Two frequencies 1 octave apart have a frequency ratio of 2:1; one has twice the frequency of the other. A half octave is halfway between them on a logarithmic scale. That is, some ratio R such that f1 * R * R = f2. Since f2 = 2 * f1, R is the square root of 2, or about 1.414. Sanity check: 40 * 1.414 = 56.6, and 56.6 * 1.414 = 80. Thus 56.6 Hz is a half-octave above 40, and a half-octave below 80. Even though 60 Hz is the arithmetic half-way point between 40 and 80 Hz, to our ears 56.6 sounds like the half-way point between them.

More generally, the ratio for the fractional octave 1/N, is 2^(1/N). Above, N=2 so the half-octave ratio is 1.414. If N=3 we have 1/3 octave ratio which is 2^(1/3) = 1.260. Here is a sequence taken to 4 significant figures:

  • 1 octave = 2.000
  • 3/4 octave = 1.682
  • 1/2 octave = 1.414
  • 1/3 octave = 1.260
  • 1/4 octave = 1.189
  • 1/5 octave = 1.149
  • 1/6 octave = 1.122
  • 1/7 octave = 1.104
  • 1/8 octave = 1.091
  • 1/9 octave = 1.080
  • 1/10 octave = 1.072
  • 1/11 octave = 1.065
  • 1/12 octave = 1.059

The last is special because in western music there are 12 notes in an octave. With equal temperament tuning, every note has equally spaced frequency ratios. Thus the frequency ratio between any 2 notes is the 12th root of 2, which is 1.059:1. Every note is about 5.9% higher in frequency than the prior note.

Bandwidth with Q

Another way to express the frequency range or bandwidth of a parametric filter is Q. Narrow filters have big Q values, wide filters have small Q values. A filter 2 octaves wide (1 octave on each side of the center frequency) has Q = 2/3 = 0.667.

For a total bandwidth of N octaves (N/2 on each side of center frequency), the formula is:

Q = sqrt(2^N) / (2^N - 1)

Here are some example values. You can check them by plugging into the formula.

  • N=2, Q=0.667
  • N=1.5, Q=0.920
  • N=1, Q=1.414
  • N=2/3, Q=2.145
  • N=1/2, Q=2.871

Note that these N octave fractions are total width, which is twice the above table which shows octave on each side of the center frequency.


Whatever tool you’re using for this, make sure you know whether it expects total bandwidth around the center frequency, or bandwidth on each side. And make sure you know whether it expects frequency ranges as raw ratios, fractions of an octave, or Q.


Suppose you are analyzing frequency response and see a peak between frequencies f1 and f2. You want to apply a parametric EQ at the center point that tapers to zero by f1 and f2.

First, find the logarithmic midpoint. Compute the ratio f2 / f1 and take its square root to get R. Multiple f1 by R, or divide f2 by R and you’ll have the logarithmic midpoint.

For example if f1 is 600 Hz and f2 is 1700 Hz, the ratio is 2.83:1, so R = sqrt(2.83) = 1.683. Double check our work: 600 * 1.683 = 1010 and 1010 * 1.683 = 1699. Close enough.

So 1,010 Hz is the logarithmic midpoint between 600 and 1700 Hz. We center our frequency here and we want it to taper to zero by 600, and 1700. That range is a ratio of 1.683 on each side, which in the above list is 3/4 octave, or Q=0.920. So now we know the center frequency and width of our parametric EQ.

Room EQ Wizard – A Great Tool!

Today I learned how to use Room EQ Wizard to tune my audio room. I had already done room tuning on my own and was happy with the results. But REW enabled me to get it even better.

Here’s the final FR measured from the listener position, 1/6 octave smoothed. Note this is 2 dB per division. The grey line is before EQ (but with room treatment), the red line is after EQ.

The dark reference line shows a linear 1 db / octave slope. Deviations are +/-3 dB of slope, but for the narrow null at 72 Hz, which resists room treatment and EQ. I’m quite happy with this. I didn’t fix every little bump, but applied a few strategically located bands. The parametric EQ to get here is pretty mild. Each EQ band has amplitude of 4 dB or less, and widths range from 1 to 1/4 octave on each side of the center freq. In other words, gentle corrections and slopes. I’d rather have a few little bumps in the response, than perfectly flat response with bloated phasey sound from extreme EQ settings. Don’t let the cure be worse than the disease!

Overall, this smoothed response throughout the range. During test listening I can switch curves instantly while the music is playing. My ears like the difference, especially noticeable on good acoustic music recordings.

