Garibaldi: Difference between revisions

| Generator tuning = 702.10

| Odd limit 1 = 9 | Mistuning 1 = 4.33 | Complexity 1 = 41

| Odd limit 2 = (2.3.5.7.19) 21 | Mistuning 2 = 4.65 | Complexity 2 = 41

'''Garibaldi''' is a [[7-limit]] (and higher) [[regular temperament|temperament]] of the [[schismatic family #Garibaldi|schismatic family]]. It is an [[extension]] of [[helmholtz (temperament)|helmholtz]] temperament beyond the 5-limit but with the same simple [[chain of fifths|chain-of-fifths]] structure (so that [[chain-of-fifths notation|standard notation]] may be used). The garibaldi temperament tempers together the Pythagorean, syntonic, and archytas commas into a singular generalized "comma", which can be used to reach intervals of 3, 5, and 7. As in helmholtz temperament, [[5/4]] is mapped to the diminished fourth (e.g. C–F♭; a comma-flat major third), and the new mapping specific to garibaldi is that [[7/4]] is mapped to the double-diminished octave (e.g. C–C𝄫; a comma-flat minor seventh). This makes garibaldi a [[marvel temperaments|marvel temperament]] and a [[hemifamity temperaments|hemifamity temperament]]. Tuning the fifth a fraction of a cent sharp gives the best tunings.  

Immediate 11-limit extensions include cassandra ({{nowrap| 41 & 53 }}), mapping 11/8 to +23 fifths, andromeda ({{nowrap| 29 & 41 }}), mapping 11/8 to −18 fifths, and helenus ({{nowrap| 53 & 65d }}), mapping 11/8 to −30 fifths. Garibaldi is most naturally a 2.3.5.7.19-[[subgroup]] temperament due to its immediate availability of [[19/16]] at the minor third (C–E♭). This is sometimes known as ''garibaldi nestoria''.

Garibaldi is very naturally considered as a [[detemperament]] of the [[12edo|12 equal temperament]]. The table below shows a 53-tone detempered scale, with a generator range of -26 to +26. Each interval category of the 12 equal temperament is further divided into "double-sub", "sub", "plain", "super" and "double-super" qualities, separated by an [[diesis (scale theory)|enharmonic diesis]], which represents the syntonic~septimal comma; the "plain" type here consists of a [[5L 7s]] scale in 6|5 mode. Combining this division with the minor and major qualities of the 12 equal temperament, and calling the "double-sub major" and "double-super minor" qualities ''artoneutral'' and ''tendoneutral'', respectively, garibaldi gives us at least ''eight'' qualities for each diatonic category: subminor, minor, supraminor, artoneutral, tendoneutral, submajor, major, and supermajor.

Notice also the little comma between artoneutral and tendoneutral. This interval spans 41 generator steps. 41edo tempers it out so that it merges artoneutral and tendoneutral into one neutral interval whereas 53edo exaggerates it to the size of the syntonic~septimal comma. 94edo tunes it to one half the size of the syntonic~septimal comma, which can be seen as a good compromise.

| style="border-left: double;" | 7 || 114.7 || 15/14~16/15

| style="border-left: double;" | −10 || 178.9 || 10/9

| style="border-left: double;" | 2 || 204.2 || 9/8

| style="border-left: double;" | 14 || 229.5 || 8/7

| 3

| style="border-left: double;" | −13 || 472.6 || 21/16

| style="border-left: double;" | 11 || 523.2 || 19/14

| style="border-left: double;" | 23 || 548.4 || 11/8

| A4, d5

| style="border-left: double;" | −12 || 1174.7 || 63/32~65/33

See the diagrams on the right for isomorphic versions.

Using schismic can be a challenge because it defies the tradition of tertian harmony in [[chain-of-fifths notation]]. The just major triad on C is C–Fb–G, for example. Due to the generalized comma of garibaldi, a natural choice is to adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as C–vE–G.  

{| class="wikitable center-1 center-3"

|+ style="font-size: 105%; white-space: nowrap;" | Cassandra nomenclature for<br />selected intervals

|-

| Perfect fifth

| C–G

| Downmajor third

| Downminor seventh

| Double-up fourth

| C–^^F

| Double-up minor sixth

| C–^^Ab

| Minor third

| C–Eb

| Down-diminished fifth<br>Double-down augmented fourth

| 13/8

 

| C–vvGb<br>C–v³F#

* 1–5/4–3/2 (C–vE–G)

* 1–6/5–3/2 (C–^Eb–G)

* 1–9/7–3/2 (C–^E-G)

* 1–7/6–3/2 (C–vEb-G)