This table contains all known systems, with information on how they worked, and how you can play them today. The stylus modulations in a phonograph record groove are shown, so you can visualize the phase relationships (tape and stereo radio used the same encoding means, but the phase relationships are not as obvious with them). The Poincaré Energy Sphere is used here as a tool, to show the distance (and separation) between two modulations.
In 1970, the page author read in a magazine about the Scheiber system of encoding 4 channels of sound into 2 channels, to be used for records, tapes, and FM stereo broadcasts. The system then recovered 4 channels to send them to speakers. The details had not yet been revealed. He sat down and quickly sketched out three likely possibilities:
HOW STEREO IS RECORDED  

Stylus Modulation  Description  Poincaré Sphere  How To Play  
Vertical and Lateral Recording 
The first phonograph recordings were made with the vertical, or hill and dale method. This caused the groove made by the recording cutter to be modulated vertically. The groove became deeper and shallower as the sound waves modulated it. This causes the playback stylus to vibrate in a vertical direction, as shown by the violet arrow. Of course, the arrow is exaggerated in size  the stylus could not possibly vibrate more than a third of the depth of the groove without causing groove tracking trouble. Later recordings (including most 78 rpm records, and all mono LPs and 45s) were recorded laterally. The stylus vibrates from side to side, as shown by the green arrow. This modulation is also used for sounds centered between the speakers in a stereo recording. The head of each arrow shows the modulation direction on an increase in sound pressure. A decrease in pressure moved the stylus the other way. 
Poincaré Sphere 
Lateral can be played on any stereo phonograph, provided the proper size stylus is used. To play vertical records, either a phase reversal of one stereo channel is needed, or the surround output of a Dolby Surround system can be used.
Capital letters indicate stereo channels. Lowercase letters indicate quadraphonic channels. 

Westrex 4545 system 
Stereo phonograph recordings use stylus motions at 45° angles to horizontal. With a Vshaped groove, the left wall of the groove (as seen from the front of the phonograph) carries the left channel. This is the wall closest to the spindle. The right wall of the groove (the wall closest to the record rim) carries the right channel. For sounds between the speakers, the modulations become shallower, becoming lateral for sounds centered between the speakers. On the Poincaré sphere, the left channel modulation is on the far side of the sphere. The right channel modulation is on the near side. A small circle shows a modulation plotted on the near (right channel) side of the sphere. A larger circle denotes a modulation plotted on the far side (left channel) of the sphere. 
Poincaré Sphere 
Stereo records can be played on any stereo phonograph, provided the proper size stylus is used. A "Three Channel" recording used three mics. The third mic (f) was fed equally to both channels of the record, producing a lateral modulation. This plays on any stereo.


THE POINCARÉ ENERGY SPHERE
AND THE SIX ORTHOGONAL MOTIONS 

Stylus Modulation  Description  Poincaré Sphere  How To Play  
Six Orthogonal Modulations Hexaphonic Tridee modulation letters shown 
The Poincaré Sphere (also called the Stokes Sphere, the Foucault Sphere, and the Fresnel Sphere) was originally conceived by Henri Poincaré in 1892 to describe polarized light, and by Foucault to describe freeswinging pendulum motion. It was adapted by Peter Scheiber in 1970 to describe the phase relationships between the stereo channels of a recording, and was independently discovered by the page author for the same purpose. All modulations are on the surface of the sphere. The six orthogonal modulations are shown as follows:
Modulations 180° apart on the sphere have complete separation. Modulations 90° apart have 3 dB separation. 
Six Orthogonal Poincaré Modulations 
The stereo pickup can play back any modulation on the Poincaré Sphere. Some kind of decoder is then used to recover the wanted signals and route them to the proper speakers.
This image represents the listener's head in speaker location diagrams. 

REGULAR MATRIX: THE SCHEIBER AND SANSUI SYSTEMS  
Stylus Modulation  Description  Poincaré Sphere  How To Play  
Regular Matrix 
In 1969, Peter Scheiber evenly divided up the spaces between the linear orthogonal motions already described, arriving at the modulations shown. No commercial recordings are known to be recorded in it. By 1971, Sansui had independently developed what is essentially the same system, which they called QS. It has the second largest discography of surround material, including one movie. In both cases, speakers are provided for the left front (lf), left back (lb), right front (rf), and right back (rb) signals. The QS system encoder produced the n (nadir  straight down) modulation when equal signals were presented at all inputs. The z (zenith  straight up) modulation is also possible, but is not normally used.

