====================================================================== MIDI Tuning Standard summary ====================================================================== Besides GS/XG scale tuning which adjusts the pitch of 12 tones in an octave individually, TiMidity++ supports MIDI Tuning Standard in Universal SysEx. MIDI Tuning Standard has the following advantages compared with GS/XG scale tuning: - Support for microtonal sound other than 12 tones - The pitch can be adjusted in 1/100 cent or less accuracy - Temperaments can be rationally setup based on the tonality For details, please refer to the recommended practice. (1) Bulk Tuning Dump Request (Non Real-Time) ---------------------------------------------------------------------- F0 7E <device ID> 08 00 tt F7 F0 7E Universal Non-Real Time SysEx header <device ID> ID of target device (7F = all devices) 08 sub-ID#1 = "MIDI Tuning Standard" 00 sub-ID#2 = "bulk tuning dump request (Non Real-Time)" tt tuning program number (0 - 127) F7 EOX ---------------------------------------------------------------------- (2) Bulk Tuning Dump (Non Real-Time) ---------------------------------------------------------------------- F0 7E <device ID> 08 01 tt <tuning name> [xx yy zz] ... chksum F7 F0 7E Universal Non-Real Time SysEx header <device ID> ID of target device (7F = all devices) 08 sub-ID#1 = "MIDI Tuning Standard" 01 sub-ID#2 = "bulk tuning dump (Non Real-Time)" tt tuning program number (0 - 127) <tuning name> 16 ASCII characters [xx yy zz] frequency data for one note (repeated 128 times) chksum checksum (XOR of all bytes excluding F0, F7, and chksum) F7 EOX ---------------------------------------------------------------------- (3) Single Note Tuning Change (Real-Time) ---------------------------------------------------------------------- F0 7F <device ID> 08 02 tt ll [kk xx yy zz] F7 F0 7F Universal Real Time SysEx header <device ID> ID of target device (7F = all devices) 08 sub-ID#1 = "MIDI Tuning Standard" 02 sub-ID#2 = "single note tuning change (Real-Time)" tt tuning program number (0 - 127) ll number of changes (1 change = 1 set of [kk xx yy zz]) [kk MIDI key number xx yy zz] frequency data for that key (repeated 'll' number of times) F7 EOX ---------------------------------------------------------------------- (4) Bulk Tuning Dump Request (Bank) (Non Real-Time) ---------------------------------------------------------------------- F0 7E <device ID> 08 03 bb tt F7 F0 7E Universal Non-Real Time SysEx header <device ID> ID of target device (7F = all devices) 08 sub-ID#1 = "MIDI tuning standard" 03 sub-ID#2 = "bulk tuning dump request (Bank) (Non Real-Time)" bb tuning bank number (0 - 127) (described as 1-128 in MIDI Tuning Specification) tt tuning program number (0 - 127) F7 EOX ---------------------------------------------------------------------- (5) Key-Based Tuning Dump (Non Real-Time) ---------------------------------------------------------------------- F0 7E <device ID> 08 04 bb tt <tuning name> [xx yy zz] ... chksum F7 F0 7E Universal Non-Real Time SysEx header <device ID> ID of target device (7F = all devices) 08 sub-ID#1 = "MIDI tuning standard" 04 sub-ID#2 = "key-based tuning dump (Non Real-Time)" bb tuning bank number (0 - 127) (described as 1-128 in MIDI Tuning Specification) tt tuning program number (0 - 127) <tuning name> 16 ASCII characters [xx yy zz] frequency data for one note (repeated 128 times) chksum checksum (XOR of all bytes excluding F0, F7, and chksum) F7 EOX ---------------------------------------------------------------------- (6) Scale/Octave