Equipment & Details

  • Test audio files created by REW version 5.2 beta 4, burned to DVD-A
  • Oppo BDP-83 toslink PCM output
  • Behringer DEQ2496 digital EQ, toslink input and output
  • Oppo HA-1 DAC-preamp, toslink input, XLR output
  • Adcom 5800 amp (27 years old!), XLR input
  • Magnepan 3.6/R speakers (18 years old!)
  • Room treatments (floor-ceiling tube traps, RPG acoustic foam, etc.)
  • MiniDSP UMK-1 calibrated measurement mic
  • Recorded from the listener position

Here are the rest of the REW plots:

Total distortion averaged about -50 dB (0.3%); higher in the bass, lower in the treble. That seems surprisingly low, considering it’s measured at the listener position and includes all distortion from the power amp, microphone & recorder. Many headphones, even some tube amps, have more distortion than this.

The bad news is that distortion at 40 Hz is about 10%. Yikes! But it’s down to 1% by 60 Hz, and higher bass distortion is typical of speakers, the exception being planar magnetic headphones.

I’ve always been happy with the bass response in this room. 25 Hz is audible, even if attenuated. But seeing these measurements, I’ll bet that if I got a subwoofer to handle everything below 60 Hz, it might reduce overall distortion. I don’t want more bass, but tighter cleaner bass is always A GOOD THING. I’ll have to look into that!

Initial impulse response is near zero in about 3 milliseconds, and you can see the reflections at 5 and 10 ms.

Total impulse energy is about -40 dB in the first 100 ms, from the listener position which includes room reverb. Room treatment damps the room, but it’s not completely dead. The grey is minimum phase IR, which is very close to the actual response.

Group Delay looks pretty flat too, but for that 70 Hz null. Planar speakers are typically much flatter & cleaner than conventional speakers here. I had to zoom the Y azis to 10 ms per division to see the curve:

The CSD looks linear (no obvious ringing frequencies) and decently fast. The room treatment certainly helps here:

Here’s the Spectrogram, again looks linear, no obvious ringing spots except down at 30 Hz. Even that decays quickly at first, then takes longer after the initial decay. That’s the tube traps at work!

This was a fun day. It’s neat to be able to get some measurements to quantify the sound I’m getting.

Balanced vs. Unbalanced Conversion

Generally speaking, balanced and differential signaling are two different things. They’re often (but not always) used together, and in audio, the term “balanced” refers to this.

Speakers and Headphones

A speaker or headphone responds to the voltage difference between its 2 input wires. It doesn’t assume either is ground, though one might be, it doesn’t matter. So connecting a speaker or headphone to a balanced output is easy. Just wire (-) to (-) and (+) to (+) whether or not the (-) is a ground (unbalanced output) or carries a signal (balanced output). If the unbalanced output has a common ground for both channels (like a headphone), you can split it to both L and R (-) in parallel.

Converting a balanced speaker or headphone output to an unbalanced connector is not as simple. An unbalanced headphone cable (a standard 1/4″ or 1/8″) has 3 wires: L (+), R (+), and a single wire that is a common ground for the L and R. You can’t connect a balanced output’s (-) wires to this ground. That would mix the channels, and allow the amp’s output stages to drive each other, which is bad because they usually have very low output impedance, so it can overdrive the output stages. Also, you can’t just ignore the output’s (-) wires and connect the headphone (-) wires together; this will give a common floating ground. In short, you need a transformer to do this conversion.


If the balanced/unbalanced conversion is between components like a preamp (not a speaker or headphone), it gets more complex because unbalanced components assume the (-) is a ground, but the balanced (-) carries a signal and its ground is a separate (3rd) wire. You can’t connect a balanced output (-) signal to ground; it will overdrive the balanced output as it tries to swing a voltage over a 0 ohm load. Also, you need to ensure the (-) wire has the same impedance to ground as the (+) wire.

So the best way to convert unbalanced to balanced between components is to use a transformer.

However, you can wire unbalanced output directly to balanced input. Connect the unbalanced (-) output to both pins 1 and 3 on the balanced side (negative & ground), and the unbalanced (+) output to pin 2 on the balanced side (positive). That is, carry the unbalanced source ground through to the balanced input. Since unbalanced (consumer) output is at a lower voltage than balanced (pro), the downstream balanced component will be receiving a lower level signal than it expects. This may or may not be a problem, depending on how clean is the input signal and the balanced device’s input voltage sensitivity and gain.