Regular Matrix 
A Dolby Surround system in the Surround mode plays these recordings, as does any QS or RM decoder. Actually, Pro Logic also does a nice job, as do any of the Dynaco or EV systems.
In the BBC trials, the basic QS matrix was Matrix A, and the QS Variomatrix was Matrix C. 

REGULAR MATRIX: THE DYNACO DIAMOND  
Stylus Modulation  Description  Poincaré Sphere  How To Play  
Dynaco Matrix 
In 1970, Dynaco developed a passive matrix decoder, using only the way the speakers are connected to the amplifiers to do the decoding. It uses the four linear orthogonal motions listed above. The Dynaco Diamond is the simplest and cheapest system available. Anybody can build one with nothing but speakers and resistors. The major disadvantage at the time was the location of the speakers on the faces of the listening room walls, instead of in the corners. This caused conflicts in room decor and in positioning speakers so they sound best. Several commercial recordings are known to be recorded in it. The Dynaco Diamond is actually better suited to classical music than most of the other systems. The page author used it in 1971 to produce a set of sound effects for a stage theatrical production.

Dynaco Matrix 
A Dolby Surround system in the Surround mode plays these recordings, as does any Dynaco Diamond, QS, or RM decoder. Pro Logic also does a nice job, as do any of the Dynaquad or EV systems.


REGULAR MATRIX: ELECTROVOICE STEREO4  
Stylus Modulation  Description  Poincaré Sphere  How To Play  
EV Stereo4 Encode EV Stereo4 Decode 
In 1970, Leonard Feldman and Jon Fixler developed what would become the ElectroVoice Stereo4 system. This was the first system to use different sets of coefficients for encoding and decoding. (Most other systems have decode coefficients that mirror the encode coefficients.) It was originally developed as a way to make headphones work with surround sound. Afterward, it was extended to work with speakers. This is a frontoriented system, because it emphasizes front separation and fronttoback separation, at the expense of rear separation. This makes it better suited to classical music than most of the other systems. Many commercial recordings are recorded with it. It has the 5th largest discography. I have normalized the equations, so all of them have the same amplitudes (the original equations did not).

EV Stereo4 Encode EV Stereo4 Decode 
A Dolby Surround system in the Surround mode plays these recordings, as does any ElectroVoice decoder. Pro Logic also does a nice job, as do any of the other Regular Matrix decoders.


REGULAR MATRIX: DYNAQUAD  
Stylus Modulation  Description  Poincaré Sphere  How To Play  
Dynaquad Encode Dynaquad Decode 
In 1971, Dynaco changed the Dynaco Diamond into the Dynaquad system, which has the speakers in the corners of the room. It is still a passive system. This system also uses different sets of coefficients for encoding and decoding. This is a frontoriented system, because it emphasizes front separation and fronttoback separation, at the expense of rear separation. Few recordings were made in this system. It is better suited to classical music than most of the other systems. I have normalized the equations, so all of them have the same amplitudes (the original equations did not).

Dynaquad Encode Dynaquad Decode 
Dynaquad QuadaptorDynaco sold a $20 decoder called a Quadaptor, to be connected between the amplifier and the speakers. A Dolby Surround system in the Surround mode plays these recordings, as does any Dynaquad decoder. Pro Logic also does a nice job, as do any of the other Regular Matrix decoders. Quadaptor equations


PHASE MATRIX: CBS SQ  
Stylus Modulation  Description  Poincaré Sphere  How To Play  
CBS SQ Panpot Encode & Decode 
In 1971, Benjamin Bauer, Columbia Records, CBS, and Sony developed the SQ system after experimenting with systems similar to QS and UMX. The system preserves lefttoright separation, at the expense of fronttoback separation. It is not well suited to ambiance recordings of concert halls. This system is unusual in that the back channels are recorded with circular stylus motions. CBS SQ has the third largest discography. There are two encoders, panpotted and 4corners, each with different modulations for sounds from the sides. There are two decoders. The 1040 decoder is cheap, with no logic. The expensive decoder uses the main modulations with signalsteering logic.