Tuning Dump 1-Byte Form (Non Real-Time) ---------------------------------------------------------------------- F0 7E <device ID> 08 05 bb tt <tuning name> [xx] ... chksum F7 F0 7E Universal Non-Real Time SysEx header <device ID> ID of target device (7F = all devices) 08 sub-ID#1 = "MIDI tuning standard" 05 sub-ID#2 = "scale/octave tuning dump 1-byte form (Non Real-Time)" bb tuning bank number (0 - 127) (described as 1-128 in MIDI Tuning Specification) tt tuning program number (0 - 127) <tuning name> 16 ASCII characters [xx] frequency data for C,C#,... B (12 bytes total) 00H means -64 Cent 40H means +/- 0 Cent 7FH means +63 Cent chksum checksum (XOR of all bytes excluding F0, F7, and chksum) F7 EOX ---------------------------------------------------------------------- (7) Scale/Octave Tuning Dump 2-Byte Form (Non Real-Time) ---------------------------------------------------------------------- F0 7E <device ID> 08 06 bb tt <tuning name> [xx yy] ... chksum F7 F0 7E Universal Non-Real Time SysEx header <device ID> ID of target device (7F = all devices) 08 sub-ID#1 = "MIDI tuning standard" 06 sub-ID#2 = "scale/octave tuning dump 2-byte form (Non Real-Time)" bb tuning bank number (0 - 127) (described as 1-128 in MIDI Tuning Specification) tt tuning program number (0 - 127) <tuning name> 16 ASCII characters [xx yy] frequency data for C,C#,... B (24 bytes total) 00H 00H means -100 cents (8,192 steps of .012207 cents) 40H 00H means 0 cents (equal temperament) 7FH 7FH means +100 cents (8,191 steps of .012207 cents) chksum checksum (XOR of all bytes excluding F0, F7, and chksum) F7 EOX ---------------------------------------------------------------------- (8) Single Note Tuning Change (Bank) (Real-Time) ---------------------------------------------------------------------- F0 7F <device ID> 08 07 bb tt ll [kk xx yy zz] ... F7 F0 7F Universal Real Time SysEx header <device ID> ID of target device (7F = all devices) 08 sub-ID#1 = "MIDI tuning standard" 07 sub-ID#2 = "single note tuning change (Bank) (Real-Time)" bb tuning bank number (0 - 127) (described as 1-128 in MIDI Tuning Specification) tt tuning program number (0 - 127) ll number of changes (1 change = 1 set of [kk xx yy zz]) [kk MIDI key number xx yy zz] frequency data for that key (repeated 'll' number of times) F7 EOX ---------------------------------------------------------------------- (9) Single Note Tuning Change (Bank) (Non Real-Time) ---------------------------------------------------------------------- F0 7E <device ID> 08 07 bb tt ll [kk xx yy zz] ... F7 F0 7E Universal Non-Real Time SysEx header <device ID> ID of target device (7F = all devices) 08 sub-ID#1 = "MIDI tuning standard" 07 sub-ID#2 = "single note tuning change (Bank) (Non Real-Time)" bb tuning bank number (0 - 127) (described as 1-128 in MIDI Tuning Specification) tt tuning program number (0 - 127) ll number of changes (1 change = 1 set of [kk xx yy zz]) [kk MIDI key number xx yy zz] frequency data for that key (repeated 'll'number of times) F7 EOX ---------------------------------------------------------------------- (10) Scale/Octave Tuning 1-Byte Form (Real-Time) ---------------------------------------------------------------------- F0 7F <device ID> 08 08 ff gg hh [ss] ... F7 F0 7F Universal Real-Time SysEx header <device ID> ID of target device (7F = all devices) 08 sub-ID#1 = "MIDI Tuning Standard" 08 sub-ID#2 = "scale/octave tuning 1-byte form (Real-Time)" ff channel/options byte 1 bits 0 to 1 = channel 15 to 16 bits 2 to 6 = reserved for future expansion gg channel byte 2 - bits 0 to 6 = channel 8 to 14 hh channel byte 3 - bits 0 to 6 = channel 1 to 7 [ss] 12 byte tuning offset of 12 semitones from C to B 00H means -64 cents 40H means 0 cents (equal temperament) 7FH means +63 cents F7 EOX ---------------------------------------------------------------------- (11) Scale/Octave Tuning 1-Byte Form (Non Real-Time) ---------------------------------------------------------------------- F0 7E <device ID> 08 08 ff gg hh [ss] ... F7 F0 7E Universal Non Real-Time SysEx header <device ID> ID of target device (7F = all devices) 08 sub-ID#1 = "MIDI Tuning Standard" 08 sub-ID#2 = "scale/octave tuning 1-byte form (Non Real-Time)" ff channel/options byte 1 bits 0 to 1 = channel 15 to 16 bits 2 to 6 = reserved for future expansion gg channel byte 2 - bits 0 to 6 = channel 8 to 14 hh channel byte 3 - bits 0 to 6 = channel 1 to 7 [ss] 12 byte tuning offset of 12 semitones from C to B 00H means -64 cents 40H means 0 cents (equal temperament) 7FH means +63 cents F7 EOX ---------------------------------------------------------------------- (12) Scale/Octave Tuning 2-Byte Form (Real-Time) ---------------------------------------------------------------------- F0 7F <device ID> 08 09 ff gg hh [ss tt] ... F7 F0 7F Universal Real-Time SysEx header <device ID> ID of target device (7F = all devices) 08 sub-ID#1 = "MIDI Tuning Standard" 09 sub-ID#2 = "scale/octave tuning 2-byte form (Real-Time)" ff channel/options byte 1 bits 0 to 1 = channel 15 to 16 bits 2 to 6 = reserved for future expansion gg channel byte 2 - bits 0 to 6 = channel 8 to 14 hh channel byte 3 - bits 0 to 6 = channel 1 to 7 [ss tt] 24 byte tuning offset of 2 bytes per semitone from C to B 00H 00H means -100 cents (8,192 steps of .012207 cents) 40H 00H means 0 cents (equal temperament) 7FH 7FH means +100 cents (8,191 steps of .012207 cents) F7 EOX ---------------------------------------------------------------------- (13) Scale/Octave Tuning 2-Byte Form (Non Real-Time) ---------------------------------------------------------------------- F0 7E <device ID> 08 09 ff gg hh [ss tt] ... F7 F0 7E Universal Non Real Time SysEx header <device ID> ID of target device (7F = all devices) 08 sub-ID#1 = "MIDI Tuning Standard" 09 sub-ID#2 = "scale/octave tuning 2-byte form (Non Real-Time)" ff channel/options byte 1 bits 0 to 1 = channel 15 to 16 bits 2 to 6 = reserved for future expansion gg channel byte 2 - bits 0 to 6 = channel 8 to 14 hh channel byte 3 - bits 0 to 6 = channel 1 to 7 [ss tt] 24 byte tuning offset of 2 bytes per semitone from C to B 00H 00H means -100 cents (8,192 steps of .012207 cents) 40H 00H means 0 cents (equal temperament) 7FH 7FH means +100 cents (8,191 steps of .012207 cents) F7 EOX ---------------------------------------------------------------------- (14) Temperament Tonality Control Tuning (Real-Time) ---------------------------------------------------------------------- F0 7F <device ID> 08 0A sf mi F7 F0 7F Universal Real-Time SysEx header <device ID> ID of target device (7F = all devices) 08 sub-ID#1 = "MIDI Tuning Standard" 0A sub-ID#2 = "temperament tonality control tuning (Real-Time)" sf number of sharp/flat (1 byte) 39H means 7 flats 3FH means 1 flat 40H means key of C 41H means 1 sharp 47H means 7 sharps mi major/minor (1 byte) 00H means major key 01H means minor key 02H means passing major key 03H means passing minor key F7 EOX ---------------------------------------------------------------------- (15) Temperament Tonality Control Tuning (Non