📄 readme.mts
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gg channel byte 2 - bits 0 to 6 = channel 8 to 14hh channel byte 3 - bits 0 to 6 = channel 1 to 7tt temperament type (1 byte) 00H means equal temperament 01H means Pythagoras tuning 02H means mean-tone tuning 03H means pure intonation 40H means user-defined temperament #0 41H means user-defined temperament #1 42H means user-defined temperament #2 43H means user-defined temperament #3F7 EOX----------------------------------------------------------------------(18) User-defined Temperament Entry (Non Real-Time)----------------------------------------------------------------------F0 7E <device ID> 08 0C tt <temper name> ll [fh fl bh bl aa bb cc dd ee ff] ... F7F0 7E Universal Non Real Time SysEx header<device ID> ID of target device (7F = all devices)08 sub-ID#1 = "MIDI Tuning Standard"0C sub-ID#2 = "user-defined temperament entry (Non Real-Time)"tt temperament program number (0 - 63)<temper name> 16 ASCII charactersll number of formula (1 formula = 1 set of [fh fl bh bl aa bb cc dd ee ff])[fh applying pitch bit mask byte 1 bits 0 to 3 = circle of fifth forward 8 to 11 bits 4 to 5 = reserved for future expansion bit 6 = major flag (reversal) fl applying pitch bit mask byte 2 bits 0 to 6 = circle of fifth forward 1 to 7 bh applying pitch bit mask byte 3 bits 0 to 3 = circle of fifth backward 8 to 11 bits 4 to 5 = reserved for future expansion bit 6 = minor flag (reversal) bl applying pitch bit mask byte 4 bits 0 to 6 = circle of fifth backward 1 to 7 aa bb fraction (aa/bb) cc dd ee ff] power ((cc/dd)^(ee/ff)) (repeated 'll' number of times)F7 EOX----------------------------------------------------------------------======================================================================The major/minor in the temperament tonality======================================================================The basic chords used in general music in C major are not only C, G, Fbut also Am, Em, Dm which appear frequently. There may also be Cm,Gm, Fm, A, E, D, and so on. Since these chords are not supported onlyin pure intonation (C major), players need to change temperamentsaccording to progress of music.To solve the issue, TiMidity++ prepares(1) pure intonation (C major) based on the pitch of C in Pythagoras tuning (C major)(2) pure intonation (A minor) based on the pitch of A in Pythagoras tuning (A minor)(3) pure intonation (passing C major) based on the pitch of A in Pythagoras tuning (C major)(4) pure intonation (passing A minor) based on the pitch of C in Pythagoras tuning (A minor)I will explain more precisely. The following table gives the lattice(Cartesian model) of the scale system:-----------------------------------------------------------------------------D-- A-- E-- B-- F#-- C#-- G#-- D#-- A#-- E#-- B#-- F##-- C##--Bb- F- C- G- D- A- E- B- F#- C#- G#- D#- A#- Gb Db Ab Eb Bb F C G D A E B F# Ebb+ Bbb+ Fb+ Cb+ Gb+ Db+ Ab+ Eb+ Bb+ F+ C+ G+ D+ Cbb++ Gbb++ Dbb++ Abb++ Ebb++ Bbb++ Fb++ Cb++ Gb++ Db++ Ab++ Eb++ Bb++ -----------------------------------------------------------------------------The