SQ Panpot Encode 
Only an SQ decoder or an EVUniversal decoder can decode this right. Dolby Surround and RM decoders decode all but the rear corners correctly. They place the rear corners in the middle of the room.
The 1040 decoder has a 10% blend between the front channels, and a 40% blend between the back channels. In the BBC trials, the basic SQ matrix was Matrix B, and the SQ Full Logic was Matrix D. 

CBS SQ 1040 Decode 
CBS SQ 4Corner Encode 
SQ 1040 Decode 
CBS SQ 4Corner Encode 

UNIFORM MATRIX: DENON UMX, BMX, and UD4  
Stylus Modulation  Description  Poincaré Sphere  How To Play  
Denon UMX 
In 1972, D. H. Cooper, T. Shiga, and Denon released the UMX (Uniform Matrix) system. It is not compatible with any other matrix system, and has some stereo compatibility problems. It was used in Japan and eastern Asia, but practically nowhere else. This was originally billed as UMX. In 1974, Denon divided it into the 2channel BMX, and the carrier disc UD4, which plays BMX without the decoder. Peter Scheiber offered the same matrix as an alternative, and CBS studied it as one of the "New Orleans" matrix ideas before settling on SQ. I normalized the equations, so all of them have the same amplitudes (the original equations did not).
UD4 carrier contents (limited to 3 KHz bandwidth):

Denon UMX 
Three 3 ways to play this correctly:
In the BBC trials, the basic BMX matrix was Matrix F. 

CARRIER DISCRETE: JVC RCA CD4  
Stylus Modulation  Description  Poincaré Sphere  How To Play  
CD4 Ultrasonic carriers 
In 1971, JVC announced a discrete system for phono records that recorded ultrasonic carriers in record grooves. RCA used it in 1974. It has the following disadvantages:
It has the 4th largest discography, but most extant discs are unusable, due to groove damage. Suggestions were made to matrix discrete discs, using RM, SQ, UMX (UD4), and H (UHJ). None except UD4 and UHJ were ever released.

CD4 Ultrasonic 
Suggestions on playing these:


PHASE MATRIX: ELECTROVOICE UNIVERSAL  
Stylus Modulation  Description  Poincaré Sphere  How To Play  
EV Universal 
In 1973, Electro Voice signed an agreement with Columbia, and produced the EVX44 Universal Decoder. The idea here was to make a decoder that can play the EV, QS, and SQ records mixed on a record changer without having to change decoders. They later suggested the same coefficients as an encoder, but no records were ever produced. The decoder has a frontoriented automatic variable blend that reduces rear separation when center front material is present. It can also be set to be blended all the time, or never blended. The lb and rb dots farthest apart on the sphere are not blended, the ones closest together are blended. The Metrotec decoder is the EV universal decoder without the blend in the "synthesized quad" position.

EV Universal 
Records were never released in this format. The decoder plays:


BBC MATRIX H  
Stylus Modulation  Description  Poincaré Sphere  How To Play  
BBC Matrix H 
In 1974, the BBC started investigating matrix broadcasting. They tried several matrices, which they labeled with letters, and settled on matrix H. One requirement was that it have full mono compatibility, which it does. But the matrix does not match any other matrix. Few recordings are "officially" released in Matrix H, but the page author has some that behave as though they were recorded in matrix H, and several more that behave as though they were recorded in matrix H with the rotations reversed (named HR by the page author) A small unofficial discography exists in H and HR, and BHJ and UHJ (see below) are also derived from H.
For HR, reverse the signs of the j terms, reverse the rotations of the stylus motions, and flip the Poincaré Sphere image over vertically. The upper left Poincaré Sphere positions for the left and right pair are panpot positions; the lower right ones are 4corners encoding. 
BBC Matrix H 
Without an H decoder, the best way is to combine an RM and an SQ decoder with six speakers, placed as follows:
Reverse the phase of the SQ lb speaker. For HR, trade the SQ speakers. An RM decoder works, but with less separation.


UNIQUAD EQUAL SEPARATION  
Stylus Modulation  Description  Poincaré Sphere  How To Play  
UniQuad ES 
This was mentioned by Peter Scheiber, but never used. The page author built an encoder and a decoder, but never produced any recordings with it. By the time it was operational, Dolby Surround had become the standard matrix. The idea was to design a matrix with equal separations between all channels.

UniQuad ES 
Only a few experimental recordings ever existed in this format. They no longer exist.