Real-Time) ---------------------------------------------------------------------- F0 7E <device ID> 08 0A sf mi F7 F0 7E Universal Non Real-Time SysEx header <device ID> ID of target device (7F = all devices) 08 sub-ID#1 = "MIDI Tuning Standard" 0A sub-ID#2 = "temperament tonality control tuning (Non Real-Time)" sf number of sharp/flat (1 byte) 39H means 7 flats 3FH means 1 flat 40H means key of C 41H means 1 sharp 47H means 7 sharps mi major/minor (1 byte) 00H means major key 01H means minor key 02H means passing major key 03H means passing minor key F7 EOX ---------------------------------------------------------------------- (16) Temperament Type Control Tuning (Real-Time) ---------------------------------------------------------------------- F0 7F <device ID> 08 0B ff gg hh tt F7 F0 7F Universal Real-Time SysEx header <device ID> ID of target device (7F = all devices) 08 sub-ID#1 = "MIDI Tuning Standard" 0B sub-ID#2 = "temperament type control tuning (Real-Time)" ff channel/options byte 1 bits 0 to 1 = channel 15 to 16 bit 2 = port A/B bits 3 to 6 = reserved for future expansion gg channel byte 2 - bits 0 to 6 = channel 8 to 14 hh channel byte 3 - bits 0 to 6 = channel 1 to 7 tt 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 #3 F7 EOX ---------------------------------------------------------------------- (17) Temperament Type Control Tuning (Non Real-Time) ---------------------------------------------------------------------- F0 7E <device ID> 08 0B ff gg hh tt F7 F0 7E Universal Non Real Time SysEx header <device ID> ID of target device (7F = all devices) 08 sub-ID#1 = "MIDI Tuning Standard" 0B sub-ID#2 = "temperament type control tuning (Non Real-Time)" ff channel/options byte 1 bits 0 to 1 = channel 15 to 16 bit 2 = port A/B bits 3 to 6 = reserved for future expansion gg channel byte 2 - bits 0 to 6 = channel 8 to 14 hh channel byte 3 - bits 0 to 6 = channel 1 to 7 tt 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 #3 F7 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] ... F7 F0 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 characters ll 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, F but 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 only in pure intonation (C major), players need to change temperaments according 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. The notation "+", "-", "++" and "--" mean 1sc higher, 1sc lower, 2sc higher and 2sc lower respectively. A certain pure intonation is given as 12 sounds arranged by the rectangle 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 of parallel key is slightly lower for major music, and slightly higher for minor music. ====================================================================== Preset temperament of Temperament Type Control Tuning ====================================================================== First, Pythagoras tuning (major) chromatic scale is expressed by the following recurrence relations. Here, the index [] is a offset of the tonic. The operation results are surely settled between 1 and 2, so they will be made into half or double if necessary. [Pythagoras tuning (major) chromatic scale] pytha_maj[ 0] = 1 # C 1 pytha_maj[ 7] = pytha_maj[ 0] * 3/2 # G 3/2 pytha_maj[ 2] = pytha_maj[ 7] * 3/2 # D 9/8 pytha_maj[ 9] = pytha_maj[ 2] * 3/2 # A 27/16 pytha_maj[ 4] = pytha_maj[ 9] * 3/2 # E 81/64 pytha_maj[11] = pytha_maj[ 4] * 3/2 # B 243/128 pytha_maj[ 6] = pytha_maj[11] * 3/2 # F# 729/512 -- pytha_maj[ 5] = pytha_maj[ 0] * 2/3 # F 4/3 pytha_maj[10] = pytha_maj[ 5] * 2/3 # Bb 16/9 pytha_maj[ 3] = pytha_maj[10] * 2/3 # Eb 32/27 pytha_maj[ 8] = pytha_maj[ 3] * 2/3 # Ab 128/81 pytha_maj[ 1] = pytha_maj[ 8] * 2/3 # Db 256/243 On the other hand, pure intonation (major) chromatic scale can be expressed by the following recurrence relations. Here, sc means a syntonic comma (81/80). [pure intonation (major) chromatic scale] pure_maj[ 0] = 1 # C 1 pure_maj[ 7] = pure_maj[ 0] * 3/2 # G 3/2 pure_maj[ 2] = pure_maj[ 7] * 3/2 # D 9/8 pure_maj[ 9] = pure_maj[ 2] * 3/2 / sc # A 5/3 pure_maj[ 4] = pure_maj[ 9] * 3/2 # E 5/4 pure_maj[11] = pure_maj[ 4] * 3/2 # B 15/8 pure_maj[ 6] = pure_maj[11] * 3/2 # F# 45/32 -- pure_maj[ 5] = pure_maj[ 0] * 2/3 # F 4/3 pure_maj[10] = pure_maj[ 5] * 2/3 * sc # Bb 9/5 pure_maj[ 3] = pure_maj[10] * 2/3 # Eb 6/5 pure_maj[ 8] = pure_maj[ 3] * 2/3 # Ab 8/5 pure_maj[ 1] = pure_maj[ 8] * 2/3 # Db 16/15 It can be understood that pure intonation is similar to Pythagoras tuning fundamentally except descending with 1sc at A and rising with 1sc at B flat while going up and down respectively from the tonic in the circle of fifths. Similarly, expressed Pythagoras tuning and pure intonation (minor) chromatic scale by the following recurrence relations. Although the fractions written to right-hand side is terrible values, the recurrence relations themselves are very simple. [Pythagoras tuning (minor) chromatic scale] pytha_min[ 0] = 1 # C 1 pytha_min[ 7] = pytha_min[ 0] * 3/2 # G 3/2 pytha_min[ 2] = pytha_min[ 7] * 3/2 # D 9/8 pytha_min[ 9] = pytha_min[ 2] * 3/2 # A 27/16 pytha_min[ 4] = pytha_min[ 9] * 3/2 # E 81/64 pytha_min[11] = pytha_min[ 4] * 3/2 # B 243/128 pytha_min[ 6] = pytha_min[11] * 3/2 # F# 729/512 pytha_min[ 1] = pytha_min[ 6] * 3/2 # C# 2187/2048 pytha_min[ 8] = pytha_min[ 1] * 3/2 # G# 6561/4096 pytha_min[ 3] = pytha_min[ 8] * 3/2 # D# 19683/16384 -- pytha_min[ 5] = pytha_min[ 0] * 2/3 # F 4/3 pytha_min[10] = pytha_min[ 5] * 2/3 # Bb 16/9 [pure intonation (minor) chromatic scale] pure_min[ 0] = 1 * sc # C 1 * sc pure_min[ 7] = pure_min[ 0] * 3/2 # G 3/2 * sc pure_min[ 2] = pure_min[ 7] * 3/2 / sc # D 10/9 * sc pure_min[ 9] = pure_min[ 2] * 3/2 # A 5/3 * sc pure_min[ 4] = pure_min[ 9] * 3/2 # E 5/4 * sc pure_min[11] = pure_min[ 4] * 3/2 # B 15/8 * sc pure_min[ 6] = pure_min[11] * 3/2 / sc # F# 25/18 * sc pure_min[ 1] = pure_min[ 6] * 3/2 # C# 25/24 * sc pure_min[ 8] = pure_min[ 1] * 3/2 # G# 25/16 * sc pure_min[ 3] = pure_min[ 8] * 3/2 # D# 75/64 * sc -- pure_min[ 5] = pure_min[ 0] * 2/3 # F 4/3 * sc pure_min[10] = pure_min[ 5] * 2/3 # Bb 16/9 * sc The differences from the major tuning are that the boundary of Pythagoras tuning goes up three positions, that the positions of descending with syntonic comma are changed, and that pure intonation is adjusted 1sc higher so that melodic parts' tonic (Pythagoras tuning) and harmonic parts' tonic (pure intonation) are overlapped. By the way, mean-tone tuning is also prepared besides Pythagoras tuning and pure intonation as preset temperament of TiMidity++. While mean-tone tuning (major) is based on the general one whose major thirds are pure, mean-tone tuning (minor) is based on Salinas tuning whose minor thirds are pure. Both mean-tone tuning (major) chromatic scale and mean-tone tuning (minor) chromatic scale can be expressed by the following recurrence relations. [mean-tone tuning (major) chromatic scale] mt_maj[ 0] = 1 # C 1 mt_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) / 2 mt_maj[ 9] = mt_maj[ 2] * 5^(1/4) # A 5^(3/4) / 2 mt_maj[ 4] = mt_maj[ 9] * 5^(1/4) # E 5/4 mt_maj[11] = mt_maj[ 