notation "ABCDEFG" is according to Pythagoras tuning. Thenotation "+", "-", "++" and "--" mean 1sc higher, 1sc lower, 2schigher and 2sc lower respectively.A certain pure intonation is given as 12 sounds arranged by therectangle of 4x3 from the lattice. For example, C tuning, A tuning,A- tuning and C+ tuning are given as following tables respectively:[C tuning (C major)]----------------------A- E- B- F#- F C G D Db+ Ab+ Eb+ Bb+ ----------------------[A tuning (A minor)]----------------------F#- C#- G#- D#- D A E B Bb+ F+ C+ G+ ----------------------[C+ tuning (passing C major)]----------------------A E B F# F+ C+ G+ D+ Db++ Ab++ Eb++ Bb++----------------------[A- tuning (passing A minor)]----------------------F#-- C#-- G#-- D#--D- A- E- B- Bb F C G ----------------------I think it is nice to select the tuning combination whose pitch ofparallel key is slightly lower for major music, and slightly higherfor minor music.======================================================================Preset temperament of Temperament Type Control Tuning======================================================================First, Pythagoras tuning (major) chromatic scale is expressed by thefollowing recurrence relations. Here, the index [] is a offset of thetonic. The operation results are surely settled between 1 and 2, sothey will be made into half or double if necessary.[Pythagoras tuning (major) chromatic scale]pytha_maj[ 0] = 1 # C 1pytha_maj[ 7] = pytha_maj[ 0] * 3/2 # G 3/2pytha_maj[ 2] = pytha_maj[ 7] * 3/2 # D 9/8pytha_maj[ 9] = pytha_maj[ 2] * 3/2 # A 27/16pytha_maj[ 4] = pytha_maj[ 9] * 3/2 # E 81/64pytha_maj[11] = pytha_maj[ 4] * 3/2 # B 243/128pytha_maj[ 6] = pytha_maj[11] * 3/2 # F# 729/512--pytha_maj[ 5] = pytha_maj[ 0] * 2/3 # F 4/3pytha_maj[10] = pytha_maj[ 5] * 2/3 # Bb 16/9pytha_maj[ 3] = pytha_maj[10] * 2/3 # Eb 32/27pytha_maj[ 8] = pytha_maj[ 3] * 2/3 # Ab 128/81pytha_maj[ 1] = pytha_maj[ 8] * 2/3 # Db 256/243On the other hand, pure intonation (major) chromatic scale can beexpressed by the following recurrence relations. Here, sc means asyntonic comma (81/80).[pure intonation (major) chromatic scale]pure_maj[ 0] = 1 # C 1pure_maj[ 7] = pure_maj[ 0] * 3/2 # G 3/2pure_maj[ 2] = pure_maj[ 7] * 3/2 # D 9/8pure_maj[ 9] = pure_maj[ 2] * 3/2 / sc # A 5/3pure_maj[ 4] = pure_maj[ 9] * 3/2 # E 5/4pure_maj[11] = pure_maj[ 4] * 3/2 # B 15/8pure_maj[ 6] = pure_maj[11] * 3/2 # F# 45/32--pure_maj[ 5] = pure_maj[ 0] * 2/3 # F 4/3pure_maj[10] = pure_maj[ 5] * 2/3 * sc # Bb 9/5pure_maj[ 3] = pure_maj[10] * 2/3 # Eb 6/5pure_maj[ 8] = pure_maj[ 3] * 2/3 # Ab 8/5pure_maj[ 1] = pure_maj[ 8] * 2/3 # Db 16/15It can be understood that pure intonation is similar to Pythagorastuning fundamentally except descending with 1sc at A and rising with1sc at B flat while going up and down respectively from the tonic inthe circle of fifths.Similarly, expressed Pythagoras tuning and pure intonation (minor)chromatic scale by the following recurrence relations. Although thefractions written to right-hand side is terrible values, therecurrence relations themselves are very simple.