REGULAR MATRIX: DOLBY SURROUND  
Stylus Modulation  Description  Poincaré Sphere  How To Play  
Dolby Surround 4.1 
This was originally developed to make "Star Wars" under the name Dolby Stereo. It quickly became the preferred surround system for movies. Home movies were encoded in it, as were many phonograph records, cassette tapes, and CDs. Most soundtrack recordings are encoded in Dolby Surround because the movies they came from were so encoded. There are more recordings in this matrix than any other. One difference from other systems is that the decoder adds a delay in the back channel before it goes to the speakers, and that back speakers (called surround speakers) are placed on both sides. This removes the problem of localizing the side images that other 4corners matrices have. Also, applying identical signals to all of the inputs produces the nadir (n) modulation.

Dolby Surround 4.1 
New Dolby Surround decoders are widely available. Other regular matrix decoders do an acceptable job.


CIRCLESURROUND  
Stylus Modulation  Description  Poincaré Sphere  How To Play  
CircleSurround 
This was patented and announced, but it makes no sense. This was intended as an alternate way to encode for Dolby Surround, but it looks like nothing but a way to get around patents. It does have a resemblance to the Stereo 180 miking system's output. But there are some discontinuities in the encoding patterns as a sound is panned all the way around the listener.

CircleSurround 
A small discography exists in this format. It is intended to be decoded with Dolby Surround. SQ can be used with the lb and rb speakers traded.


DENON BHJ AND UHJ  
Stylus Modulation  Description  Poincaré Sphere  How To Play  
Denon BHJ and UHJ 
This is a modification of the matrix H, used by the Ambisonic system. BHJ is the 2channel matrix. UHJ is a 3 or 4 channel version for DVD that increases separation. The 4channel version adds a height dimension. For the Dolby Surround channel positions, the BHJ matrix is:

BHJ and UHJ 
A small discography exists. Various decoder configurations exist. The following work:


HEXAPHONIC TRIDEE  
Stylus Modulation  Description  Poincaré Sphere  How To Play  
Hexaphonic Tridee 
This was a modification to Dolby Surround used for a few movies. It adds a third dimension of height, using the zenith (z) and nadir (n) modulations. It is also encoded into the soundtrack albums of those movies. The name is the page author's suggestion, because no mention of it by name is known. Some versions trade the n and z modulations.

Hexaphonic Tridee 
A few movies and records were released in this format, but not labeled. It is decoded with Dolby Surround and an SQ decoder. Use Dolby Surround normally, SQ lb for n, and SQ rb for z.


TETRAPHONIC 3D  
Stylus Modulation  Description  Poincaré Sphere  How To Play  
Tetraphonic 3D 
This was suggested in several articles, but never used. The page author built a decoder and used it to play the Hexaphonic Tridee system with 4 speakers. It is basically the UniQuad ES system with the speakers rearranged (the rb speaker is hung from the ceiling in the back (zb), and the lb speaker is placed on the floor (nb). So the only difference is the speaker designation. Some versions trade the nb and zb modulations. To save download bandwidth, the UniQuad ES illustrations are used.

Tetraphonic 3D 
A few records were released in this format. The SQ 1040 or the EV Universal can be used with the same speaker rearrangement. It plays the Hexaphonic Tridee system.


COMPATIQUAD  
Stylus Modulation  Description  Poincaré Sphere  How To Play  
Compatiquad 
This was not an encoding matrix. It was developed so that QS, EV, and SQ quadraphonic records could be intermixed on a record changer. It is a better match for all three matrix systems than that of the ElectroVoice Universal decoder. Since there is no encoder, no encoding equations are shown. 
Compatiquad 
No recordings exist in this format, because it is not a recording format.