4] * 5^(1/4) # B 5^(5/4) / 4 mt_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 8/5 mt_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 * sc mt_min[ 7] = mt_min[ 0] * (10/3)^(1/3) # G (10/3)^(1/3) * sc mt_min[ 2] = mt_min[ 7] * (10/3)^(1/3) # D (10/3)^(2/3) / 2 * sc mt_min[ 9] = mt_min[ 2] * (10/3)^(1/3) # A 5/3 * sc mt_min[ 4] = mt_min[ 9] * (10/3)^(1/3) # E (10/3)^(4/3) / 4 * sc mt_min[11] = mt_min[ 4] * (10/3)^(1/3) # B (10/3)^(5/3) / 4 * sc mt_min[ 6] = mt_min[11] * (10/3)^(1/3) # F# 25/18 * sc mt_min[ 1] = mt_min[ 6] * (10/3)^(1/3) # C# (10/3)^(7/3) / 16 * sc mt_min[ 8] = mt_min[ 1] * (10/3)^(1/3) # G# (10/3)^(8/3) / 16 * sc mt_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) * sc mt_min[10] = mt_min[ 5] / (10/3)^(1/3) # Bb 4 / (10/3)^(2/3) * sc The point that the boundary of mean-tone tuning goes up three positions, and that mean-tone tuning is adjusted 1sc higher, are the same situation as Pythagoras tuning and pure intonation. Now, I think that mean-tone tuning could use for a harmony-melody because of the characteristic that is more harmony-like than Pythagoras tuning, and a scale is not uneven like pure intonation. ====================================================================== User-defined temperament entry ====================================================================== The function of user-defined temperament entry (experimental) is implemented in TiMidity++. This corresponds to (18) of MIDI Tuning Standard summary (see the top of this document). For example, it can generate various temperaments by the following SysEx's. [equal temperament] f0 7e 00 08 0c 00 ; temper prog number 65 71 75 61 6c 00 00 00 00 00 00 00 00 00 00 00 ; "equal" 01 ; number of formula 0f 7f 00 00 01 01 02 01 07 0c ; (both) 2^(7/12) f7 [Pythagoras tuning] f0 7e 00 08 0c 01 ; temper prog number 50 79 74 68 61 67 6f 72 61 73 00 00 00 00 00 00 ; "Pythagoras" 02 ; number of formula 00 3f 40 1f 03 02 01 01 00 01 ; (maj) 3/2 43 7f 00 03 03 02 01 01 00 01 ; (min) 3/2 f7 [mean-tone tuning] f0 7e 00 08 0c 02 ; temper prog number 6d 65 61 6e 2d 74 6f 6e 65 00 00 00 00 00 00 00 ; "mean-tone" 02 ; number of formula 00 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 number 70 75 72 65 20 69 6e 74 6f 6e 61 74 69 6f 6e 00 ; "pure intonation" 04 ; number of formula 00 3f 40 1f 03 02 01 01 00 01 ; (maj) 3/2 00 04 40 02 05 01 02 03 04 01 ; (maj) 5*(2/3)^4 43 7f 00 03 03 02 01 01 00 01 ; (min) 3/2 40 22 00 00 05 01 02 03 04 01 ; (min) 5*(2/3)^4 f7 [Kirnberger-3] f0 7e 00 08 0c 00 ; temper prog number 4b 69 72 6e 62 65 72 67 65 72 2d 33 00 00 00 00 ; "Kirnberger-3" 02 ; number of formula 00 0f 00 00 01 01 05 01 01 04 ; (both) 5^(1/4) 00 30 00 1f 03 02 01 01 00 01 ; (both) 3/2 f7 [Hirashima temperament] f0 7e 00 08 0c 01 ; temper prog number 48 69 72 61 73 68 69 6d 61 00 00 00 00 00 00 00 ; "Hirashima" 02 ; number of formula 00 1f 00 03 01 01 05 01 01 04 ; (both) 5^(1/4) 00 00 00 3c 03 02 01 01 00 01 ; (both) 3/2 f7 [Werckmeister-3] f0 7e 00 08 0c 02 ; temper prog number 57 65 72 63 6b 6d 65 69 73 74 65 72 2d 33 00 00 ; "Werckmeister-3" 02 ; number of formula 00 07 00 00 01 09 02 01 0f 04 ; (both) 2^(15/4)/9 00 18 00 3f 03 02 01 01 00 01 ; (both) 3/2 f7 [well-temperament] f0 7e 00 08 0c 03 ; temper prog number 77 65 6c 6c 2d 74 65 6d 70 65 72 00 00 00 00 00 ; "well-temper" 02 ; number of formula 00 07 00 00 01 09 02 01 0f 04 ; (both) 2^(15/4)/9 00 00 01 7f 03 02 01 01 00 01 ; (both) 3/2 f7 ---- TAMUKI Shoichi <tamuki@linet.gr.jp>
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