[Pythagoras tuning (minor) chromatic scale]pytha_min[ 0] = 1 # C 1pytha_min[ 7] = pytha_min[ 0] * 3/2 # G 3/2pytha_min[ 2] = pytha_min[ 7] * 3/2 # D 9/8pytha_min[ 9] = pytha_min[ 2] * 3/2 # A 27/16pytha_min[ 4] = pytha_min[ 9] * 3/2 # E 81/64pytha_min[11] = pytha_min[ 4] * 3/2 # B 243/128pytha_min[ 6] = pytha_min[11] * 3/2 # F# 729/512pytha_min[ 1] = pytha_min[ 6] * 3/2 # C# 2187/2048pytha_min[ 8] = pytha_min[ 1] * 3/2 # G# 6561/4096pytha_min[ 3] = pytha_min[ 8] * 3/2 # D# 19683/16384--pytha_min[ 5] = pytha_min[ 0] * 2/3 # F 4/3pytha_min[10] = pytha_min[ 5] * 2/3 # Bb 16/9[pure intonation (minor) chromatic scale]pure_min[ 0] = 1 * sc # C 1 * scpure_min[ 7] = pure_min[ 0] * 3/2 # G 3/2 * scpure_min[ 2] = pure_min[ 7] * 3/2 / sc # D 10/9 * scpure_min[ 9] = pure_min[ 2] * 3/2 # A 5/3 * scpure_min[ 4] = pure_min[ 9] * 3/2 # E 5/4 * scpure_min[11] = pure_min[ 4] * 3/2 # B 15/8 * scpure_min[ 6] = pure_min[11] * 3/2 / sc # F# 25/18 * scpure_min[ 1] = pure_min[ 6] * 3/2 # C# 25/24 * scpure_min[ 8] = pure_min[ 1] * 3/2 # G# 25/16 * scpure_min[ 3] = pure_min[ 8] * 3/2 # D# 75/64 * sc--pure_min[ 5] = pure_min[ 0] * 2/3 # F 4/3 * scpure_min[10] = pure_min[ 5] * 2/3 # Bb 16/9 * scThe differences from the major tuning are that the boundary ofPythagoras tuning goes up three positions, that the positions ofdescending with syntonic comma are changed, and that pure intonationis adjusted 1sc higher so that melodic parts' tonic (Pythagorastuning) and harmonic parts' tonic (pure intonation) are overlapped.By the way, mean-tone tuning is also prepared besides Pythagorastuning and pure intonation as preset temperament of TiMidity++. Whilemean-tone tuning (major) is based on the general one whose majorthirds are pure, mean-tone tuning (minor) is based on Salinas tuningwhose minor thirds are pure. Both mean-tone tuning (major) chromaticscale and mean-tone tuning (minor) chromatic scale can be expressed bythe following recurrence relations.[mean-tone tuning (major) chromatic scale]mt_maj[ 0] = 1 # C 1mt_maj[ 7] = mt_maj[ 0] * 5^(1/4) # G 5^(1/4)mt_maj[ 2] = mt_maj[ 7] * 5^(1/4) # D 5^(1/2) / 2mt_maj[ 9] = mt_maj[ 2] * 5^(1/4) # A 5^(3/4) / 2mt_maj[ 4] = mt_maj[ 9] * 5^(1/4) # E 5/4mt_maj[11] = mt_maj[ 4] * 5^(1/4) # B 5^(5/4) / 4mt_maj[ 6] = mt_maj[11] * 5^(1/4) # F# 5^(3/2) / 8--mt_maj[ 5] = mt_maj[ 0] / 5^(1/4) # F 2 / 5^(1/4)mt_maj[10] = mt_maj[ 5] / 5^(1/4) # Bb 4 / 5^(1/2)mt_maj[ 3] = mt_maj[10] / 5^(1/4) # Eb 4 / 5^(3/4)mt_maj[ 8] = mt_maj[ 3] / 5^(1/4) # Ab 8/5mt_maj[ 1] = mt_maj[ 8] / 5^(1/4) # Db 8 / 5^(5/4)[mean-tone tuning (minor) chromatic scale]mt_min[ 0] = 1 * sc # C 1 * scmt_min[ 7] = mt_min[ 0] * (10/3)^(1/3) # G (10/3)^(1/3) * scmt_min[ 2] = mt_min[ 7] * (10/3)^(1/3) # D (10/3)^(2/3) / 2 * scmt_min[ 9] = mt_min[ 2] * (10/3)^(1/3) # A 5/3 * scmt_min[ 4] = mt_min[ 9] * (10/3)^(1/3) # E (10/3)^(4/3) / 4 * scmt_min[11] = mt_min[ 4] * (10/3)^(1/3) # B (10/3)^(5/3) / 4 * scmt_min[ 6] = mt_min[11] * (10/3)^(1/3) # F# 25/18 * scmt_min[ 1] = mt_min[ 6] * (10/3)^(1/3) # C# (10/3)^(7/3) / 