OTHER SYSTEMS  
Stylus Modulation  Description  Poincaré Sphere  How To Play  
Dynaco Matrix vectors used in Studio 4
Dynaquad vectors used in HP1 
Utah Studio4This was not an encoding matrix. It was a device to fake a surround sound experience. It did not do a very good job. It was essentially the same as the Dynaco Diamond, except that the speakers were placed in different places. This provided a strangely folded listening field. It surrounded the listener with sound, but produced no useful instrument localization. Since there is no encoder, no encoding equations are shown. By moving the speakers, the owner of a Studio4 could use it as a Dynaco Diamond decoder. The following passive decoder variants circuits were made by the page author. All of them required only two channels of amplification: UniQuad HP1HP1 was made from wire and 8 RCA jacks, for 6 speakers from compact stereos. It provided the Dynaquad arrangement, plus center front and center back speakers, for hexaphonic sound. UniQuad UQ1UQ1 had a variable width blend control on the front speakers, variable depth control on the back speakers, AutoVary separation enhancement, and a disco bass booster. UniQuad UQ44UQ44 is the only passive decoder ever made that decodes all of the regular and phase matrices, including SQ. It has switch positions for preset and adjustable matrices, AutoVary separation enhancement, a disco bass booster switch, and a headphone surround circuit. The disadvantage was the box larger than one foot high, wide, and deep needed to house the parts. Quadruplex, Composer, Quadrasizer, Etc.These were variants on the Dynaquad Quadaptor. They had passive decoder circuits that were just enough different that they didn't infringe on any patents. They decoded any Regular Matrix recordings. Reverb SystemsSeveral systems were sold to send synthetic reverberation to the back speakers. These can't decode encoded recordings. 
Dynaco Matrix vectors Dynaquad vectors 
Utah Studio4
Speaker placement of Dynaco Diamond signals in Studio 4 arrangement: l r
UniQuad HP1
Speaker placement of HP1: lf f rf
UniQuad UQ44Switch positions on UQ44


PHASE LOCATION (experimental)  
Stylus Modulation  Description  Poincaré Sphere  How To Play  
Phase Location 
This was proposed by Denon, but never used beyond a few experiments. The idea was that the two stereo channels always have the same amplitude, but different phase. The phase angle is equal to the direction the sound comes from.

Phase Location 
All channels can be derived using an SQ decoder and resistor summing networks. Reverse the phase of the left speaker:
Decoding equations:


SURROUND FIELD (new)  
Stylus Modulation  Description  Poincaré Sphere  How To Play  
Surround Field 
This was discovered experimentally by the page author, but not yet used commercially. The idea is that the information needed for the listener's ears to find the correct direction for each sound must be in the recording. Because of this, no separation enhancement is needed for this matrix system.
A version exists with 3D (height) information. See Surround Field. 
Surround Field 
Use a nonlogic Dolby Surround decoder, the surround mode of a Dolby Surround decoder, or any Regular Matrix decoder:
An octophonic arrangement shown on the Surround Field page works quite well. For 3D decoding, use the Hexaphonic Tridee or the Tetraphonic 3D decoding methods. 
THE BBC MATRIX TRIALS  

The BBC ran a series of trial experiments using several matrix systems. They were lettered from A to H as follows:


BBC Matrix E  
Stylus Modulation  Description  Poincaré Sphere  How To Play 
BBC Matrix E 
This was the other tetrahedral matrix proposed by Peter Scheiber. Other than for these trials, it was never used. It made the stereo image sound very "phasey".

BBC Matrix E 
The matrix decoding equations of all of the matrices in the BBC trials are the complex conjugates of the encoding equations. 
BBC Matrix G  
Stylus Modulation  Description  Poincaré Sphere  How To Play 
BBC Matrix G 
This is a modification of Matrix F (UMX) intended to get rid of the phasey play in stereo and mono. It was never used for released recordings.

BBC Matrix G 
The matrix decoding equations of all of the matrices in the BBC trials are the complex conjugates of the encoding equations. 
BBC Matrix GX (G Modified)  
Stylus Modulation  Description  Poincaré Sphere  How To Play 
BBC Matrix GX 
This is a modification of Matrix G, intended to improve separation. Matrix H is based on Matrix G. It was never used for actual released recordings

BBC Matrix GX 
The matrix decoding equations of all of the matrices in the BBC trials are the complex conjugates of the encoding equations. 
BBC Matrix HX (experimental H)  
Stylus Modulation  Description  Poincaré Sphere  How To Play 
BBC Matrix HX 
This is a modification of Matrix G, intended to improve play in stereo and mono. It is an early version of Matrix H. It was never used for released recordings.

BBC Matrix HX 
The matrix decoding equations of all of the matrices in the BBC trials are the complex conjugates of the encoding equations. 
QUADRAPHONIC AND SURROUND INDEX PAGE
UniQuad UQ1A Quadraphonic Decoder
Encode surround sound with your mixer