16 * scmt_min[ 8] = mt_min[ 1] * (10/3)^(1/3) # G# (10/3)^(8/3) / 16 * scmt_min[ 3] = mt_min[ 8] * (10/3)^(1/3) # D# 125/108 * sc--mt_min[ 5] = mt_min[ 0] / (10/3)^(1/3) # F 2 / (10/3)^(1/3) * scmt_min[10] = mt_min[ 5] / (10/3)^(1/3) # Bb 4 / (10/3)^(2/3) * scThe point that the boundary of mean-tone tuning goes up threepositions, and that mean-tone tuning is adjusted 1sc higher, are thesame situation as Pythagoras tuning and pure intonation.Now, I think that mean-tone tuning could use for a harmony-melodybecause of the characteristic that is more harmony-like thanPythagoras tuning, and a scale is not uneven like pure intonation.======================================================================User-defined temperament entry======================================================================The function of user-defined temperament entry (experimental) isimplemented in TiMidity++. This corresponds to (18) of MIDI TuningStandard summary (see the top of this document). For example, it cangenerate various temperaments by the following SysEx's.[equal temperament]f0 7e 00 08 0c 00 ; temper prog number65 71 75 61 6c 00 00 00 00 00 00 00 00 00 00 00 ; "equal"01 ; number of formula0f 7f 00 00 01 01 02 01 07 0c ; (both) 2^(7/12)f7[Pythagoras tuning]f0 7e 00 08 0c 01 ; temper prog number50 79 74 68 61 67 6f 72 61 73 00 00 00 00 00 00 ; "Pythagoras"02 ; number of formula00 3f 40 1f 03 02 01 01 00 01 ; (maj) 3/243 7f 00 03 03 02 01 01 00 01 ; (min) 3/2f7[mean-tone tuning]f0 7e 00 08 0c 02 ; temper prog number6d 65 61 6e 2d 74 6f 6e 65 00 00 00 00 00 00 00 ; "mean-tone"02 ; number of formula00 3f 40 1f 01 01 05 01 01 04 ; (maj) 5^(1/4)43 7f 00 03 01 01 0a 03 01 03 ; (min) (10/3)^(1/3)f7[pure intonation]f0 7e 00 08 0c 03 ; temper prog number70 75 72 65 20 69 6e 74 6f 6e 61 74 69 6f 6e 00 ; "pure intonation"04 ; number of formula00 3f 40 1f 03 02 01 01 00 01 ; (maj) 3/200 04 40 02 05 01 02 03 04 01 ; (maj) 5*(2/3)^443 7f 00 03 03 02 01 01 00 01 ; (min) 3/240 22 00 00 05 01 02 03 04 01 ; (min) 5*(2/3)^4f7[Kirnberger-3]f0 7e 00 08 0c 00 ; temper prog number4b 69 72 6e 62 65 72 67 65 72 2d 33 00 00 00 00 ; "Kirnberger-3"02 ; number of formula00 0f 00 00 01 01 05 01 01 04 ; (both) 5^(1/4)00 30 00 1f 03 02 01 01 00 01 ; (both) 3/2f7[Hirashima temperament]f0 7e 00 08 0c 01 ; temper prog number48 69 72 61 73 68 69 6d 61 00 00 00 00 00 00 00 ; "Hirashima"02 ; number of formula00 1f 00 03 01 01 05 01 01 04 ; (both) 5^(1/4)00 00 00 3c 03 02 01 01 00 01 ; (both) 3/2f7[Werckmeister-3]f0 7e 00 08 0c 02 ; temper prog number57 65 72 63 6b 6d 65 69 73 74 65 72 2d 33 00 00 ; "Werckmeister-3"02 ; number of formula00 07 00 00 01 09 02 01 0f 04 ; (both) 2^(15/4)/900 18 00 3f 03 02 01 01 00 01 ; (both) 3/2f7[well-temperament]f0 7e 00 08 0c 03 ; temper prog number77 65 6c 6c 2d 74 65 6d 70 65 72 00 00 00 00 00 ; "well-temper"02 ; number of formula00 07 00 00 01 09 02 01 0f 04 ; (both) 2^(15/4)/900 00 01 7f 03 02 01 01 00 01 ; (both) 3/2f7----TAMUKI Shoichi <tamuki@linet.gr.jp>⌨️ 